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Lincoln Public Schools
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Torrance, California
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Authors
The Number System
Critical Area
Critical Area Completing understanding of division of fractions and extending the
notion of number to the system of rational numbers, which includes negative numbers
Project
1
Sweet Success . . . . . . . . . . . . . . . . . . . . . . . . . . 2
Whole Numbers and Decimals
3
Domain The Number System
COMMON CORE STATE STANDARDS 6.NS.B.2, 6.NS.B.3, 6.NS.B.4
1
2
3
4
5
6
7
8
9
Show What You Know . . . . . . . . . . . . . . . . . . . . .
Vocabulary Builder . . . . . . . . . . . . . . . . . . . . . . .
Divide Multi-Digit Numbers. . . . . . . . . . . . . . . . . .
Prime Factorization . . . . . . . . . . . . . . . . . . . . . . .
Least Common Multiple . . . . . . . . . . . . . . . . . . . .
Greatest Common Factor . . . . . . . . . . . . . . . . . . .
Problem Solving • Apply the Greatest Common Factor .
Mid-Chapter Checkpoint . . . . . . . . . . . . . . . . . . .
Add and Subtract Decimals . . . . . . . . . . . . . . . . . .
Multiply Decimals . . . . . . . . . . . . . . . . . . . . . . . .
Divide Decimals by Whole Numbers . . . . . . . . . . . .
Divide with Decimals . . . . . . . . . . . . . . . . . . . . . .
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. .3
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. 35
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. 43
. 49
. 55
Chapter 1 Review/Test . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61
2
Fractions
Domain The Number System
COMMON CORE STATE STANDARDS 6.NS.A.1, 6.NS.B.4, 6.NS.C.6c
1
2
3
4
© Houghton Mifflin Harcourt Publishing Company
67
5
6
7
8
9
10
Show What You Know . . . . . . . . . . . . . .
Vocabulary Builder . . . . . . . . . . . . . . . .
Fractions and Decimals . . . . . . . . . . . . .
Compare and Order Fractions and Decimals
Multiply Fractions . . . . . . . . . . . . . . . . .
Simplify Factors . . . . . . . . . . . . . . . . . .
Mid-Chapter Checkpoint . . . . . . . . . . . .
Investigate • Model Fraction Division. . . . .
Estimate Quotients . . . . . . . . . . . . . . . .
Divide Fractions . . . . . . . . . . . . . . . . . .
Investigate • Model Mixed Number Division
Divide Mixed Numbers . . . . . . . . . . . . . .
Problem Solving • Fraction Operations . . .
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. . 67
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. . 95
. 101
. 107
. 113
. 119
. 125
Chapter 2 Review/Test . . . . . . . . . . . . . . . . . . . . . . . . . . . 131
Go online! Your math
lessons are interactive.
Use iTools, Animated
Math Models, the
Multimedia eGlossary,
and more.
Chapter 1 Overview
In this chapter, you will
explore and discover
answers to the following
Essential Questions:
• How do you solve
real-world problems
involving whole numbers
and decimals?
• How does estimation
help you solve problems
involving decimals and
whole numbers?
• How can you use the
GCF and the LCM to
solve problems?
Chapter 2 Overview
In this chapter, you will
explore and discover
answers to the following
Essential Questions:
• How can you use the
relationship between
multiplication and
division to divide
fractions?
• What is a mixed
number?
• How can you estimate
products and quotients
of fractions and mixed
numbers?
v
Chapter 3 Overview
In this chapter, you will
explore and discover
answers to the following
Essential Questions:
• How do you write,
interpret, and use
rational numbers?
• How do you calculate
the absolute value of a
number?
• How do you graph an
ordered pair?
3
Rational Numbers
137
Domain The Number System
COMMON CORE STATE STANDARDS 6.NS.C.5, 6.NS.C.6a, 6.NS.C.6b, 6.NS.C.6c, 6.NS.C.7a,
6.NS.C.7b, 6.NS.C.7c, 6.NS.C.7d, 6.NS.C.8
1
2
3
4
5
6
7
8
9
10
Show What You Know . . . . . . . . . . . . . .
Vocabulary Builder . . . . . . . . . . . . . . . .
Understand Positive and Negative Numbers.
Compare and Order Integers . . . . . . . . . .
Rational Numbers and the Number Line . . .
Compare and Order Rational Numbers . . . .
Mid-Chapter Checkpoint . . . . . . . . . . . .
Absolute Value . . . . . . . . . . . . . . . . . . .
Compare Absolute Values . . . . . . . . . . . .
Rational Numbers and the Coordinate Plane
Ordered Pair Relationships . . . . . . . . . . .
Distance on the Coordinate Plane . . . . . . .
Problem Solving • The Coordinate Plane. . .
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. 137
. 138
. 139
. 145
. 151
. 157
. 163
. 165
. 171
. 177
. 183
. 189
. 195
© Houghton Mifflin Harcourt Publishing Company
Chapter 3 Review/Test . . . . . . . . . . . . . . . . . . . . . . . . . . . 201
vi
Ratios and Rates
Critical Area
Critical Area Connecting ratio and rate to whole number multiplication and division
and using concepts of ratio and rate to solve problems
Project
4
Meet Me in St. Louis . . . . . . . . . . . . . . . . . . . . . 208
Ratios and Rates
209
Domain Ratios and Proportional Relationships
COMMON CORE STATE STANDARDS 6.RP.A.1, 6.RP.A.2, 6.RP.A.3a, 6.RP.A.3b
1
2
3
4
5
6
7
8
Show What You Know . . . . . . . . . . . . . . . . .
Vocabulary Builder . . . . . . . . . . . . . . . . . . .
Investigate • Model Ratios. . . . . . . . . . . . . . .
Ratios and Rates . . . . . . . . . . . . . . . . . . . . .
Equivalent Ratios and Multiplication Tables . . . .
Problem Solving • Use Tables to Compare Ratios
Algebra • Use Equivalent Ratios . . . . . . . . . . .
Mid-Chapter Checkpoint . . . . . . . . . . . . . . .
Find Unit Rates . . . . . . . . . . . . . . . . . . . . . .
Algebra • Use Unit Rates . . . . . . . . . . . . . . . .
Algebra • Equivalent Ratios and Graphs . . . . . .
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. 209
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. 217
. 223
. 229
. 235
. 241
. 243
. 249
. 255
Chapter 4 Review/Test . . . . . . . . . . . . . . . . . . . . . . . . . . . 261
5
Percents
267
Domain Ratios and Proportional Relationships
Go online! Your math
lessons are interactive.
Use iTools, Animated
Math Models, the
Multimedia eGlossary,
and more.
Chapter 4 Overview
In this chapter, you will
explore and discover
answers to the following
Essential Questions:
• How can you use ratios
to express relationships
and solve problems?
• How can you write a
ratio?
• What are equivalent
ratios?
• How are rates related
to ratios?
COMMON CORE STATE STANDARDS 6.RP.A.3c
© Houghton Mifflin Harcourt Publishing Company
1
2
3
4
5
6
Show What You Know . . . . . . . . . . . .
Vocabulary Builder . . . . . . . . . . . . . .
Investigate • Model Percents . . . . . . . .
Write Percents as Fractions and Decimals
Write Fractions and Decimals as Percents
Mid-Chapter Checkpoint . . . . . . . . . .
Percent of a Quantity . . . . . . . . . . . . .
Problem Solving • Percents . . . . . . . . .
Find the Whole from a Percent . . . . . . .
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. 267
. 268
. 269
. 275
. 281
. 287
. 289
. 295
. 301
Chapter 5 Review/Test . . . . . . . . . . . . . . . . . . . . . . . . . . . 307
Chapter 5 Overview
In this chapter, you will
explore and discover
answers to the following
Essential Questions:
• How can you use ratio
reasoning to solve
percent problems?
• How can you write a
percent as a fraction?
• How can you use a ratio
to find a percent of a
number?
Personal Math Trainer
Online Assessment
and Intervention
vii
Chapter 6 Overview
6
Units of Measure
313
Domain Ratios and Proportional Relationships
COMMON CORE STATE STANDARDS 6.RP.A.3d
1
2
3
4
5
Show What You Know . . . . . . . . . . . . . . . . . . . . .
Vocabulary Builder . . . . . . . . . . . . . . . . . . . . . . .
Convert Units of Length . . . . . . . . . . . . . . . . . . . .
Convert Units of Capacity . . . . . . . . . . . . . . . . . . .
Convert Units of Weight and Mass . . . . . . . . . . . . . .
Mid-Chapter Checkpoint . . . . . . . . . . . . . . . . . . .
Transform Units . . . . . . . . . . . . . . . . . . . . . . . . .
Problem Solving • Distance, Rate, and Time Formulas .
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. 313
. 314
. 315
. 321
. 327
. 333
. 335
. 341
Chapter 6 Review/Test . . . . . . . . . . . . . . . . . . . . . . . . . . . 347
© Houghton Mifflin Harcourt Publishing Company
In this chapter, you will
explore and discover
answers to the following
Essential Questions:
• How can you use
measurements to
help you describe and
compare objects?
• Why do you need to
convert between units
of measure?
• How can you use a ratio
to convert units?
• How do you transform
units to solve problems?
viii
Expressions and Equations
Critical Area
Critical Area Writing, interpreting, and using expressions and equations
Project
7
The Great Outdoors . . . . . . . . . . . . . . . . . . . . . 354
Algebra: Expressions
355
Domain Expressions and Equations
COMMON CORE STATE STANDARDS 6.EE.A.1, 6.EE.A.2a, 6.EE.A.2b, 6.EE.A.2c, 6.EE.A.3,
6.EE.A.4, 6.EE.B.6
1
2
3
4
5
6
7
8
9
Show What You Know . . . . . . . . . . . . . . .
Vocabulary Builder . . . . . . . . . . . . . . . . .
Exponents . . . . . . . . . . . . . . . . . . . . . . .
Evaluate Expressions Involving Exponents. . .
Write Algebraic Expressions . . . . . . . . . . . .
Identify Parts of Expressions. . . . . . . . . . . .
Evaluate Algebraic Expressions and Formulas
Mid-Chapter Checkpoint . . . . . . . . . . . . .
Use Algebraic Expressions . . . . . . . . . . . . .
Problem Solving • Combine Like Terms . . . .
Generate Equivalent Expressions . . . . . . . .
Identify Equivalent Expressions . . . . . . . . .
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. 355
. 356
. 357
. 363
. 369
. 375
. 381
. 387
. 389
. 395
. 401
. 407
Chapter 7 Review/Test . . . . . . . . . . . . . . . . . . . . . . . . . . . 413
Go online! Your math
lessons are interactive.
Use iTools, Animated
Math Models, the
Multimedia eGlossary,
and more.
Chapter 7 Overview
In this chapter, you will
explore and discover
answers to the following
Essential Questions:
• How do you write,
interpret, and use
algebraic expressions?
• How can you use
expressions to
represent real-world
situations?
• How do you use the
order of operations to
evaluate expressions?
• How can you
tell whether two
expressions are
equivalent?
© Houghton Mifflin Harcourt Publishing Company
Personal Math Trainer
Online Assessment
and Intervention
ix
Chapter 8 Overview
In this chapter, you will
explore and discover
answers to the following
Essential Questions:
• How can you use
equations and
inequalities to represent
situations and solve
problems?
• How can you use
Properties of Equality
to solve equations?
• How do inequalities
differ from equations?
• Why is it useful to
describe situations by
using algebra?
8
Algebra: Equations
and Inequalities
419
Domain Expressions and Equations
COMMON CORE STATE STANDARDS 6.EE.B.5, 6.EE.B.7, 6.EE.B.8
1
2
3
4
5
6
7
8
9
10
Show What You Know . . . . . . . . . . . . . . . . . . . . .
Vocabulary Builder . . . . . . . . . . . . . . . . . . . . . . .
Solutions of Equations . . . . . . . . . . . . . . . . . . . . .
Write Equations . . . . . . . . . . . . . . . . . . . . . . . . .
Investigate • Model and Solve Addition Equations. . . .
Solve Addition and Subtraction Equations . . . . . . . . .
Investigate • Model and Solve Multiplication Equations
Solve Multiplication and Division Equations . . . . . . .
Problem Solving • Equations with Fractions . . . . . . .
Mid-Chapter Checkpoint . . . . . . . . . . . . . . . . . . .
Solutions of Inequalities . . . . . . . . . . . . . . . . . . . .
Write Inequalities . . . . . . . . . . . . . . . . . . . . . . . .
Graph Inequalities. . . . . . . . . . . . . . . . . . . . . . . .
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. 419
. 420
. 421
. 427
. 433
. 439
. 445
. 451
. 457
. 463
. 465
. 471
. 477
Chapter 8 Review/Test . . . . . . . . . . . . . . . . . . . . . . . . . . . 483
9
Algebra: Relationships
Between Variables
489
Domain Expressions and Equations
In this chapter, you will
explore and discover
answers to the following
Essential Questions:
• How can you show
relationships between
variables?
• How can you determine
the equation that
gives the relationship
between two variables?
• How can you use
tables and graphs to
visualize the relationship
between two variables?
x
1
2
3
4
5
Show What You Know . . . . . . . . . . . .
Vocabulary Builder . . . . . . . . . . . . . .
Independent and Dependent Variables. .
Equations and Tables . . . . . . . . . . . . .
Problem Solving • Analyze Relationships
Mid-Chapter Checkpoint . . . . . . . . . .
Graph Relationships . . . . . . . . . . . . .
Equations and Graphs . . . . . . . . . . . .
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. 489
. 490
. 491
. 497
. 503
. 509
. 511
. 517
Chapter 9 Review/Test . . . . . . . . . . . . . . . . . . . . . . . . . . . 523
© Houghton Mifflin Harcourt Publishing Company
COMMON CORE STATE STANDARDS 6.EE.C.9
Chapter 9 Overview
Geometry and Statistics
Critical Area
Critical Area Solve real-world and mathematical problems involving area, surface area,
and volume; and developing understanding of statistical thinking
Project
10
This Place Is a Zoo! . . . . . . . . . . . . . . . . . . . . . . 530
Area
531
Domain Geometry
COMMON CORE STATE STANDARDS 6.G.A.1, 6.G.A.3
1
2
3
4
5
6
7
8
9
Show What You Know . . . . . . . . . . . .
Vocabulary Builder . . . . . . . . . . . . . .
Algebra • Area of Parallelograms . . . . . .
Investigate • Explore Area of Triangles . .
Algebra • Area of Triangles . . . . . . . . .
Investigate • Explore Area of Trapezoids .
Algebra • Area of Trapezoids . . . . . . . .
Mid-Chapter Checkpoint . . . . . . . . . .
Area of Regular Polygons. . . . . . . . . . .
Composite Figures. . . . . . . . . . . . . . .
Problem Solving • Changing Dimensions
Figures on the Coordinate Plane . . . . . .
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. 531
. 532
. 533
. 539
. 545
. 551
. 557
. 563
. 565
. 571
. 577
. 583
Chapter 10 Review/Test . . . . . . . . . . . . . . . . . . . . . . . . . . 589
11
Surface Area and Volume
Domain Geometry
COMMON CORE STATE STANDARDS 6.G.A.2, 6.G.A.4
1
2
3
4
© Houghton Mifflin Harcourt Publishing Company
595
5
6
7
Show What You Know . . . . . . . . . . . . . . .
Vocabulary Builder . . . . . . . . . . . . . . . . .
Three-Dimensional Figures and Nets . . . . . .
Investigate • Explore Surface Area Using Nets.
Algebra • Surface Area of Prisms . . . . . . . . .
Algebra • Surface Area of Pyramids . . . . . . .
Mid-Chapter Checkpoint . . . . . . . . . . . . .
Investigate • Fractions and Volume . . . . . . .
Algebra • Volume of Rectangular Prisms . . . .
Problem Solving • Geometric Measurements.
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. 595
. 596
. 597
. 603
. 609
. 615
. 621
. 623
. 629
. 635
Chapter 11 Review/Test . . . . . . . . . . . . . . . . . . . . . . . . . . 641
Go online! Your math
lessons are interactive.
Use iTools, Animated
Math Models, the
Multimedia eGlossary,
and more.
Chapter 10 Overview
In this chapter, you will
explore and discover
answers to the following
Essential Questions:
• How can you use
measurements
to describe twodimensional figures?
• What does area
represent?
• How are the areas
of rectangles and
parallelograms related?
• How are the areas of
triangles and trapezoids
related?
Chapter 11 Overview
In this chapter, you will
explore and discover
answers to the following
Essential Questions:
• How can you use
measurements to
describe threedimensional figures?
• How can you use a net
to find the surface area
of a three-dimensional
figure?
• How can you find the
volume of a rectangular
prism?
xi
Chapter 12 Overview
In this chapter, you will
explore and discover
answers to the following
Essential Questions:
• How can you display
data and analyze
measures of center?
• When does it make
sense to display data
in a dot plot? in a
histogram?
• What are the
differences between
the three measures of
center?
12
Data Displays and
Measures of Center
647
Domain Statistics and Probability
COMMON CORE STATE STANDARDS 6.SP.A.1, 6.SP.B.4, 6.SP.B.5a, 6.SP.B.5b, 6.SP.B.5c,
6.SP.B.5d
1
2
3
4
5
6
7
8
Show What You Know . . . . . . . . . . . . . . . . . . .
Vocabulary Builder . . . . . . . . . . . . . . . . . . . . .
Recognize Statistical Questions . . . . . . . . . . . . . .
Describe Data Collection. . . . . . . . . . . . . . . . . .
Dot Plots and Frequency Tables . . . . . . . . . . . . .
Histograms . . . . . . . . . . . . . . . . . . . . . . . . . .
Mid-Chapter Checkpoint . . . . . . . . . . . . . . . . .
Investigate • Mean as Fair Share and Balance Point .
Measures of Center . . . . . . . . . . . . . . . . . . . . .
Effects of Outliers . . . . . . . . . . . . . . . . . . . . . .
Problem Solving • Data Displays. . . . . . . . . . . . .
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. 647
. 648
. 649
. 655
. 661
. 667
. 673
. 675
. 681
. 687
. 693
Chapter 12 Review/Test . . . . . . . . . . . . . . . . . . . . . . . . . . 699
13
Variability and
Data Distributions
705
Domain Statistics and Probability
Chapter 13 Overview
In this chapter, you will
explore and discover
answers to the following
Essential Questions:
• How can you describe
the shape of a data set
using graphs, measures
of center, and measures
of variability?
• How do you calculate
the different measures
of center?
• How do you calculate
the different measures
of variability?
1
2
3
4
5
6
7
8
Show What You Know . . . . . . . . . . . . . . . . . . . . .
Vocabulary Builder . . . . . . . . . . . . . . . . . . . . . . .
Patterns in Data . . . . . . . . . . . . . . . . . . . . . . . . .
Box Plots . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Investigate • Mean Absolute Deviation . . . . . . . . . . .
Measures of Variability . . . . . . . . . . . . . . . . . . . . .
Mid-Chapter Checkpoint . . . . . . . . . . . . . . . . . . .
Choose Appropriate Measures of Center and Variability
Apply Measures of Center and Variability . . . . . . . . .
Describe Distributions . . . . . . . . . . . . . . . . . . . . .
Problem Solving • Misleading Statistics . . . . . . . . . .
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. 705
. 706
. 707
. 713
. 719
. 725
. 731
. 733
. 739
. 745
. 751
Chapter 13 Review/Test . . . . . . . . . . . . . . . . . . . . . . . . . . 757
Glossary . . . . . . . . . . . . . . . . . . . . . . .
Common Core State Standards Correlations.
Index . . . . . . . . . . . . . . . . . . . . . . . . .
Table of Measures . . . . . . . . . . . . . . . . .
xii
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. .H1
. H17
. H27
. H39
© Houghton Mifflin Harcourt Publishing Company
COMMON CORE STATE STANDARDS 6.SP.A.2, 6.SP.A.3, 6.SP.B.4, 6.SP.B.5c, 6.SP.B.5d
Critical Area
The Number
System
© Houghton Mifflin Harcourt Publishing Company • Image Credits: (bg) ©Craig Tuttle/Corbis
CRITICAL AREA Completing understanding of division of
fractions and extending the notion of number to the system
of rational numbers, which includes negative numbers
1
Project
Sweet Success
Get Started
WRITE
Math
A company sells Apple Cherry Mix. They make large
batches of the mix that can be used to fill 250 bags
each. Determine how many pounds of each ingredient
should be used to make one batch of Apple Cherry Mix. Then decide how much the company should charge
for each bag of Apple Cherry Mix, and explain how you
made your decision.
Important Facts
Ingredients in Apple Cherry Mix
(1 bag)
• ​ 3_4 ​ pound of dried apples
• ​ 1_2 ​ pound of dried cherries
• ​ 1_4 ​ pound of walnuts
Cost of Ingredients
• dried apples: $2.80 per pound
• dried cherries: $4.48 per pound
• walnuts: $3.96 per pound
Completed by
2
Chapters 1–3
© Houghton Mifflin Harcourt Publishing Company • Image Credits: (cr) ©D. Hurst/Alamy; (br) ©Foodcollection RF/Getty Images; (cl) ©Frederic Cirou and Isabelle Rozenbaum/PhotoAlto/Corbis
Businesses that sell food products need to combine ingredients in the
correct amounts. They also need to determine what price to charge for
the products they sell.
1
Chapter
Whole Numbers and Decimals
Personal Math Trainer
Show Wha t You Know
Online Assessment
and Intervention
Check your understanding of important skills.
Name
Factors Find all of the factors of the number.
1. 16 ____
2. 27 ____
3. 30 ____
4. 45 ____
Round Decimals Round to the place of the underlined digit.
5. 0.323
6. 4.096
___
7. 10.67
___
8. 5.278
___
___
Multiply 3-Digit and 4-Digit Numbers Multiply.
9.
2,143
× 6
__
___
10.
375
×
8
_
___
11.
3,762
×
__7
___
12.
603
× 9
_
___
© Houghton Mifflin Harcourt Publishing Company
Math in the
Maxwell saved $18 to buy a fingerprinting kit
that costs $99. He spent 0.25 of his savings to
buy a magnifying glass. Help Maxwell find out
how much more he needs to save to buy the
fingerprinting kit.
Chapter 1
3
Voca bula ry Builder
Visualize It
Review Words
Complete the Flow Map using the words with a ✓.
✓ compatible numbers
Estimation
decimal
✓ dividend
Division
divisible
✓ divisor
84.15
÷
18.7
=
4.5
factor
prime number
✓ quotient
Preview Words
common factor
80
÷
20
=
4
greatest common
factor
least common multiple
prime factorization
Understand Vocabulary
Complete the sentences using the preview words.
1. The least number that is a common multiple of two or more
numbers is the _____.
2. The greatest factor that two or more numbers have in common
© Houghton Mifflin Harcourt Publishing Company
is the _____.
3. A number that is a factor of two or more numbers is a
_____.
4. A number written as the product of its prime factors is the
_____ of the number.
4
™Interactive Student Edition
™Multimedia eGlossary
Chapter 1 Vocabulary
common factor
dividend
factor común
dividendo
23
12
divisor
factor
divisor
factor
33
25
greatest common
factor (GCF)
least common
multiple (LCM)
máximo común
divisor (MCD)
mínimo común
múltiplo (m.c.m.)
38
prime factorization
prime number
descomposición en
factores primos
número primo
77
48
78
A number multiplied by another number to
find a product
2 × 3 × 7 = 42
factors
The least number that is a common multiple
of two or more numbers
Example:
Multiples of 4: 4, 8, 12, 16, 20, 24, 28, 32, 36
Multiples of 6: 12, 18, 24, 30, 36, 42
The LCM of 4 and 6 is 12.
A number that has exactly two factors:
1 and itself
Examples: 2, 3, 5, 7, 11, 13, 17, and 19 are
prime numbers. 1 is not a prime number.
© Houghton Mifflin Harcourt Publishing Company
dividend
© Houghton Mifflin Harcourt Publishing Company
dividend
© Houghton Mifflin Harcourt Publishing Company
6 18
18 ÷ 6
© Houghton Mifflin Harcourt Publishing Company
© Houghton Mifflin Harcourt Publishing Company
© Houghton Mifflin Harcourt Publishing Company
© Houghton Mifflin Harcourt Publishing Company
© Houghton Mifflin Harcourt Publishing Company
The number that is to be divided in a division
problem
A number that is a factor of two or more
numbers
Example:
Factors of 16: 1,
2, 4,
8,
Factors of 20: 1,
2, 4, 5,
16
10,
20
The number that divides the dividend
6 18
18 ÷ 6
divisor
divisor
The greatest factor that two or more
numbers have in common
Example:
Factors of 18: 1, 2, 3, 6, 9, 18
Factors of 30: 1, 2, 3, 5, 6, 10, 15, 30
The GCF of 18 and 30 is 6.
A number written as the product of all its
prime factors
24
Example:
4
2
6
2
2
3
24 = 2 × 2 × 2 × 3
Going Places with
Words
Going to
Washington, DC
For 2 to 4 players
Materials
• 3 of 1 color per player: red, blue, green, and yellow connecting cubes
• 1 number cube
How to Play
Game
Game
Word Box
common factor
dividend
divisor
factor
greatest common
factor (GCM)
least common
multiple (LCM)
prime factorization
prime number
1. Put your 4 connecting cubes in the START circle of the same color.
2. To get a cube out of START, you must roll a 6.
• If you roll a 6, move 1 of your cubes to the same-colored circle on the
path. If you do not roll a 6, wait until your next turn.
3. Once you have a cube on the path, toss the number cube to take a turn.
Move the cube that many tan spaces. You must get all three of your cubes
4. If you land on a space with a question, answer it. If you are correct,
move ahead 1 space.
5. To reach FINISH move your connecting cubes up the path that is
the same color as your connecting cubes. The first player to get
all three cubes on FINISH wins.
© Houghton Mifflin Harcourt Publishing Company
Image Credits: (bg) ©Uros Zunic/Shutterstock; (b) ©Fotolia
on the path.
Chapter 1
4A
Game
Game
How can you use a list to find
the least common multiple of
two numbers?
Explain how to use
prime factorization to find
a least common multiple.
Name two
common
factors
for 32
and 40.
How
many
factors
does a
prime
number
have?
What is the
greatest
common
factor of 28
and 49?
Explain
two ways
to find
the prime
factorization
of 210.
True or false:
The LCM of 7 and 14 is 42.
Explain how you know.
How can you find the common
factors of two numbers?
4B
© Houghton Mifflin Harcourt Publishing Company
Game
Game
Explain why a prime
factorization may be used as a
secret code.
What
is the least
common
multiple of
4 and 10?
What
is the
dividend
in 48 ÷ 8?
What is a
least
common
multiple?
What is a
greatest
common
factor?
2 x 2 x 3 x 5 is the
prime factorization of
what number?
Why might you
estimate when a dividend
has a decimal?
How can you change
a divisor such as 16.3 into a
whole number?
© Houghton Mifflin Harcourt Publishing Company
Image Credits: (bg) ©Image Plan/Corbis; (tl) ©tarajane/Fotolia; (bl) ©VanHart/Fotolia; (c), (br)
©PhotoDisc/Getty Images; (tr) ©GlowImages/Alamy
Chapter 1
4C
Journal
Jo
ou
urnal
The Write Way
Reflect
Choose one idea. Write about it.
• Explain how to find the prime factorization of 140.
• Two students found the least common multiple of 6 and 16.
Tell how you know which student’s answer is correct.
Student A: The LCM of 6 and 16 is 96.
Student B: The LCM of 6 and 16 is 48.
• Describe two ways to find the greatest common factor of 36 and 42.
• Write and solve a word problem that uses a whole number as a
© Houghton Mifflin Harcourt Publishing Company
Image Credits: (bg) ©Uros Zunic/Shutterstock; (t) ©PhotoDisc/Getty Images
divisor and a decimal as a dividend.
4D
Lesson 1.1
Name
Divide Multi-Digit Numbers
The Number System—
6.NS.B.2
MATHEMATICAL PRACTICES
MP1, MP2, MP6
Essential Question How do you divide multi-digit numbers?
Unlock
Unlock the
the Problem
Problem
When you watch a cartoon, the frames of film seem to blend together to form
a moving image. A cartoon lasting just 92 seconds requires 2,208 frames. How
many frames do you see each second when you watch a cartoon?
Divide 2,208 ÷ 92.
Estimate using compatible numbers. _ ÷ _ = _
2
92qw
2,208
− 1 84
368
−
Divide the tens.
Divide the ones.
Compare your estimate with the quotient. Since the estimate, _ ,
is close to _ , the answer is reasonable.
So, you see _ frames each second when you watch a cartoon.
Example 1 Divide 12,749 ÷ 18.
Estimate using compatible numbers. _ ÷ _ = _
© Houghton Mifflin Harcourt Publishing Company
STEP 1 Divide.
70
18qw
12,749
− 12 6
14
− 0
149
−
r5
STEP 2 Check your answer.
× 18
__
You can write a remainder
with an r, as a fractional part of
the divisor, or as a decimal. For
131 ÷ 5, the quotient can be
written as 26 r1, 26 1_5 , or 26.2.
Multiply the whole number part of the
quotient by the divisor.
+
__
+
12,749
Add the remainder.
So, 12,749 ÷ 18 = __.
Chapter 1
5
Example 2
Divide 59,990 ÷ 280. Write the remainder as a fraction.
Estimate using compatible numbers. __ ÷ _ = _
STEP 1 Divide.
STEP 2 Write the remainder as a fraction.
280qw
59,990
−
−
remainder = ____
________
divisor
280
Write the remainder over
the divisor.
70 ÷
_________
= ___
Simplify.
280 ÷
−
70
Compare your estimate with the quotient. Since the estimate, _
is close to _ , the answer is reasonable.
So, 59,990 ÷ 280 = _.
MATHEMATICAL
PRACTICE
1 Describe two ways to check your answer in Example 2.
Share
hhow
Share and
and Show
Sh
MATH
M
BOARD
B
1.
29qw
986
Think: 30 × 3 = 90
___
6
2.
37qw
3,786
___
© Houghton Mifflin Harcourt Publishing Company
Estimate. Then find the quotient. Write the remainder, if any, with an r.
Name
Estimate. Then find the quotient. Write the remainder, if any, as a fraction.
3. 6,114 ÷ 63
4. 11,050 ÷ 26
___
___
Math
Talk
MATHEMATICAL PRACTICES 6
Explain why you can use
multiplication to check a
division problem.
On
On Your
Your Own
Own
Estimate. Then find the quotient. Write the remainder, if any, as a fraction.
5. 3,150 ÷ 9
___
6. 2,115 ÷ 72
7. 20,835 ÷ 180
___
8. Find the least whole number that can replace ■
to make the statement true.
9.
110 < ■ ÷ 47
© Houghton Mifflin Harcourt Publishing Company
DEEPER
The 128 employees of a company
volunteer 12,480 hours in 26 weeks. On average,
how many hours do they all volunteer per
week? On average, how many hours does each
employee volunteer per week?
____
MATHEMATICAL
PRACTICE
2
Use Reasoning Name two
whole numbers that can replace ■ to make
both statements true.
2 × ■ < 1,800 ÷ 12
■ > 3,744 ÷ 52
____
____
10.
___
11.
DEEPER
A factory produces 30,480 bolts in
12 hours. If the same number of bolts are
produced each hour, how many bolts does the
factory produce in 5 hours?
____
Chapter 1 • Lesson 1 7
MATHEMATICAL PRACTICES .0%&-t3&"40/tM",&4&/4&
OqnakdlRnkuhmf¤@ookhb`shnmr
OqnakdlRnkuhmf¤@ookhb`shnmr
Use the table for 12–15.
12. A Smooth Flight jet carried 6,045 passengers
last week, and all of its flights were full. How
many flights did the jet make last week?
14.
15.
16.
8
DEEPER
Last month an airline made 6,322
reservations for flights from Newark, New
Jersey, to Frankfurt, Germany. If there were
21 full flights and 64 reservations cancelled,
which airplane made the flights?
SMARTER
An airline carries about
750 passengers from Houston to Chicago each
day. How many Blue Sky jets would be needed
to carry this many passengers, and how many
empty seats would there be?
Pose a Problem Refer back
to Problem 12. Use the information in the
table to write a similar problem involving
airplane passenger seats.
SMARTER
SMARTER
For numbers 16a–16d, choose
Yes or No to indicate whether the equation
is correct.
16a. 1,350 ÷ 5 = 270
Yes
No
16b. 3,732 ÷ 4 = 933
Yes
No
16c. 4,200 ÷ 35 = 12
Yes
No
16d. 1,586 ÷ 13 = 122
Yes
No
Airplane Passenger Seats
Type of Plane
Seats
Jet Set
298
Smooth Flight
403
Blue Skies
160
Cloud Mover
70
WRITE
Math t Show Your Work
© Houghton Mifflin Harcourt Publishing Image Credits: Company (t) ©PhotoDisc/Getty Images
13.
Practice and Homework
Name
Lesson 1.1
Divide Multi-Digit Numbers
COMMON CORE STANDARD—6.NS.B.2
Compute fluently with multi-digit numbers and
find common factors and multiples.
Estimate. Then find the quotient. Write the remainder, if any,
with an r.
13
2. 19qw
800
1. 55qw
715
55
165
165
0
3. 68qw
1,025
Estimate:
700 ÷ 50 = 15
Estimate. Then find the quotient. Write the remainder, if any, as a fraction.
4. 20qw
1,683
5. 14,124 ÷ 44
Find the least whole number that can replace
7.
÷ 7 > 800
8.
6. 11,629 ÷ 29
to make the statement true.
÷ 21 > 13
9. 15 <
÷ 400
© Houghton Mifflin Harcourt Publishing Company
Problem
Problem Solving
Solving
10. A plane flew a total of 2,220 miles. Its average
speed was 555 miles per hour. How many hours
did the plane fly?
12.
WRITE
11. A van is carrying 486 pounds. There are
27 boxes in the van. What is the average weight of
each box in the van?
Math Find 56,794 ÷ 338. Write the quotient twice,
once with the remainder as a fraction and once with an r.
Chapter 1
9
Lesson Check (6.NS.B.2)
1. A caterer’s fee is based on the number of meals
she provides. How much is the price per meal if
the total fee is $1,088 for 64 meals?
2. Amelia needs 24 grams of beads to make a
bracelet. She has 320 grams of beads. How
many bracelets can she make?
Spiral Review (5.NBT.A.2, 5.NBT.A.3b, 5.NBT.B.7)
4. Gavin bought 4 packages of cheese. Each
package weighed 1.08 kilograms. How many
kilograms of cheese did Gavin buy?
5. Mr. Thompson received a water bill for $85.98.
The bill covered three months of service. He
used the same amount of water each month.
How much does Mr. Thompson pay for
water each month?
6. Layla used 0.482 gram of salt in her experiment.
Maurice use 0.51 gram of salt. Who used the
greater amount of salt?
© Houghton Mifflin Harcourt Publishing Company
3. Hank bought 2.4 pounds of apples. Each pound
cost $1.95. How much did Hank spend on the
apples?
FOR MORE PRACTICE
GO TO THE
10
Personal Math Trainer
Lesson 1.2
Name
Prime Factorization
The Number System—
6.NS.B.4
MATHEMATICAL PRACTICES
MP1, MP3, MP6, MP7
Essential Question How do you write the prime factorization of
a number?
Unlock
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the Problem
Problem
Secret codes are often used to send information over the
Internet. Many of these codes are based on very large
numbers. For some codes, a computer must determine
the prime factorization of these numbers to decode
the information.
The prime factorization of a number is the number written
as a product of all of its prime factors.
One Way Use a factor tree.
The key for a code is based on the prime factorization of 180. Find
the prime factorization of 180.
A prime number is a whole
number greater than 1 that has
exactly two factors: itself and 1.
Choose any two factors whose product is 180. Continue finding
factors until only prime factors are left.
© Houghton Mifflin Harcourt Publishing Company Image Credits: (r) ©PhotoDisc/Getty Images
A Use a basic fact.
B Use a divisibility rule.
Think: 10 times what number is equal to 180?
Think: 180 is even, so it is divisible by 2.
10 × _ = 180
2 × _ = 180
180
180
2
10
2
2
2
6
2
3
3
180 = _ × _ × _ × _ × _ List the prime factors from least to greatest.
So, the prime factorization of 180 is _ × _ × _ × _ × _.
Math
Talk
MATHEMATICAL PRACTICES 3
Apply How can you apply divisibility
rules to make the factor tree on the left?
Chapter 1
11
Another Way Use a ladder diagram.
The key for a code is based on the prime factorization of 140. Find the prime
factorization of 140.
Choose a prime factor of 140. Continue dividing by prime factors until the
quotient is 1.
A Use the divisibility rule for 2.
B Use the divisibility rule for 5.
Think: 140 is even, so 140 is divisible by 2.
Think: The last digit is 0, so 140 is divisible by 5.
5 140
2 140
140 ÷ 2 = 70
2
7 70
prime factors
14
10
2
2
1
140 = _ × _ × _ × _
List the prime factors from least to greatest.
So, the prime factorization of 140 is _ × _ × _ × _.
Math
Talk
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and Show
Sh
MATH
M
BOARD
B
MATHEMATICAL PRACTICES 1
Describe a different strategy
you could use to find the
prime factorization of 140.
Find the prime factorization.
2. 42
18
6
2 42
3
7
18 = _ × _ × _
12
42 = _ × _ × _
© Houghton Mifflin Harcourt Publishing Company
1. 18
Name
Find the prime factorization.
3. 75
4. 12
5. 65
Math
Talk
MATHEMATICAL PRACTICES 6
Explain why a prime number
cannot be written as a
product of prime factors.
On
On Your
Your Own
Own
Write the number whose prime factorization is given.
6. 2 × 2 × 2 × 7
7. 2 × 2 × 5 × 5
8. 2 × 2 × 2 × 2 × 3 × 3
© Houghton Mifflin Harcourt Publishing Company
Practice: Copy and Solve Find the prime factorization.
9. 45
10. 50
11. 32
12. 76
13. 108
14. 126
15. The area of a rectangle is the product of its length
and width. A rectangular poster has an area of
260 square inches. The width of the poster is
greater than 10 inches and is a prime number.
What is the width of the poster?
16.
MATHEMATICAL
PRACTICE
7 Look for Structure Dani says she
is thinking of a secret number. As a clue, she
says the number is the least whole number that
has three different prime factors. What is Dani’s
secret number? What is its prime factorization?
Chapter 1 • Lesson 2 13
MATHEMATICAL PRACTICES
COMMUNICA5&t1&34E7&3&tCONSTRUCT ARGUMENTS
OqnakdlRnkuhmf¤@ookhb`shnmr
OqnakdlRnkuhmf¤@ookhb`shnmr
Use the table for 17–19. Agent Sanchez must enter a
code on a keypad to unlock the door to her office.
17. In August, the digits of the code number are the
prime factors of 150. What is the code number for
the office door in August?
18.
DEEPER
In September, the fourth digit of the
code number is 2 more than the fourth digit of the
code number based on the prime factors of 225.
The prime factors of what number were used for the
code in September?
Code Number Rules
1. The code is a 4-digit number.
2. Each digit is a prime number.
19.
3. The prime numbers are entered from least to
greatest.
SMARTER
One day in October,
Agent Sanchez enters the code 3477.
How do you know that this code is
incorrect and will not open the door?
4. The code number is changed at the beginning
of each month.
WRITE
20.
Math t Show Your Work
SMARTER
Use the numbers to complete
the factor tree. You may use a number more
than once.
2
3
6
9
18
6
Write the prime factorization of 36.
14
© Houghton Mifflin Harcourt Publishing Company
36
Practice and Homework
Name
Lesson 1.2
Prime Factorization
COMMON CORE STANDARD—6.NS.B.4
Compute fluently with multi-digit numbers and
find common factors and multiples.
Find the prime factorization.
1. 44
4
2
3. 48
2. 90
44
11
2
11
2 3 2 3 11
4. 204
____
5. 400
____
____
6. 112
____
____
Problem
Problem Solving
Solving
© Houghton Mifflin Harcourt Publishing Company
7. A computer code is based on the prime
factorization of 160. Find the prime factorization
of 160.
9.
WRITE
8. The combination for a lock is a 3-digit number.
The digits are the prime factors of 42 listed from
least to greatest. What is the combination for the
lock?
Math Describe two methods for
finding the prime factorization of a number.
Chapter 1
15
Lesson Check (6.NS.B.4)
1. Maritza remembers her PIN because it is the
product of two consecutive prime numbers that
is between 1000 and 1500. What is her PIN?
2. Brent knows that the 6-digit number he uses to
open his computer is the prime factorization of
5005. If each digit of the code increases from left
to right, what is his code?
Spiral Review (5.OA.A.2, 5.NBT.A.1, 5.NBT.B.6)
4. A jet plane costs an airline $69,500,000. What is
the place value of the digit 5 in this number?
5. A museum has 13,486 butterflies, 1,856 ants,
and 13,859 beetles. What is the order of the
insects from least number to greatest number?
6. Juan is reading a 312-page book for school. He
reads 12 pages each day. How long will it take
him to finish the book?
© Houghton Mifflin Harcourt Publishing Company
3. Piano lessons cost $15. What expressions could
be used to find the cost in dollars of 5 lessons?
FOR MORE PRACTICE
GO TO THE
16
Personal Math Trainer
Lesson 1.3
Name
Least Common Multiple
The Number System—
6.NS.B.4
MATHEMATICAL PRACTICES
MP2, MP3, MP4, MP6
Essential Question How can you find the least common multiple of two
whole numbers?
Unlock
Unlock the
the Problem
Problem
In an experiment, each flowerpot will get one seed. If the
flowerpots are in packages of 6 and the seeds are in
packets of 8, what is the least number of plants that can
be grown without any seeds or pots left over?
• Explain why you cannot buy the same
number of packages of each item.
The least common multiple, or LCM, is the least number
that is a common multiple of two or more numbers.
One Way Use a list.
Make a list of the first eight nonzero multiples of 6
and 8. Circle the common multiples. Then find the
least common multiple.
Multiples of 6: 6, 12, 18, _ ,_ ,_ ,_ ,_
Multiples of 8: 8, 16, 24, _ ,_ ,_ ,_ ,_
© Houghton Mifflin Harcourt Publishing Company Image Credits: (cr) ©C Squared Studios/Photodisc/Getty Images
The least common multiple, or LCM, is _ .
Another Way Use prime factorization and a Venn diagram.
Write the prime factorization of
each number.
6=2×_
8=2×_×_
List the common prime factors of the
numbers, if any.
6 and 8 have one prime factor of _ in common.
Prime factors of 6
Place the prime factors of the numbers in
the appropriate parts of the Venn diagram.
To find the LCM, find the product of all of
the prime factors in the Venn diagram.
Prime factors of 8
2
3
3×2×2×2=_
The LCM is _.
So, the least number of plants is _.
Common prime factors
Math
Talk
MATHEMATICAL PRACTICES 4
Model How does the diagram
help you find the LCM of 6 and 8?
Chapter 1
17
Example Use prime factorization to find the LCM of 12 and 18.
Write the prime factorization
of each number.
12 = 2 × 2 × _
Line up the common factors.
18 = 2
Multiply one number from
each column.
×
3
×_
2×2×
3
×
3 = 36
The factors in the prime
factorization of a number are
usually listed in order from least
to greatest.
So, the LCM of 12 and 18 is _.
Try This! Find the LCM.
A 10, 15, and 25
B 3 and 12
Use prime factorization.
Use a list.
10 = ___
Multiples of 3: ___
15 = ___
Multiples of 12: ___
25 = ___
The LCM is _.
____
The LCM is _.
1. How can you tell whether the LCM of a pair of numbers is one of the
numbers? Give an example.
2.
MATHEMATICAL
PRACTICE
6 Explain one reason why you might use prime factorization
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1. List the first six nonzero multiples of 6 and 9. Circle the common
multiples. Then find the LCM.
Multiples of 6: ____
Multiples of 9: ____
18
The LCM of 6 and 9 is _.
© Houghton Mifflin Harcourt Publishing Company
instead of making a list of multiples to find the LCM of 10, 15, and 25.
Name
Find the LCM.
3. 3, 9
2. 3, 5
__
4. 9, 15
__
__
Math
Talk
On
On Your
Your Own
Own
MATHEMATICAL PRACTICES 6
Explain what the LCM of
two numbers represents.
Find the LCM.
5. 5, 10
6. 3, 8
__
MATHEMATICAL
PRACTICE
2
7. 9, 12
__
__
Use Reasoning Algebra Write the unknown number for the ■.
LCM: ■
8. 5, 8
9.
■ = __
10.
© Houghton Mifflin Harcourt Publishing Company
11.
■, 6
LCM: 42
■ = __
SMARTER
How can you tell when the LCM of two numbers will
equal one of the numbers or equal the product of the numbers?
MATHEMATICAL
PRACTICE
Verify the Reasoning of Others
Mr. Haigwood is shopping for a school picnic.
Veggie burgers come in packages of 15, and buns
come in packages of 6. He wants to serve veggie
burgers on buns and wants to have no items left
over. Mr. Haigwood says that he will have to buy
at least 90 of each item, since 6 × 15 = 90. Do
you agree with his reasoning? Explain.
3
12.
DEEPER
A deli has a special one-day event to
celebrate its anniversary. On the day of the event,
every eighth customer receives a free drink.
Every twelfth customer receives a free sandwich.
If 200 customers show up for the event, how
many of the customers will receive both a free
drink and a free sandwich?
Chapter 1 • Lesson 3 19
MATHEMATICAL PRACTICES
ANALYZEt-00,'034536$563&t13&$*4*0/
Unlock
Unlock the
the Problem
Problem
13. Katie is making hair clips to sell at the craft fair. To make each hair clip, she
uses 1 barrette and 1 precut ribbon. The barrettes are sold in packs of 12, and
the precut ribbons are sold in packs of 9. How many packs of each item does
she need to buy to make the least number of hair clips with no supplies left over?
a. What information are you given?
b. What problem are you being asked to solve?
c. Show the steps you use to solve
the problem.
d. Complete the sentences.
The least common multiple of
12 and 9 is _ .
Katie can make _ hair clips with no
supplies left over.
To get 36 barrettes and 36 ribbons, she
needs to buy _ packs of barrettes
and _ packs of precut ribbons.
20
SMARTER
Reptile stickers
come in sheets of 6 and fish
stickers come in sheets of 9.
Antonio buys the same number
of both types of stickers and he
buys at least 100 of each type.
What is the least number of
sheets of each type he might buy?
15.
SMARTER
For numbers 15a–15d,
choose Yes or No to indicate whether the
LCM of the two numbers is 16.
15a.
2, 8
Yes
No
15b.
2, 16
Yes
No
15c.
4, 8
Yes
No
15d.
8, 16
Yes
No
© Houghton Mifflin Harcourt Publishing Company
14.
Practice and Homework
Name
Lesson 1.3
Least Common Multiple
COMMON CORE STANDARD—6.NS.B.4
Compute fluently with multi-digit numbers and
find common factors and multiples.
Find the LCM.
1. 2, 7
2. 4, 12
3. 6, 9
Multiples of 2: 2, 4, 6, 8, 10, 12, 14
Multiples of 7: 7, 14
14
LCM: _
4. 5, 4
LCM: _
LCM: _
5. 5, 8, 4
LCM: _
Write the unknown number for the
7. 3,
LCM: 21
=_
6. 12, 8, 24
LCM: _
LCM: _
.
8.
,7
9. 10, 5
LCM: 63
=_
LCM:
=_
Problem
Problem Solving
Solving
© Houghton Mifflin Harcourt Publishing Company
10. Juanita is making necklaces to give as presents.
She plans to put 15 beads on each necklace.
Beads are sold in packages of 20. What is the
least number of packages she can buy to make
necklaces and have no beads left over?
_______
12.
WRITE
11. Pencils are sold in packages of 10, and erasers are
sold in packages of 6. What is the least number of
pencils and erasers you can buy so that there is
one pencil for each eraser with none left over?
_______
Math Explain when you would use each method
(finding multiples or prime factorization) for finding the LCM
and why.
Chapter 1
21
Lesson Check (6.NS.B.4)
1. Martha is buying hot dogs and buns for the
class barbecue. The hot dogs come in packages
of 10. The buns come in packages of 12. What
is the least number she can buy of each so that
she has exactly the same number of hot dogs
and buns? How many packages of each should
she buy?
2. Kevin makes snack bags that each contain a box
of raisins and a granola bar. Each package of
raisins contains 9 boxes. The granola bars come
12 to a package. What is the least number he
can buy of each so that he has exactly the same
number of granola bars and boxes of raisins?
How many packages of each should he buy?
3. John has 2,456 pennies in his coin collection.
He has the same number of pennies in each of
3 boxes. Estimate to the nearest hundred the
number of pennies in each box.
4. What is the distance around a triangle that has
sides measuring 2 1_8 feet, 3 1_2 feet, and 2 1_2 feet?
5. The 6th grade class collects $1,575. The class
wants to give the same amount of money
to each of 35 charities. How much will each
charity receive?
6. Jean needs 1_3 cup of walnuts for each serving
of salad she makes. She has 2 cups of walnuts.
How many servings can she make?
FOR MORE PRACTICE
GO TO THE
22
Personal Math Trainer
© Houghton Mifflin Harcourt Publishing Company
Spiral Review (5.NBT.B.6, 5.NF.B.7c)
Lesson 1.4
Name
Greatest Common Factor
The Number System—
6.NS.B.4
MATHEMATICAL PRACTICES
MP1, MP4, MP7
Essential Question How can you find the greatest common factor of two
whole numbers?
A common factor is a number that is a factor of two or more numbers.
The numbers 16 and 20 have 1, 2, and 4 as common factors.
Factors of 16: 1, 2, 4,
8,
16
Factors of 20: 1, 2, 4, 5, 10,
A number that is multiplied by
another number to find a
product is a factor.
20
Factors of 6: 1, 2, 3, 6
The greatest common factor, or GCF, is the greatest factor that two
or more numbers have in common. The greatest common factor of
16 and 20 is 4.
Factors of 9: 1, 3, 9
Every number has 1 as a factor.
Unlock
Unlock the
the Problem
Problem
Jim is cutting two strips of wood to make picture
frames. The wood strips measure 12 inches and
18 inches. He wants to cut the strips into equal
lengths that are as long as possible. Into what
lengths should he cut the wood?
12 inches
18 inches
Find the greatest common factor, or GCF, of 12 and 18.
One Way Use a list.
Factors of 12: 1, 2, _ , _ , _ , 12
Factors of 18: 1, _ , _ , _ , _ , _
The greatest common factor, or GCF, is _ .
Math
Talk
MATHEMATICAL PRACTICES 1
Analyze Into what other lengths
could Jim cut the wood to obtain
equal lengths?
Another Way Use prime factorization.
Write the prime factorization of each number.
Prime factors of 18
Prime factors of 12
© Houghton Mifflin Harcourt Publishing Company
12 = 2 × _ × 3
18 = _ × 3 × _
Place the prime factors of the numbers in the
appropriate parts of the Venn diagram.
2
3
To find the GCF, find the product of the common prime factors.
2×3=_
The GCF is _.
Common prime
factors
So, Jim should cut the wood into _-inch lengths.
Chapter 1
23
Distributive Property
Multiplying a sum by a number is the same as multiplying
5 × (8 + 6) = (5 × 8) + (5 × 6)
each addend by the number and then adding the products.
You can use the Distributive Property to express the sum of two whole
numbers as a product if the numbers have a common factor.
Example Use the GCF and the Distributive
Property to express 36 + 27 as a product.
Find the GCF of 36 and 27.
Write each number as the product
of the GCF and another factor.
GCF: _
36 + 27
(9 × _ ) + (9 × _ )
Use the Distributive Property to
write 36 + 27 as a product.
9 × (4 + _ )
Check your answer.
36 + 27 = _
9 × (4 + _ ) = 9 × _ = _
So, 36 + 27 = _ × ( _ + _ ).
2.
MATHEMATICAL
PRACTICE
4
Use Diagrams Describe how the figure at the right
shows that 36 + 27 = 9 × (4 + 3) .
4
3
9×4
9×3
9
24
© Houghton Mifflin Harcourt Publishing Company
1. Explain two ways to find the GCF of 36 and 27.
Name
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1. List the factors of 12 and 20. Circle the GCF.
Factors of 12: ___
Factors of 20: ___
Find the GCF.
2. 16, 18
3. 25, 40
__
__
4. 24, 40
5. 14, 35
__
__
Use the GCF and the Distributive Property to express the sum as a product.
6. 21 + 28
7. 15 + 27
__
__
8. 40 + 15
9. 32 + 20
__
Math
Talk
__
MATHEMATICAL PRACTICES 1
Analyze Describe how to use the
prime factorization of two numbers
to find their GCF.
On
On Your
Your Own
Own
Find the GCF.
10. 8, 25
11. 31, 32
__
__
12. 56, 64
__
13. 150, 275
__
© Houghton Mifflin Harcourt Publishing Company
Use the GCF and the Distributive Property to express the sum as a product.
14. 24 + 30
15. 49 + 14
__
18.
MATHEMATICAL
PRACTICE
__
16. 63 + 81
__
17. 60 + 12
__
1
Describe the difference between the LCM and the
GCF of two numbers.
Chapter 1 • Lesson 4 25
MATHEMATICAL PRACTICES .0%&-t3&"40/tM",&4&/4&
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OqnakdlRnkuhmf¤@ookhb`shnmr
Use the table for 19–22. Teachers at the Scott School of
Music teach only one instrument in each class.
19. Francisco teaches group lessons to all of the violin and
viola students at the Scott School of Music. All of his
classes have the same number of students. What is the
greatest number of students he can have in each class?
20.
21.
22.
DEEPER
Amanda teaches all of the bass and viola
students. All her classes have the same number of
students. Each class has the greatest possible number
of students. How many of these classes does she teach?
Scott School of Music
Instrument
SMARTER
Mia teaches jazz classes. She has
9 students in each class, and she teaches all the students
who play two instruments. How many students does
she have, and which two instruments does she teach?
WRITE
Number of
Students
Bass
20
Cello
27
Viola
30
Violin
36
Math Explain how you could use the GCF and the Distributive
23.
SMARTER
6=2×3
12 = 2 × 2 × 3
The prime factorization of each number is shown.
Prime factors of 6
Prime factors of 12
Using the prime factorization, complete the Venn
diagram and write the GCF of 6 and 12.
GCF = ______
26
Common prime factors
© Houghton Mifflin Harcourt Publishing Company • Image Credits: (t) ©mediacolours/Alamy Images
Property to express the sum of the number of bass students and the number
of violin students as a product.
Practice and Homework
Name
Lesson 1.4
Greatest Common Factor
COMMON CORE STANDARD—6.NS.B.4
Compute fluently with multi-digit numbers and
find common factors and multiples.
List the common factors. Circle the greatest common factor.
2. 36 and 90
1. 25 and 10
1, 5
____
3. 45 and 60
____
____
Find the GCF.
4. 14, 18
5. 6, 48
____
6. 16, 100
____
____
Use the GCF and the Distributive Property to express the sum as a product.
8. 18 + 27
7. 20 + 35
____
9. 64 + 40
____
____
Problem
Problem Solving
Solving
© Houghton Mifflin Harcourt Publishing Company
10. Jerome is making prizes for a game at the school
fair. He has two bags of different pins, one with
15 pins and one with 20 pins. Every prize will
have one kind of pin and will have the greatest
number of pins possible. How many pins should
be in each prize?
13.
WRITE
11. There are 24 sixth graders and 40 seventh
graders. Mr. Chan wants to divide both grades
into groups of equal size, with the greatest
possible number of students in each group.
How many students should be in each group?
Math Write a short paragraph to explain how
to use prime factorization and the Distributive Property to
express the sum of two whole numbers as a product.
Chapter 1
27
Lesson Check (6.NS.B.4)
1. There are 15 boys and 10 girls in Miss Li’s class.
She wants to group all the students so that each
group has the same number of boys and girls as
the other groups. What is the greatest number
of groups she can have?
2. A pet shop manager wants the same number
of birds in each cage. He wants to use as few
cages as possible, but can only have one type
of bird in each cage. If he has 42 parakeets and
18 canaries, how many birds will he put in
each cage?
3. There are 147 people attending a dinner party.
If each table can seat 7 people, how many tables
are needed for the dinner party?
4. Sammy has 3 pancakes. He cuts each one in
half. How many pancake halves are there?
5. The Cramer Company had a profit of $8,046,890
and the Coyle Company had a profit of
$8,700,340 last year. Which company had the
greater profit?
6. There are 111 guests attending a party. There
are 15 servers. Each server has the same
number of guests to serve. Jess will serve any
extra guests. How many guests will Jess be
serving?
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Spiral Review (5.NBT.B.5, 5.NBT.B.6, 5.NF.B.7c, 6.NS.B.2)
PROBLEM SOLVING
Name
Lesson 1.5
Problem Solving • Apply the Greatest
Common Factor
The Number System—
6.NS.B.4
MATHEMATICAL PRACTICES
MP1, MP5, MP6
Essential Question How can you use the strategy draw a diagram to
help you solve problems involving the GCF and the Distributive Property?
Unlock
Unlock the
the Problem
Problem
A trophy case at Riverside Middle School holds 18 baseball
trophies and 24 soccer trophies. All shelves hold the same
number of trophies. Only one sport is represented on each shelf.
What is the greatest number of trophies that can be on each
shelf? How many shelves are there for each sport?
Use the graphic organizer to help you solve the problem.
Read the Problem
What do I need to find?
Solve the Problem
Total trophies = baseball + soccer
18 + 24
I need to find
GCF: _
Find the GCF of 18 and 24.
© Houghton Mifflin Harcourt Publishing Company • Image Credits: (r) ©Randy Faris/Corbis
What information do I need
to use?
18 + 24
Write each number as the
product of the GCF and
another factor.
(6 × _) + (6 × _)
6 × (_ + _)
Use the Distributive Property
to write 18 + 24 as a product.
I need to use
How will I use the information?
Use the product to draw a
diagram of the trophy case.
Use B’s to represent baseball
trophies. Use S’s to represent
soccer trophies.
BBBBBB
SSSSSS
I can find the GCF of __ and
use it to draw a diagram representing
the __ of the trophy case.
Math
Talk
So, there are _ trophies on each shelf. There are _ shelves of
MATHEMATICAL PRACTICES 5
Use Tools Explain how the
Distributive Property helped you
solve the problem.
baseball trophies and _ shelves of soccer trophies.
Chapter 1
29
Try Another Problem
Delia is bagging 24 onion bagels and 16 plain bagels for her bakery
customers. Each bag will hold only one type of bagel. Each bag will hold the
same number of bagels. What is the greatest number of bagels she can put in
each bag? How many bags of each type of bagel will there be?
Use the graphic organizer to help you solve the problem.
Read the Problem
Solve the Problem
What do I need to find?
How will I use the information?
So, there will be _ bagels in each bag. There will be
_ bags of onion bagels and _ bags of plain bagels.
•
30
MATHEMATICAL
PRACTICE
6
Explain how knowing that the GCF of 24 and 16 is 8 helped
you solve the bagel problem.
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What information do I need to use?
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1. Toby is packaging 21 baseball cards and 12 football cards to
sell at a swap meet. Each packet will have the same number
of cards. Each packet will have cards for only one sport.
What is the greatest number of cards he can place in each
packet? How many packets will there be for each sport?
First, find the GCF of 21 and 12.
Unlock the Problem
√
√
Circle important facts.
√
Check your answer.
Check to make sure you answered
the question.
WRITE
Math
Show Your Work
Next, use the Distributive Property to write 21 + 12 as a
product, with the GCF as one of the factors.
So, there will be _ packets of baseball cards and
_ packets of football cards. Each packet will
contain _ cards.
© Houghton Mifflin Harcourt Publishing Company
2.
What if Toby had decided to keep one
baseball card for himself and sell the rest? How would your
answers to the previous problem have changed?
SMARTER
3. Melissa bought 42 pine seedlings and 30 juniper seedlings
to plant in rows on her tree farm. She wants each row to
have the same number of seedlings. She wants only one
type of seedling in each row. What is the greatest number of
seedlings she can plant in each row? How many rows of each
type of tree will there be?
Chapter 1 • Lesson 5 31
MATHEMATICAL PRACTICES
COMMUNICA5&t1&34E7&3&tCONSTRUCT ARGUMENTS
On
On Your
Your Own
Own
4.
MATHEMATICAL
PRACTICE
1 Make Sense of Problems A drum and bugle
WRITE
Math t Show Your Work
marching band has 45 members who play bugles and
27 members who play drums. When they march, each row has
the same number of players. Each row has only bugle players
or only drummers. What is the greatest number of players
there can be in each row? How many rows of each type of
player can there be?
5.
6.
SMARTER
The “color guard” of a drum and bugle band
consists of members who march with flags, hoops, and other
props. How would your answers to Exercise 4 change if there
were 21 color guard members marching along with the bugle
players and drummers?
DEEPER
If you continue the pattern below so that you
write all of the numbers in the pattern less than 500, how
many even numbers will you write?
4, 9, 14, 19, 24, 29…
Personal Math Trainer
32
SMARTER
Mr. Yaw’s bookcase holds
20 nonfiction books and 15 fiction books. Each shelf
holds the same number of books and contains only one type
of book. How many books will be on each shelf if each shelf
has the greatest possible number of books? Show your work.
© Houghton Mifflin Harcourt Publishing Company
7.
Practice and Homework
Name
Lesson 1.5
Problem Solving • Apply the Greatest
Common Factor
COMMON CORE STANDARD—6.NS.B.4
Compute fluently with multi-digit numbers
and find common factors and multiples.
Read the problem and solve.
1. Ashley is bagging 32 pumpkin muffins and 28 banana muffins for
some friends. Each bag will hold only one type of muffin. Each bag
will hold the same number of muffins. What is the greatest number
of muffins she can put in each bag? How many bags of each type
of muffin will there be?
GCF: 4
32 = 4 3 8
28 = 4 3 7
32 + 28 = 4 × (8 + 7)
8 bags of pumpkin muffins and _
7 bags of banana muffins,
So, there will be _
4 muffins in each bag.
with _
2. Patricia is separating 16 soccer cards and 22 baseball cards
into groups. Each group will have the same number of cards,
and each group will have only one kind of sports card. What is
the greatest number of cards she can put in each group? How
many groups of each type will there be?
© Houghton Mifflin Harcourt Publishing Company
3. Bryan is setting chairs in rows for a graduation ceremony. He has
50 black chairs and 60 white chairs. Each row will have the same
number of chairs, and each row will have the same color chair.
What is the greatest number of chairs that he can fit in each row?
How many rows of each color chair will there be?
4. A store clerk is bagging spices. He has 18 teaspoons of cinnamon
and 30 teaspoons of nutmeg. Each bag needs to contain the same
number of teaspoons, and each bag can contain only one spice.
What is the maximum number of teaspoons of spice the clerk
can put in each bag? How many bags of each spice will there be?
5.
WRITE
Math Write a problem in which you need to put
as many of two different types of objects as possible into equal
groups. Then use the GCF, Distributive Property, and a diagram
to solve your problem.
Chapter 1
33
Lesson Check (6.NS.B.4)
1. Fred has 36 strawberries and 42 blueberries. He
wants to use them to garnish desserts so that
each dessert has the same number of berries,
but only one type of berry. He wants as much
fruit as possible on each dessert. How many
berries will he put on each dessert? How many
desserts with each type of fruit will he have?
2. Dolores is arranging coffee mugs on shelves in
her shop. She wants each shelf to have the same
number of mugs. She only wants one color of
mug on each shelf. If she has 49 blue mugs and
56 red mugs, what is the greatest number she
can put on each shelf? How many shelves does
she need for each color?
3. A rectangle is 3 1_3 feet long and 2 1_3 feet wide.
What is the distance around the rectangle?
4. Lowell bought 4 1_4 pounds of apples and
3 3_5 pounds of oranges. How many pounds of
fruit did Lowell buy?
5. How much heavier is a 9 1_8 pound box than
a 2 5_6 pound box?
6. The combination of Clay’s locker is the prime
factors of 102 in order from least to greatest.
What is the combination of Clay’s locker?
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Spiral Review (5.NF.A.1, 5.NF.A.2, 6.NS.B.4)
Name
Mid-Chapter Checkpoint
Personal Math Trainer
Online Assessment
and Intervention
Vocabulary
Vocabulary
Vocabulary
greatest common
Choose the best term from the box to complete the sentence.
factor
1. The _____ of two numbers is greater than
or equal to the numbers. (p. 17)
least common multiple
prime number
2. The _____ of two numbers is less than or
equal to the numbers. (p. 23)
Concepts
Concepts and
and Skills
Skills
Estimate. Then find the quotient. Write the remainder,
if any, with an r. (6.NS.B.2)
3. 2,800 ÷ 25
4. 19,129 ÷ 37
___
___
5. 32,111 ÷ 181
___
Find the prime factorization. (6.NS.B.4)
6. 44
___
7. 36
___
8. 90
___
Find the LCM. (6.NS.B.4)
© Houghton Mifflin Harcourt Publishing Company
9. 8, 10
___
10. 4, 14
___
11. 6, 9
___
Find the GCF. (6.NS.B.4)
12. 16, 20
___
13. 8, 52
___
14. 36, 54
___
Chapter 1
35
15. A zookeeper divided 2,440 pounds of food equally among 8 elephants.
How many pounds of food did each elephant receive? (6.NS.B.2)
16. DVD cases are sold in packages of 20. Padded mailing envelopes are
sold in packets of 12. What is the least number of cases and envelopes
you could buy so that there is one case for each envelope with none left
over? (6.NS.B.4)
17.
DEEPER
Max bought two deli sandwich rolls measuring 18 inches
and 30 inches. He wants them to be cut into equal sections that are as
long as possible. Into what lengths should the rolls be cut? How many
sections will there be in all? (6.NS.B.4)
18. Susan is buying supplies for a party. If spoons only come in bags of 8 and
forks only come in bags of 6, what is the least number of spoons and the
least number of forks she can buy so that she has the same number of
each? (6.NS.B.4)
36
DEEPER
Tina is placing 30 roses and 42 tulips in vases for table
decorations in her restaurant. Each vase will hold the same number of
flowers. Each vase will have only one type of flower. What is the greatest
number of flowers she can place in each vase? If Tina has 24 tables in
her restaurant, how many flowers can she place in each vase? (6.NS.B.4)
© Houghton Mifflin Harcourt Publishing Company
19.
Lesson 1.6
Name
Add and Subtract Decimals
The Number System—
6.NS.B.3
MATHEMATICAL PRACTICES
MP3, MP6, MP7
Essential Question How do you add and subtract multi-digit decimals?
connect The place value of a digit in a number shows
the value of the digit. The number 2.358 shows 2 ones,
3 tenths, 5 hundredths, and 8 thousandths.
Place Value
Thousands
Hundreds
Tens
Ones
2
Tenths
Hundredths
Thousandths
3
5
8
•
Unlock
Unlock the
the Problem
Problem
Amanda and three of her friends volunteer at the
local animal shelter. One of their jobs is to weigh
the puppies and kittens and chart their growth.
Amanda’s favorite puppy weighed 2.358 lb last
month. If it gained 1.08 lb, how much does it weigh
this month?
• How do you know whether to add or
subtract the weights given in the problem?
Add 2.358 + 1.08.
Estimate the sum. _ + _ = _
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2.358
+
1.08
__
Compare your estimate with the sum. Since the estimate,
_, is close to __, the answer is reasonable.
So, the puppy weighs __ lb this month.
1.
MATHEMATICAL
PRACTICE
7 Look for Structure Is it necessary to add a zero after 1.08
to find the sum? Explain.
2. Explain how place value can help you add decimals.
Chapter 1
37
Example 1
A bee hummingbird, the world’s smallest bird, has a mass of
1.836 grams. A new United States nickel has a mass of
5 grams. What is the difference in grams between the mass
of a nickel and the mass of a bee hummingbird?
You can place zeros to the right
of a decimal without changing
its value.
4.91 = 4.910 = 4.9100
Subtract 5 − 1.836.
Bee hummingbird
Estimate the difference. _ − _ = _
Think: 5 = 5.__
Subtract the thousandths first.
Then subtract the hundredths, tenths, and ones.
5.
–1.836
__
Regroup as needed.
Compare your estimate with the difference. Since the estimate,
So, the mass of a new nickel is __ grams more
than the mass of a bee hummingbird.
Math
Talk
U.S. Nickel
MATHEMATICAL PRACTICES 6
Explain how to use inverse
operations to check your
answer to 5 − 1.836.
Example 2 Evaluate (6.5 − 1.97) + 3.461 using the order of operations.
Write the expression.
Perform operations in parentheses.
(6.5 – 1.97) + 3.461
6.50
–1.97
__
Add.
+ 3.461
__
So, the value of the expression is __.
38
Math
Talk
MATHEMATICAL PRACTICES 7
Look for Structure Describe
how adding and subtracting
decimals is like adding and
subtracting whole numbers.
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_, is close to __ , the answer is reasonable.
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1. Find 3.42 − 1.9.
Estimate.
Subtract the
___ first.
_−_=_
3.42
–1.90
__
Estimate. Then find the sum or difference.
2. 2.3 + 5.68 + 21.047
4. Evaluate
(8.54 + 3.46) − 6.749.
3. 33.25 − 21.463
____
____
____
Math
Talk
MATHEMATICAL PRACTICES 6
Explain why it is important to align
the decimal points when you add or
subtract decimals.
On
On Your
Your Own
Own
Estimate. Then find the sum or difference.
5. 57.08 + 34.71
__
6. 20.11 − 13.27
__
7. 62 − 9.817
8. 35.1 + 4.89
__
__
Practice: Copy and Solve Evaluate using the order of operations.
© Houghton Mifflin Harcourt Publishing Company
9. 8.01 − (2.2 + 4.67)
12.
MATHEMATICAL
PRACTICE
10. 54 + (9.2 − 1.413)
11. 21.3 − (19.1 − 3.22)
3 Make Arguments A student evaluated 19.1 + (4.32 + 6.9) and got 69.2.
How can you use estimation to convince the student that this answer is not reasonable?
13.
SMARTER
Lynn paid $4.75 for cereal, $8.96 for chicken, and $3.25
for soup. Show how she can use properties and compatible numbers to
evaluate (4.75 + 8.96) + 3.25 to find the total cost.
Chapter 1 • Lesson 6 39
14.
SMARTER
For numbers 14a–14d, select True or
False for each equation.
14a. 3.76 + 2.7 = 6.46
True
False
14b.
4.14 + 1.8 = 4.32
True
False
14c.
2.01 – 1.33 = 0.68
True
False
14d.
51 – 49.2 = 1.8
True
False
Comparing Eggs
Different types of birds lay eggs of different sizes. Small
birds lay eggs that are smaller than those that are laid
by larger birds. The table shows the average lengths and
widths of five different birds’ eggs.
Bird
Length (m)
Width (m)
Canada Goose
0.086
0.058
Hummingbird
0.013
0.013
Raven
0.049
0.033
Robin
0.019
0.015
Turtledove
0.031
0.023
Use the table for 15–17.
Canada Goose
15. What is the difference in average length between the longest egg
and the shortest egg?
16.
17.
40
DEEPER
Which egg has a width that is eight thousandths of a meter shorter
than its length?
SMARTER
How many robin eggs, laid end to end, would be
about equal in length to two raven eggs? Justify your answer.
© Houghton Mifflin Harcourt Publishing Company Image Credits: (c) ©David R. Frazier Photolibrary, Inc./Alamy Images
Average Dimensions of Bird Eggs
Practice and Homework
Name
Lesson 1.6
Add and Subtract Decimals
COMMON CORE STANDARD—6.NS.B.3
Compute fluently with multi-digit numbers and
find common factors and multiples.
Estimate. Then find the sum or difference.
1. 43.53 + 27.67
2. 17 + 3.6 + 4.049
3. 3.49 − 2.75
40 + 30 = 70
43.53
+ 27.67
71.20
4. 5.07 − 2.148
____
5. 3.92 + 16 + 0.085
____
____
6. 41.98 + 13.5 + 27.338
____
____
Evaluate using the order of operations.
7. 8.4 + (13.1 − 0.6)
____
8. 34.7 − (12.07 + 4.9)
____
9. (32.45 − 4.8) − 2.06
____
Problem
Problem Solving
Solving
© Houghton Mifflin Harcourt Publishing Company
10. The average annual rainfall in Clearview is
38 inches. This year, 29.777 inches fell. How
much less rain fell this year than falls in an
average year?
11. At the theater, the Worth family spent
$18.00 on adult tickets, $16.50 on children’s
tickets, and $11.75 on refreshments. How much
did they spend in all?
_______
13.
WRITE
_______
Math Write a word problem that involves
adding or subtracting decimals. Include the solution.
Chapter 1
41
Lesson Check (6.NS.B.3)
1. Alden fills his backpack with 0.45 kg of apples,
0.18 kg of cheese, and a water bottle that weighs
1.4 kg. How heavy are the contents of his
backpack?
2. Gabby plans to hike 6.3 kilometers to see a
waterfall. She stops to rest after hiking
4.75 kilometers. How far does she have left
to hike?
3. A 6-car monorail train can carry 78 people. If
one train makes 99 trips during the day, what
is the greatest number of people the train can
carry in one day?
4. An airport parking lot has 2,800 spaces. If each
row has 25 spaces, how many rows are there?
5. Evan brought 6 batteries that cost $10 each and
6 batteries that cost $4 each. The total cost was
the same as he would have spent buying
6 batteries that cost $14 each. So, 6 × $14 =
(6 × 10) + (6 × 4). What property does the
equation illustrate?
6. Cups come in packages of 12 and lids come
in packages of 15. What is the least number of
cups and lids that Corrine can buy if she wants
to have the same number of cups and lids?
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Spiral Review (5.NBT.B.5, 5.NBT.B.6, 6.NS.B.4)
Lesson 1.7
Name
Multiply Decimals
The Number System—
6.NS.B.3
MATHEMATICAL PRACTICES
MP2, MP6, MP8
Essential Question How do you multiply multi-digit decimals?
Unlock
Unlock the
the Problem
Problem
Last summer Rachel worked 38.5 hours per week
at a grocery store. She earned $9.70 per hour.
How much did she earn in a week?
• How can you estimate the product?
Multiply $9.70 × 38.5.
First estimate the product. $10 × 40 = __
You can use the estimate to place the decimal in a product.
$9.70
× 38.5
Multiply as you would with whole numbers.
The estimate is about $ __,
so the decimal point should be
© Houghton Mifflin Harcourt Publishing Company • Image Credits: (r) ©SW Productions/PhotoDisc/Getty Images
+
$
placed after $__.
Since the estimate, __, is close to __,
the answer is reasonable.
So, Rachel earned __ per week.
1. Explain how your estimate helped you know where to place the decimal
in the product.
Try This! What if Rachel gets a raise of $1.50 per hour? How much
will she earn when she works 38.5 hours?
Chapter 1
43
Counting Decimal Places Another way to place the decimal in a
product is to add the numbers of decimal places in the factors.
Example 1
0.084
× 0.096
___
Multiply 0.084 × 0.096.
_ decimal places
_ decimal places
Multiply as you would with whole numbers.
+
___
_ 1 _, or _ decimal places
Example 2
Write the expression.
Perform operations
in parentheses.
Evaluate 0.35 × (0.48 + 1.24) using the order of operations.
0.35 × (0.48 + 1.24)
0.35 × __
Multiply.
0. 35
_ decimal places
×
__
_ decimal places
+
__
_ 1 _, or _ decimal places
So, the value of the expression is __.
2.
MATHEMATICAL
PRACTICE
8 Use Repeated Reasoning Look for a pattern. Explain.
Math
Talk
MATHEMATICAL PRACTICES 2
Reasoning Is the product of
0.5 and 3.052 greater than
or less than 3.052?
44
0.645 × 10 = 6.45
The decimal point moves _ place to the right.
0.645 × 100 = __
The decimal point moves _ places to the right.
0.645 × 1,000 = __
The decimal point moves _ places to the right.
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0.645 × 1 = 0.645
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Estimate. Then find the product.
2. 32.5 3 7.4
1. 12.42 × 28.6
_×_=_
Estimate.
Think: The estimate is
12.42
× 28.6
__
about _, so the
decimal point should be
placed after _.
MATHEMATICAL
PRACTICE
6 Attend to Precision Algebra Evaluate using the order of operations.
3. 0.24 × (7.3 + 2.1)
4. 0.075 × (9.2 − 0.8)
5. 2.83 + (0.3 × 2.16)
7. 6.95 × 12
8. 0.055 × 1.82
On
On Your
Your Own
Own
Estimate. Then find the product.
6. 29.14 × 5.2
MATHEMATICAL
PRACTICE
6 Attend to Precision Algebra Evaluate using the order of operations.
© Houghton Mifflin Harcourt Publishing Company
9. (3.62 × 2.1) − 0.749
12.
10. 5.8 − (0.25 × 1.5)
11. (0.83 + 1.27) × 6.4
DEEPER
Jamal is buying ingredients to make a large batch of granola
to sell at a school fair. He buys 3.2 pounds of walnuts for $4.40 per pound
and 2.4 pounds of cashews for $6.25 per pound. How much change will
he receive if he pays with two $20 bills?
Chapter 1 • Lesson 7 45
MATHEMATICAL PRACTICES .0%&-t3&"40/tM",&4&/4&
Unlock
Unlock the
the Problem
Problem
The table shows some currency exchange rates for 2009.
Major Currency Exchange Rates in 2009
Currency
U.S. Dollar Japanese Yen European Euro Canadian Dollar
U.S. Dollar
1
Japanese Yen
0.011
1
European Euro
1.479
130.692
Canadian Dollar
0.951
83.995
13.
88.353
0.676
1.052
0.008
0.012
1
1.556
0.643
1
SMARTER
When Cameron went to Canada in
2007, he exchanged 40 U.S. dollars for 46.52 Canadian
dollars. If Cameron exchanged 40 U.S. dollars in 2009,
did he receive more or less than he received in 2007?
How much more or less?
Denominations of Euro
a. What do you need to find?
b. How will you use the table to solve
the problem?
c. Complete the sentences.
40 U.S. dollars were worth __
Canadian dollars in 2009.
__ Canadian dollars in 2009.
Personal Math Trainer
14.
SMARTER
At a convenience store, the
Jensen family puts 12.4 gallons of gasoline in their
van at a cost of $3.80 per gallon. They also buy
4 water bottles for $1.99 each, and 2 snacks for
$1.55 each. Complete the table to find the cost
for each item.
Item
Gasoline
12.4 × $3.80
Water bottles
4 × $1.99
Snacks
2 × $1.55
Mrs. Jensen says the total cost for everything before tax
is $56.66. Do you agree with her? Explain why or why not.
46
Calculation
Cost
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So, Cameron would receive __
Practice and Homework
Name
Lesson 1.7
Multiply Decimals
COMMON CORE STANDARD—6.NS.B.3
Compute fluently with multi-digit numbers
and find common factors and multiples.
Estimate. Then find the product.
2. 3.92 × 0.051
1. 5.69 × 7.8
3. 2.365 × 12.4
4. 305.08 × 1.5
6 × 8 = 48
5.69
× 7.8
4552
39830
44.382
__
__
__
Evaluate the expression using the order of operations.
5. (61.8 × 1.7) + 9.5
__
6. 205 − (35.80 × 5.6)
__
7. 1.9 × (10.6 − 2.17)
__
Problem
Problem Solving
Solving
© Houghton Mifflin Harcourt Publishing Company
8. Blaine exchanges $100 for yen before going to
Japan. If each U.S. dollar is worth 88.353 yen,
how many yen should Blaine receive?
_______
10.
WRITE
9. A camera costs 115 Canadian dollars. If each
Canadian dollar is worth 0.952 U.S. dollars, how
much will the camera cost in U.S. dollars?
_______
Math Explain how to mentally multiply
a decimal number by 100.
Chapter 1
47
Lesson Check (6.NS.B.3)
Estimate each product. Then find the exact product for each question.
1. A gallon of water at room temperature weighs
about 8.35 pounds. Lena puts 4.5 gallons in a
bucket. How much does the water weigh?
2. Shawn’s mobile home is 7.2 meters wide and
19.5 meters long. What is its area?
3. Last week, a store sold laptops worth a total of
$10,885. Each laptop cost $1,555. How many
laptops did the store sell last week?
4. Kyle drives his truck 429 miles on 33 gallons of
gas. How many miles can Kyle drive on 1 gallon
of gas?
5. Seven busloads each carrying 35 students
arrived at the game, joining 23 students who
were already there. Evaluate the expression
23 + 7 × 35 to find the total number of students
at the game.
6. A store is giving away a $10 coupon to every
7th person to enter the store and a $25 coupon
to every 18th person to enter the store. Which
person will be the first to get both coupons?
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48
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Spiral Review (5.NBT.B.6, 5.OA.A.1, 6.NS.B.4)
Lesson 1.8
Name
Divide Decimals by Whole Numbers
Essential Question How do you divide decimals by whole numbers?
The Number System—
6.NS.B.3
MATHEMATICAL PRACTICES
MP1, MP5, MP6, MP8
Unlock
Unlock the
the Problem
Problem
Dan opened a savings account at a bank to save for
a new snowboard. He earned $3.48 interest on his
savings account over a 3-month period. What was the
average amount of interest Dan earned per month on
his savings account?
Divide $3.48 ÷ 3.
First estimate. 3 ÷ 3 = _
© Houghton Mifflin Harcourt Publishing Company • Image Credits: (t) ©Alamy Images
1.
3qw
3.48
−3
04
− 3
18
− 18
0
Quotient
↓
Think: 3.48 is shared among 3 groups.
Divisor
→
1.23
2.46
2qw
←
Dividend
Divide the ones. Place a decimal point
after the ones place in the quotient.
Divide the tenths and then the hundredths. When
the remainder is zero and there are no more digits
in the dividend, the division is complete.
Check your answer.
$
×
3
__
$3.48
Multiply the quotient by the divisor to
check your answer.
So, Dan earned an average of __ in interest per month.
1.
MATHEMATICAL
PRACTICE
Math
Talk
MATHEMATICAL PRACTICES 6
Explain how you know your
answer is reasonable.
1 Analyze Relationships What if the same amount of interest
was gained over 4 months? Explain how you would solve the problem.
Chapter 1
49
Example
Divide 42.133 ÷ 7.
First estimate. 42 ÷ 7 = _
6.0
7qw
42.133
–-42
__
01
–-0
__
13
–-7
__
63
Think: 42.133 is shared among 7 groups.
Divide the ones. Place a decimal point after the
ones place in the quotient.
Divide the tenths. Since 1 tenth cannot be shared
among 7 groups, write a zero in the quotient.
Regroup the 1 tenth as 10 hundredths. Now you
have 13 hundredths.
Continue to divide until the remainder is zero
and there are no more digits in the dividend.
–
___
Check your answer.
6.019
×
7
__
Multiply the quotient by the divisor to
check your answer.
So, 42.133 ÷ 7 = __.
2. Explain how you know which numbers to multiply when checking
your answer.
MATH
M
BOARD
B
1. Estimate 24.186 ÷ 6. Then find the quotient. Check your answer.
Estimate. _ ÷ _ = _
Think: Place a decimal point after the
ones place in the quotient.
50
6qw
24.186
×
6
__
© Houghton Mifflin Harcourt Publishing Company
Share
hhow
Share and
and Show
Sh
Name
Estimate. Then find the quotient.
2.
7qw
$17.15
3.
___
4qw
1.068
4.
___
12qw
60.84
5. 18.042 ÷ 6
___
Math
Talk
On
On Your
Your Own
Own
___
MATHEMATICAL PRACTICES 8
Generalize Explain how
you know where to place
the decimal point in the
quotient when dividing a
decimal by a whole number.
Estimate. Then find the quotient.
6. $21.24 ÷ 6
7. 28.63 ÷ 7
___
MATHEMATICAL
PRACTICE
6
____
© Houghton Mifflin Harcourt Publishing Company
___
9. 0.108 ÷ 18
___
___
Attend to Precision Algebra Evaluate using the order of operations.
10. (3.11 + 4.0) ÷ 9
13.
8. 1.505 ÷ 35
MATHEMATICAL
PRACTICE
11. (6.18 − 1.32) ÷ 3
12. (18 − 5.76) ÷ 6
____
____
5 Use Appropriate Tools Find the length of a dollar bill to the
nearest tenth of a centimeter. Then show how to use division to find the length
of the bill when it is folded in half along the portrait of George Washington.
14.
DEEPER
Emilio bought 5.65 pounds of green grapes and 3.07 pounds of
red grapes. He divided the grapes equally into 16 bags. If each bag of grapes
has the same weight, how much does each bag weigh?
Chapter 1 • Lesson 8 51
MATHEMATICAL PRACTICES
COMMUNICA5&t1&34E7&3&tCONSTRUCT ARGUMENTS
OqnakdlRnkuhmf¤@ookhb`shnmr
OqnakdlRnkuhmf¤@ookhb`shnmr
Pose a Problem
15.
SMARTER
This table shows the
average height in inches for girls and
boys at ages 8, 10, 12, and 14 years.
Average Height (in.)
To find the average growth per year for
girls from age 8 to age 12, Emma knew
she had to find the amount of growth
between age 8 and age 12, then divide
that number by the number of years
between age 8 and age 12.
Emma used this expression:
Age 8
Age 10
Age 12
Age 14
Girls
50.75
55.50
60.50
62.50
Boys
51.00
55.25
59.00
65.20
(60.50 − 50.75) ÷ 4
She evaluated the expression using the order of operations.
Write the expression.
(60.50 − 50.75) ÷ 4
Perform operations in parentheses.
9.75 ÷ 4
Divide.
2.4375
So, the average annual growth for girls ages 8 to 12 is 2.4375 inches.
Write a new problem using the information in the table for the average
height for boys. Use division in your problem.
16.
52
Solve Your Problem
SMARTER
The table shows the number
of books each of three friends bought and
the cost. On average, which friend spent the
most per book? Use numbers and words
to explain your answer.
Friend
Number
Average
Total Cost
of books
Cost
(in dollars)
Purchased
(in dollars)
Joyce
1
$10.95
Nabil
2
$40.50
Kenneth
3
$51.15
© Houghton Mifflin Harcourt Publishing Company
Pose a Problem
Practice and Homework
Name
Lesson 1.8
Divide Decimals by Whole Numbers
COMMON CORE STANDARD—6.NS.B.3
Compute fluently with multi-digit numbers
and find common factors and multiples.
Estimate. Then find the quotient.
1. 1.284 ÷ 12
2. 9qw
2.43
3. 25.65 ÷ 15
4. 12qw
2.436
1.2 ÷ 12 = 0.1
0.107
12qw
1.284
−12
−8
0
84
−84
0
__
__
__
Evaluate using the order of operations.
5. (8 − 2.96) ÷ 3
___
6. (7.772 − 2.38) ÷ 8
7. (53.2 + 35.7) ÷ 7
___
___
Problem
Problem Solving
Solving
© Houghton Mifflin Harcourt Publishing Company
8. Jake earned $10.44 interest on his savings
account for an 18-month period. What was the
average amount of interest Jake earned on his
savings account per month?
9. Gloria worked for 6 hours a day for 2 days at the
bank and earned $114.24. How much did she
earn per hour?
_______
10.
WRITE
_______
Math Explain the importance of correctly
placing the decimal point in the quotient of a division problem.
Chapter 1
53
Lesson Check (6.NS.B.3)
Estimate each quotient. Then find the exact quotient for each question.
1. Ron divided 67.6 fluid ounces of orange juice
evenly among 16 glasses. How much did he
pour into each glass?
2. The cost of a $12.95 pizza was shared evenly by
5 friends. How much did each person pay?
3. What is the value of the digit 6 in 968,743,220?
4. The Tama, Japan, monorail carries 92,700 riders
each day. If the monorail usually carries
5,150 riders per hour, how many hours does
the monorail run each day?
5. Ray paid $812 to rent music equipment that
costs $28 per hour. How many hours did he
have the equipment?
6. Jan has 35 teaspoons of chocolate cocoa mix
and 45 teaspoons of french vanilla cocoa mix.
She wants to put the same amount of mix into
each jar, and she only wants one flavor of mix
in each jar. She wants to fill as many jars as
possible. How many jars of french vanilla cocoa
mix will Jan fill?
FOR MORE PRACTICE
GO TO THE
54
Personal Math Trainer
© Houghton Mifflin Harcourt Publishing Company
Spiral Review (5.NBT.A.1, 5.NBT.B.6, 6.NS.B.2, 6.NS.B.4)
Lesson 1.9
Name
Divide with Decimals
Essential Question How do you divide whole numbers and decimals by decimals?
connect Find each quotient to discover a pattern.
The Number System—
6.NS.B.3
MATHEMATICAL PRACTICES
MP1, MP6, MP8
4÷2=_
40 ÷ 20 = _
400 ÷ 200 = _
When you multiply both the dividend and the divisor by the same
power of _, the quotient is the __. You can use this
fact to help you divide decimals.
Unlock
Unlock the
the Problem
Problem
Tami is training for a triathlon. In a triathlon, athletes compete
in three events: swimming, cycling, and running. She cycled 66.5
miles in 3.5 hours. If she cycled at a constant speed, how far did
she cycle in 1 hour?
Compatible numbers are pairs
of numbers that are easy to
compute mentally.
Divide 66.5 ÷ 3.5.
Estimate using compatible numbers.
© Houghton Mifflin Harcourt Publishing Company Image Credits: (b) ©Brand X Pictures/Getty Images
60 ÷ 3 = _
STEP 1
Make the divisor a whole number
by multiplying the divisor and
dividend by 10.
Think: 3.5 × 10 = 35
3.5qw
66.5
66.5 × 10 = 665
STEP 2
Divide.
35qw
665
−
So, Tami cycled __ in 1 hour.
•
MATHEMATICAL
PRACTICE
1 Evaluate Reasonableness Explain whether your answer is reasonable.
Chapter 1
55
Example 1
Divide 17.25
÷ 5.75. Check.
STEP 1
Make the divisor a whole number by
multiplying the divisor and dividend by _.
5.75qw
17.25
5.75 × _ = _
17.25 × _ = _
STEP 2
1, 725
575qw
−
Divide.
STEP 3
Check.
×
__
So, 17.25 ÷ 5.75 5 _.
Example 2
Divide 37.8
÷ 0.14.
Be careful to move the decimal
point in the dividend the same
number of places that you moved
the decimal point in the divisor.
STEP 1
Make the divisor a whole number by
multiplying the divisor and dividend by _.
_×_=_
_×_=_
0.14qw
37.80
Think: Add a zero to
the right of the dividend
so that you can move
the decimal point.
Divide.
14qw
3,780
−
−
−
So, 37.8 ÷ 0.14 5 _.
56
Math
Talk
MATHEMATICAL PRACTICES 6
Explain How can you
check that the quotient is
reasonable? How can you
check that it is accurate?
© Houghton Mifflin Harcourt Publishing Company
STEP 2
Name
Share
hhow
Share and
and Show
Sh
MATH
M
BOARD
B
1. Find the quotient.
14.8qw
99.456
Think: Make the divisor a whole number by
multiplying the divisor and dividend by _.
Estimate. Then find the quotient.
2. $10.80 ÷ $1.35
____
3. 26.4 ÷ 1.76
4.
____
___
Math
Talk
On
On Your
Your Own
Own
8.7qw
53.07
MATHEMATICAL PRACTICES 8
Generalize Explain how you
know how many places to
move the decimal point in
the divisor and the dividend.
Estimate. Then find the quotient.
5. 75 ÷ 12.5
____
MATHEMATICAL
PRACTICE
© Houghton Mifflin Harcourt Publishing Company
____
7.
0.78qw
0.234
___
6 Attend to Precision Algebra Evaluate using the order of operations.
8. 36.4 + (9.2 − 4.9 ÷ 7)
____
11.
6. 544.6 ÷ 1.75
9. 16 ÷ 2.5 − 3.2 × 0.043
____
10. 142 ÷ (42 − 6.5) × 3.9
____
DEEPER
Marcus can buy 0.3 pound of sliced meat from a deli for
$3.15. How much will 0.7 pound of sliced meat cost?
Chapter 1 • Lesson 9 57
12.
SMARTER
The table shows the earnings
and the number of hours worked for three
employees. Complete the table by finding the
missing values. Which employee earned the least
per hour? Explain.
Employee
Total Earned
(in dollars)
1
$34.02
2
$42.75
3
$52.65
Number
of Hours
Worked
Earnings
per Hour
(in dollars)
$9.72
4.5
$9.75
Amoebas
Amoebas are tiny one-celled organisms. Amoebas can
range in size from 0.01 mm to 5 mm in length. You can
study amoebas by using a microscope or by studying
photographic enlargements of them.
Jacob has a photograph of an amoeba that has been
enlarged 1,000 times. The length of the amoeba in the
photo is 60 mm. What is the actual length of the amoeba?
Divide 60 ÷ 1,000 by looking for a pattern.
60 ÷ 10 = 6.0
The decimal point moves _ place to the left.
60 ÷ 100 = _
The decimal point moves _ places to the left.
60 ÷ 1,000 5 _ The decimal point moves _ places to the left.
So, the actual length of the amoeba is __ mm.
13.
14.
58
SMARTER
DEEPER
Explain the pattern.
Pelomyxa palustris is an amoeba with a length of 4.9 mm.
Amoeba proteus has a length of 0.7 mm. How many Amoeba proteus
would you have to line up to equal the length of three Pelomyxa
palustris? Explain.
© Houghton Mifflin Harcourt Publishing Company Image Credits: (c) ©M. I. Walker/Science Source
60 ÷ 1 5 60
Practice and Homework
Name
Lesson 1.9
Divide with Decimals
COMMON CORE STANDARD—6.NS.B.3
Compute fluently with multi-digit numbers
and find common factors and multiples.
Estimate. Then find the quotient.
1. 43.18 ÷ 3.4
2. 4.185 ÷ 0.93
3. 6.3qw
25.83
4. 0.143 ÷ 0.55
44 ÷ 4 = 11
12.7
431.8
34qw
−34
91
−68
238
−238
0
__
__
__
Evaluate using the order of operations.
5. 4.92 ÷ (0.8 – 0.12 ÷ 0.3)
___
6. 0.86 ÷ 5 – 0.3 × 0.5
___
7. 17.28 ÷ (1.32 – 0.24) × 0.6
___
© Houghton Mifflin Harcourt Publishing Company
Problem
Problem Solving
Solving
8. If Amanda walks at an average speed of
2.72 miles per hour, how long will it take
her to walk 6.8 miles?
_______
10.
WRITE
9. Chad cycled 62.3 miles in 3.5 hours. If he cycled
at a constant speed, how far did he cycle in
1 hour?
_______
Math Explain how dividing by a decimal is different from
dividing by a whole number and how it is similar.
Chapter 1
59
Lesson Check (6.NS.B.3)
1. Elliot drove 202.8 miles and used 6.5 gallons
of gasoline. How many miles did he travel per
gallon of gasoline?
2. A package of crackers weighing 8.2 ounces
costs $2.87. What is the cost per ounce of
crackers?
Spiral Review (5.NBT.B.5, 5.NF.B.3)
4. A zebra ran at a speed of 20 feet per second.
What operation should you use to find the
distance the zebra ran in 10 seconds?
5. Nira has $13.50. She receives a paycheck for
$55. She spends $29.40. How much money does
she have now?
6. A piece of cardboard is 24 centimeters long and
15 centimeters wide. What is its area?
© Houghton Mifflin Harcourt Publishing Company
3. Four bags of pretzels were divided equally
among 5 people. How much of a bag did each
person get?
FOR MORE PRACTICE
GO TO THE
60
Personal Math Trainer
Name
Personal Math Trainer
Chapter 1 Review/Test
Online Assessment
and Intervention
1. Use the numbers to complete the factor tree. You may use a number
more than once.
54
2
3
6
9
27
9
Write the prime factorization of 54.
2. For numbers 2a–2d, choose Yes or No to indicate whether the LCM of
the two numbers is 15.
2a.
Yes
No
2b. 5, 10
Yes
No
2c. 5, 15
Yes
No
2d. 5, 20
Yes
No
5, 3
© Houghton Mifflin Harcourt Publishing Company
3. Select two numbers that have 9 as their greatest common factor.
Mark all that apply.
A
3, 9
B
3, 18
C
9, 18
D
9, 36
E
18, 27
Assessment Options
Chapter Test
Chapter 1
61
4. The prime factorization of each number is shown.
15 = 3 × 5
18 = 2 × 3 × 3
Part A
Using the prime factorization, complete the Venn diagram.
Prime factors of 15
Prime factors of 18
Common prime factors
Part B
Find the GCF of 15 and 18.
5a.
62
222.2 ÷ 11 = 22.2
Yes
No
5b. 400 ÷ 50 = 8
Yes
No
5c. 1,440 ÷ 36 = 40
Yes
No
5d. 7,236 ÷ 9 = 804
Yes
No
© Houghton Mifflin Harcourt Publishing Company
5. For numbers 5a–5d, choose Yes or No to indicate whether each
equation is correct.
Name
6. For numbers 6a–6d, select True or False for each equation.
6a. 1.7 + 4.03 = 6
True
False
6b. 2.58 + 3.5 = 6.08
True
False
6c. 3.21 − 0.98 = 2.23
True
False
6d. 14 − 1.3 = 0.01
True
False
Personal Math Trainer
7.
SMARTER
Four
friends went shopping at
a music store. The table
shows the number of CDs
each friend bought and the
total cost. Complete the
table to show the average
cost of the CDs each friend
bought.
Friend
Number of CDs
Purchased
Total Cost
(in dollars)
Lana
4
$36.68
Troy
5
$40.55
Juanita
5
$47.15
Alex
6
$54.42
Average Cost
(in dollars)
What is the average cost of
all the CDs that the four friends bought? Show your work.
8. The table shows the earnings and the number of hours worked for five
employees. Complete the table by finding the missing values.
© Houghton Mifflin Harcourt Publishing Company
Employee
Total Money Earned
Number of
(in dollars)
Hours Worked
1
$23.75
2
$28.38
3
$38.50
4
$55.00
5.5
5
$60.00
2.5
Earnings per Hour
(in dollars)
$9.50
3.3
$8.75
Chapter 1
63
9. The distance around the outside of Cedar Park is 0.8 mile. Joanie
ran 0.25 of the distance during her lunch break. How far did she run?
Show your work.
10. A one-celled organism measures 32 millimeters in length in a
photograph. If the photo has been enlarged by a factor of 100, what is
the actual length of the organism? Show your work.
DEEPER
You can buy 5 T-shirts at Baxter’s for the same price that
you can buy 4 T-shirts at Bixby’s. If one T-shirt costs $11.80 at Bixby’s,
how much does one T-shirt cost at Baxter’s? Use numbers and words to
explain your answer.
© Houghton Mifflin Harcourt Publishing Company
11.
64
Name
12. Crackers come in packages of 24. Cheese slices come in packages
of 18. Andy wants one cheese slice for each cracker. Patrick made the
statement shown.
If Andy doesn’t want any crackers or cheese slices left over, he needs
to buy at least 432 of each.
Is Patrick’s statement correct? Use numbers and words to explain
why or why not. If Patrick’s statement is incorrect, what should he
do to correct it?
13. There are 16 sixth graders and 20 seventh graders in the Robotics
Club. For the first project, the club sponsor wants to organize the club
members into equal-size groups. Each group will have only sixth graders
or only seventh graders.
Part A
How many students will be in each group if each group has the greatest
possible number of club members? Show your work.
© Houghton Mifflin Harcourt Publishing Company
Part B
If each group has the greatest possible number of club members, how
many groups of sixth graders and how many groups of seventh graders
will there be? Use numbers and words to explain your answer.
Chapter 1
65
14. The Hernandez family is going to the beach. They buy sun block for
$9.99, 5 snacks for $1.89 each, and 3 beach toys for $1.49 each. Before
they leave, they fill up the car with 13.1 gallons of gasoline at a cost of
$3.70 per gallon.
Part A
Complete the table by calculating the total cost for each item.
Item
Calculation
Gasoline
13.1 × $3.70
Snacks
5 × $1.89
Beach toys
3 × $1.49
Sun block
1 × $9.99
Total Cost
Part B
What is the total cost for everything before tax? Show your work.
Part C
Mr. Hernandez calculates the total cost for everything before tax using
this equation.
Do you agree with his equation? Use numbers and words to explain why
or why not. If the equation is not correct, write a correct equation.
66
© Houghton Mifflin Harcourt Publishing Company
Total cost = 13.1 + 3.70 × 5 + 1.89 × 3 + 1.49 × 9.99
2
Chapter
Fractions
Show Wha t You Know
Personal Math Trainer
Online Assessment
and Intervention
Check your understanding of important skills.
Name
Compare and Order Whole Numbers Compare.
Write <, >, or = for the
1. 289
.
2. 476,225
291
3. 5,823
4. 30,189
5,286
476,225
30,201
Benchmark Fractions Write whether the fraction
is closest to 0, 1__2 or 1.
3
5. _5 __
6
6. _7 __
1
7. _ __
6
1
8. _ __
3
Multiply Fractions and Whole Numbers Find the product.
Write it in simplest form.
2
9. _3 × 21
__
3
12. _4 × 14
__
1
10. _4 × 10
__
2
13. 35 × _5
__
2
11. 6 × _9
__
3
14. _ × 12
8
__
© Houghton Mifflin Harcourt Publishing Company
Math in the
Cyndi bought an extra large pizza, cut into 12 pieces, for
today’s meeting of the Mystery Club. She ate 1_6 of the pizza
yesterday afternoon. Her brother ate 1_5 of what was left
last night. Cyndi knows that she needs 8 pieces of pizza
for the club meeting. Help Cyndi figure out if she has
enough pizza left for the meeting.
Chapter 2
67
Voca bula ry Builder
Visualize It
Complete the Bubble Map using review words that are related
to fractions.
Review Words
✓ benchmark
✓ compatible numbers
denominator
✓ equivalent fractions
fractions
mixed numbers
numerator
fractions
✓ simplest form
Preview Words
✓ multiplicative inverse
✓ reciprocal
Understand Vocabulary
Complete the sentences using the checked words.
1.
_____ are numbers that are easy to compute
with mentally.
2. One of two numbers whose product is 1 is a
© Houghton Mifflin Harcourt Publishing Company
____ or a _____.
3. A ____ is a reference point that is used for
estimating fractions.
4. When the numerator and denominator of a fraction have only
1 as a common factor, the fraction is in ____.
5. Fractions that name the same amount are _____.
68
™Interactive Student Edition
™Multimedia eGlossary
Chapter 2 Vocabulary
common
denominator
compatible numbers
denominador común
números compatibles
13
11
equivalent fractions
fraction
fracciones equivalentes
fracción
35
29
mixed number
multiplicative
inverse
número mixto
inverso multiplicativo
63
60
reciprocal
simplest form
recíproco
mínima expresión
89
92
15 ÷ 5 = 3
A number that names a part of a whole or
a part of a group
2
3
Examples:
2
5
A reciprocal of a number that is multiplied by
that number resulting in a product of 1
__ × 3
__ = 1
Example: 2
3
2
A fraction is in simplest form when the
numerator and denominator have only 1
as a common factor
÷2=3
_____
__
Example: 6
8÷2
4
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6
© Houghton Mifflin Harcourt Publishing Company
4
© Houghton Mifflin Harcourt Publishing Company
__ ÷ 4 5
__
Example: 14 3
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© Houghton Mifflin Harcourt Publishing Company
© Houghton Mifflin Harcourt Publishing Company
© Houghton Mifflin Harcourt Publishing Company
© Houghton Mifflin Harcourt Publishing Company
Numbers that are easy to compute with
mentally
A common multiple of two or more
denominators
3
Example: __
7
__
8
common
denominator
8
Fractions that name the same amount
or part
2
3
Example:
4
6
A number that is made up of a whole
number and a fraction
__
Example: 3 2
5
Two numbers are reciprocals of each other
if their product equals 1.
__ × 3
__ = 1
Example: 2
3
2
Going Places with
Guess
the Word
For 3 to 4 players
Materials
• timer
How to Play
1. Take turns to play.
2. Choose a math term, but do not say it aloud.
Game
Game
Word Box
common
denominator
compatible
numbers
equivalent
fractions
fraction
mixed number
multiplicative
inverse
reciprocal
simplest form
4. Give a one-word clue about your term. Give each player one
chance to guess the term.
5. If nobody guesses correctly, repeat Step 4 with a different
clue. Repeat until a player guesses the term or
time runs out.
6. The player who guesses the term gets
1 point. If he or she can use the term in a
sentence, they get 1 more point. Then that
player gets a turn.
7. The first player to score 10 points wins.
© Houghton Mifflin Harcourt Publishing Company
Image Credits: (bg) ©Siede Preis/PhotoDisc/Getty Images; (b) ©Comstock/Getty Images
3. Set the timer for 1 minute.
Words
Chapter 2
68A
Journal
Jo
ou
urnal
The Write Way
Reflect
Choose one idea. Write about it.
• How can you determine if a fraction is greater than, less than,
or equal to another fraction? Write 2−3 sentences to explain.
• Is this fraction in simplest form? Tell how you know.
4
__
12
• Explain how to estimate an answer to this problem using
compatible numbers.
7 5_ ÷ 3_ = _____.
8
5
• Write step-by-step instructions for how to divide two mixed
© Houghton Mifflin Harcourt Publishing Company
Image Credits: (bg) ©Siede Preis/PhotoDisc/Getty Images
numbers. Be sure to use the word reciprocal in your instructions.
68B
Lesson 2.1
Name
Fractions and Decimals
The Number System—
6.NS.C.6c
MATHEMATICAL PRACTICES
MP5, MP6, MP7
Essential Question How can you convert between fractions and decimals?
connect You can use place value to write a decimal
as a fraction or a mixed number.
Place Value
Ones
1
Tenths
.
Hundredths Thousandths
2
3
4
Unlock
Unlock the
the Problem
Problem
The African pygmy hedgehog is a popular pet in North
America. The average African pygmy hedgehog weighs
between 0.5 lb and 1.25 lb. How can these weights be
written as fractions or mixed numbers?
• How do you know if a fraction is in
simplest form?
Write 0.5 as a fraction and 1.25 as a mixed
number in simplest form.
A 0.5
0.5 is five
__.
5
0.5 = ____
Simplify using the GCF.
The GCF of 5 and 10 is
_.
5 = _________
5 ÷
____
= ___
÷
© Houghton Mifflin Harcourt Publishing Company • Image Credits: (r) ©brandi ediss/Flickr/Getty Images
Divide the numerator and
the denominator by
_.
B 1.25
1.25 = 1_____
1.25 is one and
___.
Simplify using the GCF.
The GCF of 25 and 100 is
÷
_. 1_____ = 1___________
= 1___
÷
Divide the numerator and
the denominator by
_.
So, the average African pygmy hedgehog weighs between
__ lb and __ lb.
Math
Talk
MATHEMATICAL PRACTICES 7
Look for Structure Explain
how you can use place value
to write 0.05 and 0.005 as
fractions. Then write the
fractions in simplest form.
Chapter 2
69
You can use division to write a fraction or a mixed number as a decimal.
Example 1 Write 6_ as a decimal.
3
8
STEP 1
Use division to rename the fraction
part as a decimal.
The quotient has __ decimal places.
8qw
3.000
−
−
STEP 2
Add the whole number to the decimal.
−
0
Math
Talk
6 + ___ = ___
MATHEMATICAL PRACTICES 6
Explain why zeros were
placed after the decimal
point in the dividend.
So, 6 3_8 = ___.
Sometimes you can use a number line to convert between fractions and decimals.
Example 2 Graph 3_ on the number line and write it as a decimal.
3
5
3.2
3
3 51
4
1.
SMARTER
On the number line below, write decimals for the
1 and __
2.
fractions __
50
25
0
0
70
0.1
1
50
1
25
3
50
2
25
1
10
© Houghton Mifflin Harcourt Publishing Company
So, 3 3_5 = ___.
Name
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MATH
M
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B
Write as a fraction or as a mixed number in simplest form.
1.
5 =
95.5 = 95____
2. 0.6
3. 5.75
____
____
____
Write as a decimal.
4. 7_8
3
6. __
25
__
5. 13
20
____
____
____
Math
Talk
On
On Your
Your Own
Own
MATHEMATICAL PRACTICES 6
Explain how you can find the
decimal that is equivalent
to 7_8 .
Write as a fraction or as a mixed number in simplest form.
7. 0.27
8. 0.055
____
9. 2.45
____
____
Write as a decimal.
© Houghton Mifflin Harcourt Publishing Company
10. _38
11. 3 1_5
____
__
12. 2 11
20
____
____
Identify a decimal and a fraction in simplest form for the point.
D
0
A
0.2
13. Point A
___
C
0.4
B
0.6
14. Point B
___
1.0
0.8
15. Point C
16. Point D
___
___
Chapter 2 • Lesson 1 71
MATHEMATICAL PRACTICES .0%&-t3&"40/tM",&4&/4&
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Ozark Trail Hiking Club
Hiker
Use the table for 17 and 18.
17. Members of the Ozark Trail Hiking Club hiked a steep
section of the trail in June and July. The table shows the
distances club members hiked in miles. Write Maria’s July
distance as a decimal.
July
Maria
2.95
25
Devin
3.25
31
Kelsey
3.15
27
Zoey
2.85
33
8
8
8
8
DEEPER
18.
How much farther did Zoey hike in June and
July than Maria hiked in June and July? Explain how you
found your answer.
19.
What’s the Error? Tabitha’s hiking
distance in July was 2 1_5 miles. She wrote the distance as
2.02 miles. What error did she make?
20.
June
SMARTER
MATHEMATICAL
PRACTICE
5 Use Patterns Write _3 , _4 , and _5 as decimals.
8 8
8
21.
SMARTER
Identify a decimal and a fraction in
simplest form for the point.
C
0
72
0.2
A
0.4
B
0.6
D
0.8
1.0
Point A
Point B
Point C
Point D
© Houghton Mifflin Harcourt Publishing Company • Image Credits: (t) ©Don Mason/Corbis
What pattern do you see? Use the pattern to predict the
decimal form of 6_8 and 7_8 .
Practice and Homework
Name
Lesson 2.1
Fractions and Decimals
COMMON CORE STANDARD—6.NS.C.6c
Apply and extend previous understandings of
numbers to the system of rational numbers.
Write as a fraction or as a mixed number in simplest form.
1. 0.52
2. 0.02
52
___
0.52 5
100
52 4 4
13
5 ________
5 __
100 4 4
25
___
3. 4.8
___
4. 6.025
___
___
Write as a decimal. Tell whether the decimal terminates or repeats.
__
5. 17
25
6. 7_9
___
__
7. 4 13
8
8. 7 __
20
___
11
___
___
Identify a decimal and a fraction or mixed number in simplest form for
each point.
A
0
B
0.5
9. Point A
C
1
10. Point D
___
___
D
1.5
11. Point C
___
2
12. Point B
___
Problem
Problem Solving
Solving
© Houghton Mifflin Harcourt Publishing Company
13. Grace sold 5_ of her stamp collection. What is this
8
amount as a decimal?
15.
WRITE
14. What if you scored an 0.80 on a test? What
fraction of the test, in simplest form, did you
answer correctly?
Math Write an example of a fraction that is a terminating
decimal and a fraction that is a repeating decimal. Explain how you
know your examples are terminating or repeating.
Chapter 2
73
Lesson Check (6.NS.C.6c)
1. After a storm, Michael measured 6 7_8 inches of
snow. What is this amount as a decimal?
2. A recipe calls for 3.75 cups of flour. What is this
amount as a mixed number in simplest form?
Spiral Review (6.NS.B.2, 6.NS.B.3, 6.NS.B.4)
4. Ken has 4.66 pounds of walnuts, 2.1 pounds
of cashews, and 8 pounds of peanuts. He
mixes them together and divides them equally
among 18 bags. How many pounds of nuts are
in each bag?
5. Mia needs to separate 450 pens into 18 packs.
Each pack will have the same number of pens.
How many pens should be put in each pack?
6. Evan buys 19 tubes of watercolor paint for
$50.35. What is the cost of each tube of paint?
© Houghton Mifflin Harcourt Publishing Company
3. Gina bought 2.3 pounds of red apples and
2.42 pounds of green apples. They were on sale
for $0.75 a pound. How much did the apples
cost altogether?
FOR MORE PRACTICE
GO TO THE
74
Personal Math Trainer
Lesson 2.2
Name
Compare and Order Fractions and Decimals
The Number System—
6.NS.C.7a
MATHEMATICAL PRACTICES
MP3, MP6, MP8
Essential Question How can you compare and order fractions and decimals?
To compare fractions with the same denominators, compare the
numerators. To compare fractions with the same numerators,
compare the denominators.
Same Denominators
Same Numerators
2
3
2
3
1
3
2
5
Two of three equal parts is greater
than one of three equal parts.
Two of three equal parts is greater
than two of five equal parts.
So, 32_ > 31_.
So, 2_ > 52_.
3
Unlock
Unlock the
the Problem
Problem
Three new flowering dogwood trees were planted in a park in
Springfield, Missouri. The trees were 6 1_2 ft, 5 2_3 ft, and 5 5_8 ft tall.
Order the plant heights from least to greatest.
• Equivalent fractions are
fractions that name the same
amount or part.
To compare and order fractions with unlike denominators,
write equivalent fractions with common denominators.
• A common denominator is a
denominator that is the same
in two or more fractions.
One Way Order 6_, 5_, and 5_ from least to greatest.
© Houghton Mifflin Harcourt Publishing Company • Image Credits: (r) ©Craig Tuttle/Design Pics/Corbis
1
2
2
3
5
8
STEP 1
Compare the whole numbers first.
__
61
2
__
52
3
__
55
5
8
6
STEP 2
Think: __ is a multiple of 3 and 8,
If the whole numbers are the same,
compare the fractions.
so
Use common denominators to write
equivalent fractions.
2 × 8 = 5 ____
5____
__ is a common denominator.
5×
5 ______
= 5____
3×8
8×
STEP 3
Compare the numerators.
__
5 ___ < 5 ___ < 61
2
Order the fractions from least
to greatest.
So, from least to greatest, the order is __ ft,
__ ft, __ ft.
Math
Talk
MATHEMATICAL PRACTICES 3
Apply Explain a simple
method to compare 3 3_4 and 3 3_7 .
Chapter 2
75
Fractions and Decimals You can compare fractions and decimals.
One Way
Compare to 1_2 .
Compare 0.92 and 2_7 . Write <, >, or =.
__.
Compare 0.92 to 1
STEP 1
STEP 2
2
0.92
1
__
2
__ to 1
__.
Compare 2
7
2
1
__
2
2
__
7
Math
Talk
2
_
7.
So, 0.92
Another Way
MATHEMATICAL PRACTICES 8
Generalize Explain how
to compare a fraction such
as 2_7 to 1_2 .
Rewrite the fraction as a decimal.
3. Write <, >, or =.
Compare 0.8 and __
4
3
STEP 1 Write __ as a decimal.
STEP 2
4
Use ,, ., or 5 to
compare the decimals.
4qw
3.00
–
35
__
4
_
–
3
_
4.
So, 0.8
_
0.80
0
You can use a number line to order fractions and decimals.
Example Use a number line to order 0.95, __, _, and 0.45
3 1
10 4
from least to greatest.
Write each fraction as a decimal.
3 → 10 3.00
___
q
10
STEP 2
w
• Numbers read from left to
right on a number line are in
order from least to greatest.
1
__ → 4qw
1.00
4
• Numbers read from right to
left are in order from greatest
to least.
Locate each decimal on a number line.
0.45
0
0.2
0.4
0.95
0.6
0.8
1.0
So, from least to greatest, the order is __, __, __, __.
76
© Houghton Mifflin Harcourt Publishing Company
STEP 1
Name
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M
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B
Order from least to greatest.
9
1. 33_, 35_, 2__
6
8
10
Think: Compare the
whole numbers first.
3×
3 _____
= 3 ___
6×
5×
3 _____
= 3 ___
8×
_, _, _
Write <, >, or =.
4
__
2. 0.8
3. 0.22
12
1
4. __
20
1
_
4
0.06
Use a number line to order from least to greatest.
1
5. 14_, 1.25, 1 __
5
10
1
1.2
1.4
1.6
____
Math
Talk
On
On Your
Your Own
Own
2
1.8
MATHEMATICAL PRACTICES 6
Explain how to compare 3_5
and 0.37 by comparing to 1_2 .
Order from least to greatest.
6. 0.6, 4_, 0.75
5
__
7
7. _12, 2_, __
5 15
__
8. 51_, 5.05, 55_
9
2
__
5
9. _57, 5_, __
6 12
__
Write <, >, or =.
7
10. __
© Houghton Mifflin Harcourt Publishing Company
15
14.
7
__
10
11. 1_
8
0.125
12. 71_
3
62_
3
13. 12_
5
7
1__
15
Darrell spent 3 2_ hours on a project for school. Jan spent
5
3 1_ hours and Maeve spent 3.7 hours on the project. Who spent the
4
least amount of time? Show how you found your answer. Then describe
another possible method.
SMARTER
Chapter 2 • Lesson 2 77
MATHEMATICAL PRACTICES
COMMUNICA5&t1&34E7&3&tCONSTRUCT ARGUMENTS
OqnakdlRnkuhmf¤@ookhb`shnmr
OqnakdlRnkuhmf¤@ookhb`shnmr
Use the table for 15–18.
15.
DEEPER
In one week, Altoona, PA, and
Bethlehem, PA, received snowfall every day,
Monday through Friday. On which days did Altoona
receive over 0.1 inch more snow than Bethlehem?
Altoona and Bethlehem
Snowfall (inches)
Day
16.
17.
What if Altoona received an
additional 0.3 inch of snow on Thursday? How
would the total amount of snow in Altoona compare
to the amount received in Bethlehem that day?
SMARTER
MATHEMATICAL
PRACTICE
Altoona
Bethlehem
Monday
21
4
2.6
Tuesday
31
4
25
8
3.2
Thursday
43
5
4.8
Friday
43
4
2.7
Wednesday
6 Explain two ways you could compare
2.5
18.
WRITE
Math Explain how you could compare the snowfall
amounts in Altoona on Thursday and Friday.
19.
SMARTER
Write the values in order from least
to greatest.
1
_
3
78
0.45
0.39
2
_
5
© Houghton Mifflin Harcourt Publishing Company • Image Credits: (t) ©PhotoDisc/Getty Images
the snowfall amounts in Altoona and Bethlehem
on Monday.
Practice and Homework
Name
Lesson 2.2
Compare and Order Fractions and Decimals
COMMON CORE STANDARD—6.NS.C.7a
Apply and extend previous understandings of
numbers to the system of rational numbers.
Write ,, ., or 5.
7
1. 0.64 < __
10
2. 0.48
6
__
15
7
_
8
3. 0.75
4. 7 1_
7.025
8
0.64 < 0.7
Order from least to greatest.
7 , 0.75, 5
_
5. __
12
6. 0.5, 0.41, 3_
5
6
___
___
7. 3.25, 3 2_, 3 3_
5
8
___
8. 0.9, 8_, 0.86
9
___
Order from greatest to least.
9. 0.7, 7_, 7_
10. 0.2, 0.19, 3_
5
9 8
___
___
1 , 6.1, 6.07
11. 6 __
20
___
12. 2 1_, 2.4, 2.35, 2 1_
2
8
___
Problem
Problem Solving
Solving
© Houghton Mifflin Harcourt Publishing Company
13. One day it snowed 3 3_8 inches in Altoona and
3.45 inches in Bethlehem. Which city received
less snow that day?
15.
WRITE
14. Malia and John each bought 2 pounds of
sunflower seeds. Each ate some seeds. Malia has
1 1_3 pounds left, and John has 1 2_5 pounds left. Who
ate more sunflower seeds?
Math Explain how you would compare the numbers 0.4 and 3_.
8
Chapter 2
79
Lesson Check (6.NS.C.7a)
1. Andrea has 3 7_8 yards of purple ribbon, 3.7 yards
of pink ribbon, and 3 4_5 yards of blue ribbon. List
the numbers in order from least to greatest.
__ of the math homework.
2. Nassim completed 18
25
Kara completed 0.7 of it. Debbie completed 5_8
of it. List the numbers in order from greatest to
least.
Spiral Review (6.NS.C.6c, 6.NS.B.3, 6.NS.B.4)
4. At the factory, a baseball card is placed in every
9th package of cereal. A football card is placed
in every 25th package of the cereal. What is the
first package that gets both a baseball card and
a football card?
5. $15.30 is divided among 15 students. How
much does each student receive?
6. Carrie buys 4.16 pounds of apples for $5.20.
How much does 1 pound cost?
© Houghton Mifflin Harcourt Publishing Company
7
3. Tyler bought 3 __
16 pounds of oranges. Write this
amount using a decimal.
FOR MORE PRACTICE
GO TO THE
80
Personal Math Trainer
Lesson 2.3
Name
Multiply Fractions
The Number System—
6.NS.B.4
MATHEMATICAL PRACTICES
MP2, MP6, MP8
Essential Question How do you multiply fractions?
Unlock
Unlock the
the Problem
Problem
Sasha still has 4_5 of a scarf left to knit. If she finishes 1_2
of the remaining part of the scarf today, how much
of the scarf will Sasha knit today?
You can find the product of
two fractions by multiplying the
numerators and multiplying the
denominators.
1 × __
4. Write the product in
Multiply __
2
5
simplest form.
Multiply the numerators.
Multiply the denominators.
×2
1×2
2
_____
__
__ = 1
= ___
3×5
5
15
3
1
×
__ × 4
__ = 1
_____
= ____
2 5 2×
Simplify using the GCF.
The GCF of 4 and 10 is __.
Divide the numerator and the
÷
= _______
= ___
10 ÷
denominator by __.
© Houghton Mifflin Harcourt Publishing Company • Image Credits: (c) ©Anna Peisi/Corbis
1_ 4_
× = _, so Sasha will knit _ of the scarf today.
2 5
Example 1
Multiply 1 1_4 × 1 2_3 . Write the product in simplest form.
Estimate. 1 × _ = _
Write the mixed numbers as fractions
greater than 1.
Multiply the fractions.
Write the product as a fraction or
mixed number in simplest form.
__ × 12
__ = 5
__ × ___
11
4
3
4
×
_____
=5
= ____ , or __
4×3
Since the estimate is __, the answer is reasonable.
So, 11_4 × 12_ = _, or _.
3
3
Math
Talk
MATHEMATICAL PRACTICES 8
Generalize Without
multiplying, how do you
know that 1_3 × 3_4 < 3_4 ?
Chapter 2
81
Example 2
Evaluate 4_5
+ ( 6 × 3_8 ) using the order of operations.
STEP 1
Estimate using benchmarks.
STEP 2
Perform operations
in parentheses.
__ =
+ 6×1
2
(
)
+3=
(
6×3
4+ 6×3
__
__ = 4
__ + _____
5
8 5
×8
(
A benchmark is a reference
point, such as 0, 1_2 , or 1, that is
used for estimating.
)
)
__ + ____
=4
5
×5
4 × 8 + ______
= ____
STEP 3
Write equivalent
fractions using a common
denominator.
5×8
×5
___ + ____ = ____
= 32
40
Then add.
÷
________
= 122
STEP 4
Simplify using the GCF.
40 ÷
= ____ , or __
Since the estimate is __, the answer is reasonable.
So, 4_5 + 6 × 3_ = _ , or _.
1.
MATHEMATICAL
PRACTICE
8
)
2 Use Reasoning What if you did not follow the order of
operations and instead worked from left to right? How would that affect
your answer?
2.
82
MATHEMATICAL
PRACTICE
6 Explain how you used benchmarks to estimate the answer.
© Houghton Mifflin Harcourt Publishing Company
(
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M
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B
Find the product. Write it in simplest form.
1. 6 × 3_
2. _3 × 8_
8
8
3. Sam and his friends ate 3 3_4
bags of fruit snacks. If each
bag contained 2 1_2 ounces, how
many ounces of fruit snacks
did Sam and his friends eat?
9
6
__ × 3
__ = ____
1 8
÷ = ___
________
8÷
or __
MATHEMATICAL
PRACTICE
___
___
6
Attend to Precision Algebra Evaluate using the order of operations.
Write the answer in simplest form.
5. 1_ + 4_9 × 12
3
4. 3_4 − 1_ × 3_
(
2
)
5
___
___
(
5
1
2
_
_
7. 3 × __
18 + +
7 – 1_
6. 5_ × __
4
8
10
___
Math
Talk
6
)
5
___
MATHEMATICAL PRACTICES 6
Explain why the product
of two fractions has the
same value before and after
dividing the numerator and
denominator by the GCF.
On
On Your
Your Own
Own
Practice: Copy and Solve Find the product. Write it in simplest form.
8. 12_ × 25_
3
© Houghton Mifflin Harcourt Publishing Company
12.
9. _4 × 4_
8
9
DEEPER
10. _1 × 2_
5
6
5_
6
of the 90 pets in the pet show are
4
_
cats. 5 of the cats are calico cats. What fraction
of the pets are calico cats? How many of the
pets are calico cats?
______
MATHEMATICAL
PRACTICE
13.
11. 41_7 × 31_
9
3
DEEPER
Five cats each ate 1_4 cup of cat
food. Four other cats each ate 1_3 cup of cat
food. How much food did the nine cats eat?
______
6
Attend to Precision Algebra Evaluate using the order of operations.
Write the answer in simplest form.
(
3
14. _14 × _9 + 5
)
___
9 – 3_ × 1_
15. __
10
5
2
___
(
16. _4 + 1_2 – 3_7 × 2
5
3 +7
_
17. 15 × __
)
10
___
8
___
Chapter 2 • Lesson 3 83
18.
SMARTER
Write and solve a word problem for the
expression _14 × 2_. Show your work.
3
Changing Recipes
You can make a lot of recipes more healthful by reducing
the amounts of fat, sugar, and salt.
Kelly has a recipe for muffins that asks for 1 1_2 cups
of sugar. She wants to use 1_2 that amount of sugar
and more cinnamon and vanilla. How much sugar
will she use?
Find 1_2 of 1 1_2 cups to find what part of the original amount of sugar to use.
Write the mixed number as a
fraction greater than 1.
1
__ × 11
__ = 1
__ × ___WRITE
2
2 2 2
Math t Show Your Work
= ____
Multiply.
So, Kelly will use _ cup of sugar.
Michelle has a recipe that asks for 2 1_2 cups of vegetable oil.
She wants to use 2_3 that amount of oil and use applesauce to replace the
rest. How much applesauce will she use?
20.
Cara’s muffin recipe asks for 1 1_2 cups of flour for the
muffins and cup of flour for the topping. If she makes 1_2 of the original
recipe, how much flour will she use for the muffins and topping?
SMARTER
1_
4
84
© Houghton Mifflin Harcourt Publishing Company
DEEPER
19.
Practice and Homework
Name
Lesson 2.3
Multiply Fractions
COMMON CORE STANDARD—6.NS.B.4
Compute fluently with multi-digit numbers
and find common factors and multiples.
Find the product. Write it in simplest form.
28
1. _45 × 7_ = __
40
8
2. _18 × 20
7
10
= __
3. _45 × 3_
__
8
__
6. Karen raked 3_5 of the yard. Minni raked 1_3 of the
amount Karen raked. How much of the yard did
Minni rake?
5. _34 × 1_3 × 2_5
4. 1 1_ × 1_
8
9
__
__
7. 3_8 of the pets in the pet show are dogs. 2_3 of the
dogs have long hair. What fraction of the pets are
dogs with long hair?
Evaluate using the order of operations.
8. _1 + 3_ × 8
(2
8
9. _34 × 1 − 1_
)
(
__
9
)
__
3
10. 4 × 1_ × __
8
10
__
2 × 2_
11. 6 × _4 + __
3
5 10
(
)
__
Problem
Problem Solving
Solving
© Houghton Mifflin Harcourt Publishing Company
12. Jason ran 5_7 of the distance around the school
track. Sara ran 4_ of Jason’s distance. What
5
fraction of the total distance around the track did
Sara run?
14.
WRITE
13. A group of students attend a math club. Half
of the students are boys and 4_ of the boys have
9
brown eyes. What fraction of the group are boys
with brown eyes?
Math Write and solve a word problem that involves
multiplying by a fraction.
Chapter 2
85
Lesson Check (6.NS.B.4)
1. Veronica’s mom left 3_4 of a cake on the table. Her
brothers ate 1_2 of it. What fraction of the cake did
they eat?
2. One lap around the school track is 5_8 mile.
Carin ran 3 1_2 laps. How far did she run?
5 pounds of peanuts and
__
3. Tom bought 2 16
2.45 pounds of cashews. Which did he buy
more of? Explain.
4. Eve has 24 stamps each valued at $24.75. What
is the total value of her stamps?
5. Naomi went on a 6.5-mile hike. In the morning,
she hiked 1.75 miles, rested, and then hiked
2.4 more miles. She completed the hike in the
afternoon. How much farther did she hike in
the morning than in the afternoon?
6. A bookstore owner has 48 science fiction books
and 30 mysteries he wants to sell quickly. He
will make discount packages with one type
of book in each. He wants the most books
possible in each package, but all packages must
contain the same number of books. How many
packages can he make? How many packages of
each type of book does he have?
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Spiral Review (6.NS.B.3, 6.NS.B.4, 6.NS.C.7a)
Lesson 2.4
Name
Simplify Factors
The Number System—
6.NS.B.4
MATHEMATICAL PRACTICES
MP3, MP6
Essential Question How do you simplify fractional factors by using the
greatest common factor?
Unlock
Unlock the
the Problem
Problem
Some of the corn grown in the United States is used
7 of a farmer’s total crop
__
for making fuel. Suppose 10
is corn. He sells 2_5 of the corn for fuel production.
What fraction of the farmer’s total crop does he sell
for fuel production?
7
.
Multiply 2__5 × ___
10
One Way Simplify the product.
7 = _____
2 × 7 = ____
2 × ___
__
5 10 5 × 10
Multiply the numerators.
Multiply the denominators.
Write the product as a fraction in
simplest form.
÷ 2 = ____
= ________
50 ÷
So,
_2
5
7
× __
10 = _.
Another Way Simplify before multiplying.
7 = ______
2×7
2 × ___
__
5 10 5 × 10
© Houghton Mifflin Harcourt Publishing Company • Image Credits: (t) ©Bernhard Classen/Alamy
Write the problem as a single fraction.
Think: Do any numbers in the numerator
have common factors with numbers in
the denominator?
Divide the numerator and the
denominator by the GCF.
The GCF of 2 and 10 is _.
2÷2=_
2 in the numerator and _ in
the denominator have a common
factor other than 1.
1
2×7
______
5 × 10
10 ÷ 2 = _
Multiply the numerators.
Multiply the denominators.
1 × 7 = ____
______
5×
7
× __
10 = _, so the farmer sells _ of his crop
for fuel production.
_2
5
Math
Talk
MATHEMATICAL PRACTICES 3
Compare When you
multiply two fractions, will
the product be the same
whether you multiply first or
simplify first? Explain.
Chapter 2
87
Example
__
Find 5
8
___. Simplify before multiplying.
× 14
15
Divide a numerator and a denominator
by their GCF.
1
5 × 14
__
___
8 15
Be sure to divide both a
numerator and a denominator
by a common factor to write a
fraction in simplest form.
The GCF of 5 and 15 is _.
1
The GCF of 8 and 14 is _.
Multiply the numerators.
Multiply the denominators.
5
__ × 14
___
8 15
3
1 × ___ = ____
___
3
__ = _.
So, 5_8 × 14
15
Try This! Find the product. Simplify before multiplying.
A
3
__
8
× 2__
B
9
4
__
7
7
× ___
12
The GCF of 3 and 9 is _.
The GCF of 4 and 12 is _.
The GCF of 2 and 8 is _.
The GCF of 7 and 7 is _.
3
__ × 2
__ = ____
8 9
1.
MATHEMATICAL
PRACTICE
6
7 = ___
4 × ___
__
7 12
Explain why you cannot simplify before multiplying when
2.
MATHEMATICAL
PRACTICE
3 Compare Strategies What if you divided by a common
factor other than the GCF before you multiplied? How would that affect
your answer?
88
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finding _3 × 6_7.
5
Name
MATH
M
Share
hhow
Share and
and Show
Sh
BOARD
B
Find the product. Simplify before multiplying.
3
1. _5 × __
6
2. _34 × 5_
10
9
3. _2 × __
9
3
10
5
3 = ___
__ × ___
6 10
5
4. After a picnic, __
12 of the cornbread is left over. Val
eats 3_5 of the leftover cornbread. What fraction of
the cornbread does Val eat?
5. The reptile house at the zoo has an iguana that
is 5_6 yd long. It has a Gila monster that is 4_5 of
the length of the iguana. How long is the Gila
monster?
Math
Talk
On
On Your
Your Own
Own
MATHEMATICAL PRACTICES 6
Explain two ways to find the
product 1_6 × 2_3 in simplest
form.
Find the product. Simplify before multiplying.
6. _34 × 1_
6
7 × 2_
7. __
10
3
8. _5 × 2_
8
© Houghton Mifflin Harcourt Publishing Company
11. Shelley’s basketball team won 3_4 of their games
last season. In 1_6 of the games they won, they
outscored their opponents by more than 10
points. What fraction of their games did Shelley’s
team win by more than 10 points?
13.
MATHEMATICAL
PRACTICE
9 × 5_
9. __
5
10
12.
6
11 × 3
_
10. __
12 7
DEEPER
Mr. Ortiz has 3_4 pound of oatmeal.
He uses 2_3 of the oatmeal to bake muffins. How
much oatmeal does Mr. Ortiz have left?
16 × 3_ , you can multiply the
3 Compare Strategies To find __
4
27
fractions and then simplify the product or you can simplify the fractions
and then multiply. Which method do you prefer? Explain.
Chapter 2 • Lesson 4 89
MATHEMATICAL PRACTICES
ANALYZEt-00,'034536$563&t13&$*4*0/
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OqnakdlRnkuhmf¤@ookhb`shnmr
14. Three students each popped 3_4 cup of popcorn
kernels. The table shows the fraction of each
student’s kernels that did not pop. Which
1
student had __
16 cup unpopped kernels?
15.
DEEPER
The jogging track at Francine’s
school is mile long. Yesterday Francine
completed two laps on the track. If she ran 1_3
of the distance and walked the remainder of
the way, how far did she walk?
3_
4
7
At a snack store, __
12 of the
3
customers bought pretzels and __
10 of those
customers bought low-salt pretzels. Bill states
7 of the customers bought low-salt
__
that 30
pretzels. Does Bill’s statement make sense?
Explain.
Student
Fraction of Kernels not
Popped
Katie
1
__
Mirza
1
__
Jawan
1
_
9
SMARTER
WRITE
17.
12
Math t Show Your Work
SMARTER
The table shows Tonya’s
homework assignment. Tonya’s teacher
instructed the class to simplify each
expression by dividing the numerator and
denominator by the GCF. Complete the table
by simplifying each expression and then
finding the value.
Problem Expression
a
2
_
7
× 3_4
b
3
_
7
× 7_
c
5_
7
× 2_
4
__
× 3_
d
90
10
15
9
3
8
Simplified
Expression
Value
© Houghton Mifflin Harcourt Publishing Company • Image Credits: (t) ©Judith Collins/Alamy
16.
Popcorn Popping
Practice and Homework
Name
Lesson 2.4
Simplify Factors
COMMON CORE STANDARD—6.NS.B.4
Compute fluently with multi-digit numbers
and find common factors and multiples.
Find the product. Simplify before multiplying.
2 8×5
5
_____
1. _89 × __
12 = 9 × 12
3
__
2. _34 × 16
21
15 × 2_
3. __
20
5
9 × 2_
4. __
18
3
10
__
= 27
__
7
5. _34 × __
8 × 15
__
6. __
15
30
32
__
__
__
12 × 7_
7. __
21
9
__
__
18 × 8_
8. __
22
9
__
Problem
Problem Solving
Solving
9. Amber has a 4_5 -pound bag of colored sand. She
uses 1_2 of the bag for an art project. How much
sand does she use for the project?
© Houghton Mifflin Harcourt Publishing Company
11.
WRITE
10. Tyler has 3_4 month to write a book report. He
finished the report in 2_3 that time. How much
time did it take Tyler to write the report?
3 . Then tell which way is easier and
2 × __
Math Show two ways to multiply __
15 20
justify your choice.
Chapter 2
91
Lesson Check (6.NS.B.4)
Find each product. Simplify before multiplying.
1. At Susie’s school, 5_8 of all students play sports.
Of the students who play sports, 2_5 play soccer.
What fraction of the students in Susie’s school
play soccer?
__ pounds. The box
2. A box of popcorn weighs 15
16
contains 1_3 buttered popcorn and 2_3 cheesy
popcorn. How much does the cheesy popcorn
weigh?
3. Ramòn bought a dozen ears of corn for $1.80.
What was the cost of each ear of corn?
4. A 1.8-ounce jar of cinnamon costs $4.05. What
is the cost per ounce?
7
5. Rose bought __
20 kilogram of ginger candy and
0.4 kilogram of cinnamon candy. Which did she
buy more of? Explain how you know.
6. Don walked 3 3_5 miles on Friday, 3.7 miles on
Saturday, and 3 5_8 miles on Sunday. List the
distances from least to greatest.
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Spiral Review (6.NS.B.3, 6.NS.C.7a)
Name
Mid-Chapter Checkpoint
Personal Math Trainer
Vocabulary
Vocabulary
Vocabulary
Choose the best term from the box to complete the sentence.
common denominator
equivalent fractions
5 are ____. (p. 75)
1. The fractions 1_ and __
2
Online Assessment
and Intervention
mixed number
10
2. A ____ is a denominator that is the same in two
or more fractions. (p. 75)
Concepts
Concepts and
and Skills
Skills
Write as a decimal. Tell whether you used division, a number line,
or some other method. (6.NS.C.6c)
7
3. __
__
4. 839
20
5. 15_
40
8
__
6. 19
25
Order from least to greatest. (6.NS.C.7a)
© Houghton Mifflin Harcourt Publishing Company
7. _45, 3_4, 0.88
8. 0.65, 0.59, 3_
5
__
9. 11_4, 12_, 11
3 12
10. 0.9, 7_, 0.86
8
Find the product. Write it in simplest form. (6.NS.B.4)
11. _2 × 1_
3
8
12. _4 × 2_
5
5
13. 12 × 3_4
14. Mia climbs 5_8 of the height of the rock
wall. Lee climbs 4_5 of Mia’s distance. What
fraction of the wall does Lee climb?
Chapter 2
93
15.
DEEPER
In Zoe’s class, 4_5 of the students have pets. Of the students
who have pets, 1_8 have rodents. What fraction of the students in Zoe’s
class have pets that are rodents? What fraction of the students in Zoe’s
class have pets that are not rodents? (6.NS.B.4)
16. A recipe calls for 2 2_3 cups of flour. Terell wants to make 3_4 of the recipe.
How much flour should he use? (6.NS.B.4)
17. Following the Baltimore Running Festival in 2009, volunteers collected
and recycled 3.75 tons of trash. Graph 3.75 on a number line and write
the weight as a mixed number. (6.NS.C.6c)
18.
DEEPER
Four students took an exam. The
fraction of the total possible points that each
received is given. Which student had the highest
score? If students receive a whole number of
points on every exam item, can the exam be
worth a total of 80 points? Explain. (6.NS.C.7a)
4
Student
Score
Monica
22
__
25
17
__
20
4_
5
3_
4
Lily
Nikki
Sydney
94
© Houghton Mifflin Harcourt Publishing Company
3
Lesson 2.5
Name
Model Fraction Division
Essential Question How can you use a model to show division of fractions?
connect There are two types of division problems. In one type you
find how many or how much in each group, and in the other you find
how many groups.
Investigate
Investigate
The Number System—
6.NS.A.1
MATHEMATICAL PRACTICES
MP1, MP3, MP4
Hands
On
Materials ■ fraction strips
A class is working on a community project to clear a path near
the lake. They are working in teams on sections of the path.
A. Four students clear a section that is 2_3 mi long. If each student
clears an equal part, what fraction of a mile will each clear?
2 ÷ 4.
Divide __
3
• Use fraction strips to
model the division.
Draw your model.
• What are you trying to find?
2 ÷ 4 = _ , so each student will clear _ of a mile.
_
3
B. Another team clears a section of the path that is 3_4 mi long.
If each student clears 1_8 of a mile, how many students are
on the team?
__ ÷ 1
__.
Divide 3
© Houghton Mifflin Harcourt Publishing Company
4
8
• Use fraction strips to
model the division.
Draw your model.
• What are you trying to find?
3
_÷1
_ = _ , so there are _ students on the team.
4 8
Chapter 2
95
Draw Conclusions
1.
MATHEMATICAL
PRACTICE
4 Use Models Explain how the model in problem A
shows a related multiplication fact.
2.
MATHEMATICAL
PRACTICE
1 Analyze Suppose a whole number is divided by a
fraction between 0 and 1. Is the quotient greater than or less
than the dividend? Explain and give an example.
Make
Make Connections
Connections
You can draw a model to help you solve a fraction division problem.
Jessica is making a recipe that calls for 3_4 cup of flour.
Suppose she only has a 1_2 cup-size measuring scoop.
How many 1_2 cup scoops of flour does she need?
_÷1
_.
Divide 3
2
STEP 1 Draw a model that represents the total
amount of flour.
STEP 2 Draw fraction parts that represent the
scoops of flour.
Think: Divide a whole into _.
Think: What are you trying to find?
____
Jessica needs _ cup.
There is _ full group of 1_ and _ of a group of 1_.
2
2
So, there are _ groups of 1_ in 3_4.
2
3÷1
_ = _, so Jessica will need _ scoops of flour.
_
4 2
• What if Jessica’s recipe calls for 1_4 cup flour?
How many 1_2 cup scoops of flour does she need?
96
Math
Talk
MATHEMATICAL PRACTICES 1
Describe how you used the
model to determine the
number of groups of 1_2 in 3_4 .
© Houghton Mifflin Harcourt Publishing Company
4
Name
Share
hhow
Share and
and Show
Sh
MATH
M
BOARD
B
Use the model to find the quotient.
1
1. _2 ÷ 3 = _
3
2. 3_4 ÷ _8 = _
Think: 1_2 is shared among 3 groups.
?
?
?
Use fraction strips to find the quotient. Then draw the model.
3. _1 ÷ 4 = _
3
3 =_
4. _3 ÷ __
5
10
Draw a model to solve. Then write an equation for the model. Interpret the result.
© Houghton Mifflin Harcourt Publishing Company
5. How many 1_4 cup servings of raisins are
7.
6. How many 1_ lb bags of trail mix can Josh
in _3 cup of raisins?
8
3
5
_
make from lb of trail mix?
6
____
____
____
____
Math Pose a Problem Write and solve a problem for
WRITE
3
_ ÷ 3 that represents how much in each of 3 groups.
4
Chapter 2 • Lesson 5 97
MATHEMATICAL PRACTICES .0%&-t3&"40/tM",&4&/4&
OqnakdlRnkuhmf¤@ookhb`shnmr
OqnakdlRnkuhmf¤@ookhb`shnmr
The table shows the amount of each material that
students in a sewing class need for one purse.
Use the table for 8–10. Use models to solve.
8.
DEEPER
Mrs. Brown has _1 yd of blue denim and
3
1_ yd of black denim. How many
purses can be made
2
using denim as the main fabric?
Purse Materials (yd)
9.
10.
SMARTER
One student brings _12
yd of ribbon.
If 3 students receive an equal length of the ribbon,
how much ribbon will each student receive? Will
each of them have enough ribbon for a purse?
Explain.
MATHEMATICAL
PRACTICE
3 Make Arguments There was _1 yd of
2
Ribbon
Main fabric
Trim fabric
WRITE
1
4
1
6
1
12
Math t Show Your Work
purple and pink striped fabric. Jessie said she could
1
only make __
24 of a purse using that fabric as the
trim. Is she correct? Use what you know about the
meanings of multiplication and division to defend
your answer.
11.
SMARTER
Draw a model to find
1÷4=
_
2
98
© Houghton Mifflin Harcourt Publishing Company
the quotient.
Practice and Homework
Name
Lesson 2.5
Model Fraction Division
Use the model to find the quotient.
1 4 3 5 __
1
1. _
12
4
COMMON CORE STANDARD—6.NS.A.1
Apply and extend previous understandings of
multiplication and division to divide fractions
by fractions.
2
1 4_
5 __
2. _
2 12
0
2
12
4
12
6
12
8
12
10
12
1
Use fraction strips to find the quotient.
3.
_5 4_1
6
24 4
4. _
3
2
__
__
14 6
5. _
2
__
1
1 4_
6. _
3 12
__
Use a number line to find the quotient.
1 mile each day, how many days
__
7. If Jerry runs 10
will it take for him to run 4_5 mile?
_______
© Houghton Mifflin Harcourt Publishing Company
Problem
Problem Solving
Solving
8. Mrs. Jennings has 4_3 gallon of paint for an art
project. She plans to divide the paint equally into
jars. If she puts 8_1 gallon of paint into each jar, how
many jars will she use?
_______
10.
WRITE
1
__
9. If one jar of glue weighs 12
pound, how
many jars can Rickie get from 3_2 pound
of glue?
_______
1 and 2_ ÷ 4.
Math Explain how to use a model to show 2_ ÷ __
6 12
6
Chapter 2
99
Lesson Check (6.NS.A.1)
Sketch a model to find the quotient.
1. Darcy needs 1_4 yard of fabric to make a banner.
She has 2 yards of fabric. How many banners
can she make?
__ pounds of ground beef. He
2. Lorenzo bought 15
16
3
wants to make hamburgers that weigh __
16 pound
each. How many hamburgers can he make?
3. Every night Letisha reads 22 pages. At that rate,
how long will it take her to read a book with
300 pages?
4. A principal wants to order enough notebooks
for 624 students. The notebooks come in boxes
of 28. How many boxes should he order?
5. Each block in Ton’s neighborhood is 3_8 mile
long. If he walks 4 1_2 blocks, how far does
he walk?
6. In Cathy’s garden, 5_6 of the area is planted with
3
flowers. Of the flowers, __
10 of them are red. What
fraction of Cathy’s garden is planted with red
flowers?
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GO TO THE
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Spiral Review (6.NS.B.2, 6.NS.B.4)
Lesson 2.6
Name
Estimate Quotients
Essential Question How can you use compatible numbers to
estimate quotients of fractions and mixed numbers?
The Number System—
6.NS.A.1
MATHEMATICAL PRACTICES
MP1, MP2, MP3, MP6
connect You have used compatible numbers to estimate
quotients of whole numbers and decimals. You can also use
compatible numbers to estimate quotients of fractions and
mixed numbers.
Compatible numbers are pairs
of numbers that are easy to
compute mentally.
Unlock
Unlock the
the Problem
Problem
Humpback whales have “songs” that they repeat
continuously over periods of several hours. Eric
is using an underwater microphone to record a
3 5_6 minute humpback song. He has 15 3_4 minutes of
battery power left. About how many times will he
be able to record the song?
• Which operation should you use to solve the
problem? Why?
• How do you know that the problem calls for
an estimate?
One Way Estimate 153__4 ÷ 35__6 using
compatible numbers.
Think: What whole numbers
3 and 35
__ are easy
close to 15__
4
6
to divide mentally?
© Houghton Mifflin Harcourt Publishing Company • Image Credits: (c) ©Amos Nachoum/Corbis
Rewrite the problem using
compatible numbers.
__ is close to _.
153
4
__ is close to _.
35
6
__ ÷ 35
__
153
4
6
↓
↓
16 ÷ 4 = _
Divide.
So, Eric will be able to record the complete whale song
about _ times.
1.
MATHEMATICAL
PRACTICE
3 Compare Strategies To estimate 15 _3 ÷ 3 5_ , Martin used
4
6
15 and 3 as compatible numbers. Tina used 15 and 4. Were their choices
good ones? Explain why or why not.
Chapter 2
101
Example
Estimate using compatible numbers.
A 52__ ÷ 5__
3
8
__
52
Rewrite the problem using compatible numbers.
3
÷
↓
5
__
8
↓
_÷_
Think: How many halves are there in 6?
__ = _
6÷1
2
So, 52_ ÷ 5_ is about _.
3
B
8
7÷1
__
__
4
8
7
__
8
Rewrite the problem using compatible numbers.
÷
↓
↓
_÷
Think: How many fourths are there in 1?
1=
1 ÷ __
4
1
__
4
1
__
4
_
So, 7_ ÷ 1_4 is about _.
8
2.
MATHEMATICAL
PRACTICE
2 Use Reasoning Will the actual quotient 5 2_ ÷ 5_ be greater
3
8
than or less than the estimated quotient? Explain.
4.
MATHEMATICAL
PRACTICE
9
6 Explain how you would estimate the quotient 14 _3 ÷ 3 __
4
10
using compatible numbers.
102
© Houghton Mifflin Harcourt Publishing Company
3. Will the actual quotient 7_8 ÷ 1_4 be greater than or less than the estimated
quotient? Explain.
Name
Share
hhow
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and Show
Sh
MATH
M
BOARD
B
Estimate using compatible numbers.
1. 224_
÷
5
↓
2. 12 ÷ 33_4
61_4
3. 337_ ÷ 51_
8
3
↓
_÷_=_
4. 37_ ÷ 5_
8
7 ÷ 73_
5. 34__
9
12
6. 12_ ÷ 1_
8
9
Math
Talk
On
On Your
Your Own
Own
6
MATHEMATICAL PRACTICES 6
Explain how using
compatible numbers is
different than rounding to
estimate 35 1_2 ÷ 6 5_6 .
Estimate using compatible numbers.
7. 441_4 ÷ 117_
9
SMARTER
3 ÷ 25
_
10. 21__
© Houghton Mifflin Harcourt Publishing Company
10
6
__ ÷ 83
_
8. 7111
4
9. 11_ ÷ 1_
12
6
8
Estimate to compare. Write <, >, or =.
● 35 ÷ 3
7
_
9
2
_
3
11. 294_ ÷ 51_
5
6
● 27 ÷ 6
8
_
9
5
_
8
7
12. 555_ ÷ 6__
6
10
● 11 ÷
5
_
7
5
_
8
13. Marion is making school flags. Each flag uses 2 3_4 yards of felt. Marion
has 24 1_ yards of felt. About how many flags can he make?
8
14.
DEEPER
A garden snail travels about 2 3_5 feet in 1 minute. At that
speed, about how many hours would it take the snail to travel 350 feet?
Chapter 2 • Lesson 6 103
MATHEMATICAL PRACTICES
COMMUNICA5&t1&34E7&3&tCONSTRUCT ARGUMENTS
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OqnakdlRnkuhmf¤@ookhb`shnmr
What’s the Error?
15. Megan is making pennants from a piece of butcher paper
that is 10 3_8 yards long. Each pennant requires 3_8 yard of
paper. To estimate the number of pennants she could
make, Megan estimated the quotient 10 3_8 ÷ 3_8 .
Look at how Megan solved the problem.
Find her error.
Correct the error.
Estimate the quotient.
Estimate:
103_ ÷ 3_8
8
↓
↓
10 ÷ 1_2 = 5
So, Megan can make about _ pennants.
•
MATHEMATICAL
PRACTICE
•
MATHEMATICAL
PRACTICE
1 Describe the error that Megan made.
6 Explain Tell which compatible numbers you used to
estimate 10 3_ ÷ 3_ . Explain why you chose those numbers.
8
104
SMARTER
For numbers 16a–16c, estimate to compare.
Choose <, >, or =.
16a.
3 ÷25
_
18 __
6
<
>
=
16b.
17 4_ ÷ 6 1_
6
<
>
=
19 8_ ÷ 4 5_
16c.
35 5_ ÷ 6 1_4
<
>
=
11 5_7 ÷ 2 3_4
10
5
6
30 7_ ÷ 3 1_
9
9
3
8
© Houghton Mifflin Harcourt Publishing Company
16.
8
Practice and Homework
Name
Lesson 2.6
Estimate Quotients
COMMON CORE STANDARD—6.NS.A.1
Apply and extend previous understandings of
multiplication and division to divide fractions
by fractions.
3. 22_1 ÷ 1_5
5
6
Estimate using compatible numbers.
3 ÷ 3__
9
1. 12__
16
10
↓
↓
2. 153_ ÷ 1_
8
2
12 4 4 5 3
___
4. 77_ ÷ 4_7
___
5
7. 147_ ÷ __
8
__ ÷ 1
_
6. 15
16 7
5. 181_4 ÷ 24_
5
9
___
7 ÷ 8 11
__
8. 53__
11
12
___
___
___
9. 11_ ÷ 1_
12
6
___
9
___
Problem
Problem Solving
Solving
© Houghton Mifflin Harcourt Publishing Company
10. Estimate the number of pieces Sharon will
have if she divides 15 1_3 yards of fabric into
4 4_5 yard lengths.
12.
WRITE
11. Estimate the number of 1_2 quart containers Ethan
can fill from a container with 8 7_8 quarts of water.
Math How is estimating quotients different from estimating
products?
Chapter 2
105
Lesson Check (6.NS.A.1)
1. Each loaf of pumpkin bread calls for 1 3_4 cups of
raisins. About how many loaves can be made
from 10 cups of raisins?
2. Perry’s goal is to run 2 1_4 miles each day. One lap
around the school track is 1_3 mile. About how
many laps must he run to reach his goal?
3. A recipe calls for 3_4 teaspoon of red pepper. Uri
wants to use 1_3 of that amount. How much red
pepper should he use?
4. A recipe calls for 2 2_3 cups of apple slices. Zoe
wants to use 1 1_2 times this amount. How many
cups of apples should Zoe use?
5. Edgar has 2.8 meters of rope. If he cuts it into
7 equal parts, how long will each piece be?
6. Kami has 7 liters of water to fill water bottles
that each hold 2.8 liters. How many bottles
can she fill?
FOR MORE PRACTICE
GO TO THE
106
Personal Math Trainer
© Houghton Mifflin Harcourt Publishing Company
Spiral Review (6.NS.B.3, 6.NS.B.4)
Lesson 2.7
Name
Divide Fractions
The Number System—
6.NS.A.1
MATHEMATICAL PRACTICES
MP2, MP5, MP6, MP8
Essential Question How do you divide fractions?
Unlock
Unlock the
the Problem
Problem
Toby and his dad are building a doghouse. They
need to cut a board that is 2_3 yard long into 1_6 yard
pieces. How many 1_6 yard pieces can they cut?
One Way Divide 2__3 ÷ 1__6 by using a number line.
STEP 1 Draw a number line, and shade it to
represent the total length of the board.
STEP 2 Show fraction parts that represent
the pieces of board.
Think: Divide a whole into thirds.
Think: Find the number of groups of 1__ in 2__.
6
3
2 yard, so shade 2
__.
Toby and his dad have __
3
3
1
6
0
1
3
2
3
1
So, there are _ 1_6 yard pieces in 2_3 yard.
Another Way Divide 2__3 ÷ 1__6 by using a common denominator.
© Houghton Mifflin Harcourt Publishing Company
STEP 1 Write equivalent fractions using a
common denominator.
Think:
2÷1
×
__
__ = 2
_____
__ = ___ ÷ 1
__
÷1
3 6 3×
6 6 6
_ is a multiple of 3 and 6,
so _ is a common denominator.
STEP 2 Divide.
Think: There are _ groups of 1__ in 4__.
6
6
4÷1
__
__ = _
Math
6 6
Talk
So, 2_3 ÷ 1_6 = _. Toby and his dad can cut _ 1_6 yard pieces.
MATHEMATICAL PRACTICES 2
Reasoning How can you find
the quotient 2_3 ÷ 2_9 by using
a common denominator?
Chapter 2
107
You can use reciprocals and inverse operations to divide fractions.
Two numbers whose product is 1 are reciprocals or multiplicative inverses.
2×3
_
_=1
3 2
2 and 3_ are reciprocals.
_
3
2
Activity Find a pattern.
• Complete the table by finding the products.
Division
• How are each pair of division and multiplication problems the
same, and how are they different?
Multiplication
4÷2
_
_=2
7 7
4×7
_
_=
7 2
5÷4
_
_=5
_
6 6 4
5×6
_
_=
6 4
1
_÷5
_=3
_
3 9 5
1
_×9
_=
3 5
• How could you use the pattern in the table to rewrite a division
problem involving fractions as a multiplication problem?
Example
Winnie needs pieces of string for a craft project. How many
3_
1
__
12 yd pieces of string can she cut from a piece that is 4 yd long?
1.
Divide 3_4 ÷ __
12
1
Estimate. _ ÷ __
12 = _
1 =3
3 ÷ ___
__ × ____
__
4 12 4
Use the reciprocal of the divisor
to write a multiplication problem.
4
= __
Multiply.
Check your answer.
1 3
___
12
__ 5 __ 5 __
Since the estimate is _ , the answer is reasonable.
1
So, Winnie can cut _ __
12 yd pieces of string.
108
1
Math
Talk
MATHEMATICAL PRACTICES 6
Explain how you used multiplication
to check your answer.
© Houghton Mifflin Harcourt Publishing Company
___
__ × 12
=3
Simplify the factors.
Name
Share
hhow
Share and
and Show
Sh
MATH
M
BOARD
B
Estimate. Then find the quotient.
5
1. __ ÷ 3
6
Write the whole number as
a fraction.
Use the reciprocal of the divisor to
write a multiplication problem.
Estimate. _ ÷ 3 = _
5 ÷ ___
3
__
6
5 × ___ = ____
__
6
Use a number line to find the quotient.
2. _34 ÷ 1_ = _
3 =_
3. _35 ÷ __
8
10
0
1
0
1
Estimate. Then write the quotient in simplest form.
4. _34 ÷ 5_
___
5
7. __
÷3
12
6. _12 ÷ 3_4
5. 3 ÷ 3_4
6
___
___
Math
Talk
___
MATHEMATICAL PRACTICES 5
Use Tools Explain how to find a
__ 4 1_ .
reasonable estimate for 11
4
12
On
On Your
Your Own
Own
Practice: Copy and Solve Estimate. Then write the quotient in simplest form.
9. _34 ÷ 3_
8. 2 ÷ 1_
10. _25 ÷ 5
5
8
11. 4 ÷ 1_7
© Houghton Mifflin Harcourt Publishing Company
Practice: Copy and Solve Evaluate using the order of operations.
Write the answer in simplest form.
1 ÷2
12. _3 + __
(5
15.
10
)
MATHEMATICAL
PRACTICE
1 ÷2
13. 3_5 + __
10
1
14. 3_5 + 2 ÷ __
10
8 Generalize Suppose the divisor and the dividend
of a division problem are both fractions between 0 and 1, and
the divisor is greater than the dividend. Is the quotient less than,
equal to, or greater than 1?
Chapter 2 • Lesson 7 109
MATHEMATICAL PRACTICES
ANALYZEt-00,'034536$563&t13&$*4*0/
OqnakdlRnkuhmf¤@ookhb`shnmr
OqnakdlRnkuhmf¤@ookhb`shnmr
Use the table for 16–19.
16. Kristen wants to cut ladder rungs from a 6 ft
board. How many ladder rungs can she cut?
17.
Item
Board Length
Ladder rung
3 ft
4
5 yd
8
1 yd
2
“Keep Out” sign
Pose a Problem Look back
at Problem 16. Write and solve a new problem
by changing the length of the board Kristen is
cutting for ladder rungs.
SMARTER
18. Dan paints a design that has 8 equal parts
along the entire length of the windowsill.
How long is each part of the design?
19.
Tree House Measurements
Windowsill
WRITE
Math t Show Your Work
__ yd.
Dan has a board that is 15
16
How many “Keep Out” signs can he make if
the length of the sign is changed to half of the
original length?
DEEPER
Personal Math Trainer
110
Lauren has 3_4 cup of dried
fruit. She puts the dried fruit into bags, each
holding 1_8 cup. How many bags will Lauren
use? Explain your answer using words and
numbers.
SMARTER
© Houghton Mifflin Harcourt Publishing Company
20.
Practice and Homework
Name
Lesson 2.7
Divide Fractions
COMMON CORE STANDARD—6.NS.A.1
Apply and extend previous understandings of
multiplication and division to divide fractions
by fractions.
Estimate. Then write the quotient in simplest form.
1. 5 ÷ 1_6
2. 1_2 ÷ 1_4
3. 4_5 ÷ 2_3
__ ÷ 7
4. 14
15
Estimate: 30
_
=5×6
30
= __
1
1
= 30
__
5. 8 ÷ 1_3
12 ÷ 2_
6. __
21 3
__
__
9. Joy ate 1_4 of a pizza. If she divides the rest of the
pizza into pieces equal to 1_8 pizza for her family,
how many pieces will her family get?
__
5
7. _56 ÷ __
12
__
__
8. _58 ÷ 1_2
__
10. Hideko has 3_5 yard of ribbon to tie on balloons
3
for the festival. Each balloon will need __
10 yard of
ribbon. How many balloons can Hideko tie with
ribbon?
© Houghton Mifflin Harcourt Publishing Company
Problem
Problem Solving
Solving
11. Rick knows that 1 cup of glue weighs
1
2_
__
18 pound. He has 3 pound of glue. How
many cups of glue does he have?
13.
WRITE
12. Mrs. Jennings had 5_7 gallon of paint. She gave
1_ gallon each to some students. How many
7
students received paint if Mrs. Jennings gave
away all the paint?
Math Write a word problem that involves two fractions.
Include the solution.
Chapter 2
111
Lesson Check (6.NS.A.1)
1. There was 2_3 of a pizza for 6 friends to share
equally. What fraction of the pizza did each
person get?
2. Rashad needs 2_3 pound of wax to make
a candle. How many candles can he make
with 6 pounds of wax?
Spiral Review (6.NS.A.1, 6.NS.B.3, 6.NS.B.4)
4. Ebony walked at a rate of 3 1_2 miles per hour for
1 1_3 hours. How far did she walk?
5. Penny uses 3_4 yard of fabric for each pillow she
makes. How many pillows can she make using
6 yards of fabric?
6. During track practice, Chris ran 2.5 laps in
81 seconds. What was his average time per lap?
© Houghton Mifflin Harcourt Publishing Company
3. Jeremy had 3_4 of a submarine sandwich and gave
his friend 1_3 of it. What fraction of the sandwich
did the friend receive?
FOR MORE PRACTICE
GO TO THE
112
Personal Math Trainer
Lesson 2.8
Name
Model Mixed Number Division
Essential Question How can you use a model to show division of
mixed numbers?
The Number System—
6.NS.A.1
MATHEMATICAL PRACTICES
MP4, MP5, MP8
Hands
On
Investigate
Investigate
Materials ■ pattern blocks
A science teacher has 1 2_3 cups of baking soda. She performs
an experiment for her students by mixing 1_6 cup of baking soda
with vinegar. If the teacher uses the same amount of baking
soda for each experiment, how many times can she perform
the experiment?
A. Which operation should you use to find the answer? Why?
B. Use pattern blocks to show 1 2_3 .
Draw your model.
Think: A hexagon block is
one whole, and a rhombus is
_ of a whole.
• What type and number of blocks did you use to model 1 2_3 ?
C. Cover 1 2_3 with blocks that represent 1_6 to show dividing by 1_6 . Draw
your model.
© Houghton Mifflin Harcourt Publishing Company
Think: One __
block represents _ of a
whole.
_ triangle blocks cover 12_3.
12_ ÷ 1_ = _
3
6
So, the teacher can perform the experiment _ times.
Math
Talk
MATHEMATICAL PRACTICES 5
Use Tools How can you
check that your answer is
reasonable?
Chapter 2
113
Draw Conclusions
MATHEMATICAL
PRACTICE
1.
5 Communicate Tell how your model shows a related
multiplication problem.
MATHEMATICAL
PRACTICE
2.
1 Describe Relationships Suppose a mixed number is
divided by a fraction between 0 and 1. Is the quotient greater than or less
than the dividend? Explain and give an example.
Make
Make Connections
Connections
You can use a model to divide a mixed number by a whole number.
Naomi has 2 1_4 quarts of lemonade. She wants to divide the lemonade
equally between 2 pitchers. How many quarts of lemonade should
she pour into each pitcher?
__ ÷ 2.
Divide 21
STEP 1 Draw a model that represents the total
amount of lemonade.
STEP 2 Draw parts that represent the amount
in each pitcher.
Think: Divide 3 wholes into __.
Think: What are you trying to find?
Shade _.
______
Think: In each of the two equal groups there is _ whole and _ of 1__4.
1 of 1
__
__ is
4
2
_.
So, 21_4 ÷ 2 = _. Naomi should pour _ quarts of
lemonade into each pitcher.
114
Math
Talk
MATHEMATICAL PRACTICES 8
Generalize Explain how the
quotient compares to the
dividend when dividing a
mixed number by a whole
number greater than 1.
© Houghton Mifflin Harcourt Publishing Company
4
Name
Share
hhow
Share and
and Show
Sh
MATH
M
BOARD
B
Use the model to find the quotient.
1. 31_ ÷ 1_ = _
3
3
2. 21_ ÷ 1_ = _
2
6
Use pattern blocks to find the quotient. Then draw the model.
3. 22_ ÷ 1_ = _
3
6
4. 31_ ÷ 1_ = _
2
2
Draw a model to find the quotient.
5. 31_ ÷ 3 = _
2
© Houghton Mifflin Harcourt Publishing Company
6. 141_ ÷ 2 = _
7.
MATHEMATICAL
PRACTICE
5 Use Appropriate Tools Explain how models can be used
to divide mixed numbers by fractions or whole numbers.
Chapter 2 • Lesson 8 115
MATHEMATICAL PRACTICES .0%&-t3&"40/tM",&4&/4&
OqnakdlRnkuhmf¤@ookhb`shnmr
OqnakdlRnkuhmf¤@ookhb`shnmr
Use a model to solve. Then write an equation for the model.
8.
MATHEMATICAL
PRACTICE
4 Use Models Eliza opens a box of bead kits. The
box weighs 2 2_ lb. Each bead kit weighs 1_ lb. How many kits
3
6
are in the box? What does the answer mean?
9.
DEEPER
Hassan has two boxes of trail mix. Each box holds
lb of trail mix. He eats 1_3 lb of trail mix each day. How many
days can Hassan eat trail mix before he runs out?
1 _23
11.
Sense or Nonsense? Steve made
this model to show 2 1_3 ÷ 1_6 . He says that the quotient
is 7. Is his answer sense or nonsense? Explain your
reasoning.
SMARTER
SMARTER
Eva is making muffins to sell at a fundraiser. She
cups of flour, and the recipe calls for 3_4 cup of flour for each
batch of muffins. Explain how to use a model to find the number of
batches of muffins Eva can make.
has 2 _14
116
WRITE
Math
Show Your Work
© Houghton Mifflin Harcourt Publishing Company • Image Credits: (t) ©Garry Gay/Alamy
10.
Practice and Homework
Name
Lesson 2.8
Model Mixed Number Division
COMMON CORE STANDARD—6.NS.A.1
Apply and extend previous understandings of
multiplication and division to divide fractions
by fractions.
Use the model to find the quotient.
9
1. 4 2_1 ÷ 2_1 = _
2. 3 3_1 ÷ 6_1 = _
Use pattern blocks or another model to find the quotient. Then draw the model.
3. 2 2_1 ÷ 6_1 = _
4. 2 4_3 ÷ 2 = _
Problem
Problem Solving
Solving
© Houghton Mifflin Harcourt Publishing Company
5. Marty has 2 4_5 quarts of juice. He pours the same
amount of juice into 2 bottles. How much does
he pour into each bottle?
7.
WRITE
6. How many 1_3 pound servings are in
4 2_3 pounds of cheese?
Math Write a word problem that involves dividing a mixed
number by a whole number. Solve the problem and describe how you
found the answer.
Chapter 2
117
Lesson Check (6.NS.A.1)
Sketch a model to find the quotient.
1. Emma has 4 1_2 pounds of birdseed. She wants
to divide it evenly among 3 bird feeders. How
much birdseed should she put in each?
2. A box of crackers weighs 11 1_4 ounces. Kaden
estimates that one serving is 3_4 ounce. How
many servings are in the box?
3. The Ecology Club has volunteered to clean
up 4.8 kilometers of highway. The members
are organized into 16 teams. Each team will
clean the same amount of highway. How much
highway will each team clean?
4. Tyrone has $8.06. How many bagels can he buy
if each bagel costs $0.65?
5. A nail is 0.1875 inch thick. What is its thickness
as a fraction? Is 0.1875 inch closer to 1_8 inch or
1_ inch on a number line?
4
3
4
__
6. Maria wants to find the product of 5 __
20 × 3 25
using decimals instead of fractions. How can
she rewrite the problem using decimals?
FOR MORE PRACTICE
GO TO THE
118
Personal Math Trainer
© Houghton Mifflin Harcourt Publishing Company
Spiral Review (6.NS.B.3, 6.NS.B.4, 6.NS.C.6c)
Lesson 2.9
Name
Divide Mixed Numbers
The Number System—
6.NS.A.1
MATHEMATICAL PRACTICES
MP1, MP6, MP7
Essential Question How do you divide mixed numbers?
Unlock
Unlock the
the Problem
Problem
A box weighing 9 1_3 lb contains robot kits weighing
1 1_6 lb apiece. How many robot kits are in the box?
• Underline the sentence that tells you what you
are trying to find.
• Circle the numbers you need to use to solve
_1 ÷ 1_1 .
Divide 9
3
the problem.
6
Estimate the quotient.
Write the mixed numbers
as fractions.
_÷_=_
__ ÷ 11
__ = ____ ÷ ___
91
3
3
6
6
___ × ___
= 28
Use the reciprocal of
the divisor to write a
multiplication problem.
3
___ × 6
__
= 28
3
Simplify.
= _____, or _
Multiply.
© Houghton Mifflin Harcourt Publishing Company • Image Credits: (cr) ©Photodisc/Getty Images
7
Compare your estimate with the quotient. Since the estimate, _,
is close to _, the answer is reasonable.
So, there are _ robot kits in the box.
Try This! Estimate. Then write the quotient in simplest form.
Think: Write the mixed numbers as fractions.
1 1
A 2_3 ÷ _6
B 53_ ÷ 3_
4 8
Chapter 2
119
Example
Four hikers shared 3 1_3 qt of water
equally. How much did each hiker receive?
Divide 3 1_3 ÷ 4. Check.
Estimate. _ ÷ 4 = 1
Write the mixed number
and the whole number
as fractions.
__ ÷ 4 = ____ ÷ ___
31
3
3
Use the reciprocal of
the divisor to write
a multiplication problem.
___ × ___
= 10
Simplify.
___ × 1
__
= 10
3
3
4
=_
Multiply.
Check your answer.
4 × _=
______
=_
So, each hiker received _ qt.
Math
Talk
1. Describe what you are trying to find in the Example above.
2.
MATHEMATICAL
PRACTICE
MATHEMATICAL PRACTICES 1
Evaluate whether your answer
is reasonable using the
information in the problem.
6 Compare Explain how dividing mixed numbers is similar
3.
120
SMARTER
The divisor in a division problem is between 0 and 1 and
the dividend is greater than 0. Will the quotient be greater than or less
than the dividend? Explain.
© Houghton Mifflin Harcourt Publishing Company
to multiplying mixed numbers. How are they different?
Name
MATH
M
Share
hhow
Share and
and Show
Sh
BOARD
B
Estimate. Then write the quotient in simplest form.
1.
__ ÷ 3
__ = ______ ÷ 3
__
41
3
3
4
2. Six hikers shared 4 1_2 lb of trail mix. How much
trail mix did each hiker receive?
4
___ × ___
= 13
3
= ____, or 5 ___
9
________
3. 52_ 4 3
4. 71_ 4 21_
3
2
________
2
________
Math
Talk
On
On Your
Your Own
Own
MATHEMATICAL PRACTICES 6
Explain why you write a mixed
number as a fraction before using it
as a dividend or divisor.
Estimate. Then write the quotient in simplest form.
5. 53_4 ÷ 41_
7. 63_4 ÷ 2
6. 5 ÷ 11_
2
3
___
___
10.
© Houghton Mifflin Harcourt Publishing Company
________
___
Samantha cut 6 3_4 yd of yarn into
3 equal pieces. Explain how she could use mental
math to find the length of each piece.
SMARTER
________
1 Evaluate Algebra Evaluate using the order of operations. Write the answer in simplest form.
11. 11_ × 2 ÷ 11_
2
3
___
14.
9
___
9. How many 3 1_3 yd pieces can Amanda get
from a 13 1_3 yd ribbon?
MATHEMATICAL
PRACTICE
8. 22_ ÷ 13_7
MATHEMATICAL
PRACTICE
__ + 5
_
12. 12_ ÷ 113
5
15
8
___
13. 31_ − 15_ ÷ 12_
2
6
9
___
7 Look for a Pattern Find these quotients:
20 ÷ 4 _45 , 10 ÷ 4 4_5 , 5 ÷ 4 4_5 . Describe a pattern you see.
Chapter 2 • Lesson 9 121
MATHEMATICAL PRACTICES
COMMUNICA5&t1&34E7&3&tCONSTRUCT ARGUMENTS
Unlock
Unlock the
the Problem
Problem
15.
Dina hikes 1_2 of the easy trail and
stops for a break every 3 1_4 miles. How many
breaks will she take?
DEEPER
a. What problem are you asked to solve?
Hiking Trails
Park
Trail
Cuyahoga
Valley National
Park, Ohio
Length (mi) Difficulty
Ohio and Erie
Canal Towpath
191_
easy
Brandywine
Gorge
11_
moderate
Buckeye Trail
(Jaite to Boston)
53_
difficult
2
4
5
b. How will you use the information in the table to
solve the problem?
d. What operation will you use to find how many
breaks Dina takes?
c. How can you find the distance Dina hikes? How
far does she hike?
e. How many breaks will Dina take?
WRITE
16.
M t Show Your Work
Math
SMARTER
Carlo packs 15 3_4 lb of books in 2 boxes. Each book weighs
lb. There are 4 more books in Box A than in Box B. How many books are
in Box A? Explain your work.
1 _18
122
Rex’s goal is to run 13 3_4 miles over 5 days. He wants to run
the same distance each day. Jordan said that Rex would have to run
3 3_4 miles each day to reach his goal. Do you agree with Jordan? Explain
your answer using words and numbers.
SMARTER
© Houghton Mifflin Harcourt Publishing Company
17.
Practice and Homework
Name
Lesson 2.9
Divide Mixed Numbers
COMMON CORE STANDARD—6.NS.A.1
Apply and extend previous understandings of
multiplication and division to divide fractions
by fractions.
Estimate. Then write the quotient in simplest form.
1. 21_ ÷ 21_
2
3. 2 ÷ 35_
2. 22_ ÷ 11_
3
3
3
8
Estimate: 2 4 2 5 1
_
_47
_ 4 21
_ 55
21
3
2
2
3
5
3
5_
3_
2
7
15
1
5 __
or 1__
15
____
14
__ ÷ 12
_
4. 113
15
5. 10 ÷ 62_
5
3
__
__
____
1
6. 23_ ÷ 1__
5
25
__
7. 21_ ÷ 2
5
__
8. Sid and Jill hiked 4 1_8 miles in the morning
and 1 7_8 miles in the afternoon. How many
times as far did they hike in the morning
as in the afternoon?
Problem
Problem Solving
Solving
© Houghton Mifflin Harcourt Publishing Company
9. It takes Nim 2 2_3 hours to weave a basket. He
worked Monday through Friday, 8 hours a day.
How many baskets did he make?
11.
WRITE
10. A tree grows 1 3_4 feet per year. How long will it
take the tree to grow from a height of 21 1_4 feet to a
height of 37 feet?
Math Explain how you would find how many 1 1_2 cup
servings there are in a pot that contains 22 1_2 cups of soup.
Chapter 2
123
Lesson Check (6.NS.A.1)
1. Tom has a can of paint that covers 37 1_2 square
meters. Each board on the fence has an area
3
of __
16 square meters. How many boards can he
paint?
2. A baker wants to put 3 3_4 pounds of apples in
each pie she makes. She purchased 52 1_2 pounds
of apples. How many pies can she make?
3. The three sides of a triangle measure
9.97 meters, 10.1 meters, and 0.53 meter.
What is the distance around the triangle?
4. Selena bought 5.62 pounds of meat for
$3.49 per pound. What was the total cost of
the meat?
5. Melanie prepared 7 1_2 tablespoons of a spice
mixture. She uses 1_4 tablespoon to make a batch
of barbecue sauce. Estimate the number of
batches of barbecue sauce she can make using
the spice mixture.
6. Arturo mixed together 1.24 pounds of pretzels,
0.78 pounds of nuts, 0.3 pounds of candy, and
2 pounds of popcorn. He then packaged it in
bags that each contained 0.27 pounds. How
many bags could he fill?
FOR MORE PRACTICE
GO TO THE
124
Personal Math Trainer
© Houghton Mifflin Harcourt Publishing Company
Spiral Review (6.NS.A.1, 6.NS.B.3)
PROBLEM SOLVING
Name
Lesson 2.10
Problem Solving • Fraction Operations
Essential Question How can you use the strategy use a model to help
you solve a division problem?
Unlock
Unlock the
the Problem
Problem
The Number System—
6.NS.A.1
MATHEMATICAL PRACTICES
MP1, MP2, MP4, MP6
Sam had 3_4 lb of granola. Each day he took
1_
1_
8 lb to school for a snack. If he had 4 lb left over,
how many days did Sam take granola to school?
Use the graphic organizer below to help you solve the
problem.
Read the Problem
What do I need to find?
I need to find __
© Houghton Mifflin Harcourt Publishing Company • Image Credits: (t) ©Plush Studios/Blend Images/Corbis
____.
What information do
I need to use?
How will I use the
information?
Sam started with _ lb of
I will draw a bar model to find
granola and took _ lb
how much ___
each day. He had _ lb
____
left over.
____.
Solve the Problem
3
4 lb
The model shows that Sam used _ lb
of granola.
_ groups of 1_8 are equivalent to 1_2
1
8
used
left
So, Sam took granola to school for _ days.
so 1_2 ÷ 1_8 = _ .
Math
Talk
MATHEMATICAL PRACTICES 2
Reasoning Justify your
answer by using a different
representation for the
problem.
Chapter 2
125
Try Another Problem
For a science experiment, Mr. Barrows divides 2_3 cup of salt into small jars,
1
1_
each containing __
12 cup. If he has 6 cup of salt left over, how many
jars does he fill?
Read the Problem
What do I need to find?
What information do I
need to use?
How will I use the
information?
Solve the Problem
1.
MATHEMATICAL
PRACTICE
4 Write an Expression you could use to solve the problem.
2.
MATHEMATICAL
PRACTICE
6 Explain a Method Suppose that Mr. Barrows starts with 1 _2 cups
3
of salt. Explain how you could find how many jars he fills.
126
© Houghton Mifflin Harcourt Publishing Company
So, Mr. Barrows fills _ jars.
Name
Share
hhow
Share and
and Show
Sh
MATH
M
BOARD
B
1. There is 4_5 lb of sand in the class science supplies. If one
1
scoop of sand weighs __
20 lb, how many scoops of sand
can Maria get from the class supplies and still leave 1_2 lb
in the supplies?
Unlock the Problem
•
•
•
Underline the question.
Circle important information.
Check to make sure you answered
the question.
WRITE
First, draw a bar model.
Math t Show Your Work
4
5 lb
Next, find how much sand Maria gets.
Maria will get ____
10 lb of sand.
Finally, find the number of scoops.
1
____
_ groups of __
20 are equivalent to 10
1
__
so ____
10 4 20 5 _.
So, Maria will get _ scoops of sand.
© Houghton Mifflin Harcourt Publishing Company
2.
What if Maria leaves 2_5 lb of sand in the
supplies? How many scoops of sand can she get?
SMARTER
3. There are 6 gallons of distilled water in the science
supplies. If 10 students each use an equal amount of the
distilled water and there is 1 gal left in the supplies, how
much will each student get?
Chapter 2 • Lesson 10 127
MATHEMATICAL PRACTICES
ANALYZEt-00,'034536$563&t13&$*4*0/
On
On Your
Your Own
Own
4.
5.
6.
SMARTER
The total weight of the fish in a tank
of tropical fish at Fish ‘n’ Fur was 7_8 lb. Each fish
1
weighed __
64 lb. After Eric bought some fish, the total
weight of the fish remaining in the tank was 1_2 lb.
How many fish did Eric buy?
DEEPER
Fish ‘n’ Fur had a bin containing 2 1_2 lb
of gerbil food. After selling bags of gerbil food that
each held 3_4 lb, 1_4 lb of food was left in the bin. If each
bag of gerbil food sold for $3.25, how much did the
store earn?
MATHEMATICAL
PRACTICE
1 Describe Niko bought 2 lb of dog
treats. He gave his dog 3_ lb of treats one week
5
7
and __
10 lb of treats the next week. Describe how Niko
can find how much is left.
Personal Math Trainer
7.
There were 14 1_4 cups of apple
juice in a container. Each day, Elise drank 1 1_2 cups of
apple juice. Today, there is 3_4 cup of apple juice left.
SMARTER
Derek said that Elise drank apple juice on nine
days. Do you agree with Derek? Use words and
numbers to explain your answer.
128
Math t Show Your Work
© Houghton Mifflin Harcourt Publishing Company • Image Credits: (c) ©ImageState/Alamy
WRITE
Practice and Homework
Name
Problem Solving • Fraction Operations
Lesson 2.10
COMMON CORE STANDARD—6.NS.A.1
Apply and extend previous understandings of
multiplication and division to divide fractions
by fractions.
Read each problem and solve.
1. 2_3 of a pizza was left over. A group of friends divided
1
the leftover pizza into pieces each equal to __
18 of the
original pizza. After each friend took one piece, 1_6 of
the leftover pizza remained. How many friends were
in the group?
2 pizza
3
taken
left
9
____
2. Sarah’s craft project uses pieces of yarn that are
1_ yard long. She has a piece of yarn that is 3 yards long.
8
How many 1_8 -yard pieces can she cut and still have 11_4
yards left?
____
3. Alex opens a 1-pint container of orange butter. He
1
spreads __
16 of the butter on his bread. Then he divides
the rest of the butter into 3_4 -pint containers. How
many 3_4 -pint containers is he able to fill?
© Houghton Mifflin Harcourt Publishing Company
____
9
1_
4. Kaitlin buys __
10 pound of orange slices. She eats 3 of
them and divides the rest equally into 3 bags. How
much is in each bag?
____
5.
WRITE
Math Explain how to draw a model that represents
( 1 1_4 – 1_2 ) ÷ 1_8 .
Chapter 2
129
Lesson Check (6.NS.A.1)
1. Eva wanted to fill bags with 3_4 pounds of trail
mix. She started with 11 3_8 pounds but ate
1_ pound before she started filling the bags. How
8
many bags could she fill?
2. John has a roll containing 24 2_3 feet of wrapping
paper. He wants to divide it into 11 pieces. First,
though, he must cut off 5_6 foot because it was
torn. How long will each piece be?
Spiral Review (6.NS.A.1, 6.NS.B.3, 6.NS.B.4)
4. Joseph has $32.40. He wants to buy several
comic books that each cost $2.70. How many
comic books can he buy?
5. A rectangle is 2 4_5 meters wide and 3 1_2 meters
long. What is its area?
6. A rectangle is 2.8 meters wide and 3.5 meters
long. What is its area?
© Houghton Mifflin Harcourt Publishing Company
3. Alexis has 32 2_5 ounces of beads. How many
7
necklaces can she make if each uses 2 __
10 ounces
of beads?
FOR MORE PRACTICE
GO TO THE
130
Personal Math Trainer
Name
Chapter 2 Review/Test
Personal Math Trainer
Online Assessment
and Intervention
1. Write the values in order from least to greatest.
3
_
4
0.45
5
_
8
0.5
___ ___ ___ ___
2. For numbers 2a–2d, compare. Choose <, >, or =.
<
2a. 0.75
4
_
5
2b.
>
<
3_
4
2c.
13_
5
>
=
=
<
<
>
2d.
0.325
=
7.4
>
1.9
72_
5
=
3. The table lists the heights of 4 trees.
Type of Tree
Height (feet)
Sycamore
152_
Oak
143_
Maple
153_
Birch
15.72
3
4
4
© Houghton Mifflin Harcourt Publishing Company
For numbers 3a–3d, select True or False for each statement.
3a.
The oak tree is the shortest.
True
False
3b.
The birch tree is the tallest.
True
False
3c. Two of the trees are the
same height.
True
False
3d. The sycamore tree is taller
than the maple tree.
True
False
Assessment Options
Chapter Test
Chapter 2
131
4. For numbers 4a–4d, choose Yes or No to indicate whether the statement
is correct.
B
A
0
0.2
C D
0.4
0.6
0.8
1.0
4a.
Point A represents 1.0.
Yes
No
4b.
3.
Point B represents __
10
Yes
No
Yes
No
Yes
No
4c. Point C represents 6.5.
4d.
Point D represents 4_.
5
5. Select the values that are equivalent to one twenty-fifth. Mark all
that apply.
A
1
__
B
25
C
0.04
D
0.025
25
6. The table shows Lily’s homework assignment. Lily’s teacher instructed
the class to simplify each expression by dividing the numerator and
denominator by the GCF. Complete the table by simplifying each
expression and then finding the product.
132
Expression
a
2
_×1
_
5 4
b
4×5
_
_
5 8
c
3
_×5
_
7 8
d
4 × __
3
_
9 16
Simplified Expression
Product
© Houghton Mifflin Harcourt Publishing Company
Problem
Name
7.
DEEPER
Two-fifths of the fish in Gary’s fish tank are guppies. Onefourth of the guppies are red. What fraction of the fish in Gary’s tank are
red guppies? What fraction of the fish in Gary's tank are not red guppies?
Show your work.
8. One-third of the students at Finley High School play sports. Two-fifths
of the students who play sports are girls. What fraction of all students are
girls who play sports? Use numbers and words to explain your answer.
9. Draw a model to find the quotient.
3÷2=
_
4
3 ÷ 3_ =
_
4 8
0
1
4
2
4
3
4
1
© Houghton Mifflin Harcourt Publishing Company
How are your models alike? How are they different?
10. Explain how to use a model to find the quotient.
21_ ÷ 2 =
2
Chapter 2
133
Divide. Show your work.
11. 7_ ÷ 3_ =
8
5
12.
1 ÷ 11_ =
2__
10
5
13. Sophie has 3_4 quart of lemonade. If she divides the lemonade into glasses
1 quart, how many glasses can Sophie fill? Show your work.
that hold __
16
14. Ink cartridges weigh 1_8 pound. The total weight of the cartridges in a box
is 4 1_2 pounds. How many cartridges does the box contain? Show your
work and explain why you chose the operation you did.
Personal Math Trainer
Beth had 1 yard of ribbon. She used 1_3 yard for a
project. She wants to divide the rest of the ribbon into pieces 1_6 yard long.
How many 1_6 yard pieces of ribbon can she make? Explain your solution.
SMARTER
© Houghton Mifflin Harcourt Publishing Company
15.
134
Name
16. Complete the table by finding the products. Then answer the questions
in Part A and Part B.
Division
Multiplication
1
4
_÷3
_ = __
5
4
15
1
_×4
_=
5
3
2 ÷1
10
__
_ = __
2 ×5
__
_=
13
5
13
4
_÷3
_=4
_
5
5
3
13
1
4
_×5
_=
5
3
Part A
Explain how each pair of division and multiplication problems are the
same, and how they are different.
Part B
© Houghton Mifflin Harcourt Publishing Company
Explain how to use the pattern in the table to rewrite a division problem
involving fractions as a multiplication problem.
Chapter 2
135
17. Margie hiked a 17 7_8 mile trail. She stopped every 3 2_5 miles to take a
picture. Martin and Tina estimated how many times Margie stopped.
5
←
16 ÷ 4 = 4
177_ ÷ 32_
8
5
←
8
←
177_ ÷ 32_
Tina’s Estimate
←
Martin’s Estimate
18 ÷ 3 = 6
Who made the better estimate? Use numbers and words to explain
your answer.
© Houghton Mifflin Harcourt Publishing Company
18. Brad and Wes are building a tree house. They cut a 12 1_2 foot piece of
wood into 5 of the same length pieces. How long is each piece of wood?
Show your work.
136
3
Chapter
Rational Numbers
Personal Math Trainer
Show Wha t You Know
Online Assessment
and Intervention
Check your understanding of important skills.
Name
Compare Fractions Compare. Write <, >, or =.
1. 3_
5
3. 3_
3
O
O
1_
3
3
2. _7
5_
5
6
4. _8
O
O
1_
2
2_
4
Equivalent Fractions Write an equivalent fraction.
5. _3 _
8
6. _2 _
5
__
7. 10
12 _
8. 6_ _
9
Compare Decimals Compare. Write <, >, or =.
9. 0.3
O 0.30
10. 4
11. 0.4
O 0.51
12. $2.61
Angie finds a treasure map. Use the
clues to find the location of the
treasure. Write the location as an
ordered pair.
O $6.21
Wate
Wa
W
terrrf
rfal
falllll
Bat C
B
Cavee
y-axis
© Houghton Mifflin Harcourt Publishing Company
Math in the
10
9
8
7
6
5
4
3
2
1
O 3.8
0
1 unit on th
thee ma
map
ap
repr
pres
esen
ents 1 met
eter
er..
Start at the
h p
point
halfway
y between
the baat cave and
the waterfall.
1
Briiddge
g
ge
(
:
6
Then
en walk 7
meete
ters south and
2 meters west,
and start digging.
1 2 3 4 5 6 7 8 9 10
x-axis
Chapter 3
137
Voca bula ry Builder
Visualize It
Use the checked words to complete the flow map.
What is it?
What are some
1 4
5
2 8
Review Words
compare
✓ common denominator
✓ equivalent fractions
order
(3, 5), (6, 10)
31, 2, 70, 145
4
3
and
6
6
✓ whole numbers
Preview Words
absolute value
coordinate plane
integers
-9, -26, -4
✓ negative number
opposite
✓ ordered pair
origin
Understand Vocabulary
positive number
Complete the sentences using the preview words.
quadrants
1. The _____ are the set of whole
rational number
numbers and their opposites.
2. The distance of a number from 0 on a number line is the
number’s _____.
3. Two numbers that are the same distance from zero
on the number line, but on different sides of zero, are
called _____.
© Houghton Mifflin Harcourt Publishing Company
4. A _____ is any number that can be
written as a__, where a and b are integers and b Þ 0.
b
5. The four regions of the coordinate plane that are separated by
the x- and y-axes are called _____.
138
™Interactive Student Edition
™Multimedia eGlossary
Chapter 3 Vocabulary
absolute value
coordinate plane
valor absoluto
plano de cartesiano
17
1
integers
line of symmetry
enteros
eje de simetría
51
44
line symmetry
negative integer
simetría axial
entero negativo
64
52
opposites
ordered pair
opuestos
par ordenado
68
70
Any integer less than zero
© Houghton Mifflin Harcourt Publishing Company
A line that divides a figure into two halves
that are reflections of each other
Line of symmetry
-5 -4 -3 -2 -1 0 +1 +2 +3 +4 +5
Negative integers
Positive integers
A pair of numbers used to locate a point on
a grid.
y
5
4
3
2
1
-5 -4 -3 -2 -1 0
-1
-2
-3
-4
-5
(3, 2)
x
1 2 3 4 5
© Houghton Mifflin Harcourt Publishing Company
© Houghton Mifflin Harcourt Publishing Company
x
1 2 3 4 5
© Houghton Mifflin Harcourt Publishing Company
-5 -4 -3 -2 -1 0
-1
-2
-3
-4
-5
© Houghton Mifflin Harcourt Publishing Company
5
4
3
2
1
© Houghton Mifflin Harcourt Publishing Company
y
© Houghton Mifflin Harcourt Publishing Company
© Houghton Mifflin Harcourt Publishing Company
A plane formed by a horizontal line called the
x-axis and a vertical line called the y-axis
The distance of an integer from zero
on a number line
3 units
-5 -4 -3 -2 -1
−3 = 3
0
1
2
4 = 4
3
4
5
0 = 0
The set of whole numbers and their
opposites
Opposites
-5 -4 -3 -2 -1 0 +1 +2 +3 +4 +5
Negative integers
Positive integers
A figure has line symmetry if it can be folded
about a line so that its two parts match
exactly
Two numbers that are the same distance,
but in opposite directions, from zero on a
number line
Opposites
-5 -4 -3 -2 -1 0 +1 +2 +3 +4 +5
Negative integers
−
Positive integers
3 and 3 are opposites
Chapter 3 Vocabulary
(continued)
origin
positive integer
origen
entero positivo
76
71
quadrants
rational number
cuadrantes
número racional
88
82
x-axis
x-coordinate
eje de la x
coordenada x
109
108
y-axis
y-coordinate
eje de la y
coordenada y
110
111
a
Any number that can be written as a ratio __
b
where a and b are integers and b ≠ 0.
−
7
6
3
1 = __
Examples: 6 = __
, 0.75 = __
, and −2__
3
1
4
3
The first number in an ordered pair; tells the
distance to move right or left from (0,0)
y
5
4
3
2
1
-5 -4 -3 -2 -1 0
-1
-2
-3
-4
-5
(3, 2)
x
1 2 3 4 5
The second number in an ordered pair; tells
the distance to move up or down from (0,0)
y
5
4
3
2
1
-5 -4 -3 -2 -1 0
-1
-2
-3
-4
-5
(3, 2)
x
1 2 3 4 5
© Houghton Mifflin Harcourt Publishing Company
Positive integers
© Houghton Mifflin Harcourt Publishing Company
Negative integers
© Houghton Mifflin Harcourt Publishing Company
-5 -4 -3 -2 -1 0 +1 +2 +3 +4 +5
© Houghton Mifflin Harcourt Publishing Company
© Houghton Mifflin Harcourt Publishing Company
© Houghton Mifflin Harcourt Publishing Company
© Houghton Mifflin Harcourt Publishing Company
© Houghton Mifflin Harcourt Publishing Company
Any integer greater than zero
The point where the two axes of a
coordinate plane intersect; (0,0)
y
5
4
3
2
1
x
-5 -4 -3 -2 -1 0 1 2 3 4 5
-1
- 2 Origin (0,0)
-3
-4
-5
The four regions of the coordinate plane
separated by the x- and y-axes
Quadrant II
(2, 1)
y
5
4
3
2
1
Quadrant I
(1, 1)
x
-5 -4 -3 -2 -1 0
-1
-2
Quadrant III 3
(2, 2)
-4
-5
1 2 3 4 5
Quadrant IV
(1, 2)
The horizontal number line on a coordinate
plane
y
5
4
3
2
1
-5 -4 -3 -2 -1 0
-1
-2
-3
-4
-5
x
1 2 3 4 5
The vertical number line on a coordinate
plane
y
5
4
3
2
1
-5 -4 -3 -2 -1 0
-1
-2
-3
-4
-5
x
1 2 3 4 5
Going Places with
Picture It
For 3 to 4 players
Materials
• timer
• sketch pad
How to Play
1. Take turns to play.
2. To take a turn, choose a math term from the Word Box. Do not say
the term.
3. Set the timer for 1 minute.
4. Draw pictures and numbers to give clues about the term.
5. The first player to guess the term before time runs out gets 1 point.
If he or she can use the term in a sentence, they get 1 more point.
Words
Game
Game
Word Box
absolute value
coordinate plane
integers
line of symmetry
negative integer
opposites
ordered pair
origin
positive integer
quadrants
rational number
x-axis
x-coordinate
y-axis
y-coordinate
Then that player gets a turn.
© Houghton Mifflin Harcourt Publishing Company
Image Credits: ©Mika Specta/Fotolia
6. The first player to score 10 points wins.
Chapter 3
138A
Journal
Jo
ou
urnal
The Write Way
Reflect
Choose one idea. Write about it.
• Which two numbers are opposites? Tell how you know.
4 and -4
-4 and 0
-4 and 4
4 and 0
4 and 4
• Describe a situation in which you might record both positive and
negative integers.
• Define the term absolute value in your own words. Give an example.
• Write a paragraph that uses at least three of these words.
origin
quadrants
x-axis
y-axis
© Houghton Mifflin Harcourt Publishing Company
Image Credits: ©Comstock/Getty Images
coordinate plane
138B
Lesson 3.1
Name
Understand Positive and Negative Numbers
The Number System—6.NS.C.5,
6.NS.C.6a, 6.NS.C.6c
MATHEMATICAL PRACTICES
MP5, MP6, MP7
Essential Question How can you use positive and negative numbers to represent
real-world quantities?
Integers are the set of all whole numbers and their opposites. Two
numbers are opposites if they are the same distance from 0 on the
number line, but on different sides of 0. For example, the integers
+
3 and −3 are opposites. Zero is its own opposite.
You do not need to write the
+ symbol for positive integers,
so +3 can also be written as 3.
Opposites
-5 -4 -3 -2 -1 0 +1 +2 +3 +4 +5
Negative integers
Positive integers
Positive numbers are located to the right of 0 on the number line, and
negative numbers are located to the left of 0.
Unlock
Unlock the
the Problem
Problem
The temperature at the start of a 2009 Major League
Baseball playoff game between the Colorado
Rockies and the Philadelphia Phillies was 2°C. The
temperature at the end of the game was −4°C. What
is the opposite of each temperature?
• What are you asked to find?
• Where can you find the opposite of a number
on the number line?
Graph each integer and its opposite on a
number line.
A 2
The integer 2 is on the
_ side of 0.
Graph the opposite of 2 at _.
-5 -4 -3 -2 -1 0
1
2
3
4
5
-5 -4 -3 -2 -1 0
1
2
3
4
5
So, the opposite of 2°C is _.
© Houghton Mifflin Harcourt Publishing Company
B
−4
The integer −4 is on the
_ side of 0.
Graph the opposite of −4 at _.
So, the opposite of −4°C is _.
Math
Talk
MATHEMATICAL PRACTICES 6
Explain how to find the
opposite of −8 on a number
line.
Chapter 3
139
Example 1 Name the integer that represents the situation, and tell
what 0 represents in that situation.
Situation
Integer
A team loses 10 yards on a
football play.
−
10
What Does 0 Represent?
the team neither gains nor loses yardage
A point in Yuma, Arizona, is 70 feet
above sea level.
A temperature of 40 degrees below
zero was recorded in Missouri.
Larry withdraws $30 from his
bank account.
Tricia’s golf score was 7 strokes
below par.
A chlorine ion has one more electron
than proton.
Example 2 Use a number line to find − ( −3 ), the
opposite of the opposite of 3.
STEP 1
- 5 -4 - 3 - 2 - 1
Graph 3 on the number line.
0
1
2
3
4
5
STEP 2
Use the number line to graph the opposite of 3.
STEP 3
Use the number line to graph the opposite of the number you
graphed in Step 2.
So, − ( −3 ), or the opposite of the opposite of 3, equals _.
A
•
140
+
9
_
MATHEMATICAL
PRACTICE
B
−
12
_
C 0 _
Math
Talk
Explain A plane’s altitude changes by −1,000 feet. Is the
plane going up or down? Explain.
6
MATHEMATICAL PRACTICES 7
Look for a Pattern Describe
the pattern you see when
finding the opposite of the
opposite of a number.
© Houghton Mifflin Harcourt Publishing Company
Try This! Write the opposite of the opposite of the integer.
Name
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B
Graph the integer and its opposite on a number line.
1.
−
7
opposite: _
- 10 -8 -6 - 4 - 2 0
opposite: _
2. 9
2
4
6
8
10
- 10 -8 -6 - 4 - 2 0
2
4
6
8
10
Name the integer that represents the situation, and tell what 0
represents in that situation.
Situation
Integer
What Does 0 Represent?
3. Kerri gained 24 points during a
round of a game show.
4. Ben lost 5 pounds during the
summer.
5. Marcy deposited $35 in her savings
account.
Math
Talk
On
On Your
Your Own
Own
MATHEMATICAL PRACTICES 2
Reason Abstractly Identify
a real-world situation
involving an integer and its
opposite.
Write the opposite of the integer.
6.
98 _
−
7. 0 _
53 _
−
8.
Name the integer that represents the situation, and tell what 0 represents in
that situation.
Situation
Integer
What Does 0 Represent?
9. Desmond made $850 at his
summer job.
© Houghton Mifflin Harcourt Publishing Company
10. Miguel withdraws $300 from his
checking account.
11. A barium ion has two more protons
than electrons
Write the opposite of the opposite of the integer.
12.
15.
23 _
−
13. 17 _
14.
125 _
−
DEEPER
Suppose you know a certain number’s distance from zero on
the number line. Explain how you could find the number’s distance from its opposite.
Chapter 3 • Lesson 1 141
MATHEMATICAL PRACTICES
ANALYZEt-00,'034536$563&t13&$*4*0/
OqnakdlRnkuhmf¤@ookhb`shnmr
OqnakdlRnkuhmf¤@ookhb`shnmr
Wind makes the air temperature seem colder.
The chart gives the wind chill temperature
(what the temperature seems like) at several
air temperatures and wind speeds. Use the
chart for 16–18.
Wind Chill Chart
16. At 6 a.m., the air temperature was 20°F and the
wind speed was 55 mi/hr. What was the wind
chill temperature at 6 a.m.?
18.
DEEPER
At noon, the air temperature was
15°F and the wind speed was 45 mi/hr. At what
air temperature and wind speed would the
wind chill temperature be the opposite of what
it was at noon?
25
30
25
20
15
16
9
3
−
35
14
7
0
−
45
12
5
−
−
3
−
55
11
2
4
−
4
7
9
11
SMARTER
The wind was blowing
35 mi/hr in both Ashton and Fenton. The
wind chill temperatures in the two towns were
opposites. If the air temperature in Ashton was
25°F, what was the air temperature in Fenton?
19. Sense or Nonsense? Claudia states that the
opposite of any integer is always a different
number than the integer. Is Claudia’s statement
sense or nonsense? Explain.
20.
142
SMARTER
For numbers 20a−20d, choose Yes or No to indicate
whether the situation can be represented by a negative number.
20a. Death Valley is located 282 feet below sea level.
Yes
No
20b. Austin’s golf score was 3 strokes below par.
Yes
No
20c. The average temperature in Santa Monica in August is 75°F.
Yes
No
20d. Janai withdraws $20 from her bank account.
Yes
No
© Houghton Mifflin Harcourt Publishing Company • Image Credits: (t) ©Yellow Dog Productions/Getty Images
17.
Wind (mi/hr)
Air Temperature (˚F)
Practice and Homework
Name
Lesson 3.1
Understand Positive and
Negative Numbers
COMMON CORE STANDARDS—6.NS.C.5,
6.NS.C.6a, 6.NS.C.6c Apply and extend
previous understandings of numbers to the
system of rational numbers.
Graph the integer and its opposite on a number line.
1. –6
-10 -8 -6 -4
3. 10
1
6
opposite:
-2
0
2
2. 3
4
6
8
$(' $/ $- $+
10
4. –8
opposite:
$(' $/ $- $+
$)
opposite:
'
)
+
-
/
'
)
+
-
/
('
'
)
+
-
/
('
opposite:
$(' $/ $- $+
('
$)
$)
Name the integer that represents the situation, and tell what 0
represents in that situation.
Situation
Integer
What Does 0 Represent?
5. Michael withdrew $60 from his
checking account.
6. Raquel gained 12 points while
playing a video game.
Write the opposite of the opposite of the integer.
8. 4
7. –20
9. 95
10. –63
© Houghton Mifflin Harcourt Publishing Company
Problem
Problem Solving
Solving
11. Dakshesh won a game by scoring 25 points.
Randy scored the opposite number of points
as Dakshesh. What is Randy’s score?
_______
13.
WRITE
12. When Dakshesh and Randy played the game
again, Dakshesh scored the opposite of the
opposite of his first score. What is his score?
_______
Math Give three examples of when negative numbers are
used in daily life.
Chapter 3
143
Lesson Check (6.NS.C.5, 6.NS.C.6a)
Name the integers that represent each situation.
1. During their first round of golf, Imani was
7 strokes over par and Peter was 8 strokes
below par.
2. Wyatt earned $15 baby-sitting on Saturday.
Wilson spent $12 at the movies.
3. Mr. Nolan’s code for his ATM card is a 4-digit
number. The digits of the code are the prime
factors of 84 listed from least to greatest. What is
the code for Mr. Nolan’s ATM card?
4. Over a four-year period, a tree grew 2.62 feet. If
the tree grows at a constant rate, how many feet
did the tree grow each year?
9
5. Omarion has __
10 of the pages in a book
remaining to read for school. He reads 2_3 of
the remaining pages over the weekend. What
fraction of the book does Omarion read over the
weekend?
6. Marianne has 5_8 pound of peas. She cooks 2_3 of
those peas for 5 people. If each person is served
an equal amount, how much peas did each
person get?
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Spiral Review (6.NS.A.1, 6.NS.B.3, 6.NS.B.4)
Lesson 3.2
Name
Compare and Order Integers
The Number System—6.NS.C.7a,
6.NS.C.7b
MATHEMATICAL PRACTICES
MP5, MP8
Essential Question How can you compare and order integers?
You can use a number line to compare integers.
Unlock
Unlock the
the Problem
Problem
On one play of a football game, the ball changed
position by −7 yards. On the next play, the ball
changed position by −4 yards. Compare −7 and −4.
Use a number line to compare the numbers.
STEP 1 Graph −7 and −4 on the number line.
- 10 - 9
- 8 - 7 - 6 - 5 -4 - 3 - 2 - 1 0
1
2
3
4
5
STEP 2 Note the locations of the numbers.
As you move to the right on a
horizontal number line, the values
become greater. As you move to
the left, values become less.
−
7 is to the __ of −4 on the number
© Houghton Mifflin Harcourt Publishing Company • Image Credits: (t) ©Visions of America, LLC/Alamy Images
line, so −7 is __ −4.
Try This! Use the number line to compare the numbers.
A 5 and −9
- 10 - 9
- 8 - 7 - 6 - 5 -4 - 3 - 2 - 1 0
1
2
3
4
5
6
7
8
9
10
5 is to the __ of −9 on the number line, so 5 is __ −9.
B
−
2 and 0
- 10 - 9
- 8 - 7 - 6 - 5 -4 - 3 - 2 - 1 0
1
2
3
4
5
6
7
8
9
10
_ is to the left of _ on the number line, so −2 is __ 0.
Math
Talk
MATHEMATICAL PRACTICES 1
Analyze Relationships If
r , −4 , where on a number
line is r compared to −4?
Chapter 3
145
You can also use a vertical number line to order integers.
Example
The table gives the coldest temperatures
recorded in seven cities in 2007.
Record Coldest Temperatures for 2007 (°F)
Anchorage, AK
−
17
Boise, ID
7
Duluth, MN
−
25
Los Angeles, CA
35
A Order the temperatures from least to greatest.
Memphis, TN
18
Pittsburgh, PA
−
5
Record Coldest Temperatures (°F) for 2007
STEP 1 Draw a dot on the number line to represent the
record temperature of each city. Write the first letter of
the city beside the dot.
40
30
20
STEP 2 Write the record temperatures in order from least
to greatest. Explain how you determined the order.
10
0
-10
- 20
B Use the table and the number line to answer each question.
- 30
• Which city had the colder record temperature,
Memphis or Pittsburgh? How do you know?
- 40
• What are the record temperatures for Boise, Memphis,
and Pittsburgh in order from least to greatest?
_ < _< _
• What are the record temperatures for Anchorage, Duluth,
and Los Angeles in order from greatest to least?
_>_>_
146
The symbol < means less than.
The symbol > means greater than.
Math
Talk
MATHEMATICAL PRACTICES 8
Generalize What rule can
you use to compare numbers
on a vertical number line?
© Houghton Mifflin Harcourt Publishing Company
• Which city had the warmest record temperature? How do you know?
Name
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Compare the numbers. Write < or >.
1.
−
8
Think: −8 is to the _ of 6 on the number line, so −8 is __ 6.
6
−
2. 1
3.
8
−
4
0
−
4. 3
7
Order the numbers from least to greatest.
5. 4, −3, −7
6. 0, −1, 3
_<_<_
−
7.
_<_<_
5, −3, −9
_<_<_
Order the numbers from greatest to least.
8.
−
1, −4, 2
9. 5, 0, 10
_>_>_
_>_>_
Order the numbers from least to greatest.
12.
_<_<_
−
6, −12, 30
5, −4, −3
_>_>_
Math
Talk
On
On Your
Your Own
Own
11. 2, 1, −1
−
10.
MATHEMATICAL PRACTICES 7
Look for Structure Explain
how you can use a number
line to compare numbers.
13. 15, −9, −20
_<_<_
_<_<_
Order the numbers from greatest to least.
14.
−
13, 14, −14
_>_>_
© Houghton Mifflin Harcourt Publishing Company
17.
18.
15.
−
20, −30, −40
16. 9, −37, 0
_>_>_
_>_>_
Saturday’s low temperature was −6°F. Sunday’s low
temperature was 3°F. Monday’s low temperature was −2°F. Tuesday’s low
temperature was 5°F. Which day’s low temperature was closest to 0°F?
DEEPER
MATHEMATICAL
PRACTICE
Use Symbols Write a comparison using < or > to
show that South America’s Valdes Peninsula (elevation −131 ft) is
lower than Europe’s Caspian Sea (elevation −92 ft).
4
Chapter 3 • Lesson 2 147
MATHEMATICAL PRACTICES
COMMUNICA5&t1&34E7&3&tCONSTRUCT ARGUMENTS
Problem
Solving
OqnakdlRnkuhmf¤@ookhb`shnmr
OqnakdlRnkuhmf¤@ookhb`shnmr
SMARTER
What’s the Error?
19. In the game of golf, the player with the lowest
score wins. Raheem, Erin, and Blake played a
game of miniature golf. The table shows their
scores compared to par.
Raheem
0
Erin
Blake
−
5
−
1
At the end of the game, they wanted to know
who had won.
Look at how they solved the
problem. Find their error.
Correct the error by ordering the
scores from least to greatest.
STEP 1: 0 is greater than both −1 and −5. Since
Raheem had the highest score, he did not
win.
STEP 2: −1 is less than −5, so Blake’s score was
less than Erin’s score. Since Blake had the
lowest score, he won the game.
WRITE
M t Show Your Work
Math
So, __ won. __ came in second. __ came in third.
• Describe the error that the players made.
SMARTER
Jasmine recorded the low temperatures for 3 cities.
City
Temperature (°F)
A
6
B
−4
C
2
-5 -4 -3 -2 -1 0 1 2 3 4 5 6 7
Draw a dot on the number line to represent the low temperature
of each city. Write the letter of the city above the dot.
148
© Houghton Mifflin Harcourt Publishing Company
20.
Practice and Homework
Name
Lesson 3.2
Compare and Order Integers
COMMON CORE STANDARDS—6.NS.C.7a,
6.NS.C.7b Apply and extend previous
understandings of numbers to the system of
rational numbers.
Compare the numbers. Write < or >.
1.
−
−
4 >
5
right
Think: −4 is to the
greater than
so −4 is
2. 0
−
3. 4
5. 2
−
6.
1
10
of −5 on the number line,
−
−
5.
4.
6
−
12
−
−
9
8
−
7. 1
11
−
10
Order the numbers from least to greatest.
8. 3, −2, −7
<
11.
9. 0, 2, −5
<
−
2, −3, −4
<
10.
<
<
9, −12, −10
<
<
<
<
<
13. 5, 7, 0
12. 1, −6, −13
<
−
<
Order the numbers from greatest to least.
14. 0, 13, −13
>
15.
>
−
11, 7, −5
>
16.
>
−
9, −8, 1
>
>
© Houghton Mifflin Harcourt Publishing Company
Problem
Problem Solving
Solving
17. Meg and Derek played a game. Meg scored
−11 points, and Derek scored 4 points. Write a
comparison to show that Meg’s score is less than
Derek’s score.
_______
19.
WRITE
18. Misha is thinking of a negative integer
greater than −4. What number could she
be thinking of?
_______
Math Explain how to use a number line to compare two
negative integers. Give an example.
Chapter 3
149
Lesson Check (6.NS.C.7a, 6.NS.C.7b)
The chart shows the high temperatures for seven cities on one day in January.
Chama
−5°
Chicago
2°
Denver
−8°
1. Which city had the lower temperature, Helena
or Chicago?
Fargo
−10°
Glen Spey
6°
Helena
−1°
Lansing
3°
2. Write the temperatures of the following cities
in order from greatest to least: Denver, Helena,
Lansing
3. Fiona starts at the beginning of a hiking trail
and walks 4_5 mile. She counts the mileage
1
markers that are placed every __
10 mile along the
trail. How many markers does she count?
4. If Amanda hikes at an average speed of
2.72 miles per hour, how long will it take her
to hike 6.8 miles?
5. The area of a rectangle is 5 4_5 square meters. The
width of the rectangle is 2 1_4 meter. Which is the
best estimate for the length of the rectangle?
6. Lillian bought 2.52 pounds of tomatoes and
1.26 pounds of lettuce to make a salad for
18 people. If each person got the same amount
of salad, how much salad did each person get?
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Spiral Review (6.NS.A.1, 6.NS.B.3, 6.NS.B.4, 6.NS.C.5)
Lesson 3.3
Name
Rational Numbers and the Number Line
Essential Question How can you plot rational numbers on a number line?
connect A rational number is any number that can be
written as a_b , where a and b are integers and b ≠ 0. Decimals,
fractions, and integers are all rational numbers.
The Number System—6.NS.C.6a,
6.NS.C.6c Also 6.NS.C.5
MATHEMATICAL PRACTICES
MP2, MP4, MP7
Unlock
Unlock the
the Problem
Problem
The freezing point of a liquid is the temperature at which the
liquid turns into a solid when it is cooled. The table shows the
approximate freezing points of various liquids. Graph each
temperature on a number line.
Liquid Freezing Points
Liquid
Carbonated water
−
Fizzy lemonade
−
Hydrazine
Graph the values in the table.
Freezing Point (°C)
0.3
0.5
1.4
STEP 1 Locate each number in relation to the nearest integers.
Think:
−
0.3 is the opposite of
_.
_
_
0.3 is between the integers
and
.
−
So, −0.3 is between the opposites of these integers. 0.3 is between
−
_ and _.
1.4 is between _ and _.
0.5 is between _ and _.
© Houghton Mifflin Harcourt Publishing Company • Image Credits: (t) ©Digital Vision/Getty Images
STEP 2 Graph each temperature.
−
0.3°C
Think:
−
0.5°C
1.4°C
2
2
2
1
1
1
0
0
0
-1
-1
-1
−
0.3 is 3
tenths below 0 on
the number line.
-2
-2
-2
Math
Talk
MATHEMATICAL PRACTICES 8
Draw Conclusions How can
you tell which number −0.3
is closer to, 0 or −1? Explain.
Chapter 3
151
Example
Points of Interest
Name
City Hall is located at point 0 on a map of Maple Avenue. Other
points of interest on Maple Avenue are indicated by their distances,
in miles, to the right of City Hall (positive numbers) or to the left of
City Hall (negative numbers). Graph each location on a number line.
is between _ and
_.
− _
1 12
1 1_4 is between _ and
_.
3 is between
__
4
8
City Park
− 3_
8
Fountain
− _1
12
1 1_4
Library
_3
4
Mall
STEP 1 Locate the numbers in relation to the nearest integers.
__
−3
Location
is between _ and
_.
Think:
−3
_
8
is the
opposite of
_.
_ and _.
STEP 2 Graph each location on the number line.
3
City Park: −__
8
Think:
_
−3
8
is three eighths to the left of 0 on the number line.
City Hall
-2
-1
0
1
2
1
2
1
2
City Hall
-2
-1
0
City Hall
Library: 1 1_4
-2
-1
0
City Hall
__
Mall: 3
4
-2
1.
2.
152
MATHEMATICAL
PRACTICE
2
MATHEMATICAL
PRACTICE
7
-1
0
1
2
Math
Talk
Reason Quantitatively How did you identify the two
integers that −1 1_2 is between?
Identify Relationships How do you know from looking
at the table that City Hall is between the city park and the mall?
MATHEMATICAL PRACTICES 6
Explain how you can use
a horizontal or vertical
number line to graph a
rational number.
© Houghton Mifflin Harcourt Publishing Company
Fountain:
− _
1 12
Name
Math
Talk
MATH
M
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BOARD
B
MATHEMATICAL PRACTICES 2
Reason Abstractly Two
numbers are opposites. Zero
is not one of the numbers.
Are the numbers on the
same side or opposite sides
of zero on a number line?
Explain.
Graph the number on the horizontal number line.
1.
−
2.25
-3
-2
-1
0
1
2
3
The number is between the integers _ and _.
It is closer to the integer _.
2.
3. 1_2
− _
1 58
-3
-2
-1
0
1
2
-3
3
-2
-1
0
1
2
3
On
On Your
Your Own
Own
Practice: Copy and Solve Graph the number on a vertical number line.
4. 0.6
8.
5.
−
−
6.
1.25
9. 1.4
0.7
10.
− _
1 12
−
7. 0.3
11.
0.5
− 1_
4
State whether the numbers are on the same or opposite sides of zero.
12.
−
13.
1.38 and 2.9
___
− __
9
3 10
and −0.99
14. 5_6 and −4.713
___
___
Identify a decimal and a fraction in simplest form for the point.
D A
© Houghton Mifflin Harcourt Publishing Company
-1.5
-1.0
C
-0.5
15. Point A
___
19.
B
0
0.5
1
16. Point B
___
1.5
17. Point C
18. Point D
___
___
The roots of 6 corn plants had depths of −3.54 feet, −2__4 feet,
5
−
3.86 feet, −41_ feet, −4.25 feet, and −32_ feet. How many corn plants had
5
8
roots between 3 and 4 feet deep?
DEEPER
Chapter 3 • Lesson 3 153
MATHEMATICAL PRACTICES .0%&-t3&"40/tM",&4&/4&
OqnakdlRnkuhmf¤@ookhb`shnmr
OqnakdlRnkuhmf¤@ookhb`shnmr
A star’s magnitude is a number that measures
the star’s brightness. Use the table of star
magnitudes for 20–22.
20. Between what two integers is the magnitude
of Canopus?
MATHEMATICAL
PRACTICE
Model Mathematics
Graph the magnitude of Betelgeuse on the
number line.
4
Magnitudes of Stars
Star
Magnitude
−
Arcturus
-1
- 0.5
0
0.5
Betelgeuse
1
0.7
−
Canopus
22.
What’s the Error?
Jacob graphed the magnitude of
Sirius on the number line. Explain
his error. Then graph the magnitude
correctly.
SMARTER
-2
-1
0.04
Deneb
0.72
1.25
Rigel Kentaurus A
−
Sirius
−
0.01
1.46
0
Personal Math Trainer
23.
SMARTER
The flag pole is located at point 0 on a map
of Orange Avenue. Other points of interest on Orange Avenue are
indicated by their distances, in miles to the right of the flag pole
(positive numbers) or to the left of the flag pole (negative numbers).
Graph and label each location on the number line.
Name
School
0.4
Post Office
1.8
Library
−
Fire Station
154
Location
1
−
1.3
-2
-1
0
1
2
© Houghton Mifflin Harcourt Publishing Company • Image Credits: (tr) ©Lucas Janin/Getty Images
21.
Practice and Homework
Name
Lesson 3.3
Rational Numbers and the Number Line
COMMON CORE STANDARDS—
6.NS.C.6a, 6.NS.C.6c Apply and extend
previous understandings of numbers to the
system of rational numbers. Also 6.NS.C.5
Graph the number on the number line.
1.
− _
23
4
−
3
The number is between the integers
2
It is closer to the integer
2.
3
1
−_
$)
$(
'
−
$(
'
.
-3
-2
-1
4. 1.75
)
(
(
2
0.5
5. 12_1
$)
2
0
1
2
3
.
3.
4
and
)
(
(
'
'
$(
$(
)
State whether the numbers are on the same or opposite sides of zero.
6.
−
7.
2.4 and 2.3
___
− 1
2_ and −1
8.
5
___
−
0.3 and 0.3
9. 0.44 and −2_
3
___
___
Write the opposite of the number.
10.
−
11. 5_4
5.23
___
12.
___
−
5
___
13.
− _
22
3
___
© Houghton Mifflin Harcourt Publishing Company
Problem
Problem Solving
Solving
14. The outdoor temperature yesterday reached a
low of −4.5° F. Between what two integers was the
temperature?
_______
16.
WRITE
15. Jacob needs to graph −6 2_5 on a horizontal
number line. Should he graph it to the left or
right of −6?
_______
Math Describe how to plot –3 3_ on a number line.
4
Chapter 3
155
Lesson Check (6.NS.C.6a, 6.NS.C.6c)
1. What number is the opposite of 0.2?
2. Between which two integers would you locate
–
3.4 on a number line?
Spiral Review (6.NS.A.1, 6.NS.C.6c, 6.NS.C.7a)
3. Yemi used these pattern blocks to solve a
division problem. He found a quotient of 7.
Which division problem was he solving?
4. Eric had 2 liters of water. He gave 0.42 liter to
his friend and then drank 0.32 liter. How much
5. To pass a math test, students must correctly
answer at least 0.6 of the questions. Donald’s
score is 5_8 , Karen’s score is 0.88, Gino’s score
is 3_5 and Sierra’s score is 4_5 . How many of the
6. Jonna mixes 1_4 gallon of orange juice and
1_ gallon of pineapple juice to make punch.
2
1
Each serving is __
16 gallon. How many servings
can Jonna make?
students passed the test?
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water does he have left?
Lesson 3.4
Name
Compare and Order Rational Numbers
The Number System—6.NS.C.7a,
6.NS.C.7b
MATHEMATICAL PRACTICES
MP1, MP5
Essential Question How can you compare and order rational numbers?
connect You have used a number line to compare and order
integers. You can also use a number line to compare other rational
numbers, including decimals and fractions.
Unlock
Unlock the
the Problem
Problem
Average December Temperatures
The table shows the average December
temperatures in five U.S. cities. Which city has
the greater average December temperature,
Indianapolis or Boise?
One Way
City
-2
-1
−
Boise, ID
Boston, MA
Use a number line.
0
1
−
Philadelphia, PA
2
1
0.9
Indianapolis, IN
0.6
2.1
−
Syracuse, NY
Graph the temperatures for Indianapolis and Boise.
-3
Temperature (°C)
2
3
_
Think: As you move to the
on a horizontal
number line, the numbers become greater.
0.6 is to the _ of −1.
−
So, the city whose temperature is farther to the right is __.
© Houghton Mifflin Harcourt Publishing Company
Another Way
Use place value to compare the decimals.
STEP 1 Write the temperatures with
their decimal points lined up.
Indianapolis: _
STEP 2 Compare the digits in the
ones place. If the number is
negative, include a negative
sign with the digit.
Think: 0 is __ than −1.
Boise: _
0.6 is __ than −1.
−
0.6°C is __ than −1°C, so __ has a
−
greater average December temperature than __.
Math
Talk
MATHEMATICAL PRACTICES 5
Use Appropriate Tools How
can you order the average
December temperatures of
Boston, Philadelphia, and
Syracuse from greatest to
least?
Chapter 3
157
Example 1
The elevations of objects found at a dig site are recorded in
the table. Which object was found at a lower elevation, the
fossil of the shell or the fossil of the fish?
One Way
1
Use a number line.
0
Graph the elevations for the fossil of the shell
and the fossil of the fish.
-1
Object
-2
number line, the numbers become less.
Fern
Fish
-3
is __ −3 1_4 on the number line.
Elevation (ft)
− _1
32
1_
4
− 1_
34
Shell
Think: As you move __ on a vertical
− _
3 12
Fossils
-4
Another Way
Use common denominators to compare fractions.
STEP 1 Write the elevations with a
common denominator.
− _
3 12
STEP 2 Since the whole numbers are
the same, you only need to
compare the fractions. If the
number is negative, include a
negative sign with the fraction.
_
−2
4
= −3 ___
− _
3 14
= −3 ___
is __ than − 1_4 , so
3 2 is __ than −3 1_4 .
− 1
_
elevation than the fossil of the __.
Example 2
Compare −0.1 and
−4
_.
5
Convert to all fractions or all decimals.
fractions
−
−
0.1 = ___
10
−4
_
5
−
= ___
10
8 is _ than −1, so − 4_5 is less than −0.1.
−
decimals
−
−
0.1 = −0.1
− 4_
5
= −0.
0.8 is _ than −0.1, so − 4_5 is less than −0.1.
Use a number line to check your answer.
-4
5
-1
158
- 0.1
0
Math
Talk
MATHEMATICAL PRACTICES 1
Describe Relationships
Explain how you could use
number sense to compare
−
0.1 and − 4_5 .
© Houghton Mifflin Harcourt Publishing Company • Image Credits: (t) ©PhotoDisc/Getty Images
So, the fossil of the __ was found at a lower
Name
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MATH
M
BOARD
B
Compare the numbers. Write < or >.
1.
−
0.3
0.2
Think: −0.3 is to the _ of 0.2 on the number line, so −0.3 is _ than 0.2.
_
−2
2. 1_3
3.
5
−
−
0.8
0.5
4.
_
−3
7.
− _
5 14 , −6.5, −5.3
4
−
0.7
Order the numbers from least to greatest.
5. 3.6, −7.1, −5.9
6.
_<_<_
1 −2_
_, _
−6
,
7 9
3
_<_<_
_<_<_
Math
Talk
On
On Your
Your Own
Own
MATHEMATICAL PRACTICES 2
Connect Symbols and Words
Tell what the statement
_ > − 1_ means. Explain how
−1
3
2
you know that the statement
is true.
Compare the numbers. Write < or >.
8.
_
−1
2
_
−3
7
9.
−
23.7
−
18.8
10.
− _
3 14
−
4.3
Order the numbers from greatest to least.
11.
−
2.4, 1.9, −7.6
_>_>_
© Houghton Mifflin Harcourt Publishing Company
14.
15.
12.
_ −3
_ −1
_
−2
5, 4, 2
13. 3, −6 4_5 , −3 2_3
_>_>_
_>_>_
Last week, Wednesday’s low temperature was −4.5°F,
Thursday’s low temperature was −1.2°F, Friday’s low temperature was
−
2.7°F, and Saturday’s low temperature was 0.5°F. The average low
temperature for the week was −1.5°F. How many of these days had low
temperatures less than the average low temperature for the week?
DEEPER
MATHEMATICAL
PRACTICE
Use Symbols Write a comparison using < or > to show the
relationship between an elevation of −12 1_2 ft and an elevation of −16 5_8 ft.
4
Chapter 3 • Lesson 4 159
MATHEMATICAL PRACTICES
ANALYZEt-00,'034536$563&t13&$*4*0/
OqnakdlRnkuhmf¤@ookhb`shnmr
OqnakdlRnkuhmf¤@ookhb`shnmr
Elevations, in miles, are given for the lowest
points below sea level for 4 bodies of water.
Use the table for 16–19.
16. The lowest point of which has the greater elevation,
the Arctic Ocean or Lake Tanganyika?
17. Which has a lower elevation, the lowest point of
Lake Superior or a point at an elevation of − 2_5 mi?
Lowest Points
Location
Arctic Ocean
Lake Superior
Lake Tanganyika
Red Sea
19.
20.
SMARTER
A shipwreck is
found at an elevation of –0.75 mile.
In which bodies of water could the
shipwreck have been found?
SMARTER
−
20a. 3_
5
160
Circle <, >, or =.
<
>
=
−4
_
5
−
20b. 2_5
<
>
=
20c. −6.5
<
>
=
−4.2
20d. −2.4
<
>
=
−3.7
−3
_
4
WRITE
−
0.8
− 1_
4
−
0.9
− 1_
3
Math t Show Your Work
© Houghton Mifflin Harcourt Publishing Company • Image Credits: (tr) ©imagewerks RF/Getty Images
18. List the elevations in order from least to greatest.
Elevation (mi)
Practice and Homework
Name
Lesson 3.4
Compare and Order Rational Numbers
COMMON CORE STANDARDS—6.NS.C.7a,
6.NS.C.7b Apply and extend previous
understandings of numbers to the system of
rational numbers.
Compare the numbers. Write < or >.
1.
− _
11
2 <
1
−_
2
left
Think: −1_1 is to the
2
2
−
1
−_
.
2
1
−_
3. 0.4
1.9
2
less than
so −1_1 is
2. 0.1
of −_1 on the number line,
4. _2
0.5
5
2
Order the numbers from least to greatest.
5. 0.2, −1.7, −1
<
6.
− _
23, −_3, −1_3
<
4
7.
4
5
<
−
<
0.5, −13_2, −2.7
<
<
Order the numbers from greatest to least.
8.
−
1, −_5, 0
9. 1.82, −_2, 4_
10.
5 5
6
>
<
>
>
−
2.19, −2.5, 1.1
>
>
Write a comparison using < or > to show the relationship
between the two values.
11. an elevation of −15 m and an
elevation of −20.5 m
12. a balance of $78 and a balance
of −$42
13. a score of −31 points and a
score of −30 points
© Houghton Mifflin Harcourt Publishing Company
Problem
Problem Solving
Solving
14. The temperature in Cold Town on Monday was
1°C. The temperature in Frosty Town on
Monday was −2°C. Which town was colder
on Monday?
15. Stan’s bank account balance is less than −$20.00
but greater than −$21.00. What could Stan’s
account balance be?
_______
_______
16.
WRITE
_______
Math Describe two situations in which it would be helpful to
compare or order positive and negative rational numbers.
Chapter 3
161
Lesson Check (6.NS.C.7a, 6.NS.C.7b)
1. The low temperature was —1.8 °C yesterday and
−2.1 °C today. Use the symbols < or > to show
the relationship between the temperatures.
2. The scores at the end of a game are shown. List
the scores in order from greatest to least.
–7
Vince: −0.5, Allison: 3_8 , Mariah: __
20
3. Simone bought 3.42 pounds of green apples and
2.19 pounds of red apples. She used 3 pounds to
make a pie. How many pounds of apples are left?
4. Kwan bought three rolls of regular wrapping
paper with 6.7 square meters of paper each.
He also bought a roll of fancy wrapping paper
containing 4.18 square meters. How much
paper did he have altogether?
5. Eddie needs 2 2_3 cups of flour for one batch of
pancakes. How much flour does he need for
2 1_2 batches?
6. Tommy notices that he reads 2_3 page in a
minute. At that rate, how long will it take him to
read 12 pages?
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Spiral Review (6.NS.A.1, 6.NS.B.4, 6.NS.C.6a, 6.NS.C.6c)
Name
Personal Math Trainer
Mid-Chapter Checkpoint
Online Assessment
and Intervention
Vocabulary
Vocabulary
Vocabulary
integers
Choose the best term from the box to complete the sentence.
opposites
1. Any number that can be written as a_b , where a and b are integers
rational number
and b Þ 0 is called a(n) ____. (p. 151)
2. The set of whole numbers and their opposites is the set of
____ . (p. 139)
Concepts
Concepts and
and Skills
Skills
Write the opposite of the integer. (6.NS.C.6a)
3.
−
4. 0
72
5.
_
_
−
6. 27
31
_
_
Name the integer that represents the situation, and tell
what 0 represents in that situation. (6.NS.C.5)
Situation
Integer
What Does 0 Represent?
7. Greg scored 278 points during his
turn in the video game.
8. The temperature was
8 degrees below zero.
Compare the numbers. Write < or >. (6.NS.C.7a)
9. 3
12. 1_
© Houghton Mifflin Harcourt Publishing Company
3
O
O
−
4
−1
_
2
10.
−
13.
−
O 5
3.1 O 4.3
−
6
−
O 6
14. 1 O 2
−
11. 5
3
_
4
1
− _
2
Order the numbers. (6.NS.7a)
15. 5, −2, −8
_<_<_
18. 2.5, −1.7, −4.3
_<_<_
16. 0, −3, 1
17.
_<_<_
− 5
19. _2, 1_, __
3
4 12
_<_<_
−
7, −6, −11
_>_>_
20.
−
5.2, −3.8, −9.4
_>_>_
Chapter 3
163
21. Judy is scuba diving at −7 meters, Nelda is scuba diving at
−
9 meters, and Rod is scuba diving at −3 meters. List the
divers in order from the deepest diver to the diver
who is closest to the surface. (6.NS.C.7b)
22. A football team gains 8 yards on their first play. They lose
12 yards on the next play. What two integers represent
the two plays? (6.NS.C.5)
23.
DEEPER
Game Scores
The player who scores the
closest to 0 points wins the game. The
scores of four players are given in the
table. Who won the game? (6.NS.C.7b)
Player
Points
Myra
−
Amari
− _2
13
1.93
−
Justine
1.8
− _1
12
Donovan
A
-5
164
-4
B
-3
C
-2
-1
0
1
D
2
3
4
5
© Houghton Mifflin Harcourt Publishing Company
24. Which point on the graph represents −33_4? What number does
point C represent? (6.NS.C.6c)
Lesson 3.5
Name
Absolute Value
The Number System—
6.NS.C.7c
MATHEMATICAL PRACTICES
MP2, MP3, MP4, MP8
Essential Question How can you find and interpret the absolute value
of rational numbers?
The absolute value or magnitude of a number is the number’s
distance from 0 on a number line. The absolute value of –3 is 3.
3 units
-5 -4 -3 -2 -1
0
1
2
3
4
5
The absolute value of –3 is written symbolically as | –3|.
Unlock
Unlock the
the Problem
Problem
10
In 1934, a cargo ship called the Mohican sank off the
coast of Florida. Divers today can visit the ship at an
elevation of – 32 feet. Use a number line to find | – 32|.
Graph − 32. Then find its absolute value.
- 10
Graph –32 on the number line.
- 20
Think: The distance from 0 to the point I graphed
- 30
is
_ units.
So, | –32| = _.
© Houghton Mifflin Harcourt Publishing Company • Image Credits: (tr) ©Corbis
0
- 40
- 50
1. The depth of a diver is her distance below sea level.
Because depth represents a distance, it is never negative.
Find the depth of a diver visiting the Mohican, and explain
how her depth is related to the ship’s elevation of –32 ft.
Since distance can never be
negative, the absolute value of a
number can never be negative.
Math
Talk
MATHEMATICAL PRACTICES 6
Compare the absolute
values of two numbers that
are opposites. Explain your
reasoning.
2. Explain how the expression | –32| relates to the diver’s depth.
Chapter 3
165
You can find the absolute values of decimals, fractions, and other rational
numbers just as you found the absolute values of integers.
Example 1
Food Test Results
Name
A food scientist tested a new dog food on five dogs. Each dog’s weight
was monitored during the course of the test. The results are shown in
the table. Positive values indicate weight gains in pounds. Negative
values indicate weight losses in pounds.
Buck
Goldie
Mackerel
Graph the weight changes on the number line. Then find their
absolute values.
-3
-2
-1
0
1
2
3
Think: The distance from 0 to the point I graphed is 3_4 .
| |
__
−5
8 =
3.
_
MATHEMATICAL
PRACTICE
| |
7 =
−1___
16
_
| |
__ =
21
_
8
||
| |
Weight Change (lb)
Paloma
Spike
3
_
4
_
–5
8
– __
17
16
21_
8
_
–3
8
_3_ = 3
__
4
4
__
−3
8 =
_
Interpret a Result Explain how the absolute values of the
positive and negative weight changes relate to the starting weights of the dogs.
4
Example 2 Find all integers with an absolute value of 7.
- 10 - 9 - 8 - 7 - 6 - 5 - 4 - 3 - 2
-1 0
1
2
3
4
5
6
7
8
9
10
Graph integers located 7 units from 0 on the number line.
| _ | = 7 and | _ | = 7
So, both _ and _ have an absolute value of 7.
4.
166
MATHEMATICAL
PRACTICE
Use Counterexamples Paula says that there are always
two numbers that have a given absolute value. Is she correct? Explain.
3
© Houghton Mifflin Harcourt Publishing Company
Think: The distance from 0 to integers with an absolute value of 7 is _ units.
Name
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B
Find the absolute value.
1.
| –2|
Graph –2 on the number line.
–
2 is _ units from 0.
|
-3
2| = _
–
2. |6|
| –5|
3.
4.
-2
-1
0
1
2
5. |9|
| –11|
3
6. | –15 |
Math
Talk
MATHEMATICAL PRACTICES 2
Use Reasoning Can a
number have a negative
absolute value? Explain.
On
On Your
Your Own
Own
Find the absolute value.
7.
| –37|
8. |1.8|
9.
||
_
–2
3
10.
| –6.39|
11.
| |
– _
57
8
Find all numbers with the given absolute value.
13. 5_6
© Houghton Mifflin Harcourt Publishing Company
12. 13
MATHEMATICAL
PRACTICE
14. 14.03
16. 31_7
15. 0.59
Use Reasoning Algebra Find the missing number or numbers to
make the statement true.
2
17. | ■ | = 10
18. | ■ | = 1.78
___
21.
DEEPER
___
19. | ■ | = 0
__
20. | ■ | = 15
16
___
___
Find all of the integers whose absolute value is less than | –4|.
Chapter 3 • Lesson 5 167
MATHEMATICAL PRACTICES
COMMUNICA5&t1&34E7&3&tCONSTRUCT ARGUMENTS
Unlock
Unlock the
the Problem
Problem
22. The Blue Ridge Trail starts at Park
Headquarters in Big Bear Park
and goes up the mountain. The
Green Creek Trail starts at Park
Headquarters and goes down the
mountain. The table gives elevations
of various points of interest in
relation to Park Headquarters. How
many points of interest are less than
1 kilometer above or below Park
Headquarters?
b. How can you find the number of points of
interest that are less than 1 km above or below
Park Headquarters?
23.
MATHEMATICAL
PRACTICE
Use Reasoning Name a rational
number that can replace ■ to make both
statements true.
2
■ > −3
168
| ■ | < | −3|
Elevation Compared to
Park Headquarters (km)
A
1.9
B
1.1
C
0.7
D
0.3
E
–
F
–
G
–
H
–
0.2
0.5
0.9
1.6
c. Find how far above or below Park Headquarters
each point of interest is located.
WRITE
M t Show Your Work
Math
d. How many points of interest are less than
1 kilometer above or below Park Headquarters?
24.
Laila said |4| equals | −4|.
Is Laila correct? Use the number line and
words to support your answer.
SMARTER
-4 -3
-2 -1
0
1
2
3
4
© Houghton Mifflin Harcourt Publishing Company
a. How can you find how far above or below
Park Headquarters a given point of interest
is located?
Point of Interest
Practice and Homework
Name
Lesson 3.5
Absolute Value
COMMON CORE STANDARDS—6.NS.C.7c
Apply and extend previous understandings of
numbers to the system of rational numbers.
Find the absolute value.
1. | 7|
Graph 7 on the number line.
7 is
| 7| =
7
units from 0.
1
2
3
4
5
6
7
8
9
| |
3
__
5. 420
4. | 8.65 |
3. | 16 |
2. | −8 |
0
7
10
6.
| −5,000 |
Find all numbers with the given absolute value.
7. 12
10. 36_1
9. _3
8. 1.7
5
11. 0
Find the number or numbers that make the statement true.
12.
| |
= 17
13.
| |
= 2.04
14.
| |
9
__
= 110
15.
| |
19
__
= 24
Problem
Problem Solving
Solving
© Houghton Mifflin Harcourt Publishing Company
16. Which two numbers are 7.5 units away from 0 on
a number line?
_______
18.
WRITE
17. Emilio is playing a game. He just answered a
question incorrectly, so his score will change by
−
10 points. Find the absolute value of −10.
_______
Math Write two different real-world examples. One should
involve the absolute value of a positive number, and the other should
involve the absolute value of a negative number.
Chapter 3
169
Lesson Check (6.NS.C.7c)
1. What is the absolute value of 8_9 ?
2. What two numbers have an absolute value
of 21.63?
3. Rachel earned $89.70 on Tuesday. She spent
$55.89 at the grocery store. How much money
does she have left?
__ liter of juice. Another
4. One carton contains 17
20
carton contains 0.87 liter of juice. Which carton
contains the most?
5. Maggie jogged 7_8 mile on Monday and 1_2 of that
distance on Tuesday. How far did she jog on
Tuesday?
6. Trygg has 3_4 package of marigold seeds. He
plants 1_6 of those seeds in his garden and
divides the rest equally into 10 flowerpots. What
fraction of a package of seeds is planted in each
flowerpot?
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Spiral Review (6.NS.A.1, 6.NS.B.3, 6.NS.B.4, 6.NS.C.7b)
Lesson 3.6
Name
Compare Absolute Values
Essential Question How can you interpret comparisons involving
absolute values?
The Number System—
6.NS.C.7d
MATHEMATICAL PRACTICES
MP1, MP2
Unlock
Unlock the
the Problem
Problem
Activity
Carmen is taking a one-day scuba diving class.
Completion of the class will allow her to explore
the ocean at elevations that are less than −25 feet.
Use absolute value to describe the depths to which
Carmen will be able to dive after taking the class.
• Graph an elevation of −25 feet on the number line.
• List three elevations less than −25 feet. Then graph these elevations.
Elevation (feet)
30
• Elevations less than −25 feet are found __ −25 feet.
• Because depth represents a distance below sea level, it is never
negative. In this situation, | −25| ft represents a depth of _ feet.
• Write each elevation as a depth.
© Houghton Mifflin Harcourt Publishing Company • Image Credits: (tr) ©Tim Clayton/Corbis
Elevation (ft)
−
−
−
20
10
0
-10
- 20
Depth (ft)
- 30
30
- 40
35
- 50
40
• An elevation of less than | −25| feet is a depth __ than 25 feet.
So, Carmen will be able to dive to depths __ than 25 feet after
taking the class.
1. Compare a −175-foot elevation and a 175-foot depth. Explain your reasoning.
Chapter 3
171
Example Cole has an online account for buying video games.
His account balance has always been greater than −$16. Use absolute
value to describe Cole’s account balance as a debt.
STEP 1 Graph an account balance of −$16 on the number line.
Account balance ($)
- 20
- 16
- 12
-8
-4
0
4
8
12
16
20
STEP 2 List three account balances greater than −$16. Then graph these
account balances on the number line above.
Balances greater than −$16 are found to the __ of −$16.
STEP 3 Express an account balance of −$16 as a debt.
In this situation | −$16| represents a debt of _.
STEP 4 Complete the table.
Balances Greater
Than –$16
Debt
−
$15
−
$14
$13
Each debt in the table is __ than $16.
on the account is always __ than $16.
2. Explain how you can describe a debt as an absolute value.
3.
172
Math
Talk
MATHEMATICAL PRACTICES 6
The temperature at the
North Pole was −35ºF at
noon. Explain how you can
use absolute value to express
a temperature of −35ºF.
Describe List three numbers that have a magnitude greater than −28.
Describe how you determined your answer.
MATHEMATICAL
PRACTICE
1
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Cole’s account balance is always greater than −$16, so his debt
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1. On Monday, Allie’s bank account balance was
–
$24. On Tuesday, her account balance was less
than it was on Monday. Use absolute value to
describe Allie’s balance on Tuesday as a debt.
2. Matthew scored −36 points in his turn at
a video game. In Genevieve’s turn, she scored
fewer points than Matthew. Use absolute value
to describe Genevieve’s score as a loss.
In this situation | –$24| represents a debt
of _.
On Tuesday, Allie had a debt of _
than $24.
Genevieve lost _ than 36 points.
Math
Talk
MATHEMATICAL PRACTICES 1
Analyze Relationships
Compare a negative bank
balance and the amount of
the debt owed to the bank.
Explain.
On
On Your
Your Own
Own
3.
DEEPER
One of the cats shown in the table is
a tabby. The tabby had a decrease in weight of
more than 3.3 ounces. Which cat is the tabby?
Compare. Write <, >, or =.
4.
−
8
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7. 15
10.
O|
O|
8|
5. 13
O|
−
14|
8. 34
O|
−
−
−
Weight Change
(ounces)
Cat
Missy
3.8
Angel
−
Frankie
−
Spot
−
3.2
2.6
3.4
O|
13|
6. | −23|
36|
9.
−
5
O|
−
24|
−
6|
SMARTER
Write the values in order
from least to greatest.
| −2|
|3|
| −6|
|1|
Chapter 3 • Lesson 6 173
Compare and Contrast
When you compare and contrast, you look for ways that two
or more subjects are alike (compare) and ways they are
different (contrast). This helps you to discover information
about each subject that you might not have known otherwise.
As you read the following passage, think about how the main
topics are alike and how they are different.
Trevor mows lawns after school to raise money for a new
mountain bike. Last week, it rained every day, and he couldn’t
work. While waiting for better weather, he spent some of his
savings on lawnmower repairs. As a result, his savings balance
changed by −$45. This week, the weather was better, and
Trevor returned to work. His savings balance changed by
+
$45 this week.
11. The passage has two main parts. Describe them.
12. Describe the two changes in Trevor’s savings balance.
13.
MATHEMATICAL
PRACTICE
2 Reason Quantitatively Compare the two changes in
Trevor’s savings balance. How are they alike?
14.
174
SMARTER
Contrast the two changes in Trevor’s savings
balance. How are they different?
Math t Show Your Work
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WRITE
Practice and Homework
Name
Lesson 3.6
Compare Absolute Values
COMMON CORE STANDARD—6.NS.C.7d
Apply and extend previous understandings of
numbers to the system of rational numbers.
Solve.
1. Jamie scored −5 points on her turn at a trivia
game. In Veronica’s turn, she scored more points
than Jamie. Use absolute value to describe
Veronica’s score as a loss.
In this situation, |−5| represents a loss of
The temperature on Saturday was
than 10 degrees below zero.
fewer
5
points. Veronica lost
than 5 points.
2. The low temperature on Friday was −10°F. The
low temperature on Saturday was colder. Use
absolute value to describe the temperature on
Saturday as a temperature below zero.
3. The table shows changes in the savings accounts
of five students. Which student had the greatest
increase in money? By how much did the
student’s account increase?
Student
Account Change ($)
Brett
−
Destiny
−
12
36
_______
Carissa
15
_______
Rylan
10
Compare. Write <, >, or =.
4.
−
16
| −16|
5. 20
| 20|
6. 3
| −4 |
Problem
Problem Solving
Solving
7. On Wednesday, Miguel’s bank account balance
was −$55. On Thursday, his balance was less
than that. Use absolute value to describe Miguel’s
balance on Thursday as a debt.
8. During a game, Naomi lost points. She lost fewer
than 3 points. Use an integer to describe her
possible score.
© Houghton Mifflin Harcourt Publishing Company
In this situation, −$55 represents a debt of
. On Thursday, Miguel had a debt of
than $55.
9.
WRITE
_______
Math Give two numbers that fit this description: a number
is less than another number but has a greater absolute value. Describe
how you determined the numbers.
Chapter 3
175
Lesson Check (6.NS.C.7d)
1. A temperature of –6° is colder than a
temperature of 5°F below zero. Is this statement
true or false?
2. Long Beach, California has an elevation of
−7 feet. New Orleans, Louisiana is 8 feet below
sea level. Which city has a lower elevation?
3. Dawn and Lin took off on skateboards from
the same location but traveled in opposite
directions. After 20 minutes, Dawn had
traveled 6.42 kilometers and Lin had traveled
7.7 kilometers. How far apart were they?
4. Rico and Josh took off on skateboards going in
the same direction. After 20 minutes, Rico had
traveled 5.98 kilometers and Josh had gone
8.2 kilometers. How far apart were they?
5. Etta bought 11.5 yards of fabric selling for $0.90
per yard. What was the total cost?
__ . Before he
6. Yen calculates the product 5_8 × 24
25
multiplies, he simplifies all factors. What does
the problem look like after he simplifies the
factors?
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Spiral Review (6.NS.A.1, 6.NS.B.3, 6.NS.B.4)
Lesson 3.7
Name
Rational Numbers and the Coordinate Plane
Essential Question How do you plot ordered pairs of rational numbers
on a coordinate plane?
A coordinate plane is a plane formed by a horizontal number line
called the x-axis that intersects a vertical number line called the
y-axis. The axes intersect at 0 on both number lines. The point
where the axes intersect is the origin.
An ordered pair is a pair of numbers, such as (3, 2), that can be
used to locate a point on the coordinate plane. The first number
is the x-coordinate; it tells the distance to move left or right from
the origin. The second number is the y-coordinate; it tells the
distance to move up or down from the origin. The ordered pair for
the origin is (0, 0).
The Number System—
6.NS.C.6C
MATHEMATICAL PRACTICES
MP6, MP8
y
5
4
3
2
1
-5 -4 -3 -2 -1 0
-1
-2
-3
-4
-5
(3, 2)
x
1 2 3 4 5
Origin
Unlock
Unlock the
the Problem
Problem
A screen in a video game shows a coordinate plane.
The points P, Q, R, and S represent treasure chests.
Write the ordered pair for each treasure chest’s
location.
• If a point is to the left of the y-axis, is its
x-coordinate positive or negative?
Find the coordinates of each point.
To find the coordinates of point P, start at the origin.
To find the x-coordinate, move right
(positive) or left (negative).
Move 2 units to the _.
To find the y-coordinate, move up
(positive) or down (negative).
Move _ units up.
Point P is located at ( 22, _).
Point Q is located at ( _, _).
Point R is located at (_, _).
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Point S is located at (_, _).
1.
MATHEMATICAL
PRACTICE
8 Draw Conclusions Make a conjecture about the
x-coordinate of any point that lies on the y-axis.
2. Explain why (2, 4) represents a different location than (4, 2).
Chapter 3
177
Example
A A(2,
Graph and label the point on the coordinate plane.
__
−1
y
)
2
2
Start at the origin.
1
Move _ units to the right.
The x-coordinate is positive.
x
-2
Move 1_2 unit _.
The y-coordinate is negative.
-1
0
1
2
-1
Plot the point and label it A.
-2
B B( 0.5, 0)
−
Start at the origin.
The x-coordinate is __.
Move _ unit to the _.
The y-coordinate is 0.
The point lies on the _-axis.
Plot the point and label it B.
C C (2 1__, 3__ )
2 4
D D( –1.25, –1.75)
Start at the origin.
Start at the origin.
Move _ units to the _.
Move _ units to the _.
Move _ unit _.
Move _ units _.
Plot the point and label it C.
Plot the point and label it D.
MATH
M
MATHEMATICAL PRACTICES 4
Use Symbols Describe the
location of a point that has
a positive x-coordinate and
a negative y-coordinate.
BOARD
B
y
J
1. Write the ordered pair for point J.
2
Start at the origin. Move _ units to the _
and _ units _.
-2
178
-1
0
M
1
x
2
-1
The ordered pair is __.
Write the ordered pair for the point.
2. K
1
L
3. L
4. M
-2
K
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Graph and label the point on the coordinate plane.
5. P ( 2.5, 2)
–
6. Q( –2, 1_4)
–
9. T (11_, –2)
8. S( 1, –1_2)
2
11. V ( –0.5, 0)
12. W (2, 0)
y
7. R(0, 1.5)
2
10. U(0.75, 1.25)
1
x
13. X (0, –2)
-2
-1
0
1
2
-1
-2
Math
Talk
On
On Your
Your Own
Own
Write the ordered pair for the point. Give approximate
coordinates when necessary.
14. A
15. B
16. C
MATHEMATICAL PRACTICES 3
Compare Models Explain
how graphing (3, 2) is
similar to and different from
graphing (3, –2).
y
B
17. D
18. E
20. G
19. F
21. H
C
5
4
H
3
2
J
1
Graph and label the point on the coordinate plane.
23. M 4, 0)
25. P 3, 3)
26. Q 0, –2 1_
27. R(0.5, 0.5)
28. S –5, 1_2
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(
2
)
(
29. T(0, 0)
32.
MATHEMATICAL
PRACTICE
(–
)
30. U 3 1_2 , 0
(
)
31. V( –2, –4)
Look for Structure A point lies to the left of the y-axis
and below the x-axis. What can you conclude about the coordinates
of the point?
7
x
E
y
24. N(2, 2)
(–
F
1 2 3 4 5
-5 -4 -3 -2 -1 0
-1
-2
D
-3
-4
G
-5
22. J
A
5
4
3
2
1
-5 -4 -3 -2 -1 0
-1
-2
-3
-4
-5
x
1 2 3 4 5
Chapter 3 • Lesson 7 179
MATHEMATICAL PRACTICES
COMMUNICA5&t1&34E7&3&tCONSTRUCT ARGUMENTS
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OqnakdlRnkuhmf¤@ookhb`shnmr
Many of the streets in downtown Philadelphia can
be modeled by a coordinate plane, as shown on the
map. Each unit on the map represents one block.
Use the map for 33 and 34.
DEEPER
33.
Anita works at the Historical Society.
She leaves the building and walks 3 blocks north
to a restaurant. What ordered pair represents the
restaurant?
34.
Pose a Problem Write and solve a
new problem that uses a location on the map.
36.
SMARTER
The points A, B, C, and D on a coordinate plane can
be connected to form a rectangle. Point A is located at (2, 0), point B is
located at (6, 0), and point C is located at (6, –2.5). Write the ordered pair
for point D.
MATHEMATICAL
PRACTICE
Identify Relationships Explain how you can tell that
the line segment connecting two points is vertical without graphing
the points.
37.
7
SMARTER
For numbers 37a–37d, select True or False for
each statement.
37a. Point A (2, –1) is to the right of the y-axis and below the x-axis.
True
False
37b. Point B (–5, 2) is to the left of the y-axis and below the x-axis.
True
False
37c.
True
False
True
False
Point C (3, 2) is to the right of the y-axis and above the x-axis.
37d. Point D (–2, –1) is to the left of the y-axis and below the x-axis.
180
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35.
SMARTER
Practice and Homework
Name
Lesson 3.7
Rational Numbers and
the Coordinate Plane
COMMON CORE STANDARD—6.NS.C.6c
Apply and extend previous understandings of
numbers to the system of rational numbers.
Write the ordered pair for the point. Give approximate
coordinates when necessary.
2. B
1. A
y
E
2
_)
(1, 1
2
F
1
4. D
3. C
D
-2
-1
A
0
x
1
2
-1
B
y
Graph and label the point on the coordinate plane.
5. G
6. H (0, 2.50)
7.
8. K (1, 2)
( −1_2, 11_2 )
J ( −11_, 1_ )
2 2
9. L
(
10. M (1, −0.5)
)
− _
11, −21_
2
11. N 1_4, 11_
2
(
2
2
1
x
-2
-1
0
2
-2
Problem
Problem Solving
Solving
Map of Elmwood
y
Use the map for 15–16.
2
13. What is the ordered pair for the city hall?
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1
-1
12. P (1.25, 0)
)
C
-2
City Hall
Library
1
x
-2
14.
The post office is located at ( _ ). Graph and label a point on the
−1
,2
2
map to represent the post office.
15.
WRITE
-1
0
1
2
-1
Courthouse - 2
Police Station
Math Describe how to graph the ordered pair (−1, 4.5).
Chapter 3
181
Lesson Check (6.NS.C.6c)
1. An artist uses a coordinate plane to create a
design. As part of the design, the artist wants to
graph the point (−6.5, 2). How should the artist
graph this point?
2. What are the coordinates of the campground?
y
y
2
2
1
1
-2
-1 0
-2
- 1 01
-1
Campground 2
Campground 2
Pond
Pond
Ranger
Station
Ranger
1
Station2
1
2
x
x
3. Four students volunteer at the hospital. Casey
volunteers 20.7 hours, Danielle, 20 3_4 hours,
9
18
__
Javier 18 __
10 hours, and Forrest, 20 25 hours. Who
volunteered the greatest number of hours?
4. Directions for making a quilt say to cut fifteen
squares with sides that are 3.625 inches long.
What is the side length written as a fraction?
5. Cam has a piece of plywood that is 6 7_8 feet wide.
He is going to cut shelves from the plywood that
are each 1 1_6 feet wide. Which is a good estimate
for the number of shelves Cam can make?
6. Zach has 3_4 hour to play video games. It takes
1
him __
12 hour to set up the system. Each round
of his favorite game takes 1_6 hour. How many
rounds can he play?
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Spiral Review (6.NS.A.1, 6.NS.C.6c)
Lesson 3.8
Name
Ordered Pair Relationships
The Number System—
6.NS.C.6b
MATHEMATICAL PRACTICES
MP4, MP7
Essential Question How can you identify the relationship between
points on a coordinate plane?
The four regions of the coordinate plane that are separated by the
x- and y-axes are called quadrants. Quadrants are numbered with
the Roman numerals I, II, III, and IV. If you know the signs of the
coordinates of a point, you can determine the quadrant where the
point is located.
y
Quadrant II
(2, 1)
5
4
3
2
1
-5 -4 -3 -2 -1 0
-1
-2
Quadrant III 3
(2, 2)
-4
-5
Quadrant I
(1, 1)
x
1 2 3 4 5
Quadrant IV
(1, 2)
Unlock
Unlock the
the Problem
Problem
The point ( −3, 4) represents the location of a
bookstore on a map of a shopping mall. Identify the
quadrant where the point is located.
• What is the x-coordinate of the point? _
• What is the y-coordinate of the point? _
−
Find the quadrant that contains ( 3, 4).
STEP 1 Examine the x-coordinate.
Think: The x-coordinate is
of the origin.
_ , so the point is _ units to the _
Since the point is to the left of the origin, it must be located in either
Quadrant _ or Quadrant _.
y
STEP 2 Examine the y-coordinate.
Think: The y-coordinate is _ , so the point is _ units _
from the origin.
Since the point is above the origin, it must be located in
Quadrant _.
© Houghton Mifflin Harcourt Publishing Company
Check by graphing the point ( −3, 4) on the coordinate plane.
So, the point representing the bookstore is located in
II
5
4
3
2
1
-5 -4 -3 -2 -1 0
-1
-2
-3
-4
III
-5
I
x
1 2 3 4 5
IV
Quadrant _ .
•
MATHEMATICAL
PRACTICE
Look for Structure Look at the signs of the coordinates
of points in Quadrants I and II. What do they have in common? How are
they different?
7
Chapter 3
183
A figure has line symmetry if it can be folded about a line so that its two
parts match exactly. If you cut out the isosceles triangle at the right and
fold it along the dashed line, the two parts would match. A line that divides
a figure into two halves that are reflections of each other is called a line of
symmetry.
Line of symmetry
You can use the idea of line symmetry to analyze the relationship
between points such as (5, −1) and (−5, −1) whose coordinates
differ only in their signs.
Activity
• Identify the lines of symmetry in the rectangle.
D (25, 4)
The _ -axis is a horizontal line of symmetry, and the _ -axis
is a vertical line of symmetry.
• Look at points A and B. What do you notice about the x-coordinates?
What do you notice about the y-coordinates?
y
5
4
3
2
1
-5 -4 -3 -2 -1 0
-1
-2
-3
-4
C (25, 24)
-5
A (5, 4)
x
1 2 3 4 5
B (5, 24)
• Point B is a reflection of point A across which axis? How do you know?
• Look at points A and D. What do you notice about the x-coordinates? What
do you notice about the y-coordinates?
• Which point is a reflection of point B across the x-axis and then the y-axis?
• Compare the coordinates of point B with the coordinates of point D.
184
Math
Talk
MATHEMATICAL PRACTICES 1
Describe Relationships
Describe how the coordinates
of a point change if it is
reflected across the x-axis.
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• Point D is a reflection of point A across which axis? How do you know?
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B
Identify the quadrant where the point is located.
2. (4, 1)
1. (2, −5)
To graph the point, first move to the _ from the origin.
Then move _ .
Quadrant: _
Quadrant: _
3. (−6, −2)
Quadrant: _
4. (−7, 3)
6. (1, −1)
5. (8, 8)
Quadrant: _
Quadrant: _
Quadrant: _
The two points are reflections of each other across the x- or y-axis. Identify the axis.
7. (−1, 3) and (1, 3)
8. (4, 4) and (4, −4)
9. (2, −9) and (2, 9)
10. (8, 1) and (−8, 1)
axis: _
axis: _
axis: _
axis: _
Math
Talk
On
On Your
Your Own
Own
MATHEMATICAL PRACTICES 6
Explain how you can identify
the quadrant where a given
point is located.
Identify the quadrant where the point is located.
11. (−8, −9)
Quadrant: _
13. (−13, 10)
12. (12, 1)
Quadrant: _
14. (5, −20)
Quadrant: _
Quadrant: _
© Houghton Mifflin Harcourt Publishing Company
The two points are reflections of each other across the x- or y-axis. Identify the axis.
15. (−9, −10) and (−9, 10)
axis: _
16. (21, −31) and (21, 31)
axis: _
17. (15, −20) and (−15, −20)
axis: _
Give the reflection of the point across the given axis.
18. (−7, −7), y-axis
__
19. (−15, 18), x-axis
__
20. (11, 9), x-axis
__
Chapter 3 • Lesson 8 185
MATHEMATICAL PRACTICES
ANALYZEt-00,'034536$563&t13&$*4*0/
OqnakdlRnkuhmf¤@ookhb`shnmr
OqnakdlRnkuhmf¤@ookhb`shnmr
Use the map of Gridville for 21–23.
21.
22.
DEEPER
The library’s location has opposite
x- and y-coordinates as City Hall. Across which
streets could you reflect City Hall’s location to
find the library’s location?
SMARTER
Each unit on the
map represents 1 mile. Gregory
leaves his house at (−5, 4), cycles
4 miles east, 6 miles south, and
1 mile west. In which quadrant of
the city is he now?
23. The bus station has the same x-coordinate as City Hall but the
opposite y-coordinate. In which quadrant of the city is the bus
station located?
25.
MATHEMATICAL
PRACTICE
1 Describe Relationships Describe the relationship
between the locations of the points (2, 5) and (2,−5) on the
coordinate plane.
SMARTER
Identify the quadrant where each point is
located. Write each point in the correct box.
( −1, 3)
(4, −2)
( −3, −2)
(1, −3)
( −1, 2)
(3, 4)
Quadrant I
186
Quadrant II
Quadrant III
Quadrant IV
© Houghton Mifflin Harcourt Publishing Company
24.
Practice and Homework
Name
Lesson 3.8
Ordered Pair Relationships
COMMON CORE STANDARD—6.NS.C.6b
Apply and extend previous understandings of
numbers to the system of rational numbers.
Identify the quadrant where the point is located.
1. (10, −2)
Quadrant:
IV
2. (−5, −6)
Quadrant:
3. (3, 7)
Quadrant:
The two points are reflections of each other across the
x- or y-axis. Identify the axis.
4. (5, 3) and (−5, 3)
axis:
5. (−7, 1) and (−7, −1)
axis:
6. (−2, 4) and (−2, −4)
axis:
Give the reflection of the point across the given axis.
7. (−6, −10), y-axis
9. (8, 2), x-axis
8. (−11, 3), x-axis
Problem
Problem Solving
Solving
© Houghton Mifflin Harcourt Publishing Company
10. A town’s post office is located at the point
(7, 5) on a coordinate plane. In which
quadrant is the post office located?
______
12.
WRITE
11. The grocery store is located at a point on a
coordinate plane with the same y-coordinate as
the bank but with the opposite x-coordinate. The
grocery store and bank are reflections of each
other across which axis?
______
Math Explain to a new student how a reflection across the
y-axis changes the coordinates of the original point.
Chapter 3
187
Lesson Check (6.NS.C.6b)
1. In which quadrant does the point (−4, 15) lie?
2. What are the coordinates of the point (10, −4) if
it is reflected across the y–axis?
3. Small juice bottles come in packages of 6.
Yogurt treats come in packages of 10. Paula
wants to have the exact same number of each
item. How many bottles of juice and individual
yogurt treats will she have altogether? How
many packages of each will she need?
4. Alison saves $29.26 each month. How many
months will it take her to save enough money to
buy a stereo for $339.12?
5. The library is 1.75 miles directly north of the
school. The park is 0.6 miles directly south of
the school. How far is the library from the park?
6. Tours of the art museum are offered every
_1
3 hour starting at 10 A.M. The museum closes at
4:00 P.M. How many tours are offered each day?
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Spiral Review (6.NS.A.1, 6.NS.B.3)
Lesson 3.9
Name
Distance on the Coordinate Plane
Essential Question How can you find the distance between two points
that lie on a horizontal or vertical line on a coordinate plane?
The Number System—
6.NS.C.8
MATHEMATICAL PRACTICES
MP1, MP5, MP6
Unlock
Unlock the
the Problem
Problem
The map of Foggy Mountain Park is marked on a
coordinate plane in units of 1 mile. There are two
campgrounds in the park. Camp 1 is located at
(−4, 3). Camp 2 is located at (5, 3). How far is it
from Camp 1 to Camp 2?
Find the distance from Camp 1 to Camp 2.
STEP 1
Graph the points.
Think: The points have the same _ -coordinate, so they are located on a horizontal line.
STEP 2
Find the horizontal distance from Camp 1 to the y-axis.
_, 3) and the point (0, 3).
Find the distance between the x-coordinates of the point (
The distance of a number from 0 is the
___ of the number.
| 24 | 5 4
- 6 - 5 -4 - 3 - 2 - 1
0
1
2
The distance from (−4, 3) to (0, 3) is | −4| = _ miles.
STEP 3
Find the horizontal distance from Camp 2 to the y-axis.
_ , 3) and ( _ , 3).
The distance from (5, 3) to (0, 3) is |_| = _ miles.
© Houghton Mifflin Harcourt Publishing Company
Find the distance between the x-coordinates of (
STEP 4
Remember that distance
is never negative. You can
find the distance between a
negative number and 0 by
using absolute value.
Math
Talk
Add to find the total distance: _ + _ = _ miles.
MATHEMATICAL PRACTICES 1
Evaluate Explain how you
could check that you found
the distance correctly.
So, the distance from Camp 1 to Camp 2 is _ miles.
1.
MATHEMATICAL
PRACTICE
Explain how you could use absolute value to find the distance
from Camp 2 to the Eagle Nest. What is the distance?
6
Chapter 3
189
In the problem on the previous page, you used absolute value to find the
distance between points in different quadrants. You can also use absolute
value to find the distance between points in the same quadrant.
Example Find the distance between the pair of
y
10
8
6
4
2
points on the coordinate plane.
A points A and B
STEP 1
Look at the coordinates of the points.
- 10- 8 - 6 - 4 - 2 0
-2
The _ -coordinates of the points are the same,
4
A (29, 26)
so the points lie on a horizontal line.
-6
-8
Think of the horizontal line passing through A and B as a number line.
B (24, 26) - 10
A
B
- 10 - 9 - 8 - 7 - 6 - 5 - 4 - 3 - 2 - 1
STEP 2
0
1
B
⎜−9⎟ = _ units
| 24 |
-10 - 9 - 8 - 7 - 6 - 5 - 4 - 3 - 2 - 1
STEP 3
x
2 4 6 8 10
Distance of A from 0:
| 29 |
?
D (8, 3)
2
Find the distances of A and B from 0.
A
C (8, 10)
Distance of B from 0:
⎜_⎟ = _ units
0
1
2
Subtract to find the distance from A to B: _ − _ = _ units.
So, the distance from A to B is _ units.
10
B points C and D
9
Look at the coordinates of the points.
The _ -coordinates of the points are the same, so the points lie
on a vertical line.
Think of the vertical line passing through C and D as a number line.
STEP 2
Find the distances of C and D from 0 on the vertical number line.
190
|10 |
4
2
D
|3 |
1
0
-1
Subtract to find the distance from C to D:
So, the distance between C and D is _ units.
?
6
3
Distance of D from 0: ⎜_⎜ = _ units
_ − _ = _ units.
7
5
Distance of C from 0: ⎜10⎟ = _ units
STEP 3
8
Math
Talk
MATHEMATICAL PRACTICES 7
Look for Structure Explain
how to find the distance
from M(−5, 1) to N(−5, 7).
© Houghton Mifflin Harcourt Publishing Company
STEP 1
C
Name
Share
hhow
Share and
and Show
Sh
MATH
M
BOARD
B
Find the distance between the pair of points.
1. ( −3, 1) and (2, 1)
(23, 3)
Horizontal distance from (−3, 1) to y-axis:
(23, 1)
| _|=_
Horizontal distance from (2, 1) to y-axis: |_| =
_
Distance from ( 3, 1) to (2, 1): __
−
2. (2, 1) and (2, −4)
___
6
5
4
3
2
1
3. (2, −4) and (4, −4)
(2, 1)
x
-6 -5 -4 -3 -2 -1 0
-1
-2
-3
-4
-5
-6
1 2 3 4 5 6
(2, 24)
(4, 24)
4. (−3, 3) and (−3, 1)
___
___
Math
Talk
On
On Your
Your Own
Own
y
MATHEMATICAL PRACTICES 6
Explain how you can find
the distance between two
points that have the same
y-coordinate.
Practice: Copy and Solve Graph the pair of points. Then find the distance between them.
5. (0, 5) and (0, −5)
___
© Houghton Mifflin Harcourt Publishing Company
8. ( −7, 3) and (5, 3)
___
MATHEMATICAL
PRACTICE
6. (1, 1) and (1, −3)
___
9. (3, −6) and (3, −10)
___
7. ( −2, −5) and ( −2, −1)
___
10. (8, 0) and (8, −8)
___
2 Use Reasoning Algebra Write the coordinates of a point that is the given
distance from the given point.
11. 4 units from (3, 5)
(3, )
12. 6 units from (2, 1)
(
, 1)
13. 7 units from ( −4, −1)
( −4,
)
Chapter 3 • Lesson 9 191
MATHEMATICAL PRACTICES .0%&-t3&"40/tM",&4&/4&
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OqnakdlRnkuhmf¤@ookhb`shnmr
An archaeologist is digging at an ancient city. The map
shows the locations of several important finds. Each
unit represents 1 kilometer. Use the map for 14–18.
14. How far is it from the stadium to the statue?
DEEPER
The archaeologist drives 3 km south
from the palace. How far is he from the market?
Archaeological Site
16. The archaeologist’s campsite is located at ( −9, −3) .
How far is it from the campsite to the market?
N
Royal Road
17.
18.
SMARTER
The archaeologist rode
east on a donkey from the Great Gate, at
( −11, 4), to the Royal Road. Then he rode
south to the palace. How far did the
archaeologist ride?
MATHEMATICAL
PRACTICE
SMARTER
Select the pairs of points
that have a distance of 10 between them.
Mark all that apply.
(3, −6) and (3, 4)
( −3, 8) and (7, 8)
(4, 5) and (6, 5)
(4, 1) and (4, 11)
192
W
-5 -4 -3 -2 -1 0
-1
-2
-3
Market
-4
-5
y
Stadium
Imperial
Highway
x
1 2 3 4 5
E
Statue
S
8 Generalize Explain how you could
find the distance from the palace to any point on the
Imperial Highway.
19.
Palace
5
4
3
2
1
WRITE
Math t Show Your Work
© Houghton Mifflin Harcourt Publishing Company • Image Credits: (tr) ©Danita Delimont/Alamy Images
15.
Practice and Homework
Name
Lesson 3.9
Distance on the Coordinate Plane
COMMON CORE STANDARD—6.NS.C.8
Apply and extend previous understandings of
numbers to the system of rational numbers.
Find the distance between the pair of points.
1. (1, 4) and (−3, 4)
2. (7, −2) and (11, −2)
3. (6, 4) and (6, −8)
| 1 | 5 1; | 23 | 5 3;
11354
4
units
4. (8, −10) and (5, −10)
units
units
5. (−2, −6) and (−2, 5)
units
6. (−5, 2) and (−5, −4)
units
units
Write the coordinates of a point that is the given distance from
the given point.
7. 5 units from (−1, −2)
(
, −2
8. 8 units from (2, 4)
9. 3 units from (−7, −5)
( 2, )
)
(
7,
Amusement Park
y
Ferris Wheel
6
Problem
Problem Solving
Solving
The map shows the locations of several areas in an amusement
park. Each unit represents 1 kilometer.
Bumper Cars 4
© Houghton Mifflin Harcourt Publishing Company
11. How far is the water slide from the restrooms?
_____
12.
WRITE
Rollercoaster
2
10. How far is the Ferris wheel from the rollercoaster?
_____
)
−
-6
-4
-2
x
0
2
4
6
-2
Water Slide
-4
Restrooms
-6
Math Graph the points (23, 3), (23, 7), and (4, 3)
on a coordinate plane. Explain how to find the distance from
(23, 3) to (23, 7) and from (23, 3) and (4, 3).
Chapter 3
193
Lesson Check (6.NS.C.8)
1. What is the distance between (4, −7)
and (−5, −7)?
2. Point A and point B are 5 units apart. The
coordinates of point A are (3, −9). The
y–coordinate of point B is −9. What is a
possible x–coordinate for point B?
3. An apple is cut into 10 pieces. 0.8 of the apple
is eaten. Which fraction, in simplest form,
represents the amount of apple that is left?
4. A carton contains soup cans weighing a total of
20 pounds. Each can weighs 11_4 pounds. How
many cans does the carton contain?
5. List −1, 1_4 , and −1 2_3 in order from greatest
to least.
6. The point located at (3, −1) is reflected across
the y−axis. What are the coordinates of the
reflected point?
FOR MORE PRACTICE
GO TO THE
194
Personal Math Trainer
© Houghton Mifflin Harcourt Publishing Company
Spiral Review (6.NS.A.1, 6.NS.C.6b, 6.NS.C.6c, 6.NS.C.7b)
Lesson 3.10
Name
Problem Solving • The Coordinate Plane
Essential Question How can you use the strategy draw a diagram to
help you solve a problem on the coordinate plane?
The Number System—
6.NS.C.8
MATHEMATICAL PRACTICES
MP1, MP5, MP6
Unlock
Unlock the
the Problem
Problem
An artist is using an illustration program. The program uses
a coordinate plane, with the origin (0, 0) located at the center
of the computer screen. The artist draws a dinosaur centered on the
point (4, 6). Then she moves it 10 units to the left and 12 units down.
What ordered pair represents the dinosaur’s new location?
Use the graphic organizer to help you solve the problem.
Read the Problem
What do I need to find?
I need to find the
___ for the
dinosaur’s new location.
What information do I
need to use?
How will I use the
information?
The dinosaur started at the
I can draw a diagram to
graph the information on a
point __. Then the
artist moved it __ to
___.
the left and __ down.
Solve the Problem
• Start by graphing and labeling the point __.
10
8
6
4
2
• From this point, count __ to the left.
• Then count __ down.
© Houghton Mifflin Harcourt Publishing Company
• Graph and label the point at this location, and
write its coordinates: __.
So, the dinosaur’s new location is __.
- 10- 8 - 6 - 4 - 2 0
-2
-4
-6
-8
- 10
Math
Talk
y
x
2 4 6 8 10
MATHEMATICAL PRACTICES 1
Analyze Explain how you
could check that your answer
is correct.
Chapter 3
195
Try Another Problem
Tyrone and Kyra both walk home from school. Kyra walks
4 blocks east and 3 blocks south to get home. Tyrone lives
3 blocks west and 3 blocks south of the school. How far
apart are Tyrone’s and Kyra’s homes?
Use the graphic organizer to help you solve the problem.
Read the Problem
Solve the Problem
What do I need to find?
y
x
What information do I need to use?
So, it is _ blocks from Tyrone’s house to Kyra’s house.
•
MATHEMATICAL
PRACTICE
5 Use Appropriate Tools Describe the advantages of
using a coordinate plane to solve a problem like the one above.
196
Math
Talk
MATHEMATICAL PRACTICES 6
Explain how you know that
your answer is reasonable.
© Houghton Mifflin Harcourt Publishing Company • Image Credits: (t) ©PhotoAlto/Alamy Images
How will I use the information?
Name
MATH
M
Share
hhow
Share and
and Show
Sh
1.
Unlock the Problem
√
√
BOARD
B
Draw a diagram of the situation.
Use absolute value to find distance.
DEEPER
Busby County is rectangular. A map of the county
on a coordinate plane shows the vertices of the county at
( −5, 8), (8, 8), (8, −10), and ( −5, −10). Each unit on
the map represents 1 mile. What is the county’s perimeter?
WRITE
Math t Show Your Work
First, draw a diagram of Busby County.
Busby County
10
8
6
4
2
- 10- 8 - 6 - 4 - 2 0
-2
-4
-6
-8
- 10
y
x
2 4 6 8 10
Next, use the diagram to find the length of each side of the
rectangle. Then add.
So, the perimeter of Busby County is __.
© Houghton Mifflin Harcourt Publishing Company
2.
SMARTER
What if the vertices of the county were
( 5, 8), (8, 8), (8, −6), and ( −5, −6)? What would the
perimeter of the county be?
−
3. On a coordinate map of Melville, a restaurant is located
at ( −9, −5). A laundry business is located 3 units to the left
of the restaurant on the map. What are the map coordinates
of the laundry business?
4.
DEEPER
The library is 4 blocks north and 9 blocks
east of the school. The museum is 9 blocks east and
11 blocks south of the school. How far is it from the library
to the museum?
Chapter 3 • Lesson 10 197
MATHEMATICAL PRACTICES
COMMUNICA5&t1&34E7&3&tCONSTRUCT ARGUMENTS
OqnakdlRnkuhmf¤@ookhb`shnmr
OqnakdlRnkuhmf¤@ookhb`shnmr
MATHEMATICAL
PRACTICE
5.
Make Sense of Problems Diana left her campsite at
(2, 6) on a map of Big Trees Park, hiked to Redwood Grove at ( −5, 6),
and continued on to Bass Lake at ( −5, −3). Each unit on the map
represents 1 kilometer. How far did Diana hike?
6.
Hector left his house at ( −6, 13) on
a map of Coleville and walked to the zoo at ( −6, 2).
From there he walked east to his friend’s house. He
walked a total distance of 25 blocks. If each unit on the
map represents one block, what are the coordinates of
Hector’s friend’s house?
8.
SMARTER
DEEPER
In November, the price of a cell phone was double the price
in March. In December, the price was $57, which was $29 less than the
price in November. What was the price of the cell phone in March?
SMARTER
A map of the city holding the
Olympics is placed on a coordinate plane. Olympic
Stadium is located at the origin of the map. Each unit
on the map represents 2 miles.
Graph the locations of four other Olympic buildings.
Max said the distance between the Aquatics Center
and the Olympic Village is greater than the distance
between the Media Center and the Basketball Arena.
Do you agree with Max? Use words and numbers to
support your answer.
Personal Math Trainer
WRITE
Math t Show Your Work
Building
Location
−
0MZNQJD7JMMBHF
"RVBUJDT$FOUFS
.FEJB$FOUFS
(4, −5)
#BTLFUCBMM"SFOB
−
−
10
8
6
4
2
- 10 - 8 - 6 - 4 - 2 0
-2
-4
-6
-8
- 10
198
y
2 4 6 8 10
x
© Houghton Mifflin Harcourt Publishing Company
7.
1
Practice and Homework
Name
Problem Solving • The Coordinate Plane
Lesson 3.10
COMMON CORE STANDARD—6.NS.C.8
Compute fluently with multi-digit numbers
and find common factors and multiples.
Read each problem and solve.
1. On a coordinate map of Clifton, an electronics store is located
at (6, −7). A convenience store is located 7 units north of the
electronics store on the map. What are the map coordinates of the
convenience store?
(6, 0)
____
2. Sonya and Lucas walk from the school to the library. They walk 5
blocks south and 4 blocks west to get to the library. If the school is
located at a point (9, −1) on a coordinate map, what are the map
coordinates of the library?
____
3. On a coordinate map, Sherry’s house is at the point
(10, −2) and the mall is at point (−4, −2). If each unit on
the map represents one block, what is the distance between
Sherry’s house and the mall?
____
4. Arthur left his job at (5, 4) on a coordinate map and walked to his
house at (5, −6). Each unit on the map represents 1 block. How far
did Arthur walk?
____
5. A fire station is located 2 units east and 6 units north of a hospital.
If the hospital is located at a point (−2, −3) on a coordinate map,
what are the coordinates of the fire station?
© Houghton Mifflin Harcourt Publishing Company
____
6. Xavier’s house is located at the point (4, 6). Michael’s house is 10
blocks west and 2 blocks south of Xavier’s house. What are the
coordinates of Michael’s house?
____
7.
WRITE
Math Write a problem that can be solved by
drawing a diagram on a coordinate plane.
Chapter 3
199
Lesson Check (6.NS.C.8)
1. The points (−4, −4), (−4, 4), (4, 4), and (4, −4)
form a square on a coordinate plane. How long
is a side length of the square?
2. On a coordinate map, the museum is located
at (− 5, 7). A park is located 6 units to the right
of the museum on the map. What are the
coordinates of park?
3. On a grid Joe’s house is marked at (−5, −3) and
Andy’s house is marked at (1, −3) What is the
distance, on the grid, between Joe’s house and
Andy’s house?
4. In the last two years, Mari grew 2 1_4 inches, Kim
grew 2.4 inches, and Kate grew 2 1_8 inches. Write
the amounts they grew in order from least
to greatest.
5. A jar of jelly that weighs 4.25 ounces costs $2.89.
What is the cost of one ounce of jelly?
6. Jan began with 5_ pound of modeling clay. She
6
used 1_ of the clay to make decorative magnets.
5
She divided the remaining clay into 8 equal
portions. What is the weight of the clay in each
portion?
FOR MORE PRACTICE
GO TO THE
200
Personal Math Trainer
© Houghton Mifflin Harcourt Publishing Company
Spiral Review (6.NS.A.1, 6.NS.C.6c, 6.NS.C.8)
Name
Chapter 3 Review/Test
Personal Math Trainer
Online Assessment
and Intervention
1. For numbers 1a–1d, choose Yes or No to indicate whether the situation
can be represented by a negative number.
1a. Sherri lost 100 points answering
a question wrong.
Yes
No
1b.
Yes
No
1c. Yong paid $25 for a parking
ticket.
Yes
No
1d.
Yes
No
The peak of a mountain is
2,000 feet above sea level.
A puppy gained 3 pounds.
2. The low weekday temperatures for a city are shown.
Low Temperatures
Day
Low Temperature (°F)
Monday
−
Tuesday
−
5
3
Wednesday
Thursday
Friday
2
−
7
3
Part A
Using the information in the table, order the temperatures from lowest
to highest.
© Houghton Mifflin Harcourt Publishing Company
Part B
Explain how to use a vertical number line to determine the order.
Assessment Options
Chapter Test
Chapter 3
201
3. For numbers 3a–3e, choose Yes or No to indicate whether the number is
between –1 and –2.
3a.
–_
4
5
Yes
No
3b. 12_
Yes
No
Yes
No
Yes
No
Yes
No
3
3c.
–
3d.
– _
11
3e.
1.3
4
– __
21
10
–
4. Compare 1_5 and –0.9. Use numbers and words to explain your answer.
5. Jeandre said |3| equals |–3|. Is Jeandre correct? Use a number line and
words to support your answer.
6. Write the values in order from least to greatest.
|–4|
|2|
|–12|
|8|
__
__
__
__
7a.
202
True
False
7b. Point D( –2, 1) is to the left of
the y-axis and below the x-axis.
True
False
7c. The point where the axes
intersect is the origin.
True
False
7d. If both the x- and y- coordinates
are positive, the point is to the right
of the y-axis and below the x-axis.
True
False
The x-coordinate of any point
on the y-axis is 0.
© Houghton Mifflin Harcourt Publishing Company
7. For numbers 7a–7d, select True or False for each statement.
Name
8. Mia’s house is located at point (3, 4) on a coordinate plane. The location
of Keisha’s house is the reflection of the location of Mia’s house across
the y-axis. In what quadrant is Keisha’s house in?
Personal Math Trainer
9.
Points A(3, 8) and B( –4, 8) are located on a coordinate
plane. Graph the pair of points. Then find the distance between them.
Use numbers and words to explain your answer.
SMARTER
10
y
8
6
4
2
- 10
-8
-6
-4
-2
x
0
2
4
6
8
10
-2
-4
-6
-8
© Houghton Mifflin Harcourt Publishing Company
- 10
Chapter 3
203
10.
DEEPER
10
The map shows the location J of
Jose’s house and the location F of the football
field. Jose is going to go to Tyrell’s house
and then the two of them are going to go
to the football field for practice.
Jose’s House
y
8
J
6
4
2
Part A
x
- 10
-8
-6
-4
-2
0
2
4
6
8
10
-2
Tyrell’s house is located at point T,
the reflection of point J across the
y-axis. What are the coordinates of
points T, J, and F?
-4
-6
Football Field
F
-8
- 10
Part B
If each unit on the map represents 1 block, what was the distance Tyrell
traveled to the football field and what was the distance Jose traveled to
the football field? Use numbers and words to explain your answer.
204
11a. A football team gains
3 yards on a play.
Yes
No
11b.
Yes
No
11c. A student answers a
3-point question correctly.
Yes
No
11d.
Yes
No
A golfer’s score is
3 over par.
A cat loses 3 pounds.
© Houghton Mifflin Harcourt Publishing Company
11. For numbers 11a–11d, choose Yes or No to indicate whether the
situation could be represented by the integer +3.
Name
12. Jason used a map to record the elevations of five locations.
Elevations
Location
Elevation (feet)
Nob Hill
5
Bear Creek
−
Po Valley
−
Fox Hill
18
20
8
−
Jax River
3
Jason wrote the elevations in order from lowest to highest. Is Jason
correct? Use words and numbers to explain why or why not. If Jason is
incorrect, what is the correct order?
–
3, 5, 8, –18, –20
13. For numbers 13a–13d, select True or False for each statement.
13a. 1_ is between 0 and 1.
5
13b.
13c.
13d.
– _
22 is between –1 and –2.
3
– 5
3_ is between –3 and –4.
8
4_3 is between 3 and 4.
4
True
False
True
False
True
False
True
False
© Houghton Mifflin Harcourt Publishing Company
14. Choose <, >, or =.
<
<
14a.
>
0.25
3
_
4
14c.
27_
8
<
14b.
>
=
2.875
=
=
1
_
3
>
<
0.325
14d.
–3
_
4
>
–1
_
2
=
Chapter 3
205
15. Graph 4 and −4 on the number line.
-4 -3 - 2 - 1
0
1
2
3
4
Tyler says both 4 and −4 have an absolute value of 4. Is Tyler correct?
Use the number line and words to explain why or why not.
16. Lindsay and Will have online accounts for buying music. Lindsay's
account balance is −$20 and Will's account balance is −$15. Express
each account balance as a debt and explain whose debt is greater.
17. Explain how to graph points A(–3, 0), B (0, 0), and C (0, –3) on the
coordinate plane. Then, explain how to graph point D, so that ABCD
is a square.
© Houghton Mifflin Harcourt Publishing Company
18. Point A(2, –3) is reflected across the x-axis to point B. Point B is reflected
across the y-axis to point C. What are the coordinates of point C? Use
words and numbers to explain your answer.
206
Critical Area
Ratios and Rates
© Houghton Mifflin Harcourt Publishing Company • Image Credits: (bg) ©Tannen Maury/epa/Corbis
CRITICAL AREA Connecting ratio and rate to whole number
multiplication and division and using concepts of ratio and
rate to solve problems
The St. Louis Cardinals, based in
St. Louis, Missouri, were founded
in 1882.
207
Project
Meet Me in St. Louis
Baseball teams, like the St. Louis Cardinals, record information
about each player on the team. These statistics are used to
describe a player’s performance.
Get Started
WRITE
Math
Important Facts
A batting average is calculated from the ratio of a
player’s hits to the number of at bats. Batting averages
are usually recorded as a decimal to the thousandths
place. The table shows the batting results of three
baseball players who received the Most Valuable Player
award while playing for the St. Louis Cardinals. Write
each batting ratio as a fraction. Then write the fraction
as a decimal to the thousandths place and as a percent.
Player Name
Batting Results
Albert Pujols (2008)
187 hits in 524 at
bats
Stan Musial (1948)
230 hits in 611 at
bats
Rogers Hornsby (1925)
203 hits in 504 at
bats
ALBERT PUJOLS
208
Chapters 4–6
Completed by
© Houghton Mifflin Harcourt Publishing Company • Image Credits: (b) ©Jonathan Daniel/Getty Images
The players on a baseball team take their turns batting in the same order or sequence
throughout a game. The manager sets the batting order. Suppose you are the manager
of a team that includes Pujols, Musial, and Hornsby. What batting order would you use
for those three players? Explain your answer.
4
Chapter
Ratios and Rates
Show Wha t You Know
Personal Math Trainer
Online Assessment
and Intervention
Check your understanding of important skills.
Name
Multiply or Divide to Find Equivalent Fractions Multiply or divide to
find two equivalent fractions for the given fraction.
2. 5_
6
1. _1
2
___
12
3. __
18
___
___
Extend Patterns Write a description of the pattern. Then find the
missing numbers.
4. 3, __ , 48, 192, 768, __
5. 625, 575, 525, __ , __ , 375
__
__
Multiply by 2-Digit Numbers Find the product.
© Houghton Mifflin Harcourt Publishing Company • Image Credits: (br) ©Image Source/Corbis
6.
52
×
19
_
7.
14
×
88
_
8.
37
×
21
_
9.
45
×
62
_
Math in the
The student council should have 1 representative for every
25 students. Which of these situations fits the description?
Explain your answer.
a. 5 representatives for 100 students
b. 10 representatives for 250 students
c. 15 representatives for 300 students
Chapter 4
209
Voca bula ry Builder
Visualize It
Complete the bubble map with review words that are related to
Review Words
coordinate plane
fractions.
denominator
✓ equivalent fractions
numerator
✓ ordered pair
pattern
simplify
Fractions
x-coordinate
y-coordinate
Preview Words
✓ equivalent ratios
✓ rate
Understand Vocabulary
Complete the sentences using the checked words.
✓ ratio
✓ unit rate
1. A comparison of one number to another by division is a
___________ .
2.
___________ are ratios that name the same comparison.
3.
___________ are fractions that name the same
amount or part.
4. A ratio that compares quantities with different units is a
___________ .
© Houghton Mifflin Harcourt Publishing Company
5. A ___________ is a rate that compares a quantity
to 1 unit.
6. In an ___________ the first number is the x-coordinate
and the second number is the y-coordinate.
210
™Interactive Student Edition
™Multimedia eGlossary
Chapter 4 Vocabulary
denominator
equivalent ratios
denominador
razones equivalentes
30
20
numerator
rate
numerador
tasa
86
66
ratio
unit rate
razón
tasa por unidad
103
87
x-coordinate
y-coordinate
coordenada x
coordenada y
109
111
© Houghton Mifflin Harcourt Publishing Company
4∙1
↓
5∙1
↓
Boxes
1
2
3
4
5
Cards
6
12
18
24
30
↑
↑
↑
↑
2∙6
3∙6
4∙6
5∙6
Example:
An airplane climbs 18,000 feet in 12 minutes.
The rate of climb is 18,000 ft to 12 min,
18,000 ft
________
, or 18,000 ft : 12 min
12 min
A rate expressed so that the second term in
the ratio is one unit
Example: $9 per hour is a unit rate.
The second number in an ordered pair; tells
the distance to move up or down from (0,0)
y
5
4
3
2
1
-5 -4 -3 -2 -1 0
-1
-2
-3
-4
-5
(3, 2)
x
1 2 3 4 5
© Houghton Mifflin Harcourt Publishing Company
3∙1
↓
© Houghton Mifflin Harcourt Publishing Company
2∙1
↓
© Houghton Mifflin Harcourt Publishing Company
Original
ratio
© Houghton Mifflin Harcourt Publishing Company
© Houghton Mifflin Harcourt Publishing Company
© Houghton Mifflin Harcourt Publishing Company
A ratio that compares two quantities having
different units of measure
© Houghton Mifflin Harcourt Publishing Company
Ratios that name the same comparison
The number below the bar in a fraction that
tells how many equal parts are in the whole
or in the group
3
__
4
denominator
The number above the bar in a fraction
that tells how many equal parts of the
whole are being considered
3
__
4
numerator
A comparison of two numbers, a and b,
a
that can be written as a fraction __
b
The ratio of blue squares to red squares
5
is 5 to 1, __
, or 5 : 1.
1
The first number in an ordered pair; tells the
distance to move right or left from (0,0)
y
5
4
3
2
1
-5 -4 -3 -2 -1 0
-1
-2
-3
-4
-5
(3, 2)
x
1 2 3 4 5
Going Places with
Words
Going to the
Baseball Hall
of Fame
Game
Game
Word Box
denominator
equivalent ratios
numerator
rate
ratio
unit rate
x-coordinate
y-coordinate
For 2 players
Materials
• 1 each: red and blue connecting cubes
• 1 number cube
• Clue Cards
How to Play
© Houghton Mifflin Harcourt Publishing Company
Image Credits: ©Emanuele Taroni/PhotoDisc/Getty Images
1. Choose a connecting cube and put it on START.
2. Toss the number cube to take a turn. Move your connecting cube that
many spaces.
3. If you land on these spaces:
Blue Space Follow the directions printed in the space.
Red Space Take a Clue Card from the pile. If you answer the
question correctly, keep the card. If you do not, return the card to the
bottom of the pile.
4. Collect at least 5 Clue Cards. Move around the track as often as necessary.
5. When you have 5 Clue Cards, you follow the closest center path to
reach FINISH.
6. The first player to reach FINISH wins.
Chapter 4
210A
Game
Game
TAKE A
CLUE CARD
W
to in a
Fa the tick
m
e
H
Mo e W all t
v
e a eeke of
h
ea nd.
d
1.
A
KE
D
TA
AR
EC
CLU
Get a
behind-the-scene
beh
tour of the museum.
Go back 1.
210B
© Houghton Mifflin Harcourt Publishing Company
Game
Game
RD
See your
favorite
player’s
baseball cap.
Move ahead 1.
TA
CLU KE A
E
CA
Have a sleepover
H
in
i the museum
with
w
your family.
Lose a turn.
EA
K
TA
RD
A
EC
U
CL
© Houghton Mifflin Harcourt Publishing Company
Image Credits: (bg) ©Kazuhiro Tanda/Digital Vision/Getty Images; (glove) ©D. Hurst/Alamy; (umpire),
(base) ©Comstock/Getty Images; (baseball) ©Jules Frazier/PhotoDisc/Getty Images
Chapter 4
210C
Journal
Jo
ou
urnal
The Write Way
Reflect
Choose one idea. Write about it.
• Sofia made a beaded necklace with 2 white beads for every 5 purple
beads. In all, she used 30 purple beads. Explain how to figure out the
total number of white beads she used.
• Write and solve a word problem that includes a rate.
• Tell about a time when you needed to know the unit rate of something.
© Houghton Mifflin Harcourt Publishing Company Image Credits: (bg) ©Emanuele Taroni/PhotoDisc/Getty Images; (t) ©D. Hurst/Alamy
• Are 3:7 and 6:14 equivalent ratios? Tell how you know.
210D
Lesson 4.1
Name
Model Ratios
Essential Question How can you model ratios?
The drawing shows 5 blue squares and 1 red square. You can
compare the number of blue squares to the number of red squares
by using a ratio. A ratio is a comparison of two quantities by division.
Ratios and Proportional
Relationships—6.RP.A.1 Also
6.RP.A.3a
MATHEMATICAL PRACTICES
MP5, MP7, MP8
The ratio that compares blue squares to red squares is 5 to 1.
The ratio 5 to 1 can also be written as 5:1.
Hands
On
Investigate
Investigate
Materials ■ two-color counters
Julie makes 3 bracelets for every 1 bracelet Beth makes.
Use ratios to compare the number of bracelets Julie
makes to the number Beth makes.
A. Use red and yellow counters to model the ratio
that compares the number of bracelets Julie makes
to the number of bracelets Beth makes.
Think: Julie makes _ bracelets
when Beth makes 1 bracelet.
The ratio is _:1.
B. Model the ratio that shows the number of bracelets Julie
© Houghton Mifflin Harcourt Publishing Company
makes when Beth makes 2 bracelets. Write the ratio and
explain how you modeled it.
C. How could you change the model from Part B to show
the number of bracelets Julie makes when Beth makes
3 bracelets? Write the ratio.
Math
Talk
MATHEMATICAL PRACTICES 7
Look for a Pattern For each
ratio, divide the number of
bracelets Julie makes by the
number of bracelets Beth
makes. Describe a pattern
you notice in the quotients.
Chapter 4
211
Draw Conclusions
1. Explain how you used counters to compare the number of
bracelets Julie makes to the number of bracelets Beth makes.
MATHEMATICAL
PRACTICE
Generalize Describe a rule that you can use to find the
number of bracelets Julie makes when you know the number of bracelets
Beth makes.
2.
3.
8
SMARTER
How can you use counters to find how many bracelets
Beth makes if you know the number Julie makes? Explain and give an example.
Make
Make Connections
Connections
You can use a table to compare quantities and write ratios.
A bakery uses 1 packing box for every 4 muffins. Draw a model
and make a table to show the ratio of boxes to muffins.
STEP 1 Draw a model to show the ratio that
compares boxes to muffins.
Think: There is _ box for every _ muffins.
The ratio is _ : _.
Think: Each time the number of boxes increases by 1,
the number of muffins increases by _.
What is the ratio of boxes to muffins when there are 2 boxes? _
+1
Number of
Boxes
1
Number of
Muffins
4
Math
Talk
Write another ratio shown by the table. Explain what the ratio represents.
212
2
3
4
+
MATHEMATICAL PRACTICES 5
Use Tools Describe the pattern
you see in the table comparing
the number of boxes to the
number of muffins.
© Houghton Mifflin Harcourt Publishing Company
STEP 2 Complete the table to show the ratio
of boxes to muffins.
Name
Share
hhow
Share and
and Show
Sh
MATH
M
BOARD
B
Write the ratio of yellow counters to red counters.
2.
1.
_:_
______
Draw a model of the ratio.
4. 1:5
3. 3:2
Use the ratio to complete the table.
6. On the sixth-grade field trip, there are
8 students for every 1 adult. Complete
the table to show the ratio of students
to adults.
© Houghton Mifflin Harcourt Publishing Company
5. Wen is arranging flowers in vases. For
every 1 rose she uses, she uses 6 tulips.
Complete the table to show the ratio of
roses to tulips.
7.
Roses
1
Tulips
6
2
3
4
Students
8
Adults
1
24
2
4
SMARTER
Zena adds 4 cups of flour for every 3 cups of sugar in her
recipe. Draw a model that compares cups of flour to cups of sugar.
Chapter 4 • Lesson 1 213
Draw Conclusions
The reading skill draw conclusions can help you analyze and
make sense of information.
Hikers take trail mix as a snack on long hikes because it is tasty,
nutritious, and easy to carry. There are many different recipes
for trail mix, but it is usually made from different combinations
of dried fruit, raisins, seeds, and nuts. Tanner and his dad make
trail mix that has 1 cup of raisins for
every 3 cups of sunflower seeds.
8.
MATHEMATICAL
PRACTICE
4 Model Mathematics Explain how you could model the
ratio that compares cups of raisins to cups of sunflower seeds when
Tanner uses 2 cups of raisins.
The table shows the ratio of cups of raisins to
cups of sunflower seeds for different amounts
of trail mix. Model each ratio as you complete
the table.
SMARTER
10.
MATHEMATICAL
PRACTICE
11.
DEEPER
214
Raisins (cups)
1
Sunflower Seeds
(cups)
3
2
3
4
5
Describe the pattern you see in the table.
8 Draw Conclusions What conclusion can Tanner draw from this pattern?
What is the ratio of cups of sunflower seeds to cups of trail mix when
Tanner uses 4 cups of raisins?
© Houghton Mifflin Harcourt Publishing Company
9.
Trail Mix
Practice and Homework
Name
Lesson 4.1
Model Ratios
COMMON CORE STANDARD—6.RP.A.1
Understand ratio concepts and use ratio
reasoning to solve problems.
Write the ratio of gray counters to white counters.
1.
2.
3.
gray:white 3:4
Draw a model of the ratio.
4. 5:1
5. 6:3
Use the ratio to complete the table.
6. Marc is assembling gift bags. For every 2 pencils
he places in the bag, he uses 3 stickers. Complete
the table to show the ratio of pencils to stickers.
Pencils
2
Stickers
3
4
6
8
7. Singh is making a bracelet. She uses 5 blue beads
for every 1 silver bead. Complete the table to
show the ratio of blue beads to silver beads.
Blue
5
Silver
1
10
20
3
Problem
Problem Solving
Solving
© Houghton Mifflin Harcourt Publishing Company
8. There are 4 quarts in 1 gallon. How many quarts
are in 3 gallons?
10.
WRITE
9. Martin mixes 1 cup lemonade with 4 cups
cranberry juice to make his favorite drink.
How much cranberry juice does he need if
he uses 5 cups of lemonade?
Math Suppose there was 1 centerpiece
for every 5 tables. Use counters to show the ratio of
centerpieces to tables. Then make a table to find the
number of tables if there are 3 centerpieces.
Chapter 4
215
Lesson Check (6.RP.A.1)
1. Francine is making a necklace that has 1 blue
bead for every 6 white beads. How many white
beads will she use if she uses 11 blue beads?
2. A basketball league assigns 8 players to each
team. How many players can sign up for the
league if there are 24 teams?
Spiral Review (6.NS.B.4, 6.NS.C.5, 6.NS.C.6a, 6.NS.C.7d, 6.NS.C.8)
5. Elisa made 0.44 of the free throws she attempted.
What is that amount written as a fraction in
simplest form?
4. Of the 24 students in Greg’s class, 3_ ride the bus
8
to school. How many students ride the bus?
6.
On a coordinate plane, the vertices of a
rectangle are ( –1, 1), (3, 1), ( –1, –4), and (3, –4).
What is the perimeter of the rectangle?
FOR MORE PRACTICE
GO TO THE
216
Personal Math Trainer
© Houghton Mifflin Harcourt Publishing Company
3. Louis has 45 pencils and 75 pens to divide into
gift bags at the fair. He does not want to mix the
pens and pencils. He wants to place an equal
amount in each bag. What is the greatest number
of pens or pencils he can place in each bag?
Lesson 4.2
Name
Ratios and Rates
Ratios and Proportional
Relationships—6.RP.A.1 Also
6.RP.A.2
MATHEMATICAL PRACTICES
MP1, MP2, MP5
Essential Question How do you write ratios and rates?
Unlock
Unlock the
the Problem
Problem
A bird rescue group is caring for 3 eagles, 2 hawks, and
5 owls in their rescue center.
You can compare the numbers of different types of
birds using ratios. There are three ways to write the
ratio of owls to eagles in the rescue center.
Using words
As a fraction
With a colon
5 to 3
5_
3
5:3
Ratios can be written to compare a part to a part,
a part to a whole, or a whole to a part.
Write each ratio using words, as a fraction,
and with a colon.
A Owls to hawks
© Houghton Mifflin Harcourt Publishing Company • Image Credits: (c) ©Photodisc/Getty Images
____
_ to _
_:_
Part to part
B Eagles to total birds in the rescue center
____
_ to _
_ :_
Part to whole
C Total birds in the rescue center to hawks
____
_ to _
Whole to part
_:_
1. The ratio of owls to total number of birds is 5:10. Explain what this ratio means.
Chapter 4
217
Example A restaurant sells veggie burgers at the rate of $4
for 1 burger. What rate gives the cost of 5 veggie burgers? Write the rate for
5 burgers using words, as a fraction, and with a colon.
A rate is a ratio that compares two quantities that have different units
of measure.
A rate can be written as a ratio,
a:b, where b ≠ 0.
A unit rate is a rate that makes a comparison to 1 unit. The unit rate
$4
.
for cost per veggie burger is $4 to 1 burger or ______
1 burger
Complete the table to find the rate that gives the cost of 5 veggie burgers.
Think: 1 veggie burger costs $4, so 2 veggie burgers cost $4 + _ , or 2 × _.
Unit
Rate
2 • $4
3 • $4
↓
↓
↓
Cost
$4
$8
Veggie Burgers
1
2
3
4
↑
↑
↑
2 •1
•1
• $4
• $4
↓
↑
4 •1
•1
The table shows that 5 veggie burgers cost _.
Math
Talk
So, the rate that gives the cost for 5 veggie burgers is
$
$_ to _ burgers, ___________ , or $_ : _ burgers.
burgers
MATHEMATICAL PRACTICES 2
Reasoning Describe two
other ways to say “$4 per
burger”.
Try This! Write the rate in three different ways.
A The rate that gives the cost of 3 veggie
B The rate that gives the cost of 4 veggie
$4
2. Explain why the ratio ______
is a unit rate.
1 burger
3.
218
MATHEMATICAL
PRACTICE
5 Use Patterns Explain the pattern you see in the table in the Example.
© Houghton Mifflin Harcourt Publishing Company
burgers
burgers
Name
Share
hhow
Share and
and Show
Sh
MATH
M
BOARD
B
1. Write the ratio of the number of red bars to
blue stars.
_____
Write the ratio in two different ways.
4
3. __
2. 8 to 16
4. 1:3
24
6. Marilyn saves $15 per week. Complete the
table to find the rate that gives the amount
saved in 4 weeks. Write the rate in three
different ways.
5. 7 to 9
Savings
Weeks
$30
$45
2
3
1
Math
Talk
$75
4
5
MATHEMATICAL PRACTICES 6
Explain Are the ratios 5:2 and
2:5 the same or different?
How do you know?
On
On Your
Your Own
Own
Write the ratio in two different ways.
__
7. 16
8. 8:12
40
© Houghton Mifflin Harcourt Publishing Company
11. There are 24 baseball cards in 4 packs. Complete
the table to find the rate that gives the number of
cards in 2 packs. Write this rate in three different ways.
12.
MATHEMATICAL
PRACTICE
9. 4 to 11
10. 2:13
Cards
Packs
1
2
18
24
3
4
_ cup per serving”
6 Make Connections Explain how the statement “There is 3
represents a rate.
4
Chapter 4 • Lesson 2 219
MATHEMATICAL PRACTICES .0%&-t3&"40/tM",&4&/4&
OqnakdlRnkuhmf¤@ookhb`shnmr
OqnakdlRnkuhmf¤@ookhb`shnmr
Use the diagram of a birdhouse for 13–15.
12 in.
13. Write the ratio of AB to BC in three different ways.
B
C
28 in.
24 in.
14.
15.
DEEPER
Write the ratio of the shortest side length of triangle
ABC to the perimeter of the triangle in three different ways.
MATHEMATICAL
PRACTICE
Represent a Problem Write the ratio of the
perimeter of triangle ABC to the longest side length of the
triangle in three different ways.
A
2
WRITE
Math t Show Your Work
16. Leandra places 6 photos on each page in a photo album. Find
the rate that gives the number of photos on 2 pages. Write the
rate in three different ways.
18.
19.
220
What’s the Question? The ratio of total
students in Ms. Murray’s class to students in the class who
have an older brother is 3 to 1. The answer is 1:2. What is
the question?
SMARTER
WRITE
Math What do all unit rates have in common?
SMARTER
Julia has 2 green reusable shopping bags
and 5 purple reusable shopping bags. Select the ratios that
compare the number of purple reusable shopping bags to the
total number of reusable shopping bags. Mark all that apply.
5 to 7
5:2
2 to 7
5:7
2
_
5
5
_
7
© Houghton Mifflin Harcourt Publishing Company
17.
Practice and Homework
Name
Lesson 4.2
Ratios and Rates
COMMON CORE STANDARD—6.RP.A.1
Understand ratio concepts and use ratio
reasoning to solve problems.
Write the ratio in two different ways.
1. 4_5
2. 16 to 3
__
4. 15
3. 9:13
8
4 to 5
4:5
___
___
5. There are 20 light bulbs in 5 packages. Complete the
table to find the rate that gives the number of light
bulbs in 3 packages. Write this rate in three different
ways.
___
Light Bulbs
Packages
___
8
1
2
3
16
20
4
5
Problem
Problem Solving
Solving
© Houghton Mifflin Harcourt Publishing Company
6. Gemma spends 4 hours each week playing
soccer and 3 hours each week practicing her
clarinet. Write the ratio of hours spent practicing
clarinet to hours spent playing soccer three
different ways.
8.
WRITE
7. Randall bought 2 game controllers at Electronics
Plus for $36. What is the unit rate for a game
controller at Electronics Plus?
Math Explain how to determine if a given rate is also a
unit rate.
Chapter 4
221
Lesson Check (6.RP.A.1)
1. At the grocery store, Luis bought 10 bananas
and 4 apples. What are three different ways to
write the ratio of apples to bananas?
2. Rita checked out 7 books from the library. She
had 2 non-fiction books. The rest were fiction.
What are three different ways to write the ratio
of non-fiction to fiction?
Spiral Review (6.RP.A.1, 6.NS.B.4, 6.NS.C.6c)
5. Gina draws a map of her town on a coordinate
plane. The point that represents the town’s civic
center is 1 unit to the right of the origin and
4 units above it. What are the coordinates of the
point representing the civic center?
9 pound.
4. Pedro has a bag of flour that weighs __
10
He uses 2_ of the bag to make gravy. How many
3
pounds of flour does Pedro use to make gravy?
6.
Stefan draws these shapes. What is the ratio of
triangles to stars?
© Houghton Mifflin Harcourt Publishing Company
3. McKenzie bought 1.2 pounds of coffee for
$11.82. What was the cost per pound?
FOR MORE PRACTICE
GO TO THE
222
Personal Math Trainer
Lesson 4.3
Name
Equivalent Ratios and Multiplication Tables
Essential Question How can you use a multiplication table to find equivalent ratios?
The table below shows two rows from the multiplication table: the
row for 1 and the row for 6. The ratios shown in each column of the
table are equivalent to the original ratio. Ratios that name the same
comparison are equivalent ratios.
Ratios and Proportional
Relationships—6.RP.A.3a
MATHEMATICAL PRACTICES
MP1, MP4, MP6
1
2
3
4
5
1
1
2
3
4
5
10
Original
ratio
2∙1
↓
3∙1
↓
4∙1
↓
5∙1
↓
2
2
4
6
8
3
3
6
9
12 15
Bags
1
2
3
4
5
Apples
6
12
18
24
30
4
4
8
12 16 20
↑
2∙6
↑
3∙6
↑
4∙6
↑
5∙6
5
5
10 15 20 25
6
6
12 18 24 30
You can use a multiplication table to find equivalent ratios.
Unlock
Unlock the
the Problem
Problem
1
2
3
4
5
6
7
8
9
1
1
2
3
4
5
6
7
8
9
2
2
4
6
8
10 12 14 16 18
3
3
6
9
12 15 18 21 24 27
4
4
8
12 16 20 24 28 32 36
STEP 1 Shade the rows that show the original ratio.
5
5
10 15 20 25 30 35 40 45
Think: The original ratio is ____. Shade the row for _ and
6
6
12 18 24 30 36 42 48 54
7
7
14 21 28 35 42 49 56 63
8
8
16 24 32 40 48 56 64 72
9
9
18 27 36 45 54 63 72 81
The ratio of adults to students on a field trip is 3_8 .
Write two ratios that are equivalent to 3_8 .
Use the multiplication table.
the row for
_ on the multiplication table.
STEP 2 Circle the column that shows the original ratio.
Think: There is one group of 3 adults for every group of 8 students.
© Houghton Mifflin Harcourt Publishing Company
STEP 3 Circle two columns that show equivalent ratios.
The column for 2 shows there are 2 ∙ 3 , or
there are 2 ∙ 8, or
_ adults when
_ students.
The column for 3 shows there are 3 ∙ 3, or
there are 3 ∙ 8, or
_ adults when
_ students.
So, _ and _ are equivalent to 3_8 .
Math
Talk
MATHEMATICAL PRACTICES 6
Explain whether the
multiplication table shown
represents all of the ratios
that are equivalent to 3:8.
Chapter 4
223
connect You can find equivalent ratios by using a table or by multiplying or dividing by a form of one.
One Way
Use a table.
Jessa made fruit punch by mixing 2 pints of orange juice with
5 pints of pineapple juice. To make more punch, she needs to
mix orange juice and pineapple juice in the same ratio. Write three
equivalent ratios for 2_5 .
Think: Use rows from the multiplication table to help you complete
a table of equivalent ratios.
1
2
3
4
5
1
1
2
3
4
5
2
2
4
6
8
10
3
3
6
9
12 15
4
4
8
12 16 20
5
5
10 15 20 25
Original
ratio
Orange juice (pints)
2
Pineapple juice (pints)
5
2∙2
↓
3∙2
↓
∙2
↓
8
15
↑
2∙5
↑
∙5
↑
4∙5
So, 2_5 , _, _, and _ are equivalent ratios.
Another Way
Multiply or divide by a form of one.
Write two equivalent ratios for 6_8 .
A Multiply by a form of one.
Multiply the numerator and denominator by the same number.
6

_____
= ____
8
Be sure to multiply or
divide the numerator and
the denominator by the
same number.
B Divide by a form of one.
6÷
______
= ___
8÷
So, 6_8 , _, and _ are equivalent ratios.
•
224
MATHEMATICAL
PRACTICE
Compare Explain how ratios are similar to fractions.
Explain how they are different.
6
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Divide the numerator and denominator by the same number.
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MATH
M
BOARD
B
Write two equivalent ratios.
1. Use a multiplication table to write two ratios that are equivalent to 4_7 .
Find the rows that show 4_7 .
4_
=
=
7 _ _
Find columns that show equivalent ratios.
2.
3.
3
4.
5
7
2
2
5. 4_
10
__
6. 12
5
7. 2_
30
9
Math
Talk
MATHEMATICAL PRACTICES 5
Use Tools Explain how the
multiplication table helps you
find equivalent ratios.
On
On Your
Your Own
Own
Write two equivalent ratios.
8.
9.
9
8
6
4
11. 8_
9
12. 2_
7
© Houghton Mifflin Harcourt Publishing Company
10.
5
4
13. __
6
11
Determine whether the ratios are equivalent.
8
14. _2 and __
3
12
8
6
15. __
and __
10
10
16
4
16. __
and __
60
15
3
8
17. __
and __
14
28
Chapter 4 • Lesson 3 225
MATHEMATICAL PRACTICES
COMMUNICA5&t1&34E7&3&tCONSTRUCT ARGUMENTS
OqnakdlRnkuhmf¤@ookhb`shnmr
OqnakdlRnkuhmf¤@ookhb`shnmr
Use the multiplication table for 18 and 19.
18. In Keith’s baseball games this year, the ratio of times he has
4 . Write
gotten on base to the times he has been at bat is __
14
4
.
two ratios that are equivalent to __
14
20.
21.
22.
Pose a Problem Use the
multiplication table to write a new problem
involving equivalent ratios. Then solve the problem.
SMARTER
3
4
5
6
7
8
9
1
1
2
3
4
5
6
7
8
9
2
2
4
6
8
10 12 14 16 18
3
3
6
9
12 15 18 21 24 27
4
4
8
12 16 20 24 28 32 36
5
5
10 15 20 25 30 35 40 45
6
6
12 18 24 30 36 42 48 54
7
7
14 21 28 35 42 49 56 63
8
8
16 24 32 40 48 56 64 72
9
9
18 27 36 45 54 63 72 81
MATHEMATICAL
PRACTICE
1 Describe how to write an equivalent ratio
9
__
for 27 without using a multiplication table.
DEEPER
__ .
Write a ratio that is equivalent to 6_9 and 16
24
5 , or 3_ . Write
Determine whether each ratio is equivalent to 1_3 , __
10
5
the ratio in the correct box.
SMARTER
2
__
4
3
__
9
7
___
18
___
30
21
1
_
3
226
2
10
___
30
5
__
10
6
___
10
3
_
5
1
__
2
8
___
16
© Houghton Mifflin Harcourt Publishing Company
19.
1
Practice and Homework
Name
Lesson 4.3
Equivalent Ratios and Multiplication
Tables
COMMON CORE STANDARD—6.RP.A.3a
Understand ratio concepts and use ratio
reasoning to solve problems.
Write two equivalent ratios.
1. Use a multiplication table to write two ratios that are equivalent to 5_3 .
10, 15
5
_ = __
__
3
6
9
____
2.
6
3.
7
3
6
4. __
8
___
5. 11
1
8 and 32
___
8. __
3
12
9 and 3
__
9. ___
4
12
2
Determine whether the ratios are equivalent.
5 and 1__
5
2 and __
6. __
7. ___
3
6
10
6
Problem
Problem Solving
Solving
© Houghton Mifflin Harcourt Publishing Company
10. Tristan uses 7 stars and 9 diamonds to make
a design. Write two ratios that are equivalent to 7_9 .
12.
WRITE
11. There are 12 girls and 16 boys in Javier’s math
class. There are 26 girls and 14 boys in Javier’s
choir class. Is the ratio of girls to boys in the two
classes equivalent? Explain.
Math Explain how to determine whether two ratios are equivalent.
Chapter 4
227
Lesson Check (6.RP.A.3a)
1. A pancake recipe calls for 4 cups of flour and
3 cups milk. Does a recipe calling for 2 cups
flour and 1.5 cups milk use the same ratio of
flour to milk?
2. A bracelet is made of 14 red beads and 19 gold
beads. A necklace is made of 84 red beads and
133 gold beads. Do the two pieces of jewelry
have the same ratio of red beads to gold beads?
Spiral Review (6.RP.A.1, 6.NS.A.1, 6.NS.C.6b, 6.NS.C.6c)
4. Cole had 3__ hour of free time before dinner. He
2 of4the time playing the guitar. How long
spent __
3
did he play the guitar?
5. Delia has 3__5 yards of ribbon. About how many
1__-yard-long8 pieces can she cut?
4
6. Which point is located at –1.1?
-2
A B
C
D
-1
0
1
© Houghton Mifflin Harcourt Publishing Company
3. Scissors come in packages of 3. Glue sticks
come in packages of 10. Martha wants to buy
the same number of each. What is the fewest
glue sticks Martha can buy?
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PROBLEM SOLVING
Name
Lesson 4.4
Problem Solving •
Use Tables to Compare Ratios
Ratios and Proportional
Relationships—6.RP.A.3a
MATHEMATICAL PRACTICES
MP1, MP5, MP7
Essential Question How can you use the strategy find a pattern to
help you compare ratios?
Unlock
Unlock the
the Problem
Problem
A paint store makes rose-pink paint by mixing 3 parts red paint to
8 parts white paint. A clerk mixes 4 parts red paint to 7 parts white
paint. Did the clerk mix the paint correctly to make rose-pink paint?
Use tables of equivalent ratios to support your answer.
Use the graphic organizer to help you solve the problem.
Read the Problem
What do I need to find?
I need to find whether the
ratio used by the clerk is
__ to the ratio for
What information do I
need to use?
How will I use the
information?
I need to use the rose-pink
paint ratio and the ratio used
by the clerk.
I will make tables of equivalent
ratios to compare the ratios
_ to _ and _
rose-pink paint.
© Houghton Mifflin Harcourt Publishing Company • Image Credits: (t) ©Ron Chapple Stock/Alamy Images
to _.
Solve the Problem
Rose-Pink Paint
Parts Red
3
Parts White
8
6
Clerk’s Paint Mixture
9
12
Parts Red
4
Parts White
7
14
21
28
Look for a pattern to determine whether the ratios in the first
table are equivalent to the ratios in the second table.
Think: The number 12 appears in the first row of both tables.
12 is/is not equivalent to ______
12 .
______
The ratios have the same numerator and __ denominators.
So, the clerk __ mix the paint correctly.
Math
Talk
MATHEMATICAL PRACTICES 1
Evaluate How can you check
that your answer is correct?
Chapter 4
229
Try Another Problem
In Amy’s art class, the ratio of brushes to students is 6 to 4. In
Traci’s art class, the ratio of brushes to students is 9 to 6. Is the
ratio of brushes to students in Amy’s class equivalent to the ratio
of brushes to students in Traci’s class? Use tables of equivalent
ratios to support your answer.
Read the Problem
What do I need to find?
What information do I
need to use?
How will I use the
information?
So, the ratio of brushes to students in Amy’s class is/is not equivalent
to the ratio of brushes to students in Traci’s class.
1.
MATHEMATICAL
PRACTICE
Use Patterns Explain how you used a pattern to determine whether the
ratios in the two tables are equivalent.
5
2. Tell how writing the ratios in simplest form can help you justify your answer.
230
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Solve the Problem
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and Show
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MATH
M
BOARD
B
1. In Jawan’s school, 4 out of 10 students chose basketball
as a sport they like to watch, and 3 out of 5 students
chose football. Is the ratio of students who chose
basketball (4 to 10) equivalent to the ratio of students
who chose football (3 to 5)?
√
√
√
Circle the question.
Underline important facts.
Check to make sure you answered
the question.
First, make tables to show the ratios.
Basketball
Football
Next, compare the ratios in the tables. Find a ratio in the first
table that has the same numerator as a ratio in the second table.
12
12
______ __ equivalent to ______.
So, the ratios __ equivalent.
2.
3.
SMARTER
What if 20 out of 50 students chose baseball as a sport
they like to watch? Is this ratio equivalent to the ratio for either
basketball or football? Explain.
MATHEMATICAL
PRACTICE
7 Look for Structure The table shows the results of the
© Houghton Mifflin Harcourt Publishing Company
quizzes Hannah took in one week. Did Hannah get the same score on her
math and science quizzes? Explain.
Hannah’s Quiz Results
Subject
Questions Correct
Social Studies
4 out of 5
Math
8 out of 10
Science
3 out of 4
English
10 out of 12
4. Did Hannah get the same score on the quizzes in any of her classes?
Explain.
Chapter 4 • Lesson 4 231
MATHEMATICAL PRACTICES
ANALYZEt-00,'034536$563&t13&$*4*0/
On
On Your
Your Own
Own
5.
6.
DEEPER
For every $10 that Julie makes, she saves $3. For every
$15 Liam makes, he saves $6. Is Julie’s ratio of money saved to money
earned equivalent to Liam’s ratio of money saved to money earned?
SMARTER
A florist offers three
different bouquets of tulips and irises. The
list shows the ratios of tulips to irises in each
bouquet. Determine the bouquets that have
equivalent ratios.
Bouquet Ratios
Spring Mix
4 tulips to 6 irises
Morning Melody
9 tulips to 12 irises
Splash of Sun
7. The ratio of boys to girls in a school’s soccer club is 3 to 5.
The ratio of boys to girls in the school’s chess club is
13 to 15. Is the ratio of boys to girls in the soccer club
equivalent to the ratio of boys to girls in the chess club? Explain.
10 tulips to 15 irises
WRITE
Math
Show Your Work
8.
MATHEMATICAL
PRACTICE
1 Analyze Thad, Joey, and Mia ran in a race. The finishing
times were 4.56 minutes, 3.33 minutes, and 4.75 minutes. Thad did
not finish last. Mia had the fastest time. What was each runner’s time?
232
SMARTER
Fernando donates $2 to a local charity organization
for every $15 he earns. Cleo donates $4 for every $17 she earns. Is
Fernando’s ratio of money donated to money earned equivalent to
Cleo’s ratio of money donated to money earned? Explain.
© Houghton Mifflin Harcourt Publishing Company
9.
Practice and Homework
Name
Lesson 4.4
Lesson
4.4
Problem Solving • Use Tables to
Compare Ratios
COMMON CORE STANDARD—6.RP.A.3a
Understand ratio concepts and use ratio
reasoning to solve problems.
Read each problem and solve.
1. Sarah asked some friends about their favorite colors.
She found that 4 out of 6 people prefer blue, and 8 out
of 12 people prefer green. Is the ratio of friends who
chose blue to the total asked equivalent to the ratio of
friends who chose green to the total asked?
Blue
Friends who
chose blue
4
8
12
16
Total asked
6
12
18
24
Green
Friends who
chose green
8
16
24
32
Total asked
12
24
36
48
8.
4 is equivalent to __
Yes, _
6
12
_________
2. Lisa and Tim make necklaces. Lisa uses 5 red beads
for every 3 yellow beads. Tim uses 9 red beads for
every 6 yellow beads. Is the ratio of red beads to yellow
beads in Lisa’s necklace equivalent to the ratio in
Tim’s necklace?
_________
3. Mitch scored 4 out of 5 on a quiz. Demetri scored 8 out
of 10 on a quiz. Did Mitch and Demetri get equivalent
scores?
© Houghton Mifflin Harcourt Publishing Company
_________
4.
WRITE
Math Use tables to show which of these
6.
___, and ___
ratios are equivalent: 4__, 10
6 25
15
Chapter 4
233
Lesson Check (6.RP.A.3a)
1. Mrs. Sahd distributes pencils and paper to
students in the ratio of 2 pencils to 10 sheets of
2.
paper. Three of these ratios are equivalent to ___
10
Which one is NOT equivalent?
8
7
4
1__
___
___
___
5
15
20
40
2. Keith uses 18 cherries and 3 peaches to make
a pie filling. Lena uses an equivalent ratio
of cherries to peaches when she makes pie
filling. Can Lena use a ratio of 21 cherries to
6 peaches? Explain.
Spiral Review (6.RP.A.1, 6.NS.A.1, 6.NS.C.7a, 6.NS.C.8)
7
3 ÷ ___
?
3. What is the quotient ___
20 10
4. Which of these numbers is greater than –2.25
but less than –1?
1
5. Alicia plots a point at (0, 5) and (0, –2). What is
the distance between the points?
–
1.5
0
–
2.5
6. Morton sees these stickers at a craft store. What
is the ratio of clouds to suns?
y
5
4
3
2
1
x
© Houghton Mifflin Harcourt Publishing Company
-5 -4 -3 -2 -1 0
-1
-2
-3
-4
-5
1 2 3 4 5
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ALGEBRA
Name
Lesson 4.5
Use Equivalent Ratios
Essential Question How can you use tables to solve problems involving
equivalent ratios?
Ratios and Proportional
Relationships—6.RP.A.3a
MATHEMATICAL PRACTICES
MP4, MP8
Unlock
Unlock the
the Problem
Problem
In warm weather, the Anderson family likes to
spend time on the family’s boat. The boat uses 2
gallons of gas to travel 12 miles on the lake. How
much gas would the boat use to travel
48 miles?
• What are you asked to find?
Solve by finding equivalent ratios.
Let ■ represent the unknown number of gallons.
gallons
_______
miles
2 = ___
■ ← gallons
_______
→ ___
→ 12 48 ← miles
Make a table of equivalent ratios.
Original
ratio
Gas used (gallons)
2
Distance (miles)
12
2∙2
∙2
∙2
↓
↓
↓
6
24
↑
© Houghton Mifflin Harcourt Publishing Company • Image Credits: (c) ©Andersen Ross/Getty Images
∙ 12
48
↑
3 ∙ 12
↑
∙ 12
2
__
The ratios 12
and _ are equivalent ratios,
2
12
so ___ = ___.
48
So, the boat will use _ gallons of gas to travel
48 miles.
• What if the boat uses 14 gallons of gas? Explain how you can use
equivalent ratios to find the number of miles the boat travels when it
uses 14 gallons of gas.
Chapter 4
235
Example Use equivalent ratios to find the unknown value.
3 = ___
■
__
4
20
Use common denominators to write equivalent ratios.
A
3 = ___
■
__
4 20
_ is a multiple of 4, so _ is a common denominator.
___
Multiply the
and denominator by
write the ratios using a common denominator.
3×
■
_______
= ___
_ to
4×
___ are the same, so the ___
The
are equal to each other.
20
■
____ = ___
20
20
So, the unknown value is _ and 3_4 = ____
20 .
Check your answer by making a table of equivalent ratios.
Original ratio
∙3
∙3
↓
3
6
4
8
↑
∙3
↓
↓
↑
↑
↑
∙4
∙4
∙4
∙4
56 = __
8
___
42
■
56 = __
8
___
Write an equivalent ratio with 8 in the numerator.
42
Think: Divide 56 by _ to get 8.
56
÷
8
________
= __
■
42 ÷
8 = __
8
___
So, divide the denominator by _ as well.
___ are the same, so the ___
The
are equal to each other.
8 .
__ = ____
So, the unknown value is _ and 56
42
■
Math
Talk
Check your answer by making a table of equivalent ratios.
Original ratio
∙8
↓
8
16
6
12
↑
∙6
236
∙8
∙8
∙8
∙8
∙8
↓
↓
↓
↓
↓
↑
↑
↑
↑
↑
∙6
∙6
∙6
∙6
■
∙6
MATHEMATICAL PRACTICES 8
Use Repeated Reasoning
Give an example of two
equivalent ratios. Explain
how you know that they are
equivalent.
© Houghton Mifflin Harcourt Publishing Company
B
∙3
↓
Name
MATH
M
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and Show
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BOARD
B
Use equivalent ratios to find the unknown value.
■ =4
__
1. ____
18 6
2. ____ = __
24 ■
6
18
÷
__
____________
=
■
24 ÷
6
6
__
_____
10
5
■ =4
•
____
________
10 5 •
=■
■ = _____
____
10
10
So, the unknown value is _.
__
3. _36 = 15
■
__
So, the unknown value is _.
■ = __
8
4. __
10
5
■
5. 7_ = __
4
__
__ = 40
__
6. 10
■
12
__
12
__
Math
Talk
MATHEMATICAL PRACTICES 3
Apply Explain whether you
can always find an equivalent
ratio by subtracting the same
number from the numerator
and denominator. Give an
example to support your
answer.
On
On Your
Your Own
Own
Use equivalent ratios to find the unknown value.
■
7. _26 = __
30
© Houghton Mifflin Harcourt Publishing Company
__
5 = ___
55
8. __
■
110
9
__
11. Mavis walks 3 miles in 45 minutes. How many
minutes will it take Mavis to walk 9 miles?
13.
9
9. _3 = __
■
__
12.
■ = 16
__
10. __
6
24
__
DEEPER
The ratio of boys to girls in a choir is
3 to 8. There are 32 girls in the choir. How many
members are in the choir?
■ the
2 Use Reasoning Is the unknown value in _2 = __
3
18
__ ? Explain.
same as the unknown value in 3_ = 18
MATHEMATICAL
PRACTICE
2
■
Chapter 4 • Lesson 5 237
MATHEMATICAL PRACTICES .0%&-t3&"40/tM",&4&/4&
MATHEMATICAL PRACTICES
OqnakdlRnkuhmf¤@ookhb`shnmr
OqnakdlRnkuhmf¤@ookhb`shnmr
Solve by finding an equivalent ratio.
14. It takes 8 minutes for Sue to make 2 laps around the
go-kart track. How many laps can Sue complete in
24 minutes?
15.
DEEPER
The width of Jay’s original photo is
8 inches. The length of the original photo is
10 inches. He prints a smaller version that has an
equivalent ratio of width to length. The width of the
smaller version is 4 inches less than the width of the
original. What is the length of the smaller version?
16. Ariel bought 3 raffle tickets for $5. How many tickets could Ariel
buy for $15?
17.
What’s the Error? Greg used the steps shown to find
the unknown value. Describe his error and give the correct solution.
SMARTER
■
2_ = __
6 12
6+6
12
12
12
■
8 = __
__
The unknown value is 8.
18.
238
SMARTER
Courtney bought 3 maps for $10. Use the table of
equivalent ratios to find how many maps she can buy for $30.
3
6
10
20
30
© Houghton Mifflin Harcourt Publishing Company • Image Credits: (t) ©Darryl Leniuk/Corbis
■
2 + 6 = __
_____
Practice and Homework
Name
Lesson 4.5
Use Equivalent Ratios
COMMON CORE STANDARD—6.RP.A.3a
Understand ratio concepts and use ratio
reasoning to solve problems.
Use equivalent ratios to find the unknown value.
■
4 = ___
1. ___
10 40
434
______
10 3 4
3 = 33
___
2. ___
24 ■
7 = 21
___
3. __
■ 27
■ = ___
12
4. __
9 54
■
5 __
40
16 5 __
■
__
40
40
■ 5 16
__
■
4 = ___
6. __
5 40
3 = 12
___
5. __
2 ■
__
__
__
■ = ___
45
7. __
2 30
__
__
___ = 5
__
8. 45
■ 6
__
Problem
Problem Solving
Solving
© Houghton Mifflin Harcourt Publishing Company
9. Honeybees produce 7 pounds of honey for
every 1 pound of beeswax they produce. Use
equivalent ratios to find how many pounds of
honey are produced when 25 pounds of beeswax
are produced.
11.
WRITE
10. A 3-ounce serving of tuna provides
21 grams of protein. Use equivalent ratios
to find how many grams of protein are in
9 ounces of tuna.
Math Explain how using equivalent ratios is like adding
fractions with unlike denominators.
Chapter 4
239
Lesson Check (6.RP.A.3a)
1. Jaron paid $2.70 for 6 juice boxes. How much
should Jaron expect to pay for 18 juice boxes?
2. A certain shade of orange paint is made by
mixing 3 quarts of red paint with 2 quarts of
yellow paint. To make more paint of the same
shade, how many quarts of yellow paint should
be mixed with 6 quarts of red paint?
Spiral Review (6.RP.A.3a, 6.NS.A.1, 6.NS.C.7c, 6.NS.C.8)
4. What is the absolute value of –22__?
3
5. On a map, a clothing store is located at ( –2, –3).
A seafood restaurant is located 6 units to
the right of the clothing store. What are the
coordinates of the restaurant?
6. Marisol plans to make 9 mini-sandwiches for
every 2 people attending her party. Write a ratio
that is equivalent to Marisol’s ratio.
© Houghton Mifflin Harcourt Publishing Company
3. What is the quotient 2 __4 ÷ 11__?
5
3
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240
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Personal Math Trainer
Mid-Chapter Checkpoint
Online Assessment
and Intervention
Vocabulary
Vocabulary
Vocabulary
equivalent ratios
Choose the best term from the box to complete the sentence.
rate
1. A ___ is a rate that makes a comparison
to 1 unit. (p. 218)
ratio
unit rate
2. Two ratios that name the same comparison are
___. (p. 223)
Concepts
Concepts and
and Skills
Skills
3. Write the ratio of red circles to blue squares. (6.RP.A.1)
Write the ratio in two different ways. (6.RP.A.1)
4. 8 to 12
__
5. 7:2
__
6. 5_9
7. 11 to 3
__
__
Write two equivalent ratios. (6.RP.A.3a)
© Houghton Mifflin Harcourt Publishing Company
8. 2_7
9. 6_5
9
10. __
12
__
11. 18
6
Find the unknown value. (6.RP.A.3a)
5
15 = __
12. __
■ 10
__
■ = 12
__
13. __
3
9
__
48 ■
14. __
= __
16
8
__
9
3
15. __
= __
36
■
__
Chapter 4
241
16. There are 36 students in the chess club, 40 students in the drama club,
and 24 students in the film club. What is the ratio of students in the
drama club to students in the film club? (6.RP.A.3a)
17. A trail mix has 4 cups of raisins, 3 cups of dates, 6 cups of peanuts, and 2
cups of cashews. Which ingredients are in the same ratio as cashews to
raisins? (6.RP.A.3a)
18. There are 32 adults and 20 children at a school play. What is the ratio of
children to people at the school play? (6.RP.A.3a)
242
DEEPER
Sonya got 8 out of 10 questions right on a quiz. She got the
same score on a quiz that had 20 questions. How many questions did
Sonya get right on the second quiz? How many questions did she get
wrong on the second quiz? (6.RP.A.3a)
© Houghton Mifflin Harcourt Publishing Company
19.
Lesson 4.6
Name
Find Unit Rates
Ratios and Proportional
Relationships—6.RP.A.2 Also
6.RP.A.3b
MATHEMATICAL PRACTICES
MP2, MP3, MP6
Essential Question How can you use unit rates to make comparisons?
Unlock
Unlock the
the Problem
Problem
The star fruit, or carambola, is the fruit of a tree that
is native to Indonesia, India, and Sri Lanka. Slices
of the fruit are in the shape of a five-pointed star.
Lara paid $9.60 for 16 ounces of star fruit. Find the
price of 1 ounce of star fruit.
• Underline the sentence that tells you what you
are trying to find.
• Circle the numbers you need to use to solve
the problem.
Recall that a unit rate makes a comparison to
1 unit. You can find a unit rate by dividing the
numerator and denominator by the number in the
denominator.
Write the unit rate for the price of star fruit.
Write a ratio that compares
to
__
__.
Divide the numerator and denominator by
the number in the
___.
price
_____
weight
→
$
_______
oz
$9.60
÷
___________
16 oz ÷
$
_______
1 oz
© Houghton Mifflin Harcourt Publishing Company
So, the unit rate is __. The price is __
per ounce.
Math
Talk
1.
MATHEMATICAL
PRACTICE
6 Explain why the unit rate is equivalent to the original rate.
2.
MATHEMATICAL
PRACTICE
3
MATHEMATICAL PRACTICES 6
Explain What is the difference
between a ratio and a rate?
Make Arguments Explain a way to convince others that you
found the unit rate correctly.
Chapter 4
243
Example
A During migration, a hummingbird can fly 210 miles in 7 hours,
and a goose can fly 165 miles in 3 hours. Which bird flies at a
faster rate?
Write the rate for each bird.
miles
Hummingbird: _________
7 hours
165 miles
Goose: _________
hours
165 mi ÷
__________
210 mi ÷
__________
Write the unit rates.
3 hr ÷
7 hr ÷
mi
______
mi
______
1 hr
1 hr
Compare the unit rates.
_ miles per hour is faster than _ miles per hour.
So, the __ flies at a faster rate.
B A 64-ounce bottle of apple juice costs $5.76. A 15-ounce bottle of apple
juice costs $1.80. Which item costs less per ounce?
Write the rate for
each bottle.
Write the unit
rates.
64-ounce bottle: ________
64 ounces
15-ounce bottle: _________
ounces
÷
____________
$1.80 ÷
___________
oz ÷
64 oz ÷
_______
_______
1 oz
Compare the unit
rates.
1 oz
_ per ounce is less expensive than _ per ounce.
So, the _ -ounce bottle costs less per ounce.
Try This! At one grocery store, a dozen eggs cost $1.20. At another
Store 1:
Store 2:
The unit price is lower at Store _ , so a dozen eggs for __ is the
better buy.
244
© Houghton Mifflin Harcourt Publishing Company
store, 1 1_2 dozen eggs cost $2.16. Which is the better buy?
Name
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MATH
M
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B
Write the rate as a fraction. Then find the unit rate.
1. Sara drove 72 miles on 4 gallons of gas.
2. Dean paid $27.00 for 4 movie tickets.
÷
______ = __________
= ______
4 gal
4 gal ÷
1 gal
3. Amy and Mai have to read Bud, Not Buddy
for a class. Amy reads 20 pages in 2 days.
Mai reads 35 pages in 3 days. Who reads at a
faster rate?
4. An online music store offers 5 downloads for
$6.25. Another online music store offers
12 downloads for $17.40. Which store offers
the better deal?
Math
Talk
MATHEMATICAL PRACTICES 8
Generalize Explain how to
find a unit rate.
On
On Your
Your Own
Own
Write the rate as a fraction. Then find the unit rate.
5. A company packed 108 items in 12 boxes.
© Houghton Mifflin Harcourt Publishing Company
7.
DEEPER
Geoff charges $27 for 3 hours of
swimming lessons. Anne charges $31 for
4 hours. How much more does Geoff charge
per hour than Anne?
6. There are 112 students for 14 teachers.
8.
MATHEMATICAL
PRACTICE
6
Compare One florist made
16 bouquets in 5 hours. A second florist made
40 bouquets in 12 hours. Which florist makes
bouquets at a faster rate?
Tell which rate is faster by comparing unit rates.
mi and 210
mi
______
______
9. 160
2 hr
3 hr
270 ft and _____
180 ft
10. _____
9 min
9 min
m and 120
m
_____
_____
11. 250
10 s
4s
Chapter 4 • Lesson 6 245
MATHEMATICAL PRACTICES
COMMUNICA5&t1&34E7&3&tCONSTRUCT ARGUMENTS
Unlock
Unlock the
the Problem
Problem
12.
SMARTER
Ryan wants
to buy treats for his puppy.
If Ryan wants to buy the treats
that cost the least per pack,
which treat should he buy?
Explain.
Cost of Dog Treats
Name
a. What do you need to find?
b. Find the price per pack for each treat.
Cost
Number of Packs
Pup Bites
$5.76
4
Doggie Treats
$7.38
6
Pupster Snacks
$7.86
6
Nutri-Biscuits
$9.44
8
c. Complete the sentences.
The treat with the highest price per pack is
.
The treat with the lowest price per pack is
.
because _____
.
13.
MATHEMATICAL
PRACTICE
2
Reason Abstractly What
information do you need to consider in order to
decide whether one product is a better deal than
another? When might the lower unit rate not
be the best choice? Explain.
14.
SMARTER
Select the cars that get a
higher mileage per gallon of gas than a car
that gets 25 miles per gallon. Mark all that
apply.
Car A: 22 miles per 1 gallon
Car B: 56 miles per 2 gallons
Car C: 81 miles per 3 gallons
Car D: 51 miles per 3 gallons
246
© Houghton Mifflin Harcourt Publishing Company • Image Credits: (t) ©GK Hart/Vikki Hart/PhotoDisc/Getty Images
Ryan should
buy ____
WRITE
M t Show Your Work
Math
Practice and Homework
Name
Lesson 4.6
Find Unit Rates
COMMON CORE STANDARD—6.RP.A.2
Understand ratio concepts and use ratio
reasoning to solve problems.
Write the rate as a fraction. Then find the unit rate.
1. A wheel rotates through 1,800º in
5 revolutions.
2. There are 312 cards in 6 decks of
playing cards.
1,800º
______
5 revolutions
1,800º 4 5
________
360º
5 ______
5 revolutions 4 5
1 revolution
3. Bana ran 18.6 miles of a marathon in 3 hours.
4. Cameron paid $30.16 for 8 pounds
of almonds.
Compare unit rates.
5. An online game company offers a package that
includes 2 games for $11.98. They also offer a
package that includes 5 games for $24.95. Which
package is a better deal?
6. At a track meet, Samma finished the
200-meter race in 25.98 seconds. Tom finished
the 100-meter race in 12.54 seconds. Which
runner ran at a faster average rate?
© Houghton Mifflin Harcourt Publishing Company
Problem
Problem Solving
Solving
7. Sylvio’s flight is scheduled to travel 1,792 miles
in 3.5 hours. At what average rate will the plane
have to travel to complete the trip on time?
9.
WRITE
8. Rachel bought 2 pounds of apples and
3 pounds of peaches for a total of $10.45.
The apples and peaches cost the same amount
per pound. What was the unit rate?
Math Write a word problem that involves comparing
unit rates.
Chapter 4
247
TEST
PREP
Lesson Check (6.RP.A.2, 6.RP.A.3b)
1. Cran–Soy trail mix costs $2.99 for 5 ounces,
Raisin–Nuts mix costs $3.41 for 7 ounces,
Lots of Cashews mix costs $7.04 for 8 ounces,
and Nuts for You mix costs $2.40 for 6 ounces.
List the trail mix brands in order from the least
expensive to the most expensive.
2. Aaron’s heart beats 166 times in 120 seconds.
Callie’s heart beats 88 times in 60 seconds.
Emma’s heart beats 48 times in 30 seconds.
Galen’s heart beats 22 times in 15 seconds.
Which two students’ heart rates are equivalent?
Spiral Review (6.RP.A.1, 6.RP.A.3a, 6.NS.A.1, 6.NS.7d)
3. Courtlynn combines __7 cup sour cream with __1
8
2
cup cream cheese. She then divides the mixture
between 2 bowls. How much mixture does
Courtlynn put in each bowl?
4. Write a comparison using < or > to show the
−2
–5
__.
relationship between __
and
3
6
5. There are 18 tires on one truck. How many tires
are on 3 trucks of the same type?
6. Write two ratios that are equivalent to __5.
6
FOR MORE PRACTICE
GO TO THE
248
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| |
ALGEBRA
Name
Lesson 4.7
Use Unit Rates
Essential Question How can you solve problems using unit rates?
Ratios and Proportional
Relationships—6.RP.A.3b
MATHEMATICAL PRACTICES
MP1, MP3, MP5
Unlock
Unlock the
the Problem
Problem
The Champie family is traveling from Arizona to
Texas. On the first part of the trip, they drove 500
miles in 10 hours. If they continue driving at the
same rate, how many hours will it take them to
drive 750 miles?
You can use equivalent ratios to find the number of
hours it will take the Champie family to drive 750
miles. You may need to find a unit rate before you
can write equivalent ratios.
Find equivalent ratios by using a unit rate.
Write ratios that compare miles to hours.
miles
_____
750 is not a multiple of 500.
hours
Write the known ratio as a unit rate.
→
→
500
____ = 750
______
■
10
500 ÷
__________
10 ÷ 10
←
←
miles
____
hours
____
= 750
■
______
____ = 750
Write an equivalent rate by multiplying the
___ and ___ by the
1
■
50 •
______
____
= 750
■
1•
same value.
Think: Multiply 50 by _ to get 750.
© Houghton Mifflin Harcourt Publishing Company
So, multiply the denominator by
_ also.
The ___ are the same, so the
___ are equal to each other.
______
_____ = 750
15
■
The unknown value is _.
So, it will take the family _ hours to drive 750 miles.
Math
Talk
MATHEMATICAL PRACTICES 1
Analyze Why did you need
to find a unit rate first?
Chapter 4
249
Example
Kenyon earns $105 for mowing 3 lawns. How much
would Kenyon earn for mowing 10 lawns?
STEP 1 Draw a bar model to represent the situation:
$105
$?
STEP 2 Solve the problem.
The model shows that 3 units represent $105.
You need to find the value represented by _ units.
Write a unit rate:
_.
1 unit represents $
$105
$105 ÷
_____
_____ = ___________
=$
1
3
3÷
10 units are equal to 10 times 1 unit,
so 10 units 5 10 3 $
_.
_ = $__
10 × $
So, Kenyon will earn $__ for mowing 10 lawns.
Try This!
Last summer, Kenyon earned $210 for mowing 7 lawns. How much did
he earn for mowing 5 lawns last summer?
STEP 2 Solve the problem.
250
© Houghton Mifflin Harcourt Publishing Company
STEP 1 Draw a bar model to represent the situation.
Name
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MATH
M
BOARD
B
Use a unit rate to find the unknown value.
1.
10
___ = 6
__
■ 3
6÷
10
___ = _____
■ 3÷3
10
___ = ___
■
1
2•
10
___ = ______
■
1•
10 = ___
10
___
■
6 = ___
__
8 20
2.
■
6
÷
______
= ___
20
8÷8
■
_____ = ___
20
1
■
0.75 • 20 = ___
_________
20
1•
■
____ = ___
■
20
■=_
20
■=_
On
On Your
Your Own
Own
Use a unit rate to find the unknown value.
40 = 45
__
3. __
8
■
42 = ■
__
4. __
14
5
__ = 56
__
5. ■
2
8
__ = 26
__
6. ■
4
13
Practice: Copy and Solve Draw a bar model to find the unknown value.
9
4 = __
7. __
32
11.
■
MATHEMATICAL
PRACTICE
__
8. _9 = ■
3
4
■ = 9_
9. __
14
7
3 = ____
2
10. __
■
1.25
5 Communicate Explain how to find an unknown value
© Houghton Mifflin Harcourt Publishing Company
in a ratio by using a unit rate.
12.
DEEPER
Savannah is tiling her kitchen floor. She bought 8 cases
of tile for $192. She realizes she bought too much tile and returns
2 unopened cases to the store. What was her final cost for tile?
Chapter 4 • Lesson 7 251
MATHEMATICAL PRACTICES
ANALYZEt-00,'034536$563&t13&$*4*0/
OqnakdlRnkuhmf¤@ookhb`shnmr
OqnakdlRnkuhmf¤@ookhb`shnmr
Pose a Problem
13.
Adult T-Shirt Sizes
SMARTER
Josie runs a T-shirt
printing company. The table shows the
length and width of four sizes of T-shirts.
The measurements of each size T-shirt form
equivalent ratios.
What is the length of an extra-large T-shirt?
Size
Length (inches)
Width (inches)
Small
27
18
Medium
30
20
Large
?
22
X-large
?
24
Write two equivalent ratios and find the unknown value:
Length of medium → __
■
30 = __
________________
Width of medium → 20
24
C
\
e
^
k
Width _
← Length
of X-large
______________
← Width of X-large
■ → ___
■ → 1.5
■ → __
■
30 4 20 = __
1.5 = __
? 24 = __
36 = __
_______
_______
20 4 20
24
1
24
1 ? 24
24
24
24
The length of an extra-large T-shirt is 36 inches.
Write a problem that can be solved by using the information in
the table and could be solved by using equivalent ratios.
Solve Your Problem
WRIT
WR
WRITE
ITE
IT
E
Math
M
ath t Sh
Show
how Y
You
Your
ourr Work
ou
Work
Wo
k
Personal Math Trainer
14.
SMARTER
Peri earned $27 for walking her neighbor’s
dog 3 times. If Peri earned $36, how many times did she walk her
neighbor’s dog? Use a unit rate to find the unknown value.
27
3
252
36
_______ = _______
© Houghton Mifflin Harcourt Publishing Company
Pose a Problem
Practice and Homework
Name
Lesson 4.7
Use Unit Rates
COMMON CORE STANDARD—6.RP.A.3b
Understand ratio concepts and use ratio
reasoning to solve problems.
Use a unit rate to find the unknown value.
34 = ■
__
1. ___
17 7
34
4 17 5 ■
_______
__
7
17 4 17
2
■
_ 5 __
7
1
2
3
7
_____ 5 ■
__
7
137
14
■
__ 5 __
7
7
■
___ = ___
2. 16
32 14
18 = 21
___
3. ___
■ 7
■ = ___
3
4. ___
16 12
■ 5 14
__
__
__
Draw a bar model to find the unknown value.
__
6. 3__ = ■
6 7
6
___ = __
5. 15
45 ■
__
__
© Houghton Mifflin Harcourt Publishing Company
Problem
Problem Solving
Solving
7. To stay properly hydrated, a person should drink
32 fluid ounces of water for every 60 minutes of
exercise. How much water should Damon drink
if he rides his bike for 135 minutes?
9.
WRITE
8. Lillianne made 6 out of every 10 baskets she
attempted during basketball practice. If she
attempted to make 25 baskets, how many did
she make?
Math Give some examples of real-life situations in which
you could use unit rates to solve an equivalent ratio problem.
Chapter 4
253
Lesson Check (6.RP.A.3b)
1. Randi’s school requires that there are 2 adult
chaperones for every 18 students when the
students go on a field trip to the museum. If
there are 99 students going to the museum, how
many adult chaperones are needed?
2. Landry’s neighbor pledged $5.00 for every
2 miles he swims in a charity swim-a-thon. If
Landry swims 3 miles, how much money will
his neighbor donate?
Spiral Review (6.RP.A.2, 6.RP.A.3a, 6.NS.C.6a, 6.NS.C.6c)
3. Describe a situation that could be represented
by –8.
4. What are the coordinates of point G?
y
2
1
G
-3
-2
-1
H
0
1
x
2
3
-1
5. Gina bought 6 containers of yogurt for $4. How
many containers of yogurt could Gina buy for $12?
6. A bottle containing 64 fluid ounces of juice
costs $3.84. What is the unit rate?
FOR MORE PRACTICE
GO TO THE
254
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-2
ALGEBRA
Name
Lesson 4.8
Equivalent Ratios and Graphs
Essential Question How can you use a graph to represent equivalent ratios?
Ratios and Proportional
Relationships—6.RP.A.3a
MATHEMATICAL PRACTICES
MP4, MP5, MP7
Unlock
Unlock the
the Problem
Problem
A car travels at a rate of 50 miles per hour. Use
equivalent ratios to graph the distance the car travels
over time. Graph time on the x-axis and distance on
the y-axis.
• What words in the problem tell the unit rate?
Write and graph equivalent ratios.
STEP 1 Use the unit rate to write equivalent ratios.
miles
1 hour
______________
Write the unit rate.
Complete the table of equivalent ratios.
Distance (mi)
150
Time (hr)
mi × 2
1 hr × 2
1
200
2
5
Write an equivalent ratio. _______________
mi
= ____________
hr
STEP 2 Use an ordered pair to represent each ratio
in the table.
Let the x-coordinate represent time in hours and
the y-coordinate represent distance in miles.
1 hr
(1, 50)
(2, _)
The first number in an ordered
pair is the x-coordinate, and
the second number is the
y-coordinate.
(_, 150)
(_, 200)
↓
50 mi
_____
(1, _)
(5, _)
Think: The graph represents the same relationship as
the unit rate.
For every 1 hour the car travels, the distance increases by
_ miles.
Math
Talk
MATHEMATICAL PRACTICES 7
Look for a Pattern Identify a
pattern in the graph.
Distance (mi)
© Houghton Mifflin Harcourt Publishing Company
STEP 3 Use the ordered pairs to graph the car’s
distance over time.
y
400
350
300
250
200
150
100
50
0
Car Travel
1 2 3 4 5 6 7 8 x
Time (hr)
Chapter 4
255
Example During a heavy rainstorm, the waters of the
Blue River rose at a steady rate for 8 hours. The graph shows the river’s
increase in height over time. Use the graph to complete the table of
equivalent ratios. How many inches did the river rise in 8 hours?
Increase in Blue River Height
Think: On the graph, x-coordinates represent
__, and y-coordinates represent
the river’s increase in height in
The ordered pair (1,
hour, the river rose
__.
Height (in.)
time in
_) means that after _
_ inches.
y
24
21
18
15
12
9
6
3
0
Increase in
height (in.)
3
Time (hr)
1
2
4
6
1 2 3 4 5 6 7 8 x
Time (hr)
8
So, the river rose _ inches in 8 hours.
1.
MATHEMATICAL
PRACTICE
Look for a Pattern Describe the pattern you see in the
graph and the table.
7
2. Explain how you know that the ratios in the table are equivalent.
256
MATHEMATICAL
PRACTICE
Use Appropriate Tools Matt earns $12 per hour.
Explain how you could use equivalent ratios to draw a graph of his
earnings over time.
5
© Houghton Mifflin Harcourt Publishing Company
3.
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MATH
M
BOARD
B
A redwood tree grew at a rate of 4 feet per year. Use this
information for 1–3.
1. Complete the table of equivalent ratios for the first
5 years.
3. Use the ordered pairs to graph
the tree’s growth over time.
Redwood Tree Growth
Height (ft)
1
2
2. Write ordered pairs, letting the x-coordinate represent time
in years and the y-coordinate represent height in feet.
(1, _ ), (2, _ ), ( _ , _ )
Height (ft)
Time (yr)
y
( _ , _ ), ( _ , _ )
Math
Talk
On
On Your
Your Own
Own
The graph shows the rate at which Luis’s car uses gas,
in miles per gallon. Use the graph for 4–8.
4. Complete the table of equivalent ratios.
30
Gas (gal)
1
Use Graphs Explain what
the point (1, 4) represents on
the graph of the redwood
tree's growth.
Gas Usage in Luis’s Car
2
3
4
5
miles
5. Find the car’s unit rate of gas usage. ___________
gallon
6. How far can the car go on 5 gallons of gas? __
© Houghton Mifflin Harcourt Publishing Company
x
MATHEMATICAL PRACTICES 4
7. Estimate the amount of gas needed to travel 50 miles.
Distance (mi)
Distance (mi)
Time (yr)
y
180
150
120
90
60
30
0
1 2 3 4 5 6 x
Gas (gal)
_________
8.
DEEPER
Ellen’s car averages 35 miles per gallon of gas. If you
used equivalent ratios to graph her car’s gas usage, how would the
graph differ from the graph of Luis’s car’s gas usage?
Chapter 4 • Lesson 8 257
MATHEMATICAL PRACTICES .0%&-t3&"40/tM",&4&/4&
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MATHEMATICAL
PRACTICE
Look for Structure The graph shows the depth of
a submarine over time. Use equivalent ratios to find the number
of minutes it will take the submarine to descend 1,600 feet.
7
Submarine Depth
Depth (ft)
9.
y
1,200
1,000
800
600
400
200
0
1 2 3 4 5 6 x
Time (min)
10. The graph shows the distance that a plane flying at a steady rate
travels over time. Use equivalent ratios to find how far the plane
travels in 13 minutes.
Distance (mi)
Plane Travel
y
42
35
28
21
14
7
0
Time (min)
Emilio’s Typing Rate
y
Time (min)
Personal Math Trainer
SMARTER
The Tuckers drive at a rate of 20 miles per
hour through the mountains. Use the ordered pairs to graph the
distance traveled over time.
Distance (miles)
20
40
60
80
100
Time (hours)
1
2
3
4
5
Distance
(miles)
12.
100
80
60
40
20
0
258
1 2 3 4 5
Time (hours)
x
© Houghton Mifflin Harcourt Publishing Company
Sense or Nonsense?
Emilio types at a rate of 84 words per minute.
He claims that he can type a 500-word essay in
5 minutes. Is Emilio’s claim sense or nonsense?
Use a graph to help explain your answer.
SMARTER
Number of Words
11.
1 2 3 4 5 6 x
Practice and Homework
Name
Lesson 4.8
Equivalent Ratios and Graphs
Christie makes bracelets. She uses 8 charms for each bracelet. Use
this information for 1–3.
Charms
8
16
24
32
40
Bracelets
1
2
3
4
5
3. Use the ordered pairs to graph the charms and
bracelets.
Christie’s Bracelets
y
Number of Charms
1. Complete the table of equivalent ratios for the
first 5 bracelets.
2. Write ordered pairs, letting the x-coordinate
represent the number of bracelets and the
y-coordinate represent the number of charms.
16 ), ( _ , _ ),
8 ), (2, _
(1, _
( _ , _ ), ( _ , _ )
80
72
64
56
48
40
32
24
16
8
0
y
3
4
5. Find the unit rate of granola bars per box.
________
0
© Houghton Mifflin Harcourt Publishing Company
Problem
Problem Solving
Solving
6. Look at the graph for Christie’s Bracelets. How
many charms are needed for 7 bracelets?
8.
WRITE
Crunch N Go Granola Bars
Number of Bars
Bars
2
1 2 3 4 5 6 7 8 9 10
100
90
80
70
60
50
40
30
20
10
4. Complete the table of equivalent ratios.
1
x
Number of Bracelets
The graph shows the number of granola
bars that are in various numbers of boxes
of Crunch N Go. Use the graph for 4–5.
Boxes
COMMON CORE STANDARD—6.RP.A.3a
Understand ratio concepts and use ratio
reasoning to solve problems.
x
1 2 3 4 5 6 7 8 9 10
Number of Boxes
7. Look at the graph for Crunch N Go Granola Bars.
Stefan needs to buy 90 granola bars. How many
boxes must he buy?
Math Choose a real-life example of a unit rate.
Draw a graph of the unit rate. Then explain how another
person could use the graph to find the unit rate.
Chapter 4
259
Lesson Check (6.RP.A.3a)
1. A graph shows the distance a car traveled over
time. The x-axis represents time in hours, and
the y-axis represents distance in miles. The
graph contains the point (3, 165). What does
this point represent?
2. Maura charges $11 per hour to babysit. She
makes a graph comparing the amount she
charges (the y-coordinate) to the time she
babysits (the x-coordinate). Which ordered pair
shown is NOT on the graph?
(4, 44)
(11, 1)
(1, 11)
(11, 121)
Spiral Review (6.RP.A.3b, 6.NS.C.6b, 6.NS.C.7a, 6.NS.C.7c)
3. List 0, –4, and 3 from least to greatest.
4. What two numbers can be used in place of the
j to make the statement true?
5. Morgan plots the point (4, –7) on a coordinate
plane. If she reflects the point across the y-axis,
what are the coordinates of the reflected point?
6. Jonathan drove 220 miles in 4 hours. Assuming
he drives at the same rate, how far will he travel
in 7 hours?
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GO TO THE
260
Personal Math Trainer
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|j| = 8__9
Name
Chapter 4 Review/Test
Personal Math Trainer
Online Assessment
and Intervention
1. Kendra has 4 necklaces, 7 bracelets, and 5 rings. Draw a model to show
the ratio that compares rings to bracelets.
2. There are 3 girls and 2 boys taking swimming lessons. Write the ratio
that compares the girls taking swimming lessons to the total number of
students taking swimming lessons.
3. Luis adds 3 strawberries for every 2 blueberries in his fruit smoothie.
Draw a model to show the ratio that compares strawberries to
blueberries.
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4. Write the ratio 3 to 10 in two different ways.
5. Alex takes 3 steps every 5 feet he walks. As Alex continues walking, he
takes more steps and walks a longer distance. Complete the table by
writing two equivalent ratios.
Steps
3
Distance (feet)
5
Assessment Options
Chapter Test
Chapter 4
261
6. Sam has 3 green apples and 4 red apples. Select the ratios that
compare the number of red apples to the total number of apples.
Mark all that apply.
4 to 7
3 to 7
4:7
4:3
3_
7
4_
7
7. Jeff ran 2 miles in 12 minutes. Ju Chan ran 3 miles in 18 minutes. Did Jeff
and Ju Chan run the same number of miles per minute? Complete the
tables of equivalent ratios to support your answer.
Jeff
Distance (miles)
2
Time (minutes)
12
Distance (miles)
3
Time (minutes)
18
8. Jen bought 2 notebooks for $10. Write the rate as a fraction. Then find
the unit rate.
__________ = __________
2 notebooks
262
1 notebook
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Ju Chan
Name
9. Determine whether each ratio is equivalent to 12_ , _23 , or _47 . Write the ratio
in the correct box.
6
__
4
__
9
7
___
8
14
1
_
2
20
___
40
___
35
80
2
_
3
8
___
14
4
_
6
8
___
12
4
_
7
10. Amos bought 5 cantaloupes for $8. How many cantaloupes can he buy
for $24? Show your work.
__ . Check her work by making a table of
11. Camille said 45_ is equivalent to 24
30
equivalent ratios.
4
5
DEEPER
A box of oat cereal costs $3.90 for 15 ounces. A box of rice
cereal costs $3.30 for 11 ounces. Which box of cereal costs less per
ounce? Use numbers and words to explain your answer.
© Houghton Mifflin Harcourt Publishing Company
12.
Chapter 4
263
13. Scotty earns $35 for babysitting for 5 hours. If Scotty charges the same
rate, how many hours will it take him to earn $42?
_ hours
14. Use a unit rate to find the unknown value.
42
9
_______ = _______
14
15. Jenna saves $3 for every $13 she earns. Vanessa saves $6 for every $16
she earns. Is Jenna's ratio of money saved to money earned equivalent to
Vanessa's ratio of money saved to money earned?
16. The Hendersons are on their way to a national park. They are traveling
at a rate of 40 miles per hour. Use the ordered pairs to graph the distance
traveled over time.
Distance (miles)
40
80
120
160
200
1
2
3
4
5
Time (hours)
y
200
120
80
40
0
264
1
2
3
Time (hours)
4
5 x
© Houghton Mifflin Harcourt Publishing Company
Distance (miles)
160
Name
Personal Math Trainer
17.
SMARTER
Abby goes to the pool to swim laps. The graph shows
how far Abby swam over time. Use equivalent ratios to find how far Abby
swam in 7 minutes.
y
250
Distance (meters)
200
150
100
50
0
1
2
3
4
Time (minutes)
5 x
_ meters
18. Caleb bought 6 packs of pencils for $12.
Part A
How much will he pay for 9 packs of pencils? Use numbers and words to
explain your answer.
Part B
© Houghton Mifflin Harcourt Publishing Company
Describe how to use a bar model to solve the problem.
Chapter 4
265
19. A rabbit runs 35 miles per hour. Select the animals who run at a faster
unit rate per hour than the rabbit. Mark all that apply.
Reindeer: 100 miles in 2 hours
Ostrich: 80 miles in 2 hours
Zebra: 90 miles in 3 hours
Squirrel: 36 miles in 3 hours
20. Water is filling a bathtub at a rate of 3 gallons per minute.
Part A
Complete the table of equivalent ratios for the first five minutes of the
bathtub filling up.
Amount of Water (gallons)
3
Time (minutes)
1
Part B
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Emily said there will be 36 gallons of water in the bathtub after
12 minutes. Explain how Emily could have found her answer.
266
5
Chapter
Percents
Personal Math Trainer
Show Wha t You Know
Online Assessment
and Intervention
Check your understanding of important skills.
Name
Decimal Models Shade the model to show the decimal.
1. 0.31
2. 0.7
3. 1.7
Division Find the quotient.
4. 2,002 ÷ 91
5. 98qw
3,038
6. 24,487 ÷ 47
__
__
7. 22qw
2,332
__
__
© Houghton Mifflin Harcourt Publishing Company • Image Credits: (b) ©Cathy Maler Callanan/Getty Images
Multiply Whole Numbers by Decimals Find the product.
8.
2.38
×
4
__
__
9.
32.06
×
7
__
__
10.
4.60
× 18
__
__
11.
7.04
× 32
__
__
Math in the
Esmeralda likes to listen to music while she works out. She had
a playlist on her MP3 player that lasted 40 minutes, but she
accidentally deleted 25% of the music. Does Esmeralda have
enough music left on her playlist for a 30-minute workout?
Explain your answer.
Chapter 5
267
Voca bula ry Builder
Visualize It
Complete the bubble map with review and preview words that are
Review Words
decimal
related to ratios.
equivalent ratios
factor
quotient
rate
ratio
simplify
Ratios
Preview Word
percent
Understand Vocabulary
Complete the sentences using review and preview words.
1. A comparison of one number to another by division is a
___________ .
2.
___________ name the same comparison.
3. A ratio that compares quantities with different units is a
___________ .
© Houghton Mifflin Harcourt Publishing Company
4. A ___________ is a ratio, or rate, that compares a
number to 100.
5.
___________ a fraction or a ratio by dividing the
numerator and denominator by a common factor.
268
™Interactive Student Edition
™Multimedia eGlossary
Chapter 5 Vocabulary
decimal
equivalent ratios
decimal
razones equivalentes
30
19
factor
percent
factor
porcentaje
74
33
quotient
rate
cociente
tasa
86
84
ratio
simplest form
razón
mínima expresión
87
92
4∙1
↓
5∙1
↓
Boxes
1
2
3
4
5
Cards
6
12
18
24
30
↑
2∙6
↑
3∙6
↑
4∙6
↑
5∙6
A comparison of a number to 100; percent
means “per hundred”
Example:
45 = 45%
____
100
A ratio that compares two quantities having
different units of measure
Example:
An airplane climbs 18,000 feet in 12 minutes.
The rate of climb is 18,000 ft to 12 min,
18,000 ft
________
, or 18,000 ft : 12 min
12 min
A fraction is in simplest form when the
numerator and denominator have only 1
as a common factor
6÷2=3
__
Example: _____
8÷2
4
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3∙1
↓
© Houghton Mifflin Harcourt Publishing Company
2∙1
↓
© Houghton Mifflin Harcourt Publishing Company
Original
ratio
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© Houghton Mifflin Harcourt Publishing Company
© Houghton Mifflin Harcourt Publishing Company
© Houghton Mifflin Harcourt Publishing Company
© Houghton Mifflin Harcourt Publishing Company
Ratios that name the same comparison
A number with one or more digits to the
right of the decimal point
Examples: 2.6 and 7.053
A number multiplied by another number to
find a product
2 × 3 × 7 = 42
factors
The number that results from dividing
80 ÷ 4 = 20
20
4 80
quotient
quotient
A comparison of two numbers, a and b,
a
that can be written as a fraction __
b
The ratio of blue squares to red squares
5
, or 5 : 1.
is 5 to 1, __
1
Going Places with
Words
Concentration
For 2–3 players
Materials
• 1 set of word cards
How to Play
1. Place the cards face-down in even rows. Take turns to play.
Game
Game
Word Box
decimal
equivalent ratios
factor
percent
quotient
rate
ratio
simplest form
2. Choose two cards and turn them face-up.
• If the cards show a word and its meaning, it’s a match.
• If the cards do not match, turn them over again.
3. The game is over when all cards have been matched. The
players count their pairs. The player with the most pairs wins.
© Houghton Mifflin Harcourt Publishing Company
Image Credits: (bg) ©Morgan Lane Photography/Shutterstock; (b) ©Mark Andersen/Rubberball/Corbis
Keep the pair and take another turn.
Chapter 5
268A
Journal
Jo
ou
urnal
The Write Way
Reflect
Choose one idea. Write about it.
• Which is greatest: 3_5 , .70, or 65%? Tell how you know.
4 as a decimal and as a percent.
• Explain how to write __
80
• A soccer team needs to sell 200 T-shirts to raise enough money
for a trip. So far they have sold 55% percent of the shirts. Explain
how to figure out how many more shirts they need to sell.
© Houghton Mifflin Harcourt Publishing Company
Image Credits: (bg) ©OJO Images/Getty Images; (t) ©Pincasso/Shutterstock
• Tell how to solve this problem: 15 is 20% of _____.
268B
Lesson 5.1
Name
Model Percents
Essential Question How can you use a model to show a percent?
Ratios and Proportional
Relationships—6.RP.A.3c
MATHEMATICAL PRACTICES
MP2, MP3, MP5
Hands
On
Investigate
Investigate
Materials ■ 10-by-10 grids
Not many people drive electric cars today. But one expert
estimates that by 2025, 35 percent of all cars will be powered
by electricity.
A percent is a ratio, or rate, that compares a number to 100.
Percent means “per hundred.” The symbol for percent is %.
A. Model 35% on the 10-by-10 grid. Then tell what the
percent represents.
The large square represents the whole, or 100%.
Each small square represents 1%.
• Shade the grid to show 35%.
Think: 35% is _ out of 100.
• Write 35% as a ratio comparing 35 to 100.
Think: 35 out of 100 squares is _______.
100
© Houghton Mifflin Harcourt Publishing Company • Image Credits: (t) ©Mirafoto.com/Alamy
• 35% = _______
So, by 2025, _ out of _ cars may be powered by
electricity.
B. Model 52% on a 10-by-10 grid.
•
_ out of _ squares is ______
.
100
• 52% = ______
100
C. Model 18% on a 10-by-10 grid.
•
_ out of _ squares is ______
.
100
• 18% = ______
100
Chapter 5
269
Draw Conclusions
1. Explain how you would use a 10-by-10 grid to model 7%.
2. Model 1_4 on a 10-by-10 grid. What percent is shaded? Explain.
3.
MATHEMATICAL
PRACTICE
5 Use a Concrete Model Explain how you could model
0.5% on a 10-by-10 grid.
4.
SMARTER
How would you model 181% using 10-by-10 grids?
Make
Make Connections
Connections
The table shows the types of meteorites in Meg’s collection.
Shade a grid to show the ratio comparing the number of each
type to the total number. Then write the ratio as a percent.
Meg’s Meteorite Collection
Think: A percent is a ratio that compares a number to __.
Number
Iron
21
Stone
76
Stony-iron
3
Stone
Stony-iron
_ out of __
_ out of __
_ out of __
meteorites are iron.
meteorites are stone.
meteorites are stony-iron.
____ = _%
____ = _%
100
____ = _%
Math
Talk
270
MATHEMATICAL PRACTICES 2
Connect Symbols and
Words Explain what this
statement means: 13% of the
students at Harding Middle
School are left-handed.
© Houghton Mifflin Harcourt Publishing Company
Iron
Type
Name
Share
hhow
Share and
and Show
Sh
MATH
M
BOARD
B
Write a ratio and a percent to represent the shaded part.
1.
2.
ratio: _ percent: _
3.
ratio: _ percent: _
ratio: _ percent: _
Model the percent and write it as a ratio.
4. 30%
6. 75%
5. 5%
ratio: _
ratio: _
ratio: _
OqnakdlRnkuhmf¤@ookhb`shnmr
OqnakdlRnkuhmf¤@ookhb`shnmr
7.
MATHEMATICAL
PRACTICE
5 Use a Concrete Model Explain how to model 32% on a 10-by-10 grid.
© Houghton Mifflin Harcourt Publishing Company
How does the model represent the ratio of 32 to 100?
8.
DEEPER
A floor has 100 tiles. There are 24 black tiles and 35 brown tiles.
The rest of the tiles are white. What percent of the tiles are white?
Chapter 5 • Lesson 1 271
MATHEMATICAL PRACTICES
COMMUNICA5&t1&34E7&3&tCONSTRUCT ARGUMENTS
Pose a Problem
9.
SMARTER
Javier designed a mosaic wall mural using 100 tiles in
3 different colors: yellow, blue, and red. If 64 of the tiles are yellow, what
percent of the tiles are either red or blue?
To find the number of tiles that are either red or blue, count the red and
blue squares. Or subtract the number of yellow squares, 64, from the
total number of squares, 100.
36 out of 100 tiles are red or blue.
36 .
The ratio of red or blue tiles to all tiles is ___
100
So, the percent of the tiles that are either red or blue is 36%.
Write another problem involving a percent that can be solved by using
the mosaic wall mural.
Pose a Problem
272
SMARTER
Select the 10-by-10 grids that model 45%. Mark all that apply.
●
●
●
●
●
© Houghton Mifflin Harcourt Publishing Company
10.
Solve Your Problem
Practice and Homework
Name
Lesson 5.1
Model Percents
COMMON CORE STANDARD—
6.RP.A.3c Understand ratio concepts and
use ratio reasoning to solve problems.
Write a ratio and a percent to represent the shaded part.
1.
2.
3.
31
___
31%
100 percent: ________
ratio: _______
ratio: _______ percent: ________
ratio: _______ percent: ________
Model the percent and write it as a ratio.
4. 97%
5. 24%
ratio: _______
6. 50%
ratio: _______
Pens Sold
Problem
Problem Solving
Solving
The table shows the pen colors sold at the school supply store one
week. Write the ratio comparing the number of the given color sold
to the total number of pens sold. Then shade the grid.
© Houghton Mifflin Harcourt Publishing Company
7. Black
_______
9.
WRITE
ratio: _______
Color
Number
Blue
36
Black
49
Red
15
8. Not blue
_______
Math Is every percent a ratio? Is every ratio a percent? Explain.
Chapter 5
273
Lesson Check (6.RP.A.3c)
1. What percent of the large square is shaded?
2. What is the ratio of shaded squares to unshaded
squares?
3. Write a number that is less than –2 4_5 and greater
than –3 1_5 .
4. On a coordinate grid, what is the distance
between (2, 4) and (2, –3)?
5. Each week, Diana spends 4 hours playing
soccer and 6 hours babysitting. Write a ratio to
compare the time Diana spends playing soccer
to the time she spends babysitting.
6. Antwone earns money at a steady rate mowing
lawns. The points (1, 25) and (5, 125) appear on
a graph of the amount earned versus number of
lawns mowed. What are the coordinates of the
point on the graph with an x-value of 3?
FOR MORE PRACTICE
GO TO THE
274
Personal Math Trainer
© Houghton Mifflin Harcourt Publishing Company
Spiral Review (6.RP.A.3c, 6.NS.C.6a, 6.NS.C.6c, 6.NS.C.8)
Lesson 5.2
Name
Write Percents as Fractions and Decimals
Ratios and Proportional
Relationships—6.RP.A.3c
Mathematical Practices
MP2, MP5, MP7, MP8
Essential Question How can you write percents as fractions
and decimals?
To write a percent as a fraction or a decimal, first write the percent as a
ratio that compares a number to 100. For example, 37% = ___
​ 37 ​.
100
Unlock
Unlock the
the Problem
Problem
Carlos eats a banana, an orange, and a blueberry
muffin for breakfast. What fraction of the daily value
of vitamin C does each item contain?
Write each percent as a fraction.
A Write 15% as a fraction.
15% = ____
​ 
​= ____
​ ​
100
Vitamin C Content
Item
Percent of Daily Value
Banana
15%
Orange
113%
Blueberry Muffin
0.5%
15% is 15 out of 100.
Write the fraction in simplest form.
So, 15% = _.
B Write 113% as a fraction.
113% = ____
​
​+ ____
​  13 ​
100 100
= _ + ____
​  13 ​
100
So, 113% = _.
113% is 100 out of 100 plus
13 out of 100.
100
____
​ ​5 1
100
6_MNLAESE352053_C05_013T.ai
Write the sum as a mixed
number.
C Write 0.5% as a fraction.
© Houghton Mifflin Harcourt Publishing Company
0.5% = ____
​ 
​
100
= _______
​ 0.5 • 10​= _____
​ 
​
100 • 10 1,000
1 ​
= ____
​ 
0.5% is 0.5 out of 100.
6_MNLAESE352053_C05_014T.ai
Multiply the numerator and
denominator by 10 to get a whole
number in the numerator.
Write the fraction in simplest form.
So, 0.5% = _.
•
MATHEMATICAL
PRACTICE
2 Reason Quantitatively Explain why two 10-by-10 grids
were used to show 113%.
Chapter 5 275
Example
A Write 72% as a decimal.
72% = ​ ____​
72% is 72 out of 100.
100
= __
Use place value to write 72
hundredths as a decimal.
So, 72% = __.
B Write 4% as a decimal.
4% = ​ ____​
4% is 4 out of 100.
100
6_MNLAESE352053_C05_015T.ai
Use division to write 4% as a decimal.
100 ​qww
4.00 ​
–0
40
– 0
400
–400
0
Divide the ones. Since 4 ones cannot
be shared among 100 groups, write a
zero in the quotient.
Place a decimal point after the ones
place in the quotient.
When you divide decimal
numbers by powers of 10, you
move the decimal point one
place to the left for each factor
of 10.
So, 4% = __.
C Write 25.81% as a decimal.
25.81% = ​______​
100
= __
25.81% is 25.81 out of 100.
To divide by 100, move the decimal
point 2 places to the left: 0.2581
So, 25.81% = __.
Write the percent as a fraction.
1. 80%
MATH
BOARD
2. 150%
3.0.2%
80% = ____
​ 
​= ___
​  ​
100
Write the percent as a decimal.
4. 58%
276
5. 9%
Math
Talk
MATHEMATICAL PRACTICES 6
Explain How would you use
estimation to check that your
answer is reasonable when
you write a percent as a
fraction or decimal?
© Houghton Mifflin Harcourt Publishing Company
Share
Share and
and Show
Show
Name
On
On Your
Your Own
Own
Write the percent as a fraction or mixed number.
6. 17%
7. 20%
8. 125%
9. 355%
10. 0.1%
11. 2.5%
12. 89%
13. 30%
14. 2%
15. 122%
16. 3.5%
17. 6.33%
© Houghton Mifflin Harcourt Publishing Company
Write the percent as a decimal.
18.
MATHEMATICAL
PRACTICE
2 Use Reasoning Write <, >, or =.
21.6%
​1_​
5
19.
DEEPER
Georgianne completed 60% of her
homework assignment. Write the portion of her
homework that she still needs to complete as a
fraction.
Chapter 5 • Lesson 2 277
MATHEMATICAL PRACTICES
ANALYZE • LOOK FOR STRUCTURE • PRECISION
Problem
Problem Solving
Solving •• Applications
Applications
Use the table for 20 and 21.
20. What fraction of computer and video game
players are 50 years old or more?
Age of Computer and
Video Game Players
21. What fraction of computer and video game
players are 18 years old or more?
23.
278
SMARTER
Box A and Box B each contain
black tiles and white tiles. They have the same
total number of tiles. In Box A, 45% of the tiles
11
​of the tiles are white.
are black. In Box B, ​ _
20
Compare the number of black tiles in the
boxes. Explain your reasoning.
Percent
Under 18
25%
18 to 49
49%
50 or more
26%
SMARTER
Mr. Truong is organizing a summer program
for 6th grade students. He surveyed students to find the percent
of students interested in each activity. Complete the table by
writing each percent as a fraction or decimal.
Activity
Percent
Fraction
Sports
48%
12
__
​ ​
25
Cooking
23%
0.23
Music
20%
0.2
Art
9%
___
​9 ​
100
Decimal
© Houghton Mifflin Harcourt Publishing Company
22.
Age (years)
Practice and Homework
Name
Write Percents as Fractions and Decimals
COMMON CORE STANDARD—6.RP.A.3c
Understand ratio concepts and use ratio
reasoning to solve problems.
Write the percent as a fraction or mixed number.
2. 32%
1. 44%
Lesson 5.2
3. 116%
4. 250%
​ 100​
11 ​
__
5 ​ 
25
44
44% 5 ___
5. 0.3%
6. 0.4%
7. 1.5%
8. 12.5%
Write the percent as a decimal.
9. 63%
10. 110%
11. 42.15%
12. 0.1%
Problem
Problem Solving
Solving
© Houghton Mifflin Harcourt Publishing Company
13. An online bookstore sells 0.8% of its books to
foreign customers. What fraction of the books are
sold to foreign customers?
15.
WRITE
14. In Mr. Klein’s class, 40% of the students are boys.
What decimal represents the portion of the
students that are girls?
Math Explain how percents, fractions, and
decimals are related. Use a 10-by-10 grid to make a model
that supports your explanation.
Chapter 5
279
Lesson Check (6.RP.A.3c)
1. The enrollment at Sonya’s school this year is
109% of last year’s enrollment. What decimal
represents this year’s enrollment compared to
last year’s?
2. An artist’s paint set contains 30% watercolors
and 25% acrylics. What fraction represents the
portion of the paints that are watercolors or
acrylics? Write the fraction in simplest form.
​
Spiral Review (6.RP.A.3a, 6.RP.A.3c, 6.NS.C.7a, 6.NS.C.7b, 6.NS.C.8)
3. Write the numbers in order from least to
greatest.
4. On a coordinate plane, the vertices of a
rectangle are (2, 4), (2, ​ ​−​1), (​ ​−​5, ​ ​−​1), and
(​ ​−​5, 4). What is the perimeter of the rectangle?
​ ​−​5.25 1.002 ​ ​−​5.09
A
B
C
D
Width
12
15
20
16.5
Length
20
22.5
25
27.5
6. What percent represents the shaded part?
© Houghton Mifflin Harcourt Publishing Company
5. The table below shows the widths and lengths,
in feet, for different playgrounds. Which
playgrounds have equivalent ratios of width
to length?
FOR MORE PRACTICE
GO TO THE
280
Personal Math Trainer
Practice and Homework
Name
Lesson 5.3
Write Fractions and Decimals as Percents
Write the fraction or decimal as a percent.
7
1. __
3
2. __
20
50
1
3. __
25
COMMON CORE STANDARD—
6.RP.A.3c Understand ratio concepts and
use ratio reasoning to solve problems.
4. 5_
5
735
_7 5 ___
20 3 5
20
35
5 __
5 35%
100
___
5. 0.622
6. 0.303
___
___
___
7. 0.06
___
___
8. 2.45
___
Write the number in two other forms (fraction, decimal, or percent).
__
9. 19
9
10. __
20
16
___
___
11. 0.4
___
12. 0.22
___
© Houghton Mifflin Harcourt Publishing Company
Problem
Problem Solving
Solving
3
13. According to the U.S. Census Bureau, __
25 of all
adults in the United States visited a zoo in 2007.
What percent of all adults in the United States
visited a zoo in 2007?
15.
WRITE
14. A bag contains red and blue marbles. Given
__ of the marbles are red, what percent of the
that 17
20
marbles are blue?
Math Explain two ways to write 4_5 as a percent.
Chapter 5
285
Lesson Check (6.RP.A.3c)
1. The portion of shoppers at a supermarket
who pay by credit card is 0.36. What percent
of shoppers at the supermarket do NOT pay by
credit card?
__ of a lawn is planted with Kentucky
2. About 23
40
bluegrass. What percent of the lawn is planted
with Kentucky bluegrass?
3. A basket contains 6 peaches and 8 plums.
What is the ratio of peaches to total pieces
of fruit?
4. It takes 8 minutes for 3 cars to move
through a car wash. At the same rate, how
many cars can move through the car wash
in 24 minutes?
5. A 14-ounce box of cereal sells for $2.10. What is
the unit rate?
6. A model railroad kit contains curved tracks and
straight tracks. Given that 35% of the tracks are
curved, what fraction of the tracks are straight?
Write the fraction in simplest form.
FOR MORE PRACTICE
GO TO THE
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Spiral Review (6.RP.A.1, 6.RP.A.2, 6.RP.A.3a, 6.RP.A.3c)
Name
Mid-Chapter Checkpoint
Personal Math Trainer
Online Assessment
and Intervention
Vocabulary
Vocabulary
Vocabulary
percent
Choose the best term from the box to complete the sentence.
rate
1. A __ is a ratio that compares a quantity to 100. (p. 269)
Concepts
Concepts and
and Skills
Skills
Write a ratio and a percent to represent the shaded part. (6.RP.A,3c)
2.
3.
4.
5.
6.
7.
Write the number in two other forms (fraction, decimal, or percent). (6.RP.A.3c)
© Houghton Mifflin Harcourt Publishing Company
8. 0.04
12. 0.9
3
9. __
10
13. 0.5%
10. 1%
11. 11_
14. 7_8
15. 355%
5
Chapter 5
287
9 of the avocados grown in the United States are grown in
16. About __
10
California. About what percent of the avocados grown in the United
States are grown in California? (6.RP.A.3c)
17. Morton made 36 out of 48 free throws last season. What percent of his
free throws did Morton make? (6.RP.A.3c)
18. Sarah answered 85% of the trivia questions correctly. What fraction
describes this percent? (6.RP.A. 3c)
19. About 4_ of all the orange juice in the world is produced in Brazil.
5
About what percent of all the orange juice in the world is produced in
Brazil? (6.RP.A.3c)
288
DEEPER
If you eat 4 medium strawberries, you get 48% of your daily
recommended amount of vitamin C. What fraction of your daily
amount of vitamin C do you still need? (6.RP.A.3c)
© Houghton Mifflin Harcourt Publishing Company
20.
Name
Mid-Chapter Checkpoint
Personal Math Trainer
Online Assessment
and Intervention
Vocabulary
Vocabulary
Vocabulary
percent
Choose the best term from the box to complete the sentence.
rate
1. A __ is a ratio that compares a quantity to 100. (p. 269)
Concepts
Concepts and
and Skills
Skills
Write a ratio and a percent to represent the shaded part. (6.RP.A,3c)
2.
3.
4.
5.
6.
7.
Write the number in two other forms (fraction, decimal, or percent). (6.RP.A.3c)
© Houghton Mifflin Harcourt Publishing Company
8. 0.04
12. 0.9
3
9. __
10
13. 0.5%
10. 1%
11. 11_
14. 7_8
15. 355%
5
Chapter 5
287
9 of the avocados grown in the United States are grown in
16. About __
10
California. About what percent of the avocados grown in the United
States are grown in California? (6.RP.A.3c)
17. Morton made 36 out of 48 free throws last season. What percent of his
free throws did Morton make? (6.RP.A.3c)
18. Sarah answered 85% of the trivia questions correctly. What fraction
describes this percent? (6.RP.A. 3c)
19. About 4_ of all the orange juice in the world is produced in Brazil.
5
About what percent of all the orange juice in the world is produced in
Brazil? (6.RP.A.3c)
288
DEEPER
If you eat 4 medium strawberries, you get 48% of your daily
recommended amount of vitamin C. What fraction of your daily
amount of vitamin C do you still need? (6.RP.A.3c)
© Houghton Mifflin Harcourt Publishing Company
20.
Lesson 5.4
Name
Percent of a Quantity
Ratios and Proportional
Relationships—6.RP.A.3c
MATHEMATICAL PRACTICES
MP1, MP2, MP5
Essential Question How do you find a percent of a quantity?
Unlock
Unlock the
the Problem
Problem
A typical family of four uses about 400 gallons of
water each day, and 30% of this water is for outdoor
activities, such as gardening. How many gallons of
water does a typical family of four use each day for
outdoor activities?
One Way
• Will the number of gallons of water for
outdoor activities be greater than or less than
200 gallons? Explain.
Use ratio reasoning.
Draw a bar model.
0%
10%
20%
30%
40%
50%
60%
70%
80%
90%
100%
400
The model shows that 100% represents 400 gallons.
Think: 30% is 3 groups of 10%, so divide the model into 10 equal groups.
Find the value of 10% of 400.
400 = _
1 × 400 = ______
10% of 400 = ___
Find the value of 30% of 400.
30% of 400 = 3 × _ = _
Another Way
10
Multiply.
© Houghton Mifflin Harcourt Publishing Company
You can find 30% of 400 by multiplying.
Write the percent as a rate per 100.
30
30% = ____
30 of 400.
Multiply to find ____
30 × 400 =
____
100
So, 30% of 400 gallons is _ gallons.
Try This! Find 65% of 300.
100
100
_
Math
Talk
MATHEMATICAL PRACTICES 2
Reason Quantitatively How
can you find the number of
gallons of water used for
indoor activities?
65% = _
_ × 300 = _
Chapter 5
289
Example
Charla earns $4,000 per month. She spends 40% of her
salary on rent and 15% of her salary on groceries. How
much money does Charla have left for other expenses?
STEP 1 Add to find the total
percent of Charla’s salary that
is used for rent and groceries.
40% + _ % = _%
STEP 2 Subtract the total
percent from 100% to find the
percent that is left for other
expenses.
100% − _ % = 45%
STEP 3 Write the percent from
Step 2 as a rate per 100 and
multiply.
45% = _
_ × 4,000 = __
Math
Talk
So, Charla has $ __ left for other expenses.
Share
hhow
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and Show
Sh
MATHEMATICAL PRACTICES 1
Describe What is another
way you could solve the
problem?
MATH
M
BOARD
B
Find the percent of the quantity.
0%
25%
50%
75%
100%
1. 25% of 320
320
25% = 1_4 , so use _ equal
groups.
320 = _
= ______
2. 80% of 50
____
3. 175% of 24
4. 60% of 210
____
____
5. A jar contains 125 marbles. Given that 4% of the marbles are
green, 60% of the marbles are blue, and the rest are red, how
many red marbles are in the jar?
6. There are 32 students in Mr. Moreno’s class and 62.5% of the
students are girls. How many boys are in the class?
290
Math
Talk
MATHEMATICAL PRACTICES 5
Use Tools What strategy
could you use to estimate
49.3% of 3,000?
© Houghton Mifflin Harcourt Publishing Company
1
_ × 320
4
Name
On
On Your
Your Own
Own
Find the percent of the quantity.
7. 60% of 90
8. 25% of 32.4
11. A baker made 60 muffins for a cafe. By noon,
45% of the muffins were sold. How many muffins
were sold by noon?
13. A school library has 260 DVDs in its collection.
Given that 45% of the DVDs are about science
and 40% are about history, how many of the
DVDs are about other subjects?
9. 110% of 300
10. 0.2% of 6,500
12. There are 30 treasures hidden in a castle in a
video game. LaToya found 80% of them. How
many of the treasures did LaToya find?
14.
DEEPER
Mitch planted cabbage, squash, and
carrots on his 150-acre farm. He planted half the
farm with squash and 22% with carrots. How
many acres did he plant with cabbage?
Compare. Write <, >, or =.
15. 45% of 60
© Houghton Mifflin Harcourt Publishing Company
18.
19.
60% of 45
16. 10% of 90
90% of 100
17. 75% of 8
8% of 7.5
SMARTER
Sarah had 12 free throw attempts during a game and
made at least 75% of the free throws. What is the greatest number of free
throws Sarah could have missed during the game?
MATHEMATICAL
PRACTICE
3 Chrissie likes to tip a server in a restaurant a minimum
of 20%. She and her friend have a lunch bill that is $18.34. Chrissie
says the tip will be $3.30. Her friend says that is not a minimum
of 20%. Who is correct? Explain.
Chapter 5 • Lesson 4 291
MATHEMATICAL PRACTICES .0%&-t3&"40/tM",&4&/4&
Unlock
Unlock the
the Problem
Problem
20. One-third of the juniors in the Linwood High
School Marching Band play the trumpet.
The band has 50 members and the table
shows what percent of the band members are
freshmen, sophomores, juniors, and seniors.
How many juniors play the trumpet?
a. What do you need to find?
b. How can you use the table to help you solve
the problem?
Linwood High School
Marching Band
Freshmen
26%
Sophomores
30%
Juniors
24%
Seniors
20%
WRITE
M t Show Your Work
Math
c. What operation can you use to find the
number of juniors in the band?
e. Complete the sentences.
The band has _ members. There
are _ juniors in the band. The number
of juniors who play the
trumpet is _ .
21.
SMARTER
Compare. Circle <, >, or =.
<
21a. 25% of 44 > 20% of 50
=
<
21c. 35% of 60 > 60% of 35
=
292
21b.
<
10% of 30 > 30% of 100
=
© Houghton Mifflin Harcourt Publishing Company • Image Credits: (t) ©Thinkstock/Getty Images
d. Show the steps you use to solve the problem.
Practice and Homework
Name
Lesson 5.4
Percent of a Quantity
COMMON CORE STANDARD—6.RP.A.3c
Understand ratio concepts and use ratio
reasoning to solve problems.
Find the percent of the quantity.
1. 60% of 140
2. 55% of 600
3. 4% of 50
4. 10% of 2,350
60
60% = ___
100
60
___
× 140
100
= 84
___
5. 160% of 30
6. 105% of 260
___
___
___
7. 0.5% of 12
___
___
8. 40% of 16.5
___
Problem
Problem Solving
Solving
© Houghton Mifflin Harcourt Publishing Company
9. The recommended daily amount of vitamin C for
children 9 to 13 years old is 45 mg. A serving of
a juice drink contains 60% of the recommended
amount. How much vitamin C does the juice
drink contain?
11.
WRITE
10. During a 60-minute television program,
25% of the time is used for commercials and 5%
of the time is used for the opening and closing
credits. How many minutes remain for the
program itself?
Math Explain two ways you can find 35% of 700.
Chapter 5
293
Lesson Check (6.RP.A.3c)
1. A store has a display case with cherry, peach,
and grape fruit chews. There are 160 fruit chews
in the display case. Given that 25% of the fruit
chews are cherry and 40% are peach, how many
grape fruit chews are in the display case?
2. Kelly has a ribbon that is 60 inches long. She
cuts 40% off the ribbon for an art project.
While working on the project, she decides she
only needs 75% of the piece she cut off. How
many inches of ribbon does Kelly end up using
for her project?
Spiral Review (6.NS.C.7d, 6.RP.A.3b, 6.RP.A.3c)
| −12| > 1
|0| > −4
|20| >| −10|
6 < | −3|
5. Which percent represents the model?
4. Miyuki can type 135 words in 3 minutes.
How many words can she expect to type
in 8 minutes?
3
6. About __
5 of the students at Roosevelt
Elementary School live within one mile of the
school. What percent of students live within one
mile of the school?
FOR MORE PRACTICE
GO TO THE
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3. Three of the following statements are true.
Which one is NOT true?
PROBLEM SOLVING
Name
Lesson 5.5
Problem Solving • Percents
Ratios and Proportional
Relationships—6.RP.A.3c
MATHEMATICAL PRACTICES
MP1, MP4, MP5, MP6
Essential Question How can you use the strategy use a model to help you
solve a percent problem?
Unlock
Unlock the
the Problem
Problem
The recommended daily amount of protein is about 50
grams. One Super Protein Cereal Bar contains 16% of that
amount of protein. If Stefon eats one Super Protein Cereal
Bar per day, how much protein will he need to get from
other sources to meet the recommended daily amount?
Use the graphic organizer to help you solve the problem.
Read the Problem
What do I need to find?
Write what you need to find.
What information do I
need to use?
How will I use the
information?
Write the important
information.
What strategy can you use?
Solve the Problem
The model shows that 100% = 50 grams,
© Houghton Mifflin Harcourt Publishing Company
Draw a bar model.
Recommended
Daily Amount
100%
50 = _.
so 1% of 50 = ___
50 g
16% of 50 = 16 × _ = _
100
So, the cereal bar contains __
of protein.
Cereal Bar
50 − _ = _
16%
So, __ of protein should come from other sources.
Math
Talk
MATHEMATICAL PRACTICES 1
Describe How can you use
estimation to show that your
answer is reasonable?
Chapter 5
295
Try Another Problem
Lee has saved 65% of the money she needs to buy a pair of
jeans that cost $24. How much money does Lee have, and
how much more money does she need to buy the jeans?
Read the Problem
What do I need to find?
What information do I
need to use?
How will I use the
information?
Solve the Problem
2.
MATHEMATICAL
PRACTICE
6 Explain how you could solve this problem
in a different way.
Math
Talk
296
MATHEMATICAL PRACTICES 6
Compare How is the model
you used to solve this problem
similar to and/or different
from the model on page 295?
© Houghton Mifflin Harcourt Publishing Company
1. Does your answer make sense? Explain how you know.
Name
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and Show
Sh
MATH
M
BOARD
B
1. A geologist visits 40 volcanoes in Alaska and California. 15% of
the volcanoes are in California. How many volcanoes does the
geologist visit in California and how many in Alaska?
WRITE
Math
Show Your Work
First, draw a bar model.
100%
40
Total Volcanoes
California
15%
Next, find 1%.
40 = __
100% = 40, so 1% of 40 = ___
100
Then, find 15%, the number of volcanoes in California.
15% of 40 = 15 × __ = __
Finally, subtract to find the number of volcanoes in Alaska.
So, the geologist visited __ volcanoes in California
and __ volcanoes in Alaska.
© Houghton Mifflin Harcourt Publishing Company
2.
What if 30% of the volcanoes were in California?
How many volcanoes would the geologist have visited in California
and how many in Alaska?
SMARTER
3. Ricardo has $25 to spend on school supplies. He spends 72% of the
money on a backpack and the rest on a large binder. How much does
he spend on the backpack? How much does he spend on the binder?
4. Kevin is hiking on a trail that is 4.2 miles long. So far, he has hiked
80% of the total distance. How many more miles does Kevin have
to hike in order to complete the trail?
Chapter 5 • Lesson 5 297
MATHEMATICAL PRACTICES
COMMUNICA5&t1&34E7&3&tCONSTRUCT ARGUMENTS
On
On Your
Your Own
Own
5.
6.
7.
8.
DEEPER
Jordan takes 50% of the cherries from a bowl. Then Mei
takes 50% of the remaining cherries. Finally, Greg takes 50% of the
remaining cherries. There are 3 cherries left. How many cherries were in
the bowl before Jordan arrived?
SMARTER
Each week, Tasha saves 65% of the money she earns
babysitting and spends the rest. This week she earned $40. How much
more money did she save than spend this week?
SMARTER
An employee at a state park has 53 photos of animals
found at the park. She wants to arrange the photos in rows so that every
row except the bottom row has the same number of photos. She also
wants there to be at least 5 rows. Describe two different ways she can
arrange the photos.
MATHEMATICAL
PRACTICE
6 Explain a Method Maya wants to mark a length of
7 inches on a sheet of paper, but she does not have a ruler. She has
pieces of wood that are 4 inches, 5 inches, and 6 inches long. Explain
how she can use these pieces to mark a length of 7 inches.
9.
SMARTER
Pierre’s family is driving 380 miles from San Francisco to Los
Angeles. On the first day, they drive 30% of the distance. On the second day, they drive
50% of the distance. On the third day, they drive the remaining distance and arrive in Los
Angeles. How many miles did Pierre’s family drive each day? Write the number of miles in
the correct box.
76 miles
First Day
298
190 miles
Second Day
114 miles
Third Day
© Houghton Mifflin Harcourt Publishing Company
Personal Math Trainer
Practice and Homework
Name
Lesson 5.5
Problem Solving • Percents
COMMON CORE STANDARD—6.RP.A.3c
Understand ratio concepts and use ratio
reasoning to solve problems.
Read each problem and solve.
1. On Saturday, a souvenir shop had
125 customers. Sixty-four percent of
the customers paid with a credit card.
The other customers paid with cash.
How many customers paid with cash?
125
1% of 125 = ___
100 = 1.25
64% of 125 = 64 3 1.25 = 80
125 − 80 = 45 customers
2. A carpenter has a wooden stick that is
84 centimeters long. She cuts off 25% from the end
of the stick. Then she cuts the remaining stick into 6
equal pieces. What is the length of each piece?
3. A car dealership has 240 cars in the parking lot and
17.5% of them are red. Of the other 6 colors in the lot,
each color has the same number of cars. If one of the
colors is black, how many black cars are in the lot?
4. The utilities bill for the Millers’ home in April was
$132. Forty-two percent of the bill was for gas, and the
rest was for electricity. How much did the Millers pay
for gas, and how much did they pay for electricity?
© Houghton Mifflin Harcourt Publishing Company
5. Andy’s total bill for lunch is $20. The cost of the drink
is 15% of the total bill and the rest is the cost of the
food. What percent of the total bill did Andy’s food
cost? What was the cost of his food?
6.
WRITE
______
______
______
______
Math Write a word problem that involves
finding the additional amount of money needed to
purchase an item, given the cost and the percent of
the cost already saved.
Chapter 5
299
Lesson Check (6.RP.A.3c)
1. Milo has a collection of DVDs. Out of 45
DVDs, 40% are comedies and the remaining
are action-adventures. How many actionadventure DVDs does Milo own?
2. Andrea and her partner are writing a 12-page
science report. They completed 25% of the
report in class and 50% of the remaining pages
after school. How many pages do Andrea and
her partner still have to write?
Spiral Review (6.NS.C.7c, 6.RP.A.3a, 6.RP.A.3c)
4. Ricardo graphed a point by starting at the origin
and moving 5 units to the left. Then he moved
up 2 units. What is the ordered pair for the point
he graphed?
5. The population of birds in a sanctuary increases
at a steady rate. The graph of the population
over time has the points (1, 105) and (3, 315).
Name another point on the graph.
6. Alicia’s MP3 player contains 1,260 songs. Given
that 35% of the songs are rock songs and 20% of
the songs are rap songs, how many of the songs
are other types of songs?
© Houghton Mifflin Harcourt Publishing Company
−4
3. What is the absolute value of __
25 ?
FOR MORE PRACTICE
GO TO THE
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Personal Math Trainer
Lesson 5.6
Name
Find the Whole From a Percent
Ratios and Proportional
Relationships—6.RP.A.3c
MATHEMATICAL PRACTICES
MP1, MP4
Essential Question How can you find the whole given a part and
the percent?
A percent is equivalent to the ratio of a part to a whole. Suppose there
are 20 marbles in a bag and 5 of them are blue. The whole is the total
number of marbles, 20. The part is the number of blue marbles, 5.
5 , is equal to the percent of marbles
The ratio of the part to the whole, ___
20
that are blue, 25%.
part
whole
5 = ______
5 × 5 = ____
25 = 25% ← percent
part → ___
______
whole → 20
20 × 5
100
You can use the relationship among the part, the whole, and the
percent to solve problems.
Unlock
Unlock the
the Problem
Problem
© Houghton Mifflin Harcourt Publishing Company • Image Credits: (c) ©Ronnie Kaufman/Larry Hirshowitz/Blend Images/Getty Images
Emily has sent 28 text messages so far this week. That is 20% of the
total number of text messages she is allowed in one week. What is
the total number of text messages Emily can send in one week?
One Way
Use a double number line.
Think: The whole is the total number of messages Emily can send.
The part is the number of messages Emily has sent so far.
The double number line shows that 20% represents 28 messages.
Find the number of messages represented by 100%.
Think: I want to find 100%. What can
I multiply 20 by to get 100?
0%
20%
28
20 3
5 100
28 3
5
100%
Multiply 28 by the same factor.
So, 28 is 20% of __. Emily can send
__ messages in one week.
Math
Talk
MATHEMATICAL PRACTICES 1
Analyze Relationships
Explain the relationship
among the part, the whole,
and the percent using the
information in this problem.
Chapter 5
301
Another Way
Use equivalent ratios.
Write the relationship among
the percent, part, and whole.
STEP 1
Think: The percent is _%. The
part
percent = ____
whole
20% = ____
_ messages. The
_ is unknown.
part is
STEP 2
Write the percent as a ratio.
STEP 3
Simplify the known ratio.
20 = ____
28
_____
The denominator of the
percent ratio will always be
100 because 100% represents
the whole.
1
28
20 ÷ 20 = ___
_________
= ____
100 ÷
Write an equivalent ratio.
STEP 4
Think: The numerator should be _.
28
1× 28 = ____
_______
5×
28
28 = ____
_____
So, 28 is 20% of _. Emily can send _ messages in one week.
24 is 5% of what number?
Write the relationship among
the percent, part, and whole.
STEP 1
part
percent = ____
whole
Think: The percent is _%. The part
is
5% = ____
_. The _ is unknown.
STEP 2
Write the percent as a ratio.
STEP 3
Simplify the known ratio.
5 = ____
24
_____
24
1 = ____
5÷
________
= ____
100 ÷
STEP 4
Write an equivalent ratio.
Think: The numerator should be _.
24
1×
________
= ____
20 ×
24 = ____
24 Math
_____
Talk
So, 24 is 5% of _.
302
MATHEMATICAL PRACTICES 1
Evaluate What method
could you use to check your
answer to the Example?
© Houghton Mifflin Harcourt Publishing Company
Example
Name
Share
hhow
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and Show
Sh
Find the unknown value.
1. 9 is 25% of _
MATH
M
BOARD
B
0%
25%
9
2. 14 is 10% of _
25 3
5 100
9 3
5
3. 3 is 5% of _
100%
4. 12 is 60% of _
Math
Talk
MATHEMATICAL PRACTICES 8
Use Repeated Reasoning
Explain how to solve a
problem involving a part, a
whole, and a percent.
On
On Your
Your Own
Own
Find the unknown value.
5. 16 is 20% of _
100%
0%
6. 42 is 50% of _
7. 28 is 40% of _
8. 60 is 75% of _
9. 27 is 30% of _
10. 21 is 60% of _
11. 12 is 15% of _
Solve.
© Houghton Mifflin Harcourt Publishing Company
12. 40% of the students in the sixth grade at
Andrew’s school participate in sports. If
52 students participate in sports, how many
sixth graders are there at Andrew’s school?
MATHEMATICAL
PRACTICE
2 Use Reasoning
32
14. 40% = ___
__
13.
DEEPER
There were 136 students and 34
adults at the concert. If 85% of the seats were
filled, how many seats are in the auditorium?
Algebra Find the unknown value.
91
15. 65% = ___
__
54
16. 45% = ___
__
Chapter 5 • Lesson 6 303
MATHEMATICAL PRACTICES
ANALYZEt-00,'034536$563&t13&$*4*0/
OqnakdlRnkuhmf¤@ookhb`shnmr
OqnakdlRnkuhmf¤@ookhb`shnmr
Use the advertisement for 17 and 18.
17. Corey spent 20% of his savings on a printer at Louie’s Electronics. How
much did Corey have in his savings account before he bought the
printer?
18.
SMARTER
Kai spent 90% of his money on a laptop that cost $423.
Does he have enough money left to buy a scanner? Explain.
19. Maurice has completed 17 pages of the research paper he is writing.
That is 85% of the required length of the paper. What is the required
length of the paper?
20.
DEEPER
Of 250 seventh-grade students, 175 walk to school. What
percent of seventh-graders do not walk to school?
21. What's the Error? Kate has made 20 free throws in basketball
games this year. That is 80% of the free throws she has attempted. To
find the total number of free throws she attempted, Kate wrote the
80 = ___. What error did Kate make?
equation ___
100
20
Personal Math Trainer
SMARTER
Maria spent 36% of her savings to buy a smart
phone. The phone cost $90. How much money was in Maria’s savings
account before she purchased the phone? Find the unknown value.
90
36% = _______
304
© Houghton Mifflin Harcourt Publishing Company
22.
Practice and Homework
Name
Find the Whole from a Percent
COMMON CORE STANDARD—6.RP.A.3C
Understand ratio concepts and use ratio reasoning
to solve problems.
Find the unknown value.
1. 9 is 15% of
Lesson 5.6
60
2. 54 is 75% of
3. 12 is 2% of
4. 18 is 50% of
5. 16 is 40% of
6. 56 is 28% of
7. 5 is 10% of
8. 24 is 16% of
9. 15 is 25% of
15
___
100
5
9
___
15 4 5 5 ______
3 3 3 5 __
9
_______
100 4 5
20 3 3
60
Problem
Problem Solving
Solving
© Houghton Mifflin Harcourt Publishing Company
10. Michaela is hiking on a weekend camping trip.
She has walked 6 miles so far. This is 30% of the
total distance. What is the total number of miles
she will walk?
12.
WRITE
11. A customer placed an order with a bakery for
muffins. The baker has completed 37.5% of the
order after baking 81 muffins. How many muffins
did the customer order?
Math Write a question that involves
finding what number is 25% of another number.
Solve using a double number line and check using
equivalent ratios. Compare the methods.
Chapter 5
305
Lesson Check (6.RP.3c)
1. Kareem saves his coins in a jar. 30% of the coins
are pennies. If there are 24 pennies in the jar,
how many coins does Kareem have?
2. A guitar shop has 19 acoustic guitars on display.
This is 19% of the total number of guitars. What
is the total number of guitars the shop has?
3. On a coordinate grid, in which quadrant is the
point ( −5, 4) located?
4. A box contains 16 cherry fruit chews, 15 peach
fruit chews, and 12 plum fruit chews. Which
two flavors are in the ratio 5 to 4?
__ of
5. During basketball season, Marisol made 19
25
her free throws. What percent of her free throws
did Marisol make?
6. Landon is entering the science fair. He has a
budget of $115. He has spent 20% of the money
on new materials. How much does Landon
have left to spend?
FOR MORE PRACTICE
GO TO THE
306
Personal Math Trainer
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Spiral Review (6.NS.C.6b, 6.RP.A.3a, 6.RP.A.3c)
Name
Chapter 5 Review/Test
Personal Math Trainer
Online Assessment
and Intervention
1. What percent is represented by the shaded part?
A
46%
B
60%
C
64%
D
640%
2. Write a percent to represent the shaded part.
3. Rosa made a mosaic wall mural using 42 black tiles, 35 blue tiles
and 23 red tiles. Write a percent to represent the number of red tiles
in the mural.
© Houghton Mifflin Harcourt Publishing Company
4. Model 39%.
Assessment Options
Chapter Test
Chapter 5
307
5. For 5a–5d, choose Yes or No to indicate whether the percent and the
fraction represent the same amount.
5a.
5b.
5c.
5d.
50% and 1_
2
45% and 4_
5
3
_ and 37.5%
8
2 and 210%
__
10
Yes
No
Yes
No
Yes
No
Yes
No
6. The school orchestra has 25 woodwind instruments, 15 percussion
instruments, 31 string instruments, and 30 brass instruments. Select the
portion of the instruments that are percussion. Mark all that apply.
15%
1.5
3
__
20
0.15
8. Select other ways to write 0.875. Mark all that apply.
875%
87.5%
7
_
8
875
___
100
308
© Houghton Mifflin Harcourt Publishing Company
7. For a science project, 3_4 of the students chose to make a poster and
0.25 of the students wrote a report. Rosa said that more students made
a poster than wrote a report. Do you agree with Rosa? Use numbers and
words to support your answer.
Name
9. There are 88 marbles in a bin and 25% of the marbles are red.
22
There are
25
62
red marbles in the bin.
66
10. Harrison has 30 CDs in his music collection. If 40% of the CDs are
country music and 30% are pop music, how many CDs are other
types of music?
_ CDs
11. For numbers 11a–11b, choose <, >, or =.
<
11a.
30% of 90
>
35% of 80
=
<
11b. 25% of 16
>
20% of 25
=
© Houghton Mifflin Harcourt Publishing Company
12.
DEEPER
There were 200 people who voted at the town council
meeting. Of these people, 40% voted for building a new basketball court
in the park. How many people voted against building the new basketball
court? Use numbers and words to explain your answer.
Chapter 5
309
13. James and Sarah went out to lunch. The price of lunch for both of them
was $20. They tipped their server 20% of that amount. How much did
each person pay if they shared the price of lunch and the tip equally?
14. A sandwich shop has 30 stores and 60% of the stores are in California.
The rest of the stores are in Nevada.
Part A
How many stores are in California and how many are in Nevada?
Part B
The shop opens 10 new stores. Some are in California, and some are in
Nevada. Complete the table.
Locations of Sandwich Shops
Percent of Stores
Number of Stores
California
Nevada
45%
Personal Math Trainer
SMARTER
Juanita has saved 35% of the money that she needs
to buy a new bicycle. If she has saved $63, how much money does the
bicycle cost? Use numbers and words to explain your answer.
© Houghton Mifflin Harcourt Publishing Company
15.
310
Name
16. For 16a–16d, choose Yes or No to indicate whether the statement
is correct.
16a. 12 is 20% of 60.
Yes
No
16b. 24 is 50% of 48.
Yes
No
16c. 14 is 75% of 20.
Yes
No
16d. 9 is 30% of 30.
Yes
No
17. Heather and her family are going to the grand opening of a new
amusement park. There is a special price on tickets this weekend.
Tickets cost $56 each. This is 70% of the cost of a regular price ticket.
Part A
What is the cost of a regular price ticket? Show your work.
Part B
© Houghton Mifflin Harcourt Publishing Company
Heather’s mom says that they would save more than $100 if they buy
4 tickets for their family on opening weekend. Do you agree or disagree
with Heather’s mom? Use numbers and words to support your answer.
If her statement is incorrect, explain the correct way to solve it.
18. Elise said that 0.2 equals 2%. Use words and numbers to explain
her mistake.
Chapter 5
311
19. Write 18% as a fraction.
20. Noah wants to put a variety of fish in his new fish tank. His tank is large
enough to hold a maximum of 70 fish.
Part A
Complete the table.
Type of Fish
Percent of Maximum Number
Rainbow fish
20%
Swordtail
40%
Molly
30%
Number of Fish in Tank
Part B
© Houghton Mifflin Harcourt Publishing Company
Has Noah put the maximum number of fish in his tank? Use
number and words to explain how you know. If he has not put the
maximum number of fish in the tank, how many more fish could he
put in the tank?
312
6
Chapter
Units of Measure
Personal Math Trainer
Show Wha t You Know
Online Assessment
and Intervention
Check your understanding of important skills.
Name
Choose the Appropriate Unit Circle the more reasonable unit to measure
the object.
1. the length of a car
2. the length of a soccer field
inches or feet
meters or kilometers
Multiply and Divide by 10, 100, and 1,000 Use mental math.
3. 2.51 × 10
____
6. 3.25 ÷ 10
____
4. 5.3 × 100
____
7. 8.65 ÷ 100
____
5. 0.71 × 1,000
____
8. 56.2 ÷1,000
____
Convert Units Complete.
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9. 12 lb = ■ oz
Think: 1 lb = 16 oz
____
10. 8 c = ■ pt
Think: 2 c = 1 pt
____
11. 84 in. = ■ ft
Think: 12 in. = 1 ft
____
Math in the
A cheetah can run at a rate of 105,600 yards per hour.
Find the number of miles the cheetah could run at this
rate in 5 minutes.
Chapter 6
313
Voca bula ry Builder
Visualize It
Sort the review words into the Venn diagram. One preview word has
been filled in for you.
Review Words
✓ gallon
gram
✓ length
liter
✓ mass
meter
capacity
ounce
pint
pound
✓ quart
ton
✓ weight
Customary
Metric
Preview Words
Understand Vocabulary
✓ capacity
✓ conversion factor
Complete the sentences by using the checked words.
1. A rate in which the two quantities are equal but use different
units is called a ___.
2.
____ is the the amount of matter in an object.
3.
____ is the amount a container can hold.
4. The ____ of an object tells how heavy the
© Houghton Mifflin Harcourt Publishing Company
object is.
5. Inches, feet, and yards are all customary units used to measure
____.
6. A ____ is a larger unit of capacity than a quart.
314
™Interactive Student Edition
™Multimedia eGlossary
Chapter 6 Vocabulary
capacity
common factor
capacidad
factor común
12
9
conversion factor
denominator
factor de conversión
denominador
20
16
formula
numerator
fórmula
numerador
66
34
square unit
weight
unidad cuadrada
peso
96
107
4,
Factors of 20: 1,
2,
4, 5,
8,
16
10,
20
The number below the bar in a fraction that
tells how many equal parts are in the whole
or in the group
3
__
4
denominator
The number above the bar in a fraction
that tells how many equal parts of the
whole are being considered
3
__
4
numerator
How heavy an object is
Example:
soccer ball
14–16 ounces
© Houghton Mifflin Harcourt Publishing Company
2,
© Houghton Mifflin Harcourt Publishing Company
Factors of 16: 1,
© Houghton Mifflin Harcourt Publishing Company
Example:
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© Houghton Mifflin Harcourt Publishing Company
© Houghton Mifflin Harcourt Publishing Company
© Houghton Mifflin Harcourt Publishing Company
© Houghton Mifflin Harcourt Publishing Company
A number that is a factor of two or more
numbers
The amount a container can hold
__ gallon, 2 quarts
Examples: 1
2
A rate in which two quantities are equal,
but use different units
Example: 1 yard = 3 feet, so use the rate
1 yd to convert yards from feet
____
3 ft
A set of symbols that expresses a
mathematical rule
Example: Use d = r × t to find distance.
A unit used to measure area such as
square foot (ft2), square meter (m2), and
so on
Going Places with
Words
Bingo
For 3–6 players
Materials
• 1 set of word cards
• 1 Bingo board for each player
• game markers
Game
Game
Word Box
capacity
common factor
conversion factor
denominator
formula
numerator
square unit
weight
How to Play
1. The caller chooses a card and reads the definition. Then the caller puts the
card in a second pile.
2. Players put a marker on the word that matches the definition each time they
find it on their Bingo boards.
3. Repeat Steps 1 and 2 until a player marks 5 boxes in a line going down,
across, or on a slant and calls “Bingo.”
4. To check the answers, the player who said “Bingo” reads the words aloud
© Houghton Mifflin Harcourt Publishing Company
Image Credits: ©David Madison/Stone/Getty Images
while the caller checks the definitions.
Chapter 6
314A
Journal
Jo
ou
urnal
The Write Way
Reflect
Choose one idea. Write about it.
• Describe a situation in which you might use a conversion factor.
• Sumeer has a jar that holds 4 cups of water. Eliza has a bottle that holds
3 pints of water. Explain whose container has the greater capacity.
• If the area of a room is 20 feet by 6 yards, tell how to figure out the
room’s area in square feet.
© Houghton Mifflin Harcourt Publishing Company
Image Credits: ©David Madison/Stone/Getty Images
• Write and solve a word problem that uses the formula d = r × t.
314B
Lesson 6.1
Name
Convert Units of Length
Ratios and Proportional
Relationships—6.RP.A.3d
MATHEMATICAL PRACTICES
MP1, MP2, MP6
Essential Question How can you use ratio reasoning to convert from one unit
of length to another?
In the customary measurement system, some of the
common units of length are inches, feet, yards, and
miles. You can multiply by an appropriate conversion
factor to convert between units. A conversion factor
is a rate in which the two quantities are equal, but use
different units.
Customary Units of Length
1 foot (ft) 5
1 yard (yd) 5
1 yard 5
1 mile (mi) 5
1 mile 5
12 inches (in.)
36 inches
3 feet
5,280 feet
1,760 yards
Unlock
Unlock the
the Problem
Problem
In a soccer game, Kyle scored a goal. Kyle was 33 feet from
the goal. How many yards from the goal was he?
When the same unit appears
in a numerator and a
denominator, you can divide
out the common unit before
multiplying as you would with
a common factor.
Convert 33 feet to yards.
© Houghton Mifflin Harcourt Publishing Company • Image Credits: (c) ©PhotoDisc/Getty Images
Choose a conversion factor. Think: I’m converting to
yards from feet.
Multiply 33 feet by the conversion factor. Units of feet
appear in a numerator and a denominator, so you can
divide out these units before multiplying.
1 yd
3 ft
1 yard = 3 feet, so use the rate ____.
1 yd
3 ft
1 yd
3 ft
33 ft × ____ =
33 ft × ____ = ____
1
_ yd
So, Kyle was _ yards from the goal.
How many inches from the goal was Kyle?
Choose a conversion factor. Think: I’m converting to
inches from feet.
12 in..
12 inches = 1 foot, so use the rate _____
Multiply 33 ft by the conversion factor.
33 ft × _____
12 in. = ____
12 in. =
33 ft × _____
So, Kyle was __ inches from the goal.
1 ft
1
1 ft
Math
Talk
1 ft
__ in.
MATHEMATICAL PRACTICES 6
Explain How do you know which
unit to use in the numerator and
which unit to use in the denominator
of a conversion factor?
Chapter 6
315
Metric Units You can use a similar process to convert
metric units. Metric units are used throughout most of
the world. One advantage of using the metric system is
that the units are related by powers of 10.
Metric Units of Length
1,000 millimeters (mm) 5
100 centimeters (cm) 5
10 decimeters (dm) 5
1 dekameter (dam) 5
1 hectometer (hm) 5
1 kilometer (km) 5
1 meter (m)
1 meter
1 meter
10 meters
100 meters
1,000 meters
Example A passenger airplane is 73.9 meters
long. What is the length of the airplane in centimeters?
What is the length in kilometers?
One Way
Be sure to use the correct
conversion factor. The units you
are converting from should divide
out, leaving only the units you
are converting to.
Use a conversion factor.
73.9 meters =
■ centimeters
Choose a conversion factor.
cm .
100 cm = 1 m, so use the rate _________
m
Multiply 73.9 meters by the
conversion factor. Divide out
the common units before
multiplying.
73.9 m × _______
cm = __ cm
______
m
1
So, 73.9 meters is equal to __ centimeters.
Another Way
Use powers of 10.
Metric units are related to each other by factors of 10.
kilo-
hecto÷ 10
73.9 meters =
× 10
deka-
÷ 10
× 10
meter
÷ 10
× 10
deci-
÷ 10
× 10
centi-
÷ 10
milli÷ 10
■ kilometers
Use the chart.
Kilometers are 3 places to the left of
meters in the chart. Move the decimal
point 3 places to the left. This is the
same as dividing by 1,000.
73.9
So, 73.9 meters is equal to __ kilometer.
316
0.0739
Math
Talk
MATHEMATICAL PRACTICES 2
Reasoning If you convert
285 centimeters to decimeters,
will the number of decimeters be
greater or less than the number of
centimeters? Explain.
© Houghton Mifflin Harcourt Publishing Company
× 10
× 10
Name
MATH
M
Share
hhow
Share and
and Show
Sh
BOARD
B
Convert to the given unit.
1. 3 miles = ■ yards
2. 43 dm = _ hm
yd
mi
conversion factor: __________
1,760 yd
1 mi
3 mi × _______ = __ yd
3 miles = ____
1
3. 9 yd = _ in.
4. 72 ft = _ yd
5. 7,500 mm = _ dm
Math
Talk
MATHEMATICAL PRACTICES 8
Generalize How do to
convert from inches to yards
and yards to inches?
© Houghton Mifflin Harcourt Publishing Company
On
On Your
Your Own
Own
6. Rohan used 9 yards of ribbon to wrap
gifts. How many inches of ribbon did
he use?
7. One species of frog can grow to a
maximum length of 12.4 millimeters.
What is the maximum length of this frog
species in centimeters?
8. The height of the Empire State Building
measured to the top of the lightning rod
is approximately 443.1 meters. What is this
height in hectometers?
9.
DEEPER
A snail moves at a speed of
2.5 feet per minute. How many yards will
the snail have moved in half of an hour?
Practice: Copy and Solve Compare. Write <, >, or =.
10. 32 feet
● 11 yards
11. 537 cm
● 5.37 m
12. 75 inches
● 6 feet
Chapter 6 • Lesson 1 317
MATHEMATICAL PRACTICES .0%&-t3&"40/tM",&4&/4&
OqnakdlRnkuhmf¤@ookhb`shnmr
OqnakdlRnkuhmf¤@ookhb`shnmr
What’s the Error?
13.
SMARTER
The Redwood National Park
is home to some of the largest trees in the world.
Hyperion is the tallest tree in the park, with a
height of approximately 379 feet. Tom wants
to find the height of the tree in yards.
Tom converted the height this way:
3 feet = 1 yard
3 ft
conversion factor: ____
1 yd
379
ft × ____
3 ft = 1,137 yd
_____
1
1 yd
So, 379 feet = __ yards.
•
14.
MATHEMATICAL
PRACTICE
6 Explain how you knew Tom’s answer was incorrect.
SMARTER
14a. 12 yards
318
Choose <, >, or =.
<
>
=
432 inches
14b.
321 cm
<
>
=
32.1 m
© Houghton Mifflin Harcourt Publishing Company • Image Credits: (r) ©Vito Palmisano/Getty Images
Show how to correctly
convert from 379 feet to yards.
Find and describe
Tom’s error.
Practice and Homework
Name
Lesson 6.1
Convert Units of Length
COMMON CORE STANDARD—6.RP.A.3d
Understand ratio concepts and use ratio
reasoning to solve problems.
Convert to the given unit.
1. 42 ft =
yd
1 yd
conversion factor: ____
3 ft
1
yd
____
42 ft 3
3 ft
2. 2,350 m =
km
3. 18 ft =
in.
42 ft 5 14 yd
4. 289 m =
dm
5. 5 mi =
6. 35 mm =
yd
cm
Compare. Write <, >, or =.
7. 1.9 dm
10. 98 in.
1,900 mm
8 ft
8. 12 ft
9. 56 cm
4 yd
11. 64 cm
630 mm
12. 2 mi
56,000 km
10,560 ft
© Houghton Mifflin Harcourt Publishing Company
Problem
Problem Solving
Solving
13. The giant swallowtail is the largest butterfly in
the United States. Its wingspan can be as large as
16 centimeters. What is the maximum wingspan
in millimeters?
_______
15.
WRITE
14. The 102nd floor of the Sears Tower in Chicago
is the highest occupied floor. It is 1,431 feet
above the ground. How many yards above the
ground is the 102nd floor?
_______
Math Explain why units can be divided out when
measurements are multiplied.
Chapter 6
319
Lesson Check (6.RP.A.3d)
1. Justin rides his bicycle 2.5 kilometers to school.
Luke walks 1,950 meters to school. How much
farther does Justin ride to school than Luke
walks to school?
2. The length of a room is 10 1_2 feet. What is the
length of the room in inches?
Spiral Review (6.NS.C.8, 6.RP.A.3a, 6.RP.A.3c)
3. Each unit on the map represents 1 mile. What
is the distance between the campground and
the waterfall?
0
Supply Store
Playground
Campground
Waterfall
1 2 3 4 5 6 7 8 9
__ of all teens have
5. According to a 2008 survey, 29
50
sent at least one text message in their lives.
What percent of teens have sent a text message?
6. Of the students in Ms. Danver’s class, 6 walk
to school. This represents 30% of her students.
How many students are in Ms. Danver’s class?
FOR MORE PRACTICE
GO TO THE
320
Personal Math Trainer
© Houghton Mifflin Harcourt Publishing Company
9
8
7
6
5
4
3
2
1
4. On a field trip, 2 vans can carry 32 students.
How many students can go on a field trip when
there are 6 vans?
Lesson 6.2
Name
Convert Units of Capacity
Ratios and Proportional
Relationships—6.RP.A.3d
MATHEMATICAL PRACTICES
MP2, MP4, MP6, MP8
Essential Question How can you use ratio reasoning to convert from
one unit of capacity to another?
Capacity measures the amount a container can hold
when filled. In the customary measurement system,
some common units of capacity are fluid ounces, cups,
pints, quarts, and gallons. You can convert between
units by multiplying the given units by an appropriate
conversion factor.
Customary Units of Capacity
8 fluid ounces (fl oz)
2 cups
2 pints
4 cups
4 quarts
5
5
5
5
5
1 cup (c)
1 pint (pt)
1 quart (qt)
1 quart
1 gallon (gal)
Unlock
Unlock the
the Problem
Problem
• How are quarts and gallons related?
A dairy cow produces about 25 quarts of milk
each day. How many gallons of milk does the cow
produce each day?
1 gal
• Why can you multiply a quantity by ____
4 qt
without changing the value of the quantity?
Convert 25 quarts to gallons.
Choose a conversion factor. Think: I’m
converting to gallons from quarts.
1 gal
1 gallon = 4 quarts, so use the rate ____
.
4 qt
Multiply 25 qt by the conversion factor.
25 qt × ____ = _____ × ____ = 6 ____ gal
The fractional part of the answer can
be renamed using the smaller unit.
1 gal
4 qt
25 qt
1
1 gal
4 qt
4
6 ____ gal = _ gallons, _ quart
4
So, the cow produces _ gallons, _ quart of milk each day.
© Houghton Mifflin Harcourt Publishing Company
How many pints of milk does a cow produce each day?
Choose a conversion factor. Think: I’m
converting to pints from quarts.
Multiply 25 qt by the conversion factor.
pt
qt
2 pints = 1 quart, so use the rate _________.
pt
25 qt
1
pt
_____ × ______ = _ pt
25 qt × ______
qt =
qt
So, the cow produces _ pints of milk each day.
Chapter 6
321
Metric Units You can use a similar process to convert
metric units of capacity. Just like metric units of length,
metric units of capacity are related by powers of 10.
Metric Units of Capacity
1,000 milliliters (mL) 5
100 centiliters (cL) 5
10 deciliters (dL) 5
1 dekaliter (daL) 5
1 hectoliter (hL) 5
1 kiloliter (kL) 5
1 liter (L)
1 liter
1 liter
10 liters
100 liters
1,000 liters
Example
A piece of Native American pottery
has a capacity of 1.7 liters. What is the capacity of the pot in
dekaliters? What is the capacity in milliliters?
One Way
Use a conversion factor.
1.7 liters 5
Choose a conversion factor.
■ dekaliters
1 dekaliter = 10 liters, so use the rate
daL.
_________
L
daL = __ daL
1.7 L × _______
____
L
1
So, 1.7 liters is equivalent to __ dekaliter.
Another Way
× 10
× 10
kilo-
hecto÷ 10
1.7 liters 5
Use powers of 10.
× 10
deka-
÷ 10
× 10
liter
÷ 10
× 10
× 10
deci-
centi-
÷ 10
÷ 10
1700.
Math
Talk
milli÷ 10
■ milliliters
Use the chart.
Milliliters are 3 places to the right of liters. So,
move the decimal point 3 places to the right.
1.7
So, 1.7 liters is equal to __ milliliters.
•
322
MATHEMATICAL
PRACTICE
Describe a Method Describe how you would
convert kiloliters to milliliters.
6
MATHEMATICAL PRACTICES 6
Explain Why can't you convert
between units in the customary
system by moving the decimal point
left or right?
© Houghton Mifflin Harcourt Publishing Company • Image Credits: ©American School/The Bridgeman Art Library/Getty Images
Multiply 1.7 L by the conversion
factor.
Name
Share
hhow
Share and
and Show
Sh
MATH
M
BOARD
B
Convert to the given unit.
1. 5 quarts = ■ cups
2. 6.7 liters = _ hectoliters
c
conversion factor: ________
qt
5 qt
1
4c =_c
5 quarts = ____ × ____
1 qt
3. 5.3 kL = _ L
4. 36 qt = _ gal
5. 5,000 mL = _ cL
Math
Talk
Compare the customary and metric
systems. In which system is it
easier to convert from one unit to
another?
On
On Your
Your Own
Own
6. It takes 41 gallons of water for a washing
machine to wash a load of laundry.
How many quarts of water does it take
to wash one load?
8.
MATHEMATICAL
PRACTICE
7. Sam squeezed 237 milliliters of juice from
4 oranges. How many liters of juice did
Sam squeeze?
9.
2 Reason Quantitatively A
bottle contains 3.78 liters of water. Without
calculating, determine whether there are
more or less than 3.78 deciliters of water in
the bottle. Explain your reasoning.
© Houghton Mifflin Harcourt Publishing Company
MATHEMATICAL PRACTICES 6
DEEPER
Tonya has a 1-quart, a 2-quart,
and a 3-quart bowl. A recipe asks for 16
ounces of milk. If Tonya is going to triple the
recipe, what is the smallest bowl that will
hold the milk?
Practice: Copy and Solve Compare. Write <, >, or =.
10. 700,000 L
13. 10 pt
● 70 kL
● 5 qt
11. 6 gal
● 30 qt
14. 500 mL
● 50 L
12. 54 kL
15. 14 c
● 540,000 dL
● 4 qt
Chapter 6 • Lesson 2 323
MATHEMATICAL PRACTICES
COMMUNICA5&t1&34E7&3&tCONSTRUCT ARGUMENTS
Unlock
Unlock the
the Problem
Problem
16.
SMARTER
Jeffrey
is loading cases of bottled
water onto a freight elevator.
There are 24 one-pint bottles
in each case. The maximum
weight that the elevator can
carry is 1,000 pounds. If
1 gallon of water weighs 8.35 pounds, what is
the maximum number of full cases Jeffrey can
load onto the elevator?
c. How can you find the number of cases that
Jeffrey can load onto the elevator?
a. What do you need to find?
d. What is the maximum number of full cases
Jeffrey can load onto the elevator?
WRITE
17.
324
DEEPER
Monica put 1 liter, 1 deciliter,
1 centiliter, and 1 milliliter of water into a
bowl. How many milliliters of water did she
put in the bowl?
18.
M t Show Your Work
Math
SMARTER
Select the conversions that
are equivalent to 235 liters. Mark
all that apply.
A
235,000 milliliters
B
0.235 milliliters
C
235,000 kiloliters
D
0.235 kiloliters
© Houghton Mifflin Harcourt Publishing Company
b. How can you find the weight of 1 case of
bottled water? What is the weight?
Practice and Homework
Name
Lesson 6.2
Convert Units of Capacity
COMMON CORE STANDARD—6.RP.A.3d
Understand ratio concepts and use ratio
reasoning to solve problems.
Convert to the given unit.
1. 7 gallons =
2. 5.1 liters =
quarts
4 qt
conversion factor: ____
1 gal
4 qt
7 gal 3
1 gal
7 gal = 28 qt
kiloliters
Move the decimal point 3 places to the left.
_
5.1 liters = 0.0051 kiloliters
3. 20 qt =
gal
4. 40 L =
6. 29 cL =
daL
7. 7.7 kL =
5. 33 pt =
mL
8. 24 fl oz =
cL
qt
pt
pt
c
© Houghton Mifflin Harcourt Publishing Company
Problem
Problem Solving
Solving
9. A bottle contains 3.5 liters of water. A second
bottle contains 3,750 milliliters of water. How
many more milliliters are in the larger bottle
than in the smaller bottle?
_______
11.
WRITE
10. Arnie’s car used 100 cups of gasoline during
a drive. He paid $3.12 per gallon for gas.
How much did the gas cost?
_______
Math Explain how units of length and capacity are similar
in the metric system.
Chapter 6
325
Lesson Check (6.RP.A.3d)
1. Gina filled a tub with 25 quarts of water. What is
this amount in gallons and quarts?
2. Four horses are pulling a wagon. Each horse
drinks 45,000 milliliters of water each day.
How many liters of water will the horses drink
in 5 days?
Spiral Review (6.NS.C.8, 6.RP.A.2, 6.RP.A.3b, 6.RP.A.3c, 6.RP.A.3d)
3. The map shows Henry’s town. Each unit
represents 1 kilometer. After school, Henry
walks to the library. How far does he walk?
6
4. An elevator travels 117 feet in 6.5 seconds.
What is the elevator’s speed as a unit rate?
y
4
2
x
-6
-4
-2
Library
0
-2
2
4
6
School
-4
5. Julie’s MP3 player contains 860 songs. If 20% of
the songs are rap songs and 15% of the songs
are R&B songs, how many of the songs are other
types of songs?
6. How many kilometers are equivalent to
3,570 meters?
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-6
Lesson 6.3
Name
Convert Units of Weight and Mass
Ratios and Proportional
Relationships—6.RP.A.3d
MATHEMATICAL PRACTICES
MP1, MP2, MP3, MP4
Essential Question How can you use ratio reasoning to convert from
one unit of weight or mass to another?
The weight of an object is a measure of how heavy
it is. Units of weight in the customary measurement
system include ounces, pounds, and tons.
Customary Units of Weight
1 pound (lb) 5 16 ounces (oz)
1 ton (T) 5 2,000 pounds
Unlock
Unlock the
the Problem
Problem
The largest pearl ever found weighed 226 ounces.
What was the pearl’s weight in pounds?
• How are ounces and pounds related?
• Will you expect the number of pounds to be
greater than 226 or less than 226? Explain.
Convert 226 ounces to pounds.
Choose a conversion factor.
Think: I’m converting to
pounds from ounces.
Multiply 226 ounces by the
conversion factor.
Think: The fractional part of
© Houghton Mifflin Harcourt Publishing Company • Image Credits: (b) @Bon Appetit/Alamy
the answer can be renamed
using the smaller unit.
lb .
1 lb = 16 oz, so use the rate _________
oz
oz × _____
1 lb = 226
1 lb =
______
226 oz × _____
1
16 oz
16 oz
____ lb
16
____ lb = _ lb, _ oz
16
So, the largest pearl weighed _ pounds, _ ounces.
The largest emerald ever found weighed 38 pounds.
What was its weight in ounces?
Choose a conversion factor.
Think: I’m converting to
ounces from pounds.
oz .
16 oz = 1 lb, so use the rate ________
Multiply 38 lb by the
conversion factor.
oz = 38
lb × 16
oz = __ oz
_____
_____
_____
38 lb × 16
lb
1 lb
1
1 lb
So, the emerald weighed __ ounces.
1.
MATHEMATICAL
PRACTICE
4 Model Mathematics Explain how you could convert the
emerald’s weight to tons.
Chapter 6
327
Metric Units The amount of matter in an object
is called the mass. Metric units of mass are
related by powers of 10.
Metric Units of Mass
1,000 milligrams (mg) 5 1 gram (g)
100 centigrams (cg) 5 1 gram
10 decigrams (dg) 5 1 gram
1 dekagram (dag) 5 10 grams
1 hectogram (hg) 5 100 grams
1 kilogram (kg) 5 1,000 grams
Example Corinne caught a trout with a mass of 2,570 grams.
What was the mass of the trout in centigrams? What was the
mass in kilograms?
One Way Use a conversion factor.
2,570 grams to centigrams
cg
Choose a conversion factor.
100 cg = 1 g, so use the rate ________
g.
Multiply 2,570 g by the
conversion factor.
2,570 g
100 cg
_______
× ______ =
1
1g
__ cg
So, the trout’s mass was __ centigrams.
Another Way Use powers of 10.
Recall that metric units are related to each other by factors of 10.
× 10
× 10
kilo-
hecto÷ 10
× 10
deka-
÷ 10
× 10
gram
÷ 10
× 10
deci-
÷ 10
× 10
centi-
÷ 10
milli÷ 10
2,570 grams to kilograms
Use the chart.
2570.
2.570
So, 2,570 grams = __ kilograms.
2.
MATHEMATICAL
PRACTICE
1 Describe Relationships Suppose hoots and goots are
units of weight, and 2 hoots = 4 goots. Which is heavier, a hoot or a
goot? Explain.
328
Math
Talk
MATHEMATICAL PRACTICES 2
Reason Quantitatively
Compare objects with masses
of 1 dg and 1 dag. Which
has a greater mass? Explain.
© Houghton Mifflin Harcourt Publishing Company
Kilograms are 3 places to the left of grams.
Move the decimal point 3 places to the left.
Name
hhow
Share
Share and
and Show
Sh
MATH
M
BOARD
B
Convert to the given unit.
1. 9 pounds =
2. 3.77 grams = _ dekagram
ounces
oz
conversion factor: ________
lb
oz = _ oz
_____
9 pounds = 9 lb × 16
1 lb
3. Amanda’s computer weighs 56 ounces.
How many pounds does it weigh?
4. A honeybee can carry 40 mg of nectar.
How many grams of nectar can a
honeybee carry?
Math
Talk
On
On Your
Your Own
Own
MATHEMATICAL PRACTICES 3
Compare How are metric
units of capacity and
mass alike? How are they
different?
Convert to the given unit.
5. 4 lb = __ oz
6. 7.13 g = __ cg
© Houghton Mifflin Harcourt Publishing Company
8. The African Goliath frog can weigh up
to 7 pounds. How many ounces can the
Goliath frog weigh?
9.
7. 3 T = __ lb
DEEPER
The mass of a standard
hockey puck must be at least 156 grams.
What is the minimum mass of 8 hockey
pucks in kilograms?
Practice: Copy and Solve Compare. Write <, >, or =.
10. 250 lb
13.
● 0.25 T
11. 65.3 hg
● 653 dag
12. 5 T
● 5,000 lb
SMARTER
Masses of precious stones are measured in carats, where
1 carat = 200 milligrams. What is the mass of a 50-dg diamond in carats?
Chapter 6 • Lesson 3 329
MATHEMATICAL PRACTICES
ANALYZEt-00,'034536$563&t13&$*4*0/
OqnakdlRnkuhmf¤@ookhb`shnmr
OqnakdlRnkuhmf¤@ookhb`shnmr
Use the table for 14–17.
Sport Ball Weights (in ounces)
14. Express the weight range for bowling balls in
pounds.
16.
17.
DEEPER
How many more pounds does the
heaviest soccer ball weigh than the heaviest
baseball? Round your answer to the nearest
hundredth.
A manufacturer produces
3 tons of baseballs per day and packs
them in cartons of 24 baseballs each.
If all of the balls are the minimum
allowable weight, how many cartons
of balls does the company produce
each day?
MATHEMATICAL
PRACTICE
160–256
14–16
5 Communicate Explain how you
SMARTER
The Wilson family’s newborn
baby weighs 84 ounces. Choose the numbers to
show the baby’s weight in pounds and ounces.
5
6
7
330
2.1–2.3
SMARTER
could use mental math to estimate the number of
soccer balls it would take to produce a total weight
of 1 ton.
18.
5–5.25
pounds
3
4
5
ounces
WRITE
Math t Show Your Work
© Houghton Mifflin Harcourt Publishing Company
15.
Practice and Homework
Name
Lesson 6.3
Convert Units of Weight and Mass
COMMON CORE STANDARD—6.RP.A.3d
Understand ratio concepts and use ratio
reasoning to solve problems.
Convert to the given unit.
1. 5 pounds =
2. 2.36 grams =
ounces
16 oz
conversion factor: _____
Move the decimal point 2 places to the left.
1 lb
16 oz = 80oz
5 pounds = 5 lb × _____
2.36 grams = 0.0236 hectogram
1 lb
3. 30 g =
4. 17.2 hg =
dg
6. 38,600 mg =
dag
hectograms
7. 87 oz =
g
lb
oz
5. 400 lb =
T
8. 0.65 T =
lb
Problem
Problem Solving
Solving
© Houghton Mifflin Harcourt Publishing Company
9. Maggie bought 52 ounces of swordfish selling for
$6.92 per pound. What was the total cost?
11.
WRITE
10. Three bunches of grapes have masses of 1,000
centigrams, 1,000 decigrams, and 1,000 grams,
respectively. What is the total combined mass of
the grapes in kilograms?
Math Explain how you would find the number of
ounces in 0.25T.
Chapter 6
331
Lesson Check (6.RP.A.3d)
1. The mass of Denise’s rock sample is 684 grams.
The mass of Pauline’s rock sample is 29,510
centigrams. How much greater is the mass of
Denise’s sample than Pauline’s sample?
2. A sign at the entrance to a bridge reads:
Maximum allowable weight 2.25 tons. Jason’s
truck weighs 2,150 pounds. How much
additional weight can he carry?
Spiral Review (6.RP.A.1, 6.RP.A.2, 6.RP.A.3a, 6.RP.A.3b, 6.RP.A.3c)
4. Miguel hiked 3 miles in 54 minutes. At this rate,
how long will it take him to hike 5 miles?
5. Marco borrowed $150 from his brother. He has
paid back 30% so far. How much money does
Marco still owe his brother?
6. How many milliliters are equivalent to
2.7 liters?
© Houghton Mifflin Harcourt Publishing Company
3. There are 23 students in a math class. Twelve of
them are boys. What is the ratio of girls to total
number of students?
FOR MORE PRACTICE
GO TO THE
332
Personal Math Trainer
Name
Mid-Chapter Checkpoint
Vocabulary
Vocabulary
Online Assessment
and Intervention
Vocabulary
Choose the best term from the box to complete the sentence.
1. A ___ is a rate in which the two quantities
are equal, but use different units. (p. 315)
2.
Personal Math Trainer
capacity
conversion factor
metric system
___ is the amount a container can hold. (p. 321)
Concepts
Concepts and
and Skills
Skills
Convert units to solve. (6.RP.A.3d)
3. A professional football field is 160 feet wide.
What is the width of the field in yards?
4. Julia drinks 8 cups of water per day. How many
quarts of water does she drink per day?
5. The mass of Hinto’s math book is 4,458 grams.
What is the mass of 4 math books in kilograms?
6. Turning off the water while brushing your teeth
saves 379 centiliters of water. How many liters of
water can you save if you turn off the water the
next 3 times you brush your teeth?
Convert to the given unit. (6.RP.A.3d)
© Houghton Mifflin Harcourt Publishing Company
7. 34.2 mm = _ cm
10. 4 gal = _ qt
8. 42 in. = _ ft
11. 53 dL = _ daL
9. 1.4 km = _ hm
12. 28 c = _ pt
Chapter 6
333
13. Trenton’s laptop is 32 centimeters wide. What is the width of the laptop
in decimeters? (6.RP.A.3d)
14. A truck is carrying 8 cars weighing an average of 4,500 pounds each.
What is the total weight in tons of the cars on the truck? (6.RP.A.3d)
15.
DEEPER
Ben’s living room is a rectangle measuring 10 yards by
168 inches. By how many feet does the length of the room exceed
the width? (6.RP.A.3d)
17. Kaylah’s cell phone has a mass of 50,000 centigrams. What is the mass of
her phone in grams? (6.RP.A.3d)
334
© Houghton Mifflin Harcourt Publishing Company
16. Jessie served 13 pints of orange juice at her party. How many
quarts of orange juice did she serve? (6.RP.A.3d)
Lesson 6.4
Name
Transform Units
Ratios and Proportional
Relationships—6.RP.A.3d
MATHEMATICAL PRACTICES
MP1, MP3, MP5, MP6
Essential Question How can you transform units to solve problems?
You can sometimes use the units of the quantities in a problem to
help you decide how to solve the problem.
Unlock
Unlock the
the Problem
Problem
A car's gas mileage is the average distance the car
can travel on 1 gallon of gas. Maria's car has a gas
mileage of 20 miles per gallon. How many miles
can Maria travel on 9 gallons of gas?
• Would you expect the answer to be greater or
less than 20 miles? Why?
Analyze the units in the problem.
STEP 1 Identify the units.
__________
Gas mileage: 20 miles per gallon = 20
You know two quantities:
the car’s gas mileage and
the amount of gas.
1
Amount of gas: 9 __
You want to know a third quantity: Distance:
the distance the car can travel.
■ __
STEP 2 Determine the relationship among the units.
20 miles
Think: The answer needs to have units of miles. If I multiply ______
by
1 gallon
9 gallons, I can divide out units of gallons. The product will have units of
__, which is what I want.
STEP 3 Use the relationship.
9 gal
20 mi × 9 gal = 20
mi × _____ =
_____
_____
1
© Houghton Mifflin Harcourt Publishing Company
1 gal
1 gal
__
So, Maria can travel __ on 9 gallons of gas.
1. Explain why the units of gallons are crossed out in the multiplication
step above.
Chapter 6
335
Sometimes you may need to convert units before solving a problem.
Example
The material for a rectangular awning has an area of
315 square feet. If the width of the material is 5 yards,
what is the length of the material in feet? (Recall that the
area of a rectangle is equal to its length times its width.)
STEP 1
Identify the units.
You know two quantities: the area
of the material and the width of the
material.
You want to know a third quantity:
the length of the material.
STEP 2
Area: 315 sq ft = 315 ft × ft
Width: 5 _
Length: ■ ft
You can write units of area as
products.
Determine the relationship among the units.
Think: The answer needs to have units
of feet. So, I should convert the width
from yards to feet.
sq ft = ft × ft
5 yd
ft
Width: ____ × ______=
1
1 yd
ft
Think: If I divide the area by the width I can divide out units of feet. The
quotient will have units of
Use the relationship.
Divide the area by the
width to find the length.
315 sq ft ÷ _ ft
Write the division using
a fraction bar.
sq ft
__________
Write the units of area as a product
and divide out the common units.
ft × ft =
____________
15 ft
ft
So, the length of the material is __.
2.
MATHEMATICAL
PRACTICE
ft
Math
Talk
3 Apply Explain how knowing how to find the area of a
rectangle could help you solve the problem above.
3.
MATHEMATICAL
PRACTICE
6 Explain why the answer is in feet even though units of
feet are divided out.
336
MATHEMATICAL PRACTICES 1
Analyze How can examining
the units in a problem help
you solve the problem?
© Houghton Mifflin Harcourt Publishing Company • Image Credits: (t) Victoria Smith/HRW
STEP 3
_ , which is what I want.
Name
Share
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and Show
Sh
MATH
M
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B
1. A dripping faucet leaks
12 gallons of water per day.
How many gallons does the faucet
leak in 6 days?
________ and _ days
Quantities you know: 12
1
Quantity you want to know: ■
gal
________
×
__
days = __
1 day
So, the faucet leaks __ in 6 days.
2. Bananas sell for $0.44 per pound. How much will
7 pounds of bananas cost?
3. Grizzly Park is a rectangular park with an area of
24 square miles. The park is 3 miles wide. What is
its length in miles?
On
On Your
Your Own
Own
Multiply or divide the quantities.
24 kg
4. _____ × 15 min
1 min
© Houghton Mifflin Harcourt Publishing Company
7.
L × 9 hr
____
6. 17
5. 216 sq cm ÷ 8 cm
DEEPER
The rectangular rug in Marcia’s living
room measures 12 feet by 108 inches. What is the
rug’s area in square feet?
8.
1 hr
MATHEMATICAL
PRACTICE
1 Make Sense of Problems
A box-making machine makes cardboard
boxes at a rate of 72 boxes per minute. How
many minutes does it take to make 360 boxes?
Personal Math Trainer
9.
SMARTER
The area of an Olympic-size swimming pool is
1,250 square meters. The length of the pool is 5,000 centimeters.
Select True or False for each statement.
9a.
The length of the pool is 50 meters.
True
False
9b.
The width of the pool is 25 meters.
True
False
True
False
9c. The area of the pool is 1.25 square kilometers
Chapter 6 • Lesson 4 337
Make Predictions
A prediction is a guess about something in the future.
A prediction is more likely to be accurate if it is based on
facts and logical reasoning.
The Hoover Dam is one of America’s largest producers
of hydroelectric power. Up to 300,000 gallons of water
can move through the dam’s generators every second.
Predict the amount of water that moves through the
generators in half of an hour.
FACT
300,000 gallons per
second
PREDICTION
? gallons
in half of an hour
Use what you know about transforming units to make a prediction.
You know the rate of the water through the
generators, and you are given an amount of time.
__ _
Rate of flow: _____________; time: 1
You want to find the amount of water.
Amount of water: ■ gallons
Convert the amount of time to seconds to match the
units in the rate.
1
__ hr =
2
gal
1 sec
Multiply the rate by the amount of time to find the
amount of water.
1 min
__ sec
gal
sec =
_____________
× ___________
sec
1
__ gal
So, a good prediction of the amount of water that moves through the
generators in half of an hour is ____.
Transform units to solve.
10. An average of 19,230 people tour the Hoover
Dam each week. Predict the number of people
touring the dam in a year.
338
11.
SMARTER
The Hoover
Dam generates an average of
about 11,506,000 kilowatt-hours of electricity
per day. Predict the number of kilowatt-hours
generated in 7 weeks.
© Houghton Mifflin Harcourt Publishing Company • Image Credits: (t) ©Ron Chapple Stock/Alamy Images
1
Math t Show Your Work
WRITE
min
30 min × _________
sec =
______
2
Practice and Homework
Name
Lesson 6.4
Transform Units
COMMON CORE STANDARD—6.RP.A.3d
Understand ratio concepts and use ratio
reasoning to solve problems.
Multiply or divide the quantities.
62 g
1. _____× 4 days
2. 322 sq yd ÷ 23 yd
1 day
322
sq yd
________
23 yd
62 g
4 days
_____
3 _____ 5 248 g
1 day
1
322
yd 3 yd
__________
5 14 yd
23 yd
128 kg
3. ______ × 10 hr
4. 136 sq km ÷ 8 km
88 lb
5. ____
× 12 days
1 day
6. 154 sq mm ÷ 11 mm
$150
7. _____× 20 sq ft
8. 234 sq ft ÷ 18 ft
1 hr
1 sq ft
Problem
Problem Solving
Solving
© Houghton Mifflin Harcourt Publishing Company
9. Green grapes are on sale for $2.50 a pound. How
much will 9 pounds cost?
_______
11.
WRITE
10. A car travels 32 miles for each gallon of gas.
How many gallons of gas does it need to
travel 192 miles?
_______
Math Write and solve a problem in which you have to
transform units. Use the rate 45 people per hour in your question.
Chapter 6
339
Lesson Check (6.RP.A.3d)
1. A rectangular parking lot has an area of
682 square yards. The lot is 22 yards wide.
What is the length of the parking lot?
2. A machine assembles 44 key chains per hour.
How many key chains does the machine
assemble in 11 hours?
Spiral Review (6.RP.A.3a, 6.RP.A.3c)
8
3. Three of these ratios are equivalent to __
20 . Which
one is NOT equivalent?
2
_
5
12
__
24
16
__
40
4. The graph shows the money that Marco earns
for different numbers of days worked. How
much money does he earn per day?
40
___
100
720
640
560
480
400
320
240
160
80
0
1 2 3 4 5 6 7 8 9
Days Worked
FOR MORE PRACTICE
GO TO THE
340
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5. Megan answered 18 questions correctly on
a test. That is 75% of the total number of
questions. How many questions were on
the test?
Dollars
Money Earned
PROBLEM SOLVING
Name
Lesson 6.5
Problem Solving • Distance, Rate, and
Time Formulas
Ratios and Proportional
Relationships—6.RP.A.3d
MATHEMATICAL PRACTICES
MP1, MP7
Essential Question How can you use the strategy use a formula to solve
problems involving distance, rate, and time?
You can solve problems involving distance, rate, and time by using
the formulas below. In each formula, d represents distance,
r represents rate, and t represents time.
Distance, Rate, and Time Formulas
To find distance, use
d=r×t
To find rate, use
r=d÷t
To find time, use
t=d÷r
Unlock
Unlock the
the Problem
Problem
Helena drives 220 miles to visit Niagara Falls. She drives at an average
speed of 55 miles per hour. How long does the trip take?
Use the graphic organizer to help you solve the problem.
Read the Problem
What do I need to find?
© Houghton Mifflin Harcourt Publishing Company • Image Credits: (r) ©Frank Krahmer/Getty Images
I need to find the __ the trip takes.
What information do I need to use?
I need to use the __ Helena travels and
the _ of speed her car is moving.
Solve the Problem
• First write the formula for finding time.
t=d÷r
• Next substitute the values for d and r.
mi
t = _ mi ÷ ________
1 hr
• Rewrite the division as multiplication by the
mi.
_____
reciprocal of 55
1 hr
How will I use the information?
First I will choose the formula __ because I
need to find time. Next I will substitute for d and r. Then
I will __ to find the time.
mi
1 hr = _ hr
t = ________ × ________
1
Math
Talk
mi
MATHEMATICAL PRACTICES 7
Look for Structure How do
you know which formula
to use?
So, the trip takes __ hours.
Chapter 6
341
Try Another Problem
Santiago's class traveled to the Museum of Natural Science for
a field trip. To reach the destination, the bus traveled at a rate of
65 miles per hour for 2 hours. What distance did Santiago's class
travel?
Choose a formula.
d=r×t
r=d÷t
t=d÷r
Use the graphic organizer below to help you solve the problem.
Read the Problem
Solve the Problem
What do I need to find?
How will I use the information?
So, Santiago's class traveled __ miles.
Math
Talk
1. What if the bus traveled at a rate of 55 miles per hour for 2.5 hours?
How would the distance be affected?
2.
342
MATHEMATICAL
PRACTICE
MATHEMATICAL PRACTICES 1
Evaluate How could you
check your answer by solving
the problem a different way?
Identify Relationships Describe how to find the rate if you are
given the distance and time.
7
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What information do I need to use?
Name
Share
hhow
Share and
and Show
Sh
MATH
M
BOARD
B
Unlock the Problem
√
√
Choose the appropriate formula.
Include the unit in your answer.
1. Mariana runs at a rate of 180 meters per minute.
How far does she run in 5 minutes?
First, choose a formula.
WRITE
Math t Show Your Work
Next, substitute the values into the formula and solve.
So, Mariana runs ___ in 5 minutes.
2.
What if Mariana runs for 20 minutes at
the same speed? How many kilometers will she run?
SMARTER
3. A car traveled 130 miles in 2 hours. How fast did the car
travel?
4. A subway car travels at a rate of 32 feet per second. How far
does it travel in 16 seconds?
5. A garden snail travels at a rate of 2.6 feet per minute.
At this rate, how long will it take for the snail to travel
65 feet?
© Houghton Mifflin Harcourt Publishing Company
6.
7.
DEEPER
A squirrel can run at a maximum speed of
12 miles per hour. At this rate, how many seconds will it
take the squirrel to run 3 miles?
SMARTER
A cyclist rides 8 miles in 32 minutes. What
is the speed of the cyclist in miles per hour?
Chapter 6 • Lesson 5 343
MATHEMATICAL PRACTICES .0%&-t3&"40/tM",&4&/4&
On
On Your
Your Own
Own
8. A pilot flies 441 kilometers in 31.5 minutes. What is the speed
of the airplane?
9.
DEEPER
Chris spent half of his money on a pair of
headphones. Then he spent half of his remaining money on
CDs. Finally, he spent his remaining $12.75 on a book. How
much money did Chris have to begin with?
WRITE
10.
11.
SMARTER
André and Yazmeen leave at the same
time and travel 75 miles to a fair. André drives 11 miles
in 12 minutes. Yazmeen drives 26 miles in 24 minutes.
If they continue at the same rates, who will arrive at the
fair first? Explain.
MATHEMATICAL
PRACTICE
Math
Show Your Work
3 Make Arguments Bonnie says that if she drives
at an average rate of 40 miles per hour, it will take her about
2 hours to drive 20 miles across town. Does Bonnie’s statement
make sense? Explain.
Personal Math Trainer
SMARTER
Claire says that if she runs at an average
rate of 6 miles per hour, it will take her about
2 hours to run 18 miles. Do you agree or disagree with
Claire? Use numbers and words to support your answer.
© Houghton Mifflin Harcourt Publishing Company
12.
344
Practice and Homework
Name
Lesson 6.5
Problem Solving • Distance, Rate,
and Time Formulas
COMMON CORE STANDARD—6.RP.A.3d
Understand ratio concepts and use ratio
reasoning to solve problems.
Read each problem and solve.
1. A downhill skier is traveling at a rate of 0.5 mile per
minute. How far will the skier travel in 18 minutes?
d=r×t
0.5
mi
_____
d = 1 min × 18 min
d = 9 miles
2. How long will it take a seal swimming at a speed of
8 miles per hour to travel 52 miles?
______
3. A dragonfly traveled at a rate of 35 miles per hour for
2.5 hours. What distance did the dragonfly travel?
______
4. A race car travels 1,212 kilometers in 4 hours. What is
the car’s rate of speed?
© Houghton Mifflin Harcourt Publishing Company
______
5. Kim and Jay leave at the same time to travel
25 miles to the beach. Kim drives 9 miles in
12 minutes. Jay drives 10 miles in 15 minutes.
If they both continue at the same rate, who will
arrive at the beach first?
______
6.
WRITE
Math Describe the location of the variable d in the formulas
involving rate, time, and distance.
Chapter 6
345
Lesson Check (6.RP.A.3d)
1. Mark cycled 25 miles at a rate of 10 miles
per hour. How long did it take Mark to cycle
25 miles?
2. Joy ran 13 miles in 3 1_4 hours. What was her
average rate?
Spiral Review (6.RP.A.3a, 6.RP.A.3c, 6.RP.A.3d)
4. In the Chang family’s budget, 0.6% of the
expenses are for internet service. What fraction
of the family’s expenses is for internet service?
Write the fraction in simplest form.
5. How many meters are equivalent to
357 centimeters?
6. What is the product of the two quantities
shown below?
© Houghton Mifflin Harcourt Publishing Company
9
3. Write two ratios that are equivalent to __
12 .
60 mi × 12 hr
_____
1 hr
FOR MORE PRACTICE
GO TO THE
346
Personal Math Trainer
Name
Chapter 6 Review/Test
Personal Math Trainer
Online Assessment
and Intervention
1. A construction crew needs to remove 2.5 tons of river rock during
the construction of new office buildings.
800
The weight of the rocks is 2,000 pounds.
5,000
2. Select the conversions that are equivalent to 10 yards.
Mark all that apply.
A
20 feet
C
30 feet
B
240 inches
D
360 inches
3. Meredith runs at a rate of 190 meters per minute. Use the formula
d = r × t to find how far she runs in 6 minutes.
© Houghton Mifflin Harcourt Publishing Company
4. The table shows data for 4 cyclists during one day of training.
Complete the table by finding the speed for each cyclist. Use the
formula r = d ÷ t.
Cyclist
Distance (mi)
Time (hr)
Alisha
36
3
Jose
39
3
Raul
40
4
Ruthie
22
2
Assessment Options
Chapter Test
Rate (mi per hr)
Chapter 6
347
5. For numbers 5a–5c, choose <, >, or =.
<
>
5a. 5 kilometers
5,000 meters
=
<
5b. 254 centiliters
>
25.4 liters
=
<
>
5c. 6 kilogram
600 gram
=
6. A recipe calls for 16 fluid ounces of light whipping cream. If Anthony has
1 pint of whipping cream in his refrigerator, does he have enough for the
recipe? Explain your answer using numbers and words.
7. For numbers 7a–7d, choose <, >, or =.
<
7a. 43 feet
7b. 5 tons
>
348
DEEPER
7c. 10 pints
>
=
=
<
<
>
=
8.
15 yards
5000 pounds
7d. 6 miles
>
5 quarts
600 yards
=
The distance from Caleb’s house to the school is 1.5 miles,
and the distance from Ashlee’s house to the school is 3,520 feet. Who
lives closer to the school, Caleb or Ashlee? Use numbers and words to
support your answer.
© Houghton Mifflin Harcourt Publishing Company
<
Name
9. Write the mass measurements in order from least to greatest.
7.4
kilograms
7.4
decigrams
7.4
centigrams
__ __ __
10. An elephant’s heart beats 28 times per minute. Complete the product to
find how many times its heart beats in 30 minutes.
beats × _____________
minutes =
___________
1 minute
1
beats
11. The length of a rectangular football field, including both end zones, is
120 yards. The area of the field is 57,600 square feet. For numbers 11a–11d,
select True or False for each statement.
11a. The width of the field is
True
False
True
False
True
False
True
False
480 yards.
11b. The length of the field is
360 feet.
11c. The width of the field is
160 feet.
11d. The area of the field is
© Houghton Mifflin Harcourt Publishing Company
6,400 square yards.
12. Harry received a package for his birthday. The package weighed
357,000 centigrams. Select the conversions that are equivalent to
357,000 centigrams. Mark all that apply.
3.57 kilograms
357 dekagrams
3,570 grams
3,570,000 decigrams
Chapter 6
349
13. Mr. Martin wrote the following problem on the board.
Juanita’s car has a gas mileage of 21 miles per gallon. How many
miles can Juanita travel on 7 gallons of gas?
21 miles × ________
1
Alex used the expression _______
to find the answer. Explain
1 gallon
7 gallons
Alex’s mistake.
14. Mr. Chen filled his son’s wading pool with 20 gallons of water.
80
20 gallons is equivalent to 60 quarts.
40
15. Nadia has a can of vegetables with a mass of 411 grams. Write equivalent
conversions in the correct boxes.
41.1
kilograms
0.411
hectograms
dekagrams
16. Steve is driving 440 miles to visit the Grand Canyon. He drives at an
average rate of 55 miles per hour. Explain how you can find the amount
of time it will take Steve to get to the Grand Canyon.
350
© Houghton Mifflin Harcourt Publishing Company
4.11
Name
17. Lucy walks one time around the lake. She walks for 1.5 hours at
an average rate of 3 miles per hour. What is the distance, in miles,
around the lake?
__ miles
18. The parking lot at a store has a width of 20 yards 2 feet and a length
of 30 yards.
20 yards 2 feet
30 yards
Part A
Derrick says that the width could also be written as 22 feet.
Explain whether you agree or disagree with Derrick.
Part B
© Houghton Mifflin Harcourt Publishing Company
The cost to repave the parking lot is $2 per square foot. Explain how
much it would cost to repave the parking lot.
Chapter 6
351
Personal Math Trainer
19.
SMARTER
Jake is using a horse trailer to take his horses to his
new ranch.
Part A
Complete the table by finding the weight, in pounds, of Jake’s horse
trailer and each horse.
Weight
(T)
Horse
0.5
Trailer
1.25
Weight
(lb)
Part B
Jake’s truck can tow a maximum weight of 5,000 pounds. What is
the maximum number of horses he can take in his trailer at one
time without going over the maximum weight his truck can tow?
Use numbers and words to support your answer.
© Houghton Mifflin Harcourt Publishing Company
20. A rectangular room measures 13 feet by 132 inches. Tonya said the area
of the room is 1,716 square feet. Explain her mistake, then find the area
in square feet.
352
7
Chapter
Algebra: Expressions
Personal Math Trainer
Show Wha t You Know
Online Assessment
and Intervention
Check your understanding of important skills.
Name
Addition Properties Find the unknown number. Tell whether you used the Identity
(or Zero) Property, Commutative Property, or Associative Property of Addition.
1. 128 + _ = 128
2. (17 + 36 ) + 14 = 17 + ( _ + 14 )
_______
3. 23 + 15 = _ + 23
_______
4. 9 + (11 + 46 ) = ( 9 + _ ) + 46
_______
_______
Multiply with Decimals Find the product.
5. 1.5 × 7
___
6. 5.83 × 6
7. 3.7 × 0.8
___
8. 0.27 × 0.9
___
___
Use Parentheses Identify which operation to do first.
Then, find the value of the expression.
9. 5 × (3 + 6 ) ___
11. 40 ÷ (20 − 16 ) ___
10. (24 ÷ 3 ) − 2 ___
12. (7 × 6 ) + 5 ___
© Houghton Mifflin Harcourt Publishing Company
Math in the
Greg just moved into an old house and found a mysterious
trunk in the attic. The lock on the trunk has a dial numbered
1 to 60. Greg found the note shown at right lying near the
trunk. Help Greg figure out the three numbers needed to
open the lock.
ation
Lock Combin
!
Top Secret
3x
1st number:
: 5x – 1
2nd number
: x2 + 4
3rd number
Hint: x = 6
Chapter 7
355
Voca bula ry Builder
Visualize It
Review Words
Sort the review words into the bubble map.
addition
+
−
difference
division
multiplication
product
quotient
Operations
subtraction
×
÷
sum
Preview Words
algebraic expression
Understand Vocabulary
base
Complete the sentences using the preview words.
coefficient
1. An exponent is a number that tells how many times a(n)
____ is used as a factor.
evaluate
numerical expression
terms
2. In the expression 4a, the number 4 is a(n)
variable
____.
3. To ____ an expression, substitute numbers
for the variables in the expression.
4. A mathematical phrase that uses only numbers and operation
symbols is a(n) ____ .
© Houghton Mifflin Harcourt Publishing Company
5. A letter or symbol that stands for one or more numbers is a(n)
____ .
6. The parts of an expression that are separated by an addition
or subtraction sign are the ____ of the
expression.
356
™Interactive Student Edition
™Multimedia eGlossary
Chapter 7 Vocabulary
algebraic expression
base
expresión algebraica
base
6
3
coefficient
difference
coeficiente
diferencia
22
10
equivalent
expressions
expresiones
equivalentes
evaluate
evaluar
31
28
exponent
factor
exponente
factor
32
33
The answer to a subtraction problem
7−3=4
difference
To find the value of a numerical or algebraic
expression
Example: Evaluating 2x + 3y when x = 1
and y = 4 gives 2(1) + 3(4) = 2 + 12 = 14.
A number multiplied by another number to
find a product
factors
An expression that includes at least one
variable
x + 10
variable
3×y
3 × (a + 4)
variable
variable
A number that is multiplied by a variable
Example: 4k The coefficient of the term
4k is 4.
Expressions that are equal to each other
for any values of their variables
Example: 2x + 4x = 6x
A number that shows how many times the
base is used as a factor
3
5×5×5=5
exponent
6
2 × 3 × 7 = 42
© Houghton Mifflin Harcourt Publishing Company
base
© Houghton Mifflin Harcourt Publishing Company
3 repeated factors
© Houghton Mifflin Harcourt Publishing Company
exponent
© Houghton Mifflin Harcourt Publishing Company
© Houghton Mifflin Harcourt Publishing Company
© Houghton Mifflin Harcourt Publishing Company
© Houghton Mifflin Harcourt Publishing Company
3
5×5×5=5
6
© Houghton Mifflin Harcourt Publishing Company
A number used as a repeated factor
3 repeated factors
base
Chapter 7 Vocabulary
(continued)
like terms
numerical expression
términos semejantes
expresión numérica
67
49
order of operations
product
orden de las operaciones
producto
80
69
quotient
sum
cociente
suma o total
99
84
terms
variable
términos
variable
101
105
The answer to a multiplication problem
3 × 4 = 12
product
The answer to an addition problem
2+5=7
sum
A letter or symbol that stands for an
unknown number or numbers
x + 10
variable
3×y
variable
3 × (a + 4)
variable
© Houghton Mifflin Harcourt Publishing Company
3
2 +4
© Houghton Mifflin Harcourt Publishing Company
4 × (8 + 5)
© Houghton Mifflin Harcourt Publishing Company
2
3 + 16 × 2
© Houghton Mifflin Harcourt Publishing Company
© Houghton Mifflin Harcourt Publishing Company
© Houghton Mifflin Harcourt Publishing Company
© Houghton Mifflin Harcourt Publishing Company
© Houghton Mifflin Harcourt Publishing Company
A mathematical phrase that uses only
numbers and operation signs
Expressions that have the same variable with
the same exponent
Algebraic Expression
Terms
5x + 3y − 2x
5x, 3y, and 2x
8z2 + 4z + 12z2
8z , 4z, and 12z
8z2 and 12z2
15 − 3x + 5
15, 3x, and 5
15 and 5
2
Like Terms
5x and 2x
2
A set of rules which gives the order in which
calculations are done in an expression
Order of Operations
1. Perform operations in parentheses.
2. Find the values of numbers with
exponents.
3. Multiply and divide from left to right.
4. Add and subtract from left to right.
The number that results from dividing
80 ÷ 4 = 20
20
4 80
quotient
quotient
The parts of an expression that are separated
by an addition or subtraction sign
Example:
4k + 5
The expression has two terms,
4k and 5.
Going Places with
Words
Going Down
the Blue Ridge
Parkway
For 2 to 4 players
Materials
• 1 each as needed: red, blue, green, and yellow connecting cubes
• 1 number cube
• Clue Cards
How to Play
1. Each player puts a connecting cube on START.
2. To take a turn, toss the number cube. Move that many spaces.
© Houghton Mifflin Harcourt Publishing Company
Image Credits: ©Jupiterimages/Creatas/Alamy
3. If you land on these spaces:
Green Space Follow the directions in the space.
Game
Game
Word Box
algebraic
expression
base
coefficient
difference
equivalent
expressions
evaluate
exponent
factor
like terms
numerical
expression
order of operations
product
quotient
sum
terms
variable
Yellow Space Explain how to evaluate the expression. If you are
correct, move ahead 1 space.
Blue Space Use a math term to name what is shown. If you are
correct, move ahead 1 space.
Red Space The player to your right draws a Clue Card and reads you
the question. If you answer correctly, move ahead 1 space. Return the
Clue Card to the bottom of the pile.
4. The first player to reach FINISH wins.
Chapter 7
356A
Game
Game
Take a scenic
hike. Move
ahead 1.
DIRECTIONS Each player puts a connecting cube on START. •
To take a turn, toss the number cube. Move that many spaces.
• If you land on these spaces: Green Space Follow the
directions in the space. • Yellow Space Explain how to evaluate
the expression. If you are correct, move ahead 1 space. • Blue
Space Use a math term to name what is shown. If you are
correct, move ahead 1 space. • Red Space The player to your
right draws a Clue Card and reads you the question. If you
answer correctly, move ahead 1 space.
Return the Clue Card to the bottom
of the pile. • The first player to
reach FINISH wins.
CLUE
CARD
Get a flat tire.
Lose 1 turn.
4 x (5 + 1)
CLUE
CARD
⃗
CCLLUUEE
CCAARRDD
Visit
Visi
Vi
sit the
Natural Bridge.
Go back 1.
48 ÷ 42
356B
© Houghton Mifflin Harcourt Publishing Company
⃗
23
52
Game
Game
© Houghton Mifflin Harcourt Publishing Company
⃗
Climb
Chimney Rock.
Trade places
with another
player.
x+7
(tr) ©imagebroker/Alamy; (br) ©Brand X Pictures/Getty Images
Image Credits: (bg) ©Mark Karrass/Corbis; (tl) ©dmvphotos/Shutterstock; (bl) ©Wikimedia Commons;
4 × (y + 5)
for y = 11
CLUE
CARD
2 × (33 + 7)
CLUE
CARD
42 × 10 + 6
72 − (x2 + 5)
for x = 2
Ride along
the Great Smoky
Mountains
Railroad. Take
another turn.
Chapter 7
356C
Journal
Jo
ou
urnal
The Write Way
Reflect
Choose one idea. Write about it.
• Tell how to rewrite this expression so that each base has an exponent.
4×4×4×7×7
• Explain how to evaluate this expression using the order of operations:
3 × (1 + 52).
• Define the term algebraic expression in your own words. Give an example.
• Jin wrote the following expression. Explain how he can simplify the
expression by using like terms. Then write an equivalent expression.
© Houghton Mifflin Harcourt Publishing Company
Image Credits: ©Jupiterimages/Creatas/Alamy
2x + x + 9
356D
Lesson 7.1
Name
Exponents
Expressions and Equations—
6.EE.A.1
MATHEMATICAL PRACTICES
MP6, MP7, MP8
Essential Question How do you write and find the value of expressions
involving exponents?
You can use an exponent and a base to show repeated
multiplication of the same factor. An exponent is a number
that tells how many times a number called the base is used
as a repeated factor.
• 52 can be read “the 2nd power
of 5” or “5 squared.”
• 53 can be read “the 3rd power
of 5” or “5 cubed.”
6
3 ← exponent
5 × 5 × 5 = 5←
3 repeated factors base
Unlock
Unlock the
the Problem
Problem
The table shows the number of bonuses a player
can receive in each level of a video game. Use an
exponent to write the number of bonuses a player
can receive in level D.
Use an exponent to write 3 × 3 × 3 × 3 .
The number _ is used as a repeated factor.
© Houghton Mifflin Harcourt Publishing Company • Image Credits: (c) ©Granger Wootz/Blend Images/Corbis
3 is used as a factor _ times.
Write the base and exponent.
_
So, a player can receive _ bonuses in level D.
Math
Talk
Try This! Use one or more exponents to write the expression.
Bonuses
A
3
B
333
C
33333
D
3333333
MATHEMATICAL PRACTICES 6
Explain How do you know
which number to use as the
base and which number to
use as the exponent?
B 6×6×8×8×8
A 7×7×7×7×7
The number _ is used as a repeated factor.
7 is used as a factor _ times.
Write the base and exponent.
Level
_
The numbers _ and _ are used as
repeated factors.
6 is used as a factor _ times.
8 is used as a factor _ times.
Write each base with its
own exponent.
6
38
Chapter 7
357
Example 1 Find the value.
A 103
STEP 1 Use repeated multiplication to write 103.
The repeated factor is _ .
103 5 _ 3 _ 3 _
Write the factor _ times.
STEP 2 Multiply.
Multiply each pair of factors, working from left to right.
10 3 10 3 10 5 _ 3 10
B 71
The repeated factor is
Write the factor
_.
71 5 _
5 __
Math
Talk
_ time.
MATHEMATICAL PRACTICES 7
Look for a Pattern In 103,
what do you notice about
the value of the exponent
and the product? Is there
a similar pattern in other
powers of 10? Explain.
Example 2 Write 81 with an exponent by using 3 as the base.
STEP 1 Find the correct exponent.
Try 2.
32 5 3 3 3 5 _
Try 3.
33 5 _ 3 _ 3 _ 5 _
Try 4.
34 5 _ 3 _ 3 _ 3 _ 5 _
STEP 2 Write using the base and exponent.
81 5 _
2.
3.
SMARTER
MATHEMATICAL
PRACTICE
Is 52 equal to 25 ? Explain why or why not.
6 Describe a Method Describe how you could have solved the
problem in Example 2 by using division.
358
© Houghton Mifflin Harcourt Publishing Company
1. Explain how to write repeated multiplication of a factor by using an exponent.
Name
Share
hhow
Share and
and Show
Sh
MATH
M
BOARD
B
1. Write 24 by using repeated multiplication. Then find the value of 24.
24 5 2 3 2 3 _ 3 _ 5 _
Use one or more exponents to write the expression.
2. 7 × 7 × 7 × 7
3. 5 × 5 × 5 × 5 × 5
4. 3 × 3 × 4 × 4
Math
Talk
On
On Your
Your Own
Own
MATHEMATICAL PRACTICES 8
Generalize In 34, does it
matter in what order you
multiply the factors when
finding the value? Explain.
Find the value.
5. 202
6. 821
7. 35
8. Write 32 as a number with an exponent by using 2 as the base.
Complete the statement with the correct exponent.
© Houghton Mifflin Harcourt Publishing Company
9. 5
12.
= 125
MATHEMATICAL
PRACTICE
10. 16
= 16
8 Use Repeated Reasoning
1
2
3
4
5
Find the values of 4 , 4 , 4 , 4 , and 4 . Look for
a pattern in your results and use it to predict the
ones digit in the value of 46.
11. 30
13.
= 900
SMARTER
Select the expressions that
are equivalent to 32. Mark all that apply.
A 25
B 84
C 23 × 4
D 2×4×4
Chapter 7 • Lesson 1 359
Bacterial Growth
Bacteria are tiny, one-celled organisms that live almost
everywhere on Earth. Although some bacteria cause
disease, other bacteria are helpful to humans, other
animals, and plants. For example, bacteria are needed to
make yogurt and many types of cheese.
Under ideal conditions, a certain type of bacterium cell
grows larger and then splits into 2 “daughter” cells. After
20 minutes, the daughter cells split, resulting in 4 cells. This
splitting can happen again and again as long as conditions
remain ideal.
Complete the table.
Bacterial Growth
Time (min)
1
21 = 2
22 = 2 × 2 = 4
WRITE
0
20
Math t40
Show Your Work
23 = _ × _ × _ = _
60
2
80
= 2 × 2 × 2 × 2 = 16
25 = _ × _ × _ × _ × _ = _
100
2
120
=_×_×_×_×_×_=_
27 = 2 × 2 × 2 × 2 × 2 × 2 × 2 = _
_
Extend the pattern in the table above to answer 14 and 15.
14.
360
DEEPER
What power of 2 shows the
number of cells after 3 hours? How many cells
are there after 3 hours?
15.
SMARTER
How many
minutes would it take to have
a total of 4,096 cells?
© Houghton Mifflin Harcourt Publishing Company • Image Credits: (t) ©Biomedical Imaging Unit, Southampton General Hospital/Science Source
Number of Cells
Practice and Homework
Name
Lesson 7.1
Exponents
COMMON CORE STANDARD—6.EE.A.1
Apply and extend previous understandings of
arithmetic to algebraic expressions.
Use one or more exponents to write the expression.
1. 6 × 6
2. 11 × 11 × 11 × 11
62
____
____
3. 9 × 9 × 9 × 9 × 7 × 7
____
Find the value.
4. 64
5. 16
____
6. 105
____
7. Write 144 with an exponent by using 12
as the base.
____
8. Write 343 with an exponent by using
7 as the base.
Problem
Problem Solving
Solving
© Houghton Mifflin Harcourt Publishing Company
9. Each day Sheila doubles the number of push-ups
she did the day before. On the fifth day, she does
2 × 2 × 2 × 2 × 2 push-ups. Use an exponent to
write the number of push-ups Shelia does on the
fifth day.
11.
WRITE
10. The city of Beijing has a population of
more than 107 people. Write 107 without
using an exponent.
Math Explain what the expression 45 means and how to
find its value.
Chapter 7
361
TEST
PREP
Lesson Check (6.EE.A.1)
1. The number of games in the first round
of a chess tournament is equal to
2 × 2 × 2 × 2 × 2 × 2. Write the number
of games using an exponent.
2. The number of gallons of water in a tank at an
aquarium is equal to 83. How many gallons of
water are in the tank?
Spiral Review (6.RP.A.3a, 6.RP.A.3c, 6.RP.A.3d)
Strawberry juice (cups)
2
3
4
Lemonade (cups)
6
9
12
5. How many milliliters are equivalent to
2.7 liters?
__ ?
4. Which percent is equivalent to the fraction 37
50
6. Use the formula d = rt to find the distance
traveled by a car driving at an average speed of
50 miles per hour for 4.5 hours.
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3. The table shows the amounts of strawberry
juice and lemonade needed to make different
amounts of strawberry lemonade. Name
another ratio of strawberry juice to lemonade
that is equivalent to the ratios in the table.
Lesson 7.2
Name
Evaluate Expressions Involving Exponents
Essential Question How do you use the order of operations to evaluate
expressions involving exponents?
A numerical expression is a mathematical phrase that
uses only numbers and operation symbols.
3 + 16 × 22
4 × ( 8 + 51 )
23 + 4
You evaluate a numerical expression when you find its value.
To evaluate an expression with more than one operation, you
must follow a set of rules called the order of operations.
Expressions and Equations—
6.EE.A.1
MATHEMATICAL PRACTICES
MP4, MP6
Order of Operations
1. Perform operations in parentheses.
2. Find the values of numbers with
exponents.
3. Multiply and divide from left to right.
4. Add and subtract from left to right.
Unlock
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the Problem
Problem
An archer shoots 6 arrows at a target. Two arrows
hit the ring worth 8 points, and 4 arrows hit the ring
worth 4 points. Evaluate the expression 2 × 8 + 42
to find the archer’s total number of points.
Follow the order of operations.
Write the expression. There are
no parentheses.
© Houghton Mifflin Harcourt Publishing Company • Image Credits: (r) ©Herbert Kehrer/Corbis
Find the value of numbers with
exponents.
2 × 8 + 42
2×8+_
__ from left to right.
_ + 16
Then add.
_
So, the archer scores a total of _ points.
Math
Talk
Try This! Evaluate the expression 24 ÷ 2 .
3
There are no parentheses.
24 ÷ 23
Find the value of numbers with
exponents.
24 ÷ _
Then divide.
_
MATHEMATICAL PRACTICES 6
Explain In which order
should you perform the
operations to evaluate the
expression 30 − 10 + 52?
Chapter 7
363
Example 1 Evaluate the expression 72 ÷ (13 − 4) + 5 × 2 .
3
Write the expression.
72 ÷ (13 − 4) + 5 × 23
Perform operations in ___.
72 ÷ _ + 5 × 23
Find the values of numbers with ___.
72 ÷ 9 + 5 × _
Multiply and ___ from left to right.
_+5×8
8+_
_
Then add.
Example 2
Last month, an online bookstore had approximately 105 visitors to its
website. On average, each visitor bought 2 books. Approximately how
many books did the bookstore sell last month?
STEP 1 Write an expression.
Think: The number of books sold is equal to the number of visitors times the
number of books each visitor bought.
number of visitors times number of books bought
↓
10
5
↓
↓
×
_
STEP 2 Evaluate the expression.
Write the expression. There are no parentheses.
105 × 2
Find the values of numbers with ___.
___ × 2
Multiply.
___
•
364
MATHEMATICAL
PRACTICE
6 Explain why the order of operations is necessary.
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So, the bookstore sold approximately ___ books last month.
Name
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1. Evaluate the expression 9 + (52 − 10).
9 + (52 − 10)
Write the expression.
9 + ( _ − 10)
Follow the order of operations within the parentheses.
9+_
_
Add.
Evaluate the expression.
2. 6 + 33 ÷ 9
3. (15 − 3)2 ÷ 9
___
4. (8 + 92) − 4 × 10
___
___
Math
Talk
On
On Your
Your Own
Own
MATHEMATICAL PRACTICES 7
Look for Structure How does
the parentheses make the
values of these expressions
different: (22 + 8) ÷ 4 and
22 + (8 ÷ 4)?
Evaluate the expression.
5. 10 + 62 × 2 ÷ 9
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___
SMARTER
6. 62 − (23 + 5)
___
7. 16 + 18 ÷ 9 + 34
___
Place parentheses in the expression so that it equals the given value.
8. 102 − 50 ÷ 5
value: 10
___
9. 20 + 2 × 5 + 41
value: 38
___
10. 28 ÷ 22 + 3
value: 4
___
Chapter 7 • Lesson 2 365
MATHEMATICAL PRACTICES
COMMUNICA5&t1&34E7&3&tCONSTRUCT ARGUMENTS
OqnakdlRnkuhmf¤@ookhb`shnmr
OqnakdlRnkuhmf¤@ookhb`shnmr
Use the table for 11–13.
11.
MATHEMATICAL
PRACTICE
4 Write an Expression To find the
cost of a window, multiply its area in square feet
by the price per square foot. Write and evaluate
an expression to find the cost of a knot window.
DEEPER
A builder installs 2 rose windows
and 2 tulip windows. Write and evaluate an
expression to find the combined area of the
windows.
Art Glass Windows
Area
(square feet)
Price per
square foot
Knot
22
$27
Rose
32
$30
Tulip
42
$33
Type
13.
SMARTER
DeShawn bought a tulip window.
Emma bought a rose window. Write and evaluate
an expression to determine how much more
DeShawn paid for his window than Emma paid
for hers.
14. What’s the Error? Darius wrote 17 − 22 = 225.
Explain his error.
15.
366
SMARTER
Ms. Hall wrote the expression
2 × 3 + 5 ÷ 4 on the board. Shyann said
the first step is to evaluate 52. Explain Shyann’s
mistake. Then evaluate the expression.
WRITE
Math t Show Your Work
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12.
Practice and Homework
Name
Lesson 7.2
Evaluate Expressions Involving Exponents
COMMON CORE STANDARD—6.EE.A.1
Apply and extend previous understandings of
arithmetic to algebraic expressions.
Evaluate the expression.
1. 5 + 17 − 102 ÷ 5
2. 72 − 32 × 4
3. 24 ÷ (7 − 5)
5 + 17 − 100 ÷ 5
5 + 17 − 20
22 − 20
2
____
4. (82 + 36) ÷ (4 × 52)
____
5. 12 + 21 ÷ 3 + (22 × 0)
____
____
6. (12 − 8)3 − 24 × 2
____
Place parentheses in the expression so that it equals the given value.
7. 12 ÷ 2 × 23; value: 120
8. 72 × 1 − 5 ÷ 3; value: 135
_______
_______
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Problem
Problem Solving
Solving
9. Hugo is saving for a new baseball glove. He saves
$10 the first week, and $6 each week for the next
6 weeks. The expression 10 + 62 represents the
total amount in dollars he has saved. What is the
total amount Hugo has saved?
11.
WRITE
10. A scientist placed fish eggs in a tank. Each
day, twice the number of eggs from the previous
day hatch. The expression 5 × 26 represents the
number of eggs that hatch on the sixth day. How
many eggs hatch on the sixth day?
Math Explain how you could determine whether
a calculator correctly performs the order of operations.
Chapter 7
367
TEST
PREP
Lesson Check (6.EE.A.1)
1. Ritchie wants to paint his bedroom ceiling and
four walls. The ceiling and each of the walls
are 8 feet by 8 feet. A gallon of paint covers
40 square feet. Write an expression that can
be used to find the number of gallons of paint
Ritchie needs to buy.
2. A Chinese restaurant uses about 225 pairs
of chopsticks each day. The manager wants
to order a 30-day supply of chopsticks. The
chopsticks come in boxes of 750 pairs. How
many boxes should the manager order?
3. Annabelle spent $5 to buy 4 raffle tickets. How
many tickets can she buy for $20?
4. Gavin has 460 baseball players in his collection
of baseball cards, and 15% of the players
are pitchers. How many pitchers are in
Gavin’s collection?
5. How many pounds are equivalent to 40 ounces?
6. List the expressions in order from least to
greatest.
15
33
42
81
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Spiral Review (6.RP.A.3a, 6.RP.A.3c, 6.RP.A.3d, 6.EE.A.1)
Lesson 7.3
Name
Write Algebraic Expressions
Essential Question How do you write an algebraic expression to
represent a situation?
Expressions and Equations—
6.EE.A.2a
MATHEMATICAL PRACTICES
MP2, MP4, MP6
An algebraic expression is a mathematical phrase that
includes at least one variable. A variable is a letter or
symbol that stands for one or more numbers.
x + 10
↑
3×y
↑
3 × (a + 4)
↑
variable
variable
variable
There are several ways to
show multiplication with a
variable. Each expression below
represents “3 times y.”
33y
3y
3(y)
3?y
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the Problem
Problem
An artist charges $5 for each person in a cartoon
drawing. Write an algebraic expression for the cost
in dollars for a drawing that includes p people.
Write an algebraic expression for the cost.
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Think: cost for each person times number of __
↑
↑
↑
_
×
p
So, the cost in dollars is _.
Math
Talk
Try This! On Mondays, a bakery adds 2 extra muffins for free
with every muffin order. Write an algebraic expression for the
number of muffins customers will receive on Mondays when
they order m muffins.
Think: muffins ordered
↑
_
_
↑
+
MATHEMATICAL PRACTICES 2
Reasoning Why is p an
appropriate variable for
this problem? Would it
be appropriate to select a
different variable? Explain.
extra muffins on Mondays
↑
2
So, customers will receive __ muffins on Mondays.
Chapter 7
369
Example 1 The table at the right shows the number of
points that items on a quiz are worth. Write an algebraic expression
for the quiz score of a student who gets m multiple-choice items and
s short-answer items correct.
points for multiple-choice items
↑
(2 × m )
_
↑
Quiz Scoring
Item Type
Points
Multiple-choice
2
Short-answer
5
points for short-answer items
+
↑
(_)
So, the student’s quiz score is __ points.
Example 2 Write an algebraic expression for the word expression.
A 30 more than the product of 4 and x
Think: Start with the product of 4 and x. Then find 30 more than the product.
the product of 4 and x
_3_
30 more than the product
_ 1 4x
B 4 times the sum of x and 30
Think: Start with the sum of x and 30. Then find 4 times the sum.
the sum of x and 30
_+_
4 times the sum
_ 3 (x 1 30)
2.
370
One student wrote 4 + x for the word expression
“4 more than x.” Another student wrote x + 4 for the same word
expression. Are both students correct? Justify your answer.
SMARTER
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1. When you write an algebraic expression with two operations, how
can you show which operation to do first?
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1. Write an algebraic expression for the product of 6 and p.
What operation does the word “product” indicate?
___
The expression is _ × _.
Write an algebraic expression for the word expression.
3. 9 less than the quotient of n and 5
2. 11 more than e
______
______
Math
Talk
On
On Your
Your Own
Own
Write an algebraic expression for the word expression.
4. 20 divided by c
______
6. There are 12 eggs in a dozen. Write an algebraic
expression for the number of eggs in d dozen.
© Houghton Mifflin Harcourt Publishing Company
______
8.
MATHEMATICAL
PRACTICE
MATHEMATICAL PRACTICES 6
Explain why 3x is an
algebraic expression.
5. 8 times the product of 5 and t
______
7. A state park charges a $6.00 entry fee plus
$7.50 per night of camping. Write an algebraic
expression for the cost in dollars of entering the
park and camping for n nights.
______
7 Look for Structure At a bookstore, the expression 2c + 8g
gives the cost in dollars of c comic books and g graphic novels. Next month,
the store’s owner plans to increase the price of each graphic novel by $3.
Write an expression that will give the cost of c comic books and g graphic
novels next month.
Chapter 7 • Lesson 3 371
MATHEMATICAL PRACTICES
ANALYZEt-00,'034536$563&t13&$*4*0/
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the Problem
Problem
9. Martina signs up for the cell phone plan
described at the right. Write an expression
that gives the total cost of the plan in dollars
if Martina uses it for m months.
a. What information do you know about the cell
phone plan?
c. What operation can you use to show the
discount of $10 for the first month?
b. Write an expression for the monthly fee in
dollars for m months.
d. Write an expression for the total cost of the
plan in dollars for m months.
SMARTER
A group of n
friends evenly share the cost
of dinner. The dinner costs
$74. After dinner, each friend
pays $11 for a movie. Write an
expression to represent what
each friend paid for dinner
and the movie.
11.
SMARTER
A cell phone company charges
$40 per month plus $0.05 for each text message
sent. Select the expressions that represent the
cost in dollars for one month of cell phone
usage and sending m text messages. Mark all
that apply.
40m + 0.05
40 + 0.05m
40 more than the product of 0.05 and m
the product of 40 and m plus 0.05
372
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10.
Practice and Homework
Name
Lesson 7.3
Write Algebraic Expressions
COMMON CORE STANDARD—6.EE.A.2a
Apply and extend previous understandings of
arithmetic to algebraic expressions.
Write an algebraic expression for the word expression.
1. 13 less than p
p − 13
____
4. the sum of 15 and the product
of 5 and v
____
7. the quotient of m and 7
____
10. 10 less than the quotient of g
and 3
____
2. the sum of x and 9
____
5. the difference of 2 and the
product of 3 and k
____
8. 9 more than 2 multiplied
by f
____
11. the sum of 4 multiplied by
a and 5 multiplied by b
____
3. 6 more than the difference of
b and 5
____
6. 12 divided by the sum of h
and 2
____
9. 6 minus the difference of
x and 3
____
12. 14 more than the difference of
r and s
____
Problem
Problem Solving
Solving
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13. Let h represent Mark’s height in inches. Suzanne
is 7 inches shorter than Mark. Write an algebraic
expression that represents Suzanne’s height
in inches.
15.
WRITE
14. A company rents bicycles for a fee of $10 plus
$4 per hour of use. Write an algebraic expression
for the total cost in dollars for renting a bicycle
for h hours.
Math Give an example of a real-world situation involving
two unknown quantities. Then write an algebraic expression to
represent the situation.
Chapter 7
373
TEST
PREP
Lesson Check (6.EE.A.2a)
1. The female lion at a zoo weighs 190 pounds
more than the female cheetah. Let c represent
the weight in pounds of the cheetah. Write an
expression that gives the weight in pounds of
the lion.
2. Tickets to a play cost $8 each. Write an
expression that gives the ticket cost in dollars
for a group of g girls and b boys.
Spiral Review (6.RP.A.2, 6.RP.A.3a, 6.RP.A.3c, 6.RP.A.3d, 6.EE.A.1)
3. A bottle of cranberry juice contains 32 fluid
ounces and costs $2.56. What is the unit rate?
4. There are 32 peanuts in a bag. Elliott takes 25%
of the peanuts from the bag. Then Zaire takes
50% of the remaining peanuts. How many
peanuts are left in the bag?
5. Hank earns $12 per hour for babysitting. How
much does he earn for 15 hours of babysitting?
6. Write an expression that represents the area of
the figure in square centimeters.
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2 cm
7 cm
7 cm
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Lesson 7.4
Name
Identify Parts of Expressions
Expressions and Equations—
6.EE.A.2b
MATHEMATICAL PRACTICES
MP1, MP2, MP6
Essential Question How can you describe the parts of an expression?
Unlock
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the Problem
Problem
At a gardening store, seed packets cost $2 each. Martin bought
6 packets of lettuce seeds and 7 packets of pea seeds. The
expression 2 × (6 + 7) represents the cost in dollars of Martin’s
seeds. Identify the parts of the expression. Then write a word
expression for 2 × (6 + 7).
Describe the parts of the expression 2
• Explain how you could find the cost of
each type of seed.
× (6 + 7).
Identify the operations in the
expression.
multiplication and __
Describe the part of the
expression in parentheses,
and tell what it represents.
• The part in parentheses shows
the _ of 6 and _.
• The sum represents the number
of packets of __
seeds plus the number of packets
of __ seeds.
Describe the multiplication,
and tell what it represents.
• One of the factors is _. The other
© Houghton Mifflin Harcourt Publishing Company • Image Credits: (r) ©ableimages/Alamy
factor is the _ of 6 and 7.
• The product represents the _ per packet times
the number of __ Martin bought.
So, a word expression for 2 × (6 + 7) is “the __ of 2 and the
_ of _ and 7.”
•
MATHEMATICAL
PRACTICE
6
Attend to Precision Explain how the expression 2 × (6 + 7)
differs from 2 × 6 + 7. Then, write a word expression for 2 × 6 + 7.
Chapter 7
375
The terms of an expression are the parts of the expression that are
separated by an addition or subtraction sign. A coefficient is a
number that is multiplied by a variable.
4k + 5
The expression has two terms, 4k and 5. The
coefficient of the term 4k is 4.
Example
Identify the parts of the expression. Then write a
word expression for the algebraic expression.
A 2x + 8
Identify the terms in the
expression.
The expression is the sum of _ terms.
The terms are _ and 8.
Describe the first term.
The first term is the product of the coefficient
_ and the variable _.
Describe the second term.
The second term is the number _.
A word expression for 2x + 8 is “8 more than the __
Math
Talk
of _ and x. ”
MATHEMATICAL PRACTICES 6
Explain Why are the terms
of the expression 2x and 8,
not x and 8?
B 3a − 4b
Identify the terms in the
expression.
The expression is the __ of
2 terms. The terms are _ and _.
Describe the first term.
The first term is the product of the
__ 3 and the variable _.
Describe the second term.
The second term is the product of the
A word expression for the algebraic expression is “the difference of
_ times _ and 4 __ b. ”
376
Math
Talk
MATHEMATICAL PRACTICES 2
Reasoning Identify the
coefficient of y in the
expression 12 + y. Explain
your reasoning.
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coefficient _ and the variable _.
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B
Identify the parts of the expression. Then, write a word
expression for the numerical or algebraic expression.
1. 7 × (9 ÷ 3)
The part in parentheses shows the __ of _ divided by _.
One factor of the multiplication is _ , and the other factor is 9 ÷ 3.
Word expression: _______
2. 5m + 2n
Math
Talk
On
On Your
Your Own
Own
MATHEMATICAL PRACTICES 6
Attend to Precision Describe
the expression 9 × (a + b) as
a product of two factors.
Practice: Copy and Solve Identify the parts of the expression. Then
write a word expression for the numerical or algebraic expression.
3. 8 + (10 − 7)
4. 1.5 × 6 + 8.3
5. b + 12 x
6. 4a ÷ 6
Identify the terms of the expression. Then, give the coefficient of each
term.
7. k − 1_ d
© Houghton Mifflin Harcourt Publishing Company
3
9.
MATHEMATICAL
PRACTICE
8. 0.5x + 2.5y
2 Connect Symbols and Words Ava said she wrote an
expression with three terms. She said the first term has the coefficient
7, the second term has the coefficient 1, and the third term has the
coefficient 0.1. Each term involves a different variable. Write an expression
that could be the expression Ava wrote.
Chapter 7 • Lesson 4 377
MATHEMATICAL PRACTICES .0%&-t3&"40/tM",&4&/4&
OqnakdlRnkuhmf¤@ookhb`shnmr
OqnakdlRnkuhmf¤@ookhb`shnmr
Use the table for 10–12.
10.
DEEPER
A football team scored 2 touchdowns
and 2 extra points. Their opponent scored 1
touchdown and 2 field goals. Write a numerical
expression for the points scored in the game.
Football Scoring
11. Write an algebraic expression for the number of
points scored by a football team that makes
t touchdowns, f field goals, and e extra points.
12. Identify the parts of the expression you wrote in
Exercise 11.
14.
378
Points
Touchdown
6
Field Goal
3
Extra Point
1
WRITE
Math
Show Your Work
SMARTER
Give an example of an expression
involving multiplication in which one of the factors
is a sum. Explain why you do or do not need
parentheses in your expression.
SMARTER
Kennedy bought a pounds of almonds at $5
per pound and p pounds of peanuts at $2 per pound. Write
an algebraic expression for the cost of Kennedy’s purchase.
© Houghton Mifflin Harcourt Publishing Company • Image Credits: (t) ©Comstock/Getty Images
13.
Type
Practice and Homework
Name
Lesson 7.4
Identify Parts of Expressions
COMMON CORE STANDARD—6.EE.A.2a
Apply and extend previous understandings of
arithmetic to algebraic expressions.
Identify the parts of the expression. Then write a word
expression for the numerical or algebraic expression.
1. (16 − 7) ÷ 3
2. 8 + 6q + q
The subtraction is the difference of 16 and 7.
The division is the quotient of the difference
and 3. Word expression: the quotient of the
difference of 16 and 7 and 3
Identify the terms of the expression. Then give the
coefficient of each term.
3. 11r + 7s
4. 6g − h
Problem
Problem Solving
Solving
© Houghton Mifflin Harcourt Publishing Company
5. Adam bought granola bars at the store. The
expression 6p + 5n gives the number of bars
in p boxes of plain granola bars and n boxes of
granola bars with nuts. What are the terms of
the expression?
7.
WRITE
6. In the sixth grade, each student will get
4 new books. There is one class of 15 students
and one class of 20 students. The expression
4 × (15 + 20) gives the total number of new
books. Write a word expression for the
numerical expression.
Math Explain how knowing the order of operations helps
you write a word expression for a numerical or algebraic expression.
Chapter 7
379
Lesson Check (6.EE.A.2a)
1. A fabric store sells pieces of material for $5
each. Ali bought 2 white pieces and 8 blue
pieces. She also bought a pack of buttons for $3.
The expression 5 × (2 + 8) + 3 gives the cost in
dollars of Ali’s purchase. How can you describe
the term (2 + 8) in words?
TEST
PREP
2. A hotel offers two different types of rooms. The
expression k + 2f gives the number of beds
in the hotel where k is the number of rooms
with a king size bed and f is the number of
rooms with 2 full size beds. What are the terms
of the expression?
3. Meg paid $9 for 2 tuna sandwiches. At the
same rate, how much does Meg pay for
8 tuna sandwiches?
4. Jan is saving for a skateboard. She has saved $30
already, which is 20% of the total price. How
much does the skateboard cost?
5. It took Eduardo 8 hours to drive from Buffalo,
NY, to New York City, a distance of about
400 miles. Find his average speed.
6. Write an expression that represents the value, in
cents, of n nickels.
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Spiral Review (6.RP.A.3b, 6.RP.A.3c, 6.RP.A.3d, 6.EE.A.2a)
Lesson 7.5
Name
Evaluate Algebraic Expressions and Formulas
Expressions and Equations—
6.EE.A.2c
MATHEMATICAL PRACTICES
MP4, MP5, MP6
Essential Question How do you evaluate an algebraic expression or a formula?
To evaluate an algebraic expression, substitute numbers for the
variables and then follow the order of operations.
Unlock
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the Problem
Problem
Amir is saving money to buy an MP3 player that costs
$120. He starts with $25, and each week he saves $9.
The expression 25 + 9w gives the amount in dollars
that Amir will have saved after w weeks.
• Which operations does the expression 25 + 9w
include?
• In what order should you perform the
operations?
A How much will Amir have saved after
8 weeks?
Evaluate the expression for w = 8.
Write the expression.
25 + 9 w
Substitute 8 for w.
25 + 9 × _
Multiply.
25 + _
Add.
_
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So, Amir will have saved $ _ after 8 weeks.
B After how many weeks will Amir have saved
enough money to buy the MP3 player?
Make a table to find the week when the amount
saved is at least $120.
Week
9
Amount Saved
Value of 25 + 9w
25 + 9 × 9 = 25 + _ = 106
10
25 + 9 × 10 = 25 + _ = __
11
25 + 9 × 11 = 25 + _ = __
So, Amir will have saved enough money for the
MP3 player after _ weeks.
Math
Talk
MATHEMATICAL PRACTICES 6
E
Explain
What does it mean
to substitute a value for a
variable?
Chapter 7
381
Example 1 Evaluate the expression for the given value of the variable.
A 4 × (m − 8) ÷ 3 for m = 14
Write the expression.
4 × (m − 8) ÷ 3
Substitute 14 for m.
4 × ( _ − 8) ÷ 3
Perform operations in parentheses.
4×_÷3
Multiply and divide from left to right.
_÷3
_
B 3 × (y2 + 2) for y = 4
Write the expression.
3 × ( y 2 + 2)
Substitute 4 for y.
3 × ( _ 2 + 2)
Follow the order of operations within
the parentheses.
3 × ( _ + 2)
When squaring a number, be
sure to multiply the number by
itself.
42 = 4 × 4
3×_
Multiply.
_
Recall that a formula is a set of symbols that expresses a mathematical rule.
Example 2
Write the expression for the perimeter
of a rectangle.
2ℓ + 2w
Substitute 2.4 for ℓ and _ for w.
2×_+2×_
Multiply from left to right.
_ + 2 × 1.2
4.8 + _
Add.
So, the perimeter of the garden is _ meters.
382
_
Math
Talk
MATHEMATICAL PRACTICES 6
1
Compare How is evaluating
an algebraic expression
different from evaluating a
numerical expression?
© Houghton Mifflin Harcourt Publishing Company
The formula P = 2ℓ × 2w gives the perimeter P of a rectangle with length ℓ and width w.
What is the perimeter of a rectangular garden with a length of 2.4 meters and a width of
1.2 meters?
Name
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and Show
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MATH
M
BOARD
B
1. Evaluate 5k + 6 for k = 4.
Write the expression.
__
Substitute 4 for k.
5×_+6
Multiply.
_+6
Add.
_
Evaluate the expression for the given value of the variable.
2. m − 9 for m = 13
3. 16 − 3b for b = 4
4. p2 + 4 for p = 6
5. The formula A = ℓw gives the area A of a rectangle with length ℓ
and width w. What is the area in square feet of a United States flag
with a length of 12 feet and a width of 8 feet?
Math
Talk
MATHEMATICAL PRACTICES 8
Use Repeated Reasoning
What information do
you need to evaluate an
algebraic expression?
On
On Your
Your Own
Own
Practice: Copy and Solve Evaluate the expression for the given value of the variable.
© Houghton Mifflin Harcourt Publishing Company
6. 7s + 5 for s = 3
9. 6 × (2v − 3) for v = 5
12.
7. 21 − 4d for d = 5
10. 2 × ( k2 − 2) for k = 6
8. (t − 6)2 for t = 11
11. 5 × (f − 32) ÷ 9 for f = 95
DEEPER
The formula P = 4s gives the perimeter P of a square with
side length s. How much greater is the perimeter of a square with a side
length of 51_ inches than a square with a side length of 5 inches?
2
Chapter 7 • Lesson 5 383
MATHEMATICAL PRACTICES
COMMUNICA5&t1&34E7&3&tCONSTRUCT ARGUMENTS
OqnakdlRnkuhmf¤@ookhb`shnmr
OqnakdlRnkuhmf¤@ookhb`shnmr
The table shows how much a company charges for
skateboard wheels. Each pack of 8 wheels costs $50.
Shipping costs $7 for any order. Use the table for 13−15.
13. Complete the table.
14. A skateboard club has $200 to spend on new wheels
this year. What is the greatest number of packs of
wheels the club can order?
Costs for Skateboard Wheels
Packs
50 × n + 7
Cost
1
50 × 1 + 7
$57
2
15.
MATHEMATICAL
PRACTICE
1 Make Sense of Problems A sporting
goods store placed an order for 12 packs of wheels on
the first day of each month last year. How much did the
sporting goods store spend on these orders last year?
3
4
5
16.
What’s the Error? Bob used
these steps to evaluate 3m − 3 ÷ 3 for m = 8.
Explain his error.
SMARTER
WRITE
Math t Show Your Work
3 × 8 − 3 ÷ 3 = 24 − 3 ÷ 3
= 21 ÷ 3
17.
SMARTER
The surface area of a cube can be
found by using the formula 6s2, where s represents the
length of the side of the cube.
The surface area of a cube that has a side length of
54
3 meters is
108
2,916
384
meters squared.
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=7
Practice and Homework
Name
Evaluate Algebraic Expressions and Formulas
Lesson 7.5
COMMON CORE STANDARD—6.EE.A.2c
Apply and extend previous understandings of
arithmetic to algebraic expressions.
Evaluate the expression for the given values of the variables.
1. w + 6 for w = 11
2. 17 − 2c for c = 7
3. b2 − 4 for b = 5
11 1 6
17
____
4. (h − 3)2 for h = 5
____
7. 7m − 9n for m = 7 and
n=5
____
5. m + 2m + 3 for m = 12
____
8. d2 − 9k + 3 for d = 10 and
k=9
____
____
6. 4 × (21 − 3h) for h = 5
____
9. 3x + 4y ÷ 2 for x = 7 and
y = 10
____
Problem
Problem Solving
Solving
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10. The formula P = 2ℓ + 2w gives the perimeter
P of a rectangular room with length ℓ and width
w. A rectangular living room is 26 feet long and
21 feet wide. What is the perimeter of the room?
12.
WRITE
11. The formula c = 5(f − 32) ÷ 9 gives the Celsius
temperature in c degrees for a Fahrenheit
temperature of f degrees. What is the Celsius
temperature for a Fahrenheit temperature of
122 degrees?
Math Explain how the terms variable, algebraic expression, and evaluate are related.
Chapter 7
385
Lesson Check (6.EE.A.2c)
1. When Debbie baby-sits, she charges $5 to go to
the house plus $8 for every hour she is there.
The expression 5 + 8h gives the amount she
charges. How much will she charge to baby-sit
for 5 hours?
TEST
PREP
2. The formula to find the cost C of a square sheet
of glass is C = 25s2 where s represents the
length of a side in feet. How much will Ricardo
pay for a square sheet of glass that is 3 feet
on each side?
Spiral Review (6.NS.A.1, 6. RP.A.3c, 6.EE.A.1)
3_ + 5_ ÷ 2_
4 6 3
5. What is the value of 73?
4. Patricia scored 80% on a math test. She missed
4 problems. How many problems were on
the test?
6. James and his friends ordered b hamburgers
that cost $4 each and f fruit cups that cost $3
each. Write an algebraic expression for the total
cost of their purchases.
FOR MORE PRACTICE
GO TO THE
386
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3. Evaluate using the order of operations.
Name
Mid-Chapter Checkpoint
Personal Math Trainer
Online Assessment
and Intervention
Vocabulary
Vocabulary
Vocabulary
Choose the best term from the box to complete the sentence.
coefficient
1. A(n) ____ tells how many times a base is used
as a factor. (p. 357)
exponent
numerical expression
2. The mathematical phrase 5 + 2 × 18 is an example of a(n)
____ . (p. 363)
Concepts
Concepts and
and Skills
Skills
Find the value. (6.EE.A.1)
3. 54
4. 212
5. 83
7. 2 × ( 10 − 2 ) ÷ 22
8. 30 − ( 33 − 8 )
Evaluate the expression. (6.EE.A.1)
6. 92 × 2 − 42
Write an algebraic expression for the word expression. (6.EE.A.2a)
© Houghton Mifflin Harcourt Publishing Company
9. the quotient of c and 8
10. 16 more than the product of 5
and p
11. 9 less than the sum of x and 5
Evaluate the expression for the given value of the variable. (6.EE.A.2c)
12. 5 × (h + 3) for h = 7
13. 2 × (c2 − 5) for c = 4
14. 7a − 4a for a = 8
Chapter 7
387
15. The greatest value of any U.S. paper money ever printed is 105 dollars.
What is this amount written in standard form? (6.EE.A.1)
16. A clothing store is raising the price of all its sweaters by $3.00. Write an
expression that could be used to find the new price of a sweater that
originally cost d dollars. (6.EE.A.2a)
17. Kendra bought a magazine for $3 and 4 paperback books for $5 each.
The expression 3 + 4 × 5 represents the total cost in dollars of her
purchases. What are the terms in this expression? (6.EE.A.2b)
18. The expression 5c + 7m gives the number of people who can ride in
c cars and m minivans. What are the coefficients in this expression?
(6.EE.A.2b)
388
DEEPER
The formula P = a + b + c gives the perimeter P of a
triangle with side lengths a, b, and c. How much greater is the perimeter
of a triangular field with sides that measure 33 yards, 56 yards, and
65 yards than the perimeter of a triangular field with sides that measure
26 yards, 49 yards, and 38 yards? (6.EE.A.2c)
© Houghton Mifflin Harcourt Publishing Company
19.
Lesson 7.6
Name
Use Algebraic Expressions
Essential Question How can you use variables and algebraic expressions
to solve problems?
Expressions and Equations—
6.EE.B.6
MATHEMATICAL PRACTICES
MP1, MP2, MP4
Sometimes you are missing a number that you need to solve a problem.
You can represent a problem like this by writing an algebraic expression
in which a variable represents the unknown number.
Unlock
Unlock the
the Problem
Problem
Rafe’s flight from Los Angeles to New York took 5 hours. He wants to
know the average speed of the plane in miles per hour.
A Write an expression to represent the average speed of the plane.
Use a variable to represent the unknown quantity.
Think: The plane’s average speed is equal to the distance traveled divided by the
time traveled.
Use a variable to represent the unknown
quantity.
Let d represent the __
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traveled in units of __.
Write an algebraic expression for the
average speed.
d mi
____
hr
B Rafe looks up the distance between Los Angeles and New York on
the Internet and finds that the distance is 2,460 miles. Use this
distance to find the average speed of Rafe’s plane.
Evaluate the expression for d
Write the expression.
= 2,460.
d mi
___
5 hr
Substitute 2,460 for d.
mi
________
5 hr
2,460
mi ÷
Divide to find the unit rate.
mi
___________
= ______
1 hr
5 hr ÷ 5
Math
So, the plane's average speed was __ miles per hour.
Talk
MATHEMATICAL PRACTICES 1
Evaluate How could you
check whether you found
the plane’s average speed
correctly?
Chapter 7
389
In the problem on the previous page, the variable represented a single
value—the distance in miles between Los Angeles and New York. In other
situations, a variable may represent any number in a particular set of
numbers, such as the set of positive numbers.
Example
Joanna makes and sells candles online. She
charges $7 per candle, and shipping is $5 per order.
A Write an expression that Joanna can use to find the total cost for any
candle order.
Let n represent the number of __
a customer buys, where n is a whole number
greater than 0.
Think: The number of
candles a customer
buys will vary from
order to order.
The cost per order equals the charge per times the number of plus the shipping
candle
candles
charge.
↓
↓
↓
↓
↓
_
×
_
+
_
So, an expression for the total cost of a candle order is __.
B In March, one of Joanna’s customers placed an order for 4 candles. In
May, the same customer placed an order for 6 candles. What was the
total charge for both orders?
Find the charge in dollars for each order.
March
May
Write the expression.
7n + 5
7n + 5
Substitute the number of candles
ordered for n.
7×_+5
7×_+5
_+5
_
_+5
_
Follow the order of operations.
STEP 2
Find the charge in dollars for both orders.
Add the charge in dollars for
March to the charge in dollars
for May.
_+_=_
So, the total charge for both orders was _.
Math
Talk
MATHEMATICAL PRACTICES 2
Reasoning Why is the value of
the variable n in the Example
restricted to the set of whole
numbers greater than 0?
390
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STEP 1
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B
Louisa read that the highest elevation of Mount Everest is 8,848 meters.
She wants to know how much higher Mount Everest is than Mount
Rainier. Use this information for 1–2.
1. Write an expression to represent the
difference in heights of the two mountains. Tell
what the variable in your expression represents.
2. Louisa researches the highest elevation of Mount
Rainier and finds that it is 4,392 meters. Use
your expression to find the difference in the
mountains’ heights.
Math
Talk
On
On Your
Your Own
Own
A muffin recipe calls for 3 times as much flour as sugar.
Use this information for 3–5.
3. Write an expression that can be used to
find the amount of flour needed for a given
amount of sugar. Tell what the variable in
your expression represents.
5.
MATHEMATICAL
PRACTICE
MATHEMATICAL PRACTICES 2
Reason Quantitatively Explain
whether the variable in
Exercise 1 represents a single
unknown number or any
number in a particular set.
4. Use your expression to find the amount of flour
needed when 3_4 cup of sugar is used.
2 Reason Quantitatively Is the value of the variable in your expression
© Houghton Mifflin Harcourt Publishing Company
restricted to a particular set of numbers? Explain.
Practice: Copy and Solve Write an algebraic expression for each
word expression. Then evaluate the expression for these values of the
variable: 1_2 , 4, and 6.5.
6. the quotient of p and 4
7. 4 less than the sum of x and 5
Chapter 7 • Lesson 6 391
MATHEMATICAL PRACTICES
ANALYZEt-00,'034536$563&t13&$*4*0/
OqnakdlRnkuhmf¤@ookhb`shnmr
OqnakdlRnkuhmf¤@ookhb`shnmr
Use the graph for 8–10.
Top Speeds of African Animals
8. Write expressions for the distance in feet that
each animal could run at top speed in a given
amount of time. Tell what the variable in your
expressions represents.
22 Elephant
Animal
Cheetah
103
51 Giraffe
9.
DEEPER
How much farther could a
cheetah run in 20 seconds at top speed than a
hippopotamus could?
0
10.
A giraffe runs at top speed
toward a tree that is 400 feet away. Write an
expression that represents the giraffe’s
distance in feet from the tree after s seconds.
Personal Math Trainer
11.
SMARTER
A carnival charges $7 for
admission and $2 for each ride. An expression
for the total cost of going to the carnival and
riding n rides is 7 + 2n.
Complete the table by finding the total cost of
going to the carnival and riding n rides.
Number of rides, n
1
2
3
4
392
7 + 2n
Total Cost
40
60
80
100
Speed (ft/sec)
WRITE
SMARTER
20
Math t Show Your Work
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21 Hippopotamus
Practice and Homework
Name
Lesson 7.6
Use Algebraic Expressions
Jeff sold the pumpkins he grew for $7 each at the farmer’s
market.
1. Write an expression to represent the amount of
money Jeff made selling the pumpkins. Tell what
the variable in your expression represents.
COMMON CORE STANDARD—6.EE.B.6
Reason about and solve one-variable
equations and inequalities.
2. If Jeff sold 30 pumpkins, how much money
did he make?
7p, where p is the number
of pumpkins
An architect is designing a building. Each floor will be
12 feet tall.
3. Write an expression for the number of
floors the building can have for a given building
height. Tell what the variable in your
expression represents.
4. If the architect is designing a building that is
132 feet tall, how many floors can be built?
Write an algebraic expression for each word expression.
Then evaluate the expression for these values of the variable: 1, 6, 13.5.
5. the quotient of 100 and the sum of b and 24
6. 13 more than the product of m and 5
© Houghton Mifflin Harcourt Publishing Company
Problem
Problem Solving
Solving
7. In the town of Pleasant Hill, there is an average of
16 sunny days each month. Write an expression
to represent the approximate number of sunny
days for any number of months. Tell what the
variable represents.
9.
WRITE
8. How many sunny days can a resident of Pleasant
Hill expect to have in 9 months?
Math Describe a situation in which a variable could be used to represent any
whole number greater than 0.
Chapter 7
393
Lesson Check (6.EE.B.6)
1. Oliver drives 45 miles per hour. Write an
expression that represents the distance in
miles he will travel for h hours driven.
TEST
PREP
2. Socks cost $5 per pair. The expression 5p
represents the cost in dollars of p pairs of
socks. Why must p be a whole number?
3. Sterling silver consists of 92.5% silver and 7.5%
copper. What decimal represents the portion of
silver in sterling silver?
4. How many pints are equivalent to 3 gallons?
5. Which operation should be done first to
evaluate 10 + (66 – 62)?
6. Evaluate the algebraic expression h(m + n) ÷ 2
for h = 4, m = 5, and n = 6.
FOR MORE PRACTICE
GO TO THE
394
Personal Math Trainer
© Houghton Mifflin Harcourt Publishing Company
Spiral Review (6.RP.A.3c, 6.RP.A.3d, 6.EE.A.1, 6.EE.A.2c)
Lesson 7.7
Name
Problem Solving • Combine Like Terms
Expressions and Equations—
6.EE.A.3
MATHEMATICAL PRACTICES
MP1, MP4, MP5
Essential Question How can you use the strategy use a model to
combine like terms?
Like terms are terms that have the same variables with the
same exponents. Numerical terms are also like terms.
Algebraic Expression
Terms
Like Terms
5x + 3y − 2x
5x, 3y, and 2x
2
5x and 2x
2
8z2 + 4z + 12z2
8z , 4z, and 12z
8z2 and 12z2
15 − 3x + 5
15, 3x, and 5
15 and 5
Unlock
Unlock the
the Problem
Problem
Baseball caps cost $9, and patches cost $4. Shipping is $8 per order. The
expression 9n + 4n + 8 gives the cost in dollars of buying caps with patches
for n players. Simplify the expression 9n + 4n + 8 by combining like terms.
Use the graphic organizer to help you solve the problem.
Read the Problem
What do I need to find?
I need to simplify the expression
© Houghton Mifflin Harcourt Publishing Company • Image Credits: (r) ©Kevin Dodge/Corbis
__ .
What information do I
need to use?
How will I use the
information?
I need to use the like terms 9n
I can use a bar model to find the
and _ .
sum of the _ terms.
Solve the Problem
Draw a bar model to add _ and _. Each square represents n, or 1n.
9n
n
n
n
n
n
4n
n
n
n
n
n
n
n
The model shows that 9n + 4n =
_.
9n + 4n + 8 = _ + 8
So, a simplified expression for the cost in dollars is __ .
n
Math
Talk
n
MATHEMATICAL PRACTICES 4
Use Models Explain how the
bar model shows that your
answer is correct.
Chapter 7
395
Try Another Problem
Paintbrushes normally cost $5 each, but they are on sale for $1 off.
A paintbrush case costs $12. The expression 5p − p + 12 can be used to find
the cost in dollars of buying p paintbrushes on sale plus a case for them.
Simplify the expression 5p − p + 12 by combining like terms.
Use the graphic organizer to help you solve the problem.
Read the Problem
What do I need to find?
What information do I
need to use?
How will I use the
information?
Solve the Problem
So, a simplified expression for the cost in dollars is __ .
MATHEMATICAL
PRACTICE
4
Use Models Explain how the bar model shows that your answer is correct.
2. Explain how you could combine like terms without using a model.
396
© Houghton Mifflin Harcourt Publishing Company
1.
Name
Unlock the Problem
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M
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B
1. Museum admission costs $7, and tickets to the mammoth
exhibit cost $5. The expression 7p + 5p represents the
cost in dollars for p people to visit the museum and attend
the exhibit. Simplify the expression by combining like terms.
√
Read the entire problem carefully
before you begin to solve it.
√
Check your answer by using a
different method.
First, draw a bar model to combine the like terms.
WRITE
Math
Show Your Work
Next, use the bar model to simplify the expression.
So, a simplified expression for the cost in dollars is _.
2.
What if the cost of tickets to the exhibit were
reduced to $3? Write an expression for the new cost in dollars for p
people to visit the museum and attend the exhibit. Then, simplify
the expression by combining like terms.
SMARTER
© Houghton Mifflin Harcourt Publishing Company
3. A store receives tomatoes in boxes of 40 tomatoes each. About
4 tomatoes per box cannot be sold due to damage. The expression
40b − 4b gives the number of tomatoes that the store can sell from
a shipment of b boxes. Simplify the expression by combining like
terms.
4. Each cheerleading uniform includes a shirt and a skirt. Shirts
cost $12 each, and skirts cost $18 each. The expression 12u + 18u
represents the cost in dollars of buying u uniforms. Simplify the
expression by combining like terms.
5. A shop sells vases holding 9 red roses and 6 white roses. The
expression 9v + 6v represents the total number of roses needed
for v vases. Simplify the expression by combining like terms.
Chapter 7 • Lesson 7 397
MATHEMATICAL PRACTICES .0%&-t3&"40/tM",&4&/4&
On
On Your
Your Own
Own
DEEPER
Marco received a gift card. He used it to
buy 2 bike lights for $10.50 each. Then he bought a
handlebar bag for $18.25. After these purchases, he had
$0.75 left on the card. How much money was on the gift
card when Marco received it?
7. Lydia collects shells. She has 24 sea snail shells,
16 conch shells, and 32 scallop shells. She wants to
display the shells in equal rows, with only one type of
shell in each row. What is the greatest number of shells
Lydia can put in each row?
8.
9.
Sea snail shells
Scallop shell
Conch shell
SMARTER
The three sides of a triangle
measure 3x + 6 inches, 5x inches, and 6x inches.
Write an expression for the perimeter of the
triangle in inches. Then simplify the expression
by combining like terms.
MATHEMATICAL
PRACTICE
3 Verify the Reasoning of Others Karina
states that you can simplify the expression 20x + 4
by combining like terms to get 24x. Does Karina’s
statement make sense? Explain.
Personal Math Trainer
10.
SMARTER
Vincent is ordering accessories for his surfboard. A set of
fins costs $24 each and a leash costs $15. The shipping cost is $4 per order.
The expression 24b + 15b + 4 can be used to find the cost in dollars of buying
b fins and b leashes plus the cost of shipping.
For numbers 10a–10c, select True or False for each statement.
10a.
The terms are 24b, 15b and 4.
● True
● False
10b.
The like terms are 24b and 15b.
● True
● False
● True
● False
10c. The simplified expression is 43b.
398
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6.
Practice and Homework
Name
Problem Solving • Combine Like Terms
Lesson 7.7
COMMON CORE STANDARD—6.EE.A.3
Apply and extend previous understandings of
arithmetic to algebraic expressions.
Read each problem and solve.
1. A box of pens costs $3 and a box of markers costs $5. The
expression 3p + 5p represents the cost in dollars to make p
packages that includes 1 box of pens and 1 box of markers.
Simplify the expression by combining like terms.
3p + 5p = 8p
_____
2. Riley’s parents got a cell phone plan that has a $40 monthly
fee for the first phone. For each extra phone, there is a $15
phone service charge and a $10 text service charge. The
expression 40 + 15e + 10e represents the total phone bill in
dollars, where e is the number of extra phones. Simplify the
expression by combining like terms.
_____
3. A radio show lasts for h hours. For every 60 minutes of air
time during the show, there are 8 minutes of commercials.
The expression 60h – 8h represents the air time in minutes
available for talk and music. Simplify the expression by
combining like terms.
© Houghton Mifflin Harcourt Publishing Company
_____
4. A sub shop sells a meal that includes an Italian sub for
$6 and chips for $2. If a customer purchases more than
3 meals, he or she receives a $5 discount. The expression 6m
+ 2m – 5 shows the cost in dollars of the customer’s order for
m meals, where m is greater than 3. Simplify the expression by
combining like terms.
_____
5.
WRITE
Math Explain how combining like terms is similar
to adding and subtracting whole numbers. How are they different?
Chapter 7
399
Lesson Check (6.EE.A.3)
1. For each gym class, a school has 10 soccer
balls and 6 volleyballs. All of the classes share
15 basketballs. The expression 10c + 6c + 15
represents the total number of balls the
school has for c classes. What is a simpler
form of the expression?
TEST
PREP
2. A public library wants to place 4 magazines
and 9 books on each display shelf. The
expression 4s + 9s represents the number
of items that will be displayed on s shelves.
Simplify this expression.
Spiral Review (6.RP.A.3a)
4. How many kilograms are equivalent to
3,200 grams?
5. Toni earns $200 per week plus $5 for every
magazine subscription that she sells. Write
an expression, in dollars, that represents how
much she will earn in a week in which she sells
s subscriptions.
6. At a snack stand, drinks cost $1.50. Write an
expression that could be used to find the total
cost of d drinks.
© Houghton Mifflin Harcourt Publishing Company
3. A bag has 8 bagels. Three of the bagels
are cranberry. What percent of the bagels
are cranberry?
FOR MORE PRACTICE
GO TO THE
400
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Lesson 7.8
Name
Generate Equivalent Expressions
Expressions and Equations—
6.EE.A.3
MATHEMATICAL PRACTICES
MP2, MP3, MP8
Essential Question How can you use properties of operations to write
equivalent algebraic expressions?
Equivalent expressions are equal to each other for any values of their
variables. For example, x + 3 and 3 + x are equivalent. You can use
properties of operations to write equivalent expressions.
x+3
3+x
4+3
3+4
7
7
Properties of Addition
Commutative Property of Addition
12 + a = a + 12
If the order of terms changes, the sum stays the same.
Associative Property of Addition
5 + (8 + b) = (5 + 8) + b
When the grouping of terms changes, the sum stays the same.
Identity Property of Addition
0+c=c
The sum of 0 and any number is that number.
Properties of Multiplication
Commutative Property of Multiplication
d×9=9×d
If the order of factors changes, the product stays the same.
Associative Property of Multiplication
When the grouping of factors changes, the product stays the same.
11 × (3 × e) = (11 × 3) × e
Identity Property of Multiplication
1×f=f
The product of 1 and any number is that number.
Unlock
Unlock the
the Problem
Problem
Nelson ran 2 miles, 3 laps, and 5 miles. The expression 2 + 3ℓ + 5
represents the total distance in miles Nelson ran, where ℓ is the length in
miles of one lap. Write an equivalent expression with only two terms.
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Rewrite the expression 2 + 3ℓ + 5 with only two terms.
The like terms are 2 and
_. Use the
2 + 3ℓ + 5 = 3ℓ + _ + 5
__ Property to reorder the terms.
Use the
__ Property to regroup the terms.
Add within the parentheses.
= 3ℓ + ( _ + _ )
= 3ℓ + _
So, an equivalent expression for the total distance in miles is __.
Chapter 7
401
Distributive Property
Multiplying a sum by a number is the same as multiplying
each term by the number and then adding the products.
5 × (g + 9) = (5 × g) + (5 × 9)
The Distributive Property can also be used with multiplication and
subtraction. For example, 2 × (10 − h) = (2 × 10) − (2 × h).
Example 1
Use properties of operations to write an
expression equivalent to 5a + 8a − 16 by combining like terms.
Use the Commutative Property of Multiplication
to rewrite the like terms 5a and 8a.
5a + 8a − 16 = a × _ + a × _ − 16
Use the Distributive Property to rewrite
a × 5 + a × 8.
= _ × (5 + 8) − 16
Add within the parentheses.
= a × _ − 16
Use the Commutative Property of Multiplication
to rewrite a × 13.
= _ − 16
So, the expression __ is equivalent to 5a + 8a − 16.
Example 2
Use the Distributive Property to write an
equivalent expression.
A 6(y + 7)
Use the Distributive Property.
6(y + 7) = (6 × _ ) + (6 × _ )
Multiply within the parentheses.
= 6y + _
So, the expression __ is equivalent to 6(y + 7).
When one factor in a product
is in parentheses, you can leave
out the multiplication sign. So,
6 × ( y + 7) can be written as
6( y + 7).
B 12a + 8b
The GCF of 12 and 8 is _.
Write the first term, 12a, as the product 12a + 8b = 4 × 3a + 8b
of the GCF and another factor.
Write the second term, 8b, as the
product of the GCF and another factor.
= 4 × 3a + 4 × _
Use the Distributive Property.
= 4 × ( _ + 2b)
So, the expression __ is equivalent to 12a + 8b.
402
Math
Talk
MATHEMATICAL PRACTICES 8
Generalize Give a different
expression that is equivalent
to 12a + 8b. Explain what
property you used.
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Find the greatest common factor (GCF)
of the coefficients of the terms.
Name
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hhow
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and Show
Sh
MATH
M
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B
Use properties of operations to write an equivalent expression by
combining like terms.
7 r − 1___
5 r
1. 3___
10
10
2. 20a + 18 + 16a
3. 7s + 8t + 10s + 12t
Use the Distributive Property to write an equivalent expression.
4. 8(h + 1.5)
5. 4m + 4p
6. 3a + 9b
Math
Talk
On
On Your
Your Own
Own
Practice: Copy and Solve Use the Distributive Property to write an
equivalent expression.
7. 3.5(w + 7)
10. 20b + 16c
13.
MATHEMATICAL
PRACTICE
MATHEMATICAL PRACTICES 6
Compare List three
expressions with two terms
that are equivalent to 5x.
Compare and discuss your list
with a partner’s.
8. _12(f + 10)
9. 4(3z + 2)
11. 30d + 18
12. 24g − 8h
4 Write an Expression The lengths of the sides of a triangle
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are 3t, 2t + 1, and t + 4. Write an expression for the perimeter (sum of
the lengths). Then, write an equivalent expression with 2 terms.
14.
DEEPER
Use properties of operations to write an expression
equivalent to the sum of the expressions 3(g + 5) and 2(3g − 6).
Chapter 7 • Lesson 8 403
MATHEMATICAL PRACTICES
COMMUNICA5&t1&34E7&3&tCONSTRUCT ARGUMENTS
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OqnakdlRnkuhmf¤@ookhb`shnmr
15.
Sense or Nonsense Peter and Jade are
using what they know about properties to write an expression
equivalent to 2 × (n + 6) + 3. Whose answer makes sense?
Whose answer is nonsense? Explain your reasoning.
SMARTER
Peter’s Work
Jade’s Work
Expression:
2 × (n + 6) + 3
Expression:
2 × (n + 6) + 3
Associative Property
of Addition:
2 × n + (6 + 3)
Distributive Property:
(2 × n) + (2 × 6) + 3
2n + 12 + 3
Add within parentheses:
2×n+9
Multiply within
parentheses:
Multiply:
2n + 9
Associative Property
of Addition:
2n + (12 + 3)
Add within
parentheses:
2n + 15
16.
SMARTER
Write the algebraic expression in the box that shows an
equivalent expression.
6(z + 5)
6z + 5
404
6z + 5z
11z
2 + 6z + 3
6z + 30
© Houghton Mifflin Harcourt Publishing Company
For the answer that is nonsense, correct the statement.
Practice and Homework
Name
Lesson 7.8
Generate Equivalent Expressions
COMMON CORE STANDARD—6.EE.A.3
Apply and extend previous understandings of
arithmetic to algebraic expressions.
Use properties of operations to write an equivalent expression by combining like terms.
1. 7h − 3h
2. 5x + 7 + 2x
4h
____
4. y2 + 13y − 8y
____
3. 16 + 13p − 9p
____
5. 5(2h + 3) + 3h
____
6. 12 + 18n + 7 − 14n
____
____
Use the Distributive Property to write an equivalent expression.
7. 2(9 + 5k)
8. 4d + 8
____
9. 21p + 35q
____
____
Problem
Problem Solving
Solving
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10. The expression 15n + 12n + 100 represents
the total cost in dollars for skis, boots, and
a lesson for n skiers. Simplify the expression
15n + 12n + 100. Then find the total cost
for 8 skiers.
_______
12.
WRITE
11. Casey has n nickels. Megan has 4 times as many
nickels as Casey has. Write an expression for the
total number of nickels Casey and Megan have.
Then simplify the expression.
_______
Math Explain how you would use properties to write an expression equivalent to 7y + 4b – 3y.
Chapter 7
405
Lesson Check (6.EE.A.3)
1. Tickets to a museum cost $8. The dinosaur
exhibit costs $5 extra. The expression 8n + 5n
represents the cost in dollars for n people to
visit the museum and the exhibit. What is a
simpler form of the expression is 8n + 5n?
TEST
PREP
2. What is an expression that is equivalent to
3(2p – 3)?
3. A Mexican restaurant received 60 take-out
orders. The manager found that 60% of the
orders were for tacos and 25% of the orders
were for burritos. How many orders were for
other items?
4. The area of a rectangular field is 1,710 square
feet. The length of the field is 45 feet. What is
the width of the field?
5. How many terms are in 2 + 4x + 7y?
6. Boxes of cereal usually cost $4, but they are on
sale for $1 off. A gallon of milk costs $3. The
expression 4b – 1b + 3 can be used to find
the cost in dollars of buying b boxes of cereal
and a gallon of milk. Write the expression in
simpler form.
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Spiral Review (6.RP.A.3c, 6.RP.A.3d, 6.EE.A.2b, 6.EE.A.3)
Lesson 7.9
Name
Identify Equivalent Expressions
Expressions and Equations—
6.EE.A.4
MATHEMATICAL PRACTICES
MP2, MP6
Essential Question How can you identify equivalent algebraic
expressions?
Unlock
Unlock the
the Problem
Problem
Each train on a roller coaster has 10 cars, and
each car can hold 4 riders. The expression
10t × 4 can be used to find the greatest number of
riders when there are t trains on the track. Is this
expression equivalent to 14t? Use properties of
operations to support your answer.
• What is one property of operations that you
could use to write an expression equivalent to
10t × 4?
Determine whether 10t × 4 is equivalent to 14t.
The expression 14t is the product of a number
and a variable, so rewrite 10t × 4 as a product of a
number and a variable.
Use the Commutative Property of
Multiplication.
10t × 4 = 4 ×
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___
_
_) × t
Use the
Property of Multiplication.
= (4 ×
Multiply within the parentheses.
=_
Compare the expressions 40t and 14t.
Think: 40 times a number is not equal to 14 times the number, except when the
number is 0.
Check by choosing a value for t and evaluating 40t and 14t.
Write the expressions.
40t
14t
Use 2 as a value for t.
40 × _
14 ×
_
_
Multiply. The expressions have
different values.
So, the expressions 10t × 4 and 14t are ___.
_
Math
Talk
MATHEMATICAL PRACTICES 6
Explain Why are the expressions
7a and 9a not equivalent, even
though they have the same
value when a = 0?
Chapter 7
407
Example
Use properties of operations to determine
whether the expressions are equivalent.
A 7y + (x + 3y) and 10y + x
The expression 10y + x is a sum of two terms, so rewrite
7y + ( x + 3y) as a sum of two terms.
7y + (x + 3y) = 7y + ( _ + _ )
Use the Commutative Property of Addition to
rewrite x + 3y.
___ Property of
Use the
Addition to group like terms.
= ( _ + 3y) + x
Combine like terms.
=_+x
Math
Talk
Compare the expressions 10y + x and 10y + x: They are the same.
So, the expressions 7y + ( x + 3y) and 10y + x
are __.
MATHEMATICAL PRACTICES 8
Generalize Explain how
you can decide whether two
algebraic expressions are
equivalent.
B 10(m + n) and 10m + n
The expression 10m + n is a sum of two terms, so rewrite 10(m + n) as a
sum of two terms.
Use the Distributive Property.
10(m + n) = (10 ×
Multiply within the parentheses.
_ ) + (10 ×_ )
= 10m +
_
Compare the expressions 10m + 10n and 10m + n.
Think: The first terms of both expressions are
second terms are different.
__ , but the
Write the expressions.
10m + 10n
10m + n
Use 2 as a value for m and 4 as a
value for n.
10 × _ + 10 × _
10 × _ + _
Multiply.
_+_
_+_
Add. The expressions have different values.
_
_
So, the expressions 10(m + n) and 10m + n are
___.
408
Math
Talk
MATHEMATICAL PRACTICES 3
Apply How do you know that
the terms 10n and n from Part
B are not equivalent?
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Check by choosing values for m and n and evaluating 10m + 10n and
10m + n.
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hhow
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and Show
Sh
MATH
M
BOARD
B
Use properties of operations to determine whether the
expressions are equivalent.
1. 7k + 4 + 2k and 4 + 9k
Rewrite 7k + 4 + 2k. Use the Commutative
Property of Addition.
7k + 4 + 2k = 4 + _ + 2k
Use the Associative Property of Addition.
= 4 + (_ + _)
Add like terms.
=4+_
The expressions 7k + 4 + 2k and 4 + 9k are __.
2. 9a × 3 and 12a
3. 8p + 0 and 8p × 0
4. 5(a + b) and
(5a + 2b) + 3b
Math
Talk
MATHEMATICAL PRACTICES 2
Reasoning How do you
know that x + 5 is not
equivalent to x + 8?
On
On Your
Your Own
Own
Use properties of operations to determine whether the
expressions are equivalent.
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5. 3(v + 2) + 7v and 16v
8.
6. 14h + (17 + 11h) and
25h + 17
7. 4b × 7 and 28b
DEEPER
Each case of dog food contains c cans. Each case of cat food
contains 12 cans. Four students wrote the expressions below for the
number of cans in 6 cases of dog food and 1 case of cat food. Which of
the expressions are correct?
6c + 12
6c × 12
6(c + 2)
(2c + 4) × 3
Chapter 7 • Lesson 9 409
MATHEMATICAL PRACTICES
ANALYZEt-00,'034536$563&t13&$*4*0/
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OqnakdlRnkuhmf¤@ookhb`shnmr
Use the table for 9–11.
9. Marcus bought 4 packets of baseball cards and
4 packets of animal cards. Write an algebraic
expression for the total number of cards
Marcus bought.
10.
MATHEMATICAL
PRACTICE
Collectible Cards
Type
Number per Packet
Baseball
b
Cartoon
c
Movie
m
Animal
a
3 Make Arguments Is the
expression for the number of cards Marcus
bought equivalent to 4(a + b)? Justify your
answer.
WRITE
12.
SMARTER
Angelica buys 3 packets of movie
cards and 6 packets of cartoon cards
and adds these to the 3 packets of movie cards
she already has. Write three equivalent algebraic
expressions for the number of cards Angelica
has now.
SMARTER
Select the expressions that are
equivalent to 3(x + 2). Mark all that apply.
A 3x + 6
B 3x + 2
C 5x
D x+5
410
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11.
Math t Show Your Work
Practice and Homework
Name
Lesson 7.9
Identify Equivalent Expressions
Use properties of operations to determine whether the
expressions are equivalent.
1. 2s + 13 + 15s and
17s + 13
equivalent
____
4. (9w × 0) − 12 and 9w − 12
____
7. 14m + 9 − 6m and 8m + 9
____
10. 9x + 0 + 10x and 19x + 1
____
2. 5 × 7h and 35h
____
5. 11(p + q) and
11p + (7q + 4q)
____
8. (y × 1) + 2 and y + 2
____
11. 12c − 3c and 3(4c − 1)
____
COMMON CORE STANDARD—6.EE.A.4
Apply and extend previous understandings of
arithmetic to algebraic expressions.
3. 10 + 8v − 3v and 18 − 3v
____
6. 6(4b + 3d) and 24b + 3d
____
9. 4 + 5(6t + 1) and 9 + 30t
____
12. 6a × 4 and 24a
____
Problem
Problem Solving
Solving
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13. Rachel needs to write 3 book reports with b pages
and 3 science reports with s pages during the
school year. Write an algebraic expression for the
total number of pages Rachel will need to write.
15.
WRITE
14. Rachel’s friend Yassi has to write 3(b + s) pages
for reports. Use properties of operations to
determine whether this expression is equivalent
to the expression for the number of pages Rachel
has to write.
Math Use properties of operations to show whether 7y + 7b + 3y and 7(y + b) + 3b are
equivalent expressions. Explain your reasoning.
Chapter 7
411
Lesson Check (6.EE.A.4)
1. Ian had 4 cases of comic books and 6 adventure
books. Each case holds c comic books. He gave
1 case of comic books to his friend. Write an
expression that gives the total number of books
Ian has left.
TEST
PREP
2. In May, Xia made 5 flower planters with f
flowers in each planter. In June, she made
8 flower planters with f flowers in each planter.
Write an expression in simplest form that gives
the number of flowers Xia has in the planters.
3. Keisha wants to read for 90 minutes. So far, she
has read 30% of her goal. How much longer
does she need to read to reach her goal?
4. Marvyn travels 105 miles on his scooter. He
travels for 3 hours. What is his average speed?
5. The expression 5(F − 32) ÷ 9 gives the Celsius
temperature for a Fahrenheit temperature of
F degrees. The noon Fahrenheit temperature
in Centerville was 86 degrees. What was the
temperature in degrees Celsius?
6. At the library book sale, hardcover books sell for
$4 and paperbacks sell for $2. The expression
4b + 2b represents the total cost for b hardcover
books and b paperbacks. Write a simpler
expression that is equivalent to 4b + 2b.
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Spiral Review (6.RP.A.3c, 6.RP.A.3d, 6.EE.A.2c, 6.EE.A.3)
Name
Chapter 7 Review/Test
Personal Math Trainer
Online Assessment
and Intervention
1. Use exponents to rewrite the expression.
3×3×3×3×5×5
×5
3
2. A plumber charges $10 for transportation and $55 per hour for repairs.
Write an expression that can be used to find the cost in dollars for a
repair that takes h hours.
3. Ellen is 2 years older than her brother Luke. Let k represent Luke’s age.
Identify the expression that can be used to find Ellen’s age.
A
k−2
B
k+2
C
2k
D
k
_
2
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4. Write 43 using repeated multiplication. Then find the value of 43.
5. Jasmine is buying beans. She bought r pounds of red beans that
cost $3 per pound and b pounds of black beans that cost $2 per pound.
The total amount of her purchase is given by the expression 3r + 2b.
Select the terms of the expression. Mark all that apply.
A 2
B 2b
C 3
D 3r
Assessment Options
Chapter Test
Chapter 7
413
6. Choose the number that makes the sentence true.
The formula V = s3 gives the volume V of a cube with side length s.
The volume of a cube that has a side length of 8 inches
24
is
64
inches cubed.
512
7. Liang is ordering new chairs and cushions for his dining room table.
A new chair costs $88 and a new cushion costs $12. Shipping costs $34.
The expression 88c + 12c + 34 gives the total cost for buying c sets of
chairs and cushions. Simplify the expression by combining like terms.
8. Mr. Ruiz writes the expression 5 × (2 + 1)2 ÷ 3 on the board.
Chelsea says the first step is to evaluate 12. Explain Chelsea’s
mistake. Then, evaluate the expression.
the product of 7 and m
Write an algebraic expression for the word expression. Then, evaluate the
expression for m = 4. Show your work.
414
© Houghton Mifflin Harcourt Publishing Company
9. Jake writes this word expression.
Name
10. Sora has some bags that each contain 12 potatoes. She takes 3 potatoes
from each bag. The expression 12p – 3p represents the number of potatoes
p left in the bags. Simplify the expression by combining like terms. Draw a
line to match the expression with the simplified expression.
12p − 3p
© Houghton Mifflin Harcourt Publishing Company
11.
•
15p
•
13p
•
11p
•
9p
•
DEEPER
Logan works at a florist. He earns $600 per week plus $5 for
each floral arrangement he delivers. Write an expression that gives the
amount in dollars that Logan earns for delivering f floral arrangements.
Use the expression to find the amount Logan will earn if he delivers
45 floral arrangements in one week. Show your work.
12. Choose the word that makes the sentence true.
Dara wrote the expression 7 × (d + 4) in her notebook. She used the
Associative
Commutative
Property to write the equivalent expression 7d + 28.
Distributive
Chapter 7
415
13. Use properties of operations to determine whether 5(n + 1) + 2n
and 7n + 1 are equivalent expressions.
14. Alisha buys 5 boxes of peanut butter granola bars and 5 boxes of
cinnamon granola bars. Let p represent the number of peanut butter
granola bars and c represent the number of cinnamon granola bars.
Jaira and Emma each write an expression that represents the total
number of granola bars Alisha bought. Are the expressions equivalent?
Justify your answer.
Jaira
Emma
5p + 5c
5(p + c)
15. Abe is 3 inches taller than Chen. Select the expressions that represent
Abe’s height if Chen’s height is h inches. Mark all that apply.
h−3
h+3
the sum of h and 3
the difference between h and 3
3(k + 2)
6k + 5
416
3k + 2k
5k
2 + 6k + 3
3k + 6
© Houghton Mifflin Harcourt Publishing Company
16. Write the algebraic expression in the box that shows an equivalent
expression.
Name
17. Draw a line to match the property with the expression that shows the
property.
Associative Property of Addition
Commutative Property of Addition
Identity Property of Addition
•
•
•
•
•
•
0 + 14 = 14
14 + b = b + 14
6 + (8 + b) = (6 + 8) + b
Personal Math Trainer
18.
SMARTER
A bike rental company charges $10 to rent a bike plus
$2 for each hour the bike is rented. An expression for the total cost of
renting a bike for h hours is 10 + 2h. Complete the table to find the total
cost of renting a bike for h hours.
Number of Hours, h
10 + 2h
1
10 + 2 × 1
Total Cost
2
3
4
19. An online sporting goods store charges $12 for a pair of athletic socks.
Shipping is $2 per order.
Part A
Write an expression that Hana can use to find the total cost in dollars for
ordering n pairs of socks.
© Houghton Mifflin Harcourt Publishing Company
Part B
Hana orders 3 pairs of athletic socks and her friend, Charlie, orders
2 pairs of athletic socks. What is the total cost, including shipping,
for both orders? Show your work.
Chapter 7
417
20. Fernando simplifies the expression (6 + 2)2 – 4 × 3.
Part A
Fernando shows his work on the board. Use numbers and words to
explain his mistake.
(6 + 2)2 – 4 × 3
(6 + 4) – 4 × 3
10 – 4 × 3
6×3
18
Part B
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Simplify the expression (6 + 2)2 − 4 × 3 using the order of operations.
418
8
Chapter
Algebra: Equations and
Inequalities
Personal Math Trainer
Show Wha t You Know
Online Assessment
and Intervention
Check your understanding of important skills.
Name
Multiplication Properties Find the unknown number. Write which multiplication
property you used.
1. 42 × _ = 42
_______
2. 9 × 6 = _ × 9
_______
Evaluate Algebraic Expressions Evaluate the expression.
3. 4a − 2b for a = 5 and b = 3
_______
5. 8c × d − 6 for c = 10 and d = 2
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_______
4. 7x + 9y for x = 7 and y = 1
_______
6. 4s ÷ t + 10 for s = 9 and t = 3
_______
Add Fractions and Decimals Find the sum. Write the sum in simplest form.
7. 35.68 + 17.84 = __
3
9. _4 + 1_ = __
8
8. 24.38 + 25.3 = __
2
10. _5 + 1_ = __
4
Math in the
The equation m = 19.32v can be used to find the mass m in
grams of a pure gold coin with volume v in cubic centimeters.
Carl has a coin with a mass of 37.8 grams. The coin’s volume is
2.1 cubic centimeters. Could the coin be pure gold? Explain your
reasoning.
Chapter 8
419
Voca bula ry Builder
Visualize It
Review Words
Use the review words to complete the tree diagram.
You may use some words more than once.
algebraic expressions
numbers
numerical expressions
expressions
operations
variables
Preview Words
Addition Property of
Equality
equation
inequality
Understand Vocabulary
inverse operations
Draw a line to match the preview word with its definition.
Preview Words
1. Addition Property of
solution of an equation
Subtraction Property
Definitions
• operations that undo each
•
Equality
of Equality
other
2. inequality
•
• a value of a variable that makes an
equation true
4. equation
•
• property that states that if you add
•
5. solution of an equation
6. Subtraction Property
of Equality
•
•
the same number to both sides of an
equation, the two sides will remain equal
• a mathematical statement that compares
two expressions by using the symbol <,
>, ≤, ≥, or ≠
• property that states that if you subtract
the same number from both sides of an
equation, the two sides will remain equal
• a statement that two mathematical
expressions are equal
420
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™Multimedia eGlossary
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3. inverse operations
Chapter 8 Vocabulary
Addition Property
of Equality
propiedad de suma
de la igualdad
algebraic expression
expresión algebraica
3
2
Division Property
of Equality
equation
propiedad de división
de la igualdad
24
ecuación
Identity Property
of Addition
Identity Property
of Multiplication
propiedad de identidad
de la suma
40
propiedad de identidad
de la multiplicación 41
inequality
inverse operations
desigualdad
operaciones inversas
43
27
46
variable
variable
An algebraic or numerical sentence that
shows that two quantities are equal
Examples:
8 + 12 = 20
14 = a − 3
2d = 14
The property that states that the product of
any number and 1 is that number
Examples:
475 × 1 = 475
2×1=2
__
__
3
3
0.7 × 1 = 0.7
Opposite operations, or operations that
undo each other, such as addition and
subtraction or multiplication and division
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3 × (a + 4)
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variable
3×y
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x + 10
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© Houghton Mifflin Harcourt Publishing Company
© Houghton Mifflin Harcourt Publishing Company
© Houghton Mifflin Harcourt Publishing Company
© Houghton Mifflin Harcourt Publishing Company
An expression that includes at least one
variable
Addition Property of Equality
7−4=3
If you add the same number to both
sides of an equation, the two sides
will remain equal.
7−4+4=3+4
Division Property of Equality
If you divide both sides of an equation by
the same nonzero number, the two sides will
remain equal.
7+0=7
7=7
2 × 6 = 12
2 × 6 = 12
_____
___
2
2
1×6=6
6=6
The property that states that when you add
zero to a number, the result is that number
Examples:
5,026 + 0 = 5,026
2+0=2
__
__
3
3
1.5 + 0 = 1.5
A mathematical sentence that contains the
symbol <, >, ≤, ≥, or ≠
Examples:
8 < 11
9 > −4
a ≤ 50
x ≥ 3.2
Chapter 8 Vocabulary
(continued)
Multiplication
Property of Equality
reciprocal
propiedad de multiplicación
de la igualdad
recíproco
solution of an
equation
solution of an
inequality
solución de una
ecuación
solución de una
desigualdad
89
62
94
Subtraction
Property of Equality
variable
propiedad de resta de
la igualdad
98
variable
95
105
2
A value that, when substituted for the
variable, makes an inequality true
A letter or symbol that stands for an
unknown number or numbers
x + 10
variable
3×y
variable
3 × (a + 4)
variable
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3
© Houghton Mifflin Harcourt Publishing Company
__ × 3
__ = 1
Example: 2
© Houghton Mifflin Harcourt Publishing Company
© Houghton Mifflin Harcourt Publishing Company
© Houghton Mifflin Harcourt Publishing Company
© Houghton Mifflin Harcourt Publishing Company
Two numbers are reciprocals of each other
if their product equals 1.
Multiplication Property of Equality
If you multiply both sides of an equation by
the same number, the two sides will remain
equal.
12 = 3
___
4
___ = 4 × 3
4 × 12
4
1 × 12 = 12
12 = 12
A value that, when substituted for the
variable, makes an equation true.
Example:
x+3=5
x = 2 is the solution of the
equation because 2 + 3 = 5.
Subtraction Property of Equality
3+4=7
If you subtract the same number from
both sides of an equation, the two
sides will remain equal.
3+4−4=7−4
3+0=3
3=3
Going Places with
Pick It
For 3 players
Materials
• 4 sets of word cards
How to Play
1. Each player is dealt 5 cards. The remaining cards
are a draw pile.
2. To take a turn, ask any player if he or she has a word
or term that matches one of your word cards.
3. If the player has the word, he or she gives the word
• If you are correct, keep the card and put
the matching pair in front of you. Take
another turn.
• If you are wrong, return the card. Your
turn is over.
4. If the player does not have the word, he
or she answers, “Pick it.” Then you take
a card from the draw pile.
5. If the card you draw matches one of your
word cards, follow the directions for Step 3
above. If it does not, your turn is over.
Game
Game
Word Box
Addition Property
of Equality
algebraic
expression
Division Property of
Equality
equation
Identity Property of
Addition
Identity Property of
Multiplication
inequality
inverse operations
Multiplication
Property of Equality
reciprocal
solution of an
equation
solution of an
inequality
Subtraction
Property of Equality
variable
6. The game is over when one player has no
cards left. The player with the most pairs wins.
© Houghton Mifflin Harcourt Publishing Company
Image Credits: (bg) ©PhotoDisc/Getty Images; (snowflake) ©Stockdisc/Getty Images
card to you, and you must define the term.
Words
Chapter 8
420A
Journal
Jo
ou
urnal
The Write Way
Reflect
Choose one idea. Write about it.
• Which of the following is an equation? Tell how you know.
8 + 20 =
9x + 4
6d = 12
3(5 + 23)
• Explain how you can use inverse operations to solve this equation and check
your solution.
a − 7 = 15
• Suppose you write a math advice column and a reader needs help solving a
multiplication equation. Write a letter to the reader explaining how to use the
Division Property of Equality to solve the equation.
How are they different?
420B
© Houghton Mifflin Harcourt Publishing Company Image Credits: (bg) ©Scott Markewitz/Getty Images; (snowflake) ©Stockdisc/Getty Images
• Compare and contrast an equation and an inequality. How are they alike?
Lesson 8.1
Name
Solutions of Equations
Essential Question How do you determine whether a number is a
solution of an equation?
Expressions and Equations—
6.EE.B.5
MATHEMATICAL PRACTICES
MP2, MP4, MP6
An equation is a statement that two mathematical expressions are
equal. These are examples of equations:
8 + 12 = 20
14 = a − 3
2d = 14
A solution of an equation is a value of a variable that makes an
equation true.
x+3=5
x = 2 is the solution of the equation because 2 + 3 = 5.
Unlock
Unlock the
the Problem
Problem
In the 2009–2010 season, the women’s basketball team
of Duke University lost 5 of their 29 games. The equation
w + 5 = 29 can be used to find the team’s number of
wins w. Determine whether w = 14 or w = 24 is a solution
of the equation, and tell what the solution means.
Use substitution to determine the solution.
© Houghton Mifflin Harcourt Publishing Company • Image Credits: ©Sports Illustrated/Getty Images
STEP 1 Check whether w = 14 is a solution.
Write the equation.
w + 5 = 29
Substitute 14 for w.
_ + 5 ≟ 29
Add.
The symbol ≠ means
“is not equal to.”
_ ≠ 29
The equation is not true when w = 14, so w = 14 is not a solution.
STEP 2 Check whether w = 24 is a solution.
Write the equation.
w + 5 = 29
Substitute 24 for w.
_ + 5 ≟ 29
Add.
_ = 29
The equation is true when w = 24, so w = 24 is a solution.
So, the solution of the equation w + 5 = 29 is w = _ ,
which means that the team won _ games.
Math
Talk
MATHEMATICAL PRACTICES 6
Explain How is an algebraic
equation, such as x + 1 = 4,
different from a numerical
equation, such as 3 + 1 = 4?
Chapter 8
421
Example 1 Determine whether the given value of the variable is a
solution of the equation.
A x − 0.7 = 4.3; x = 3.6
x − 0.7 = 4.3
Write the equation.
Substitute the given value for the variable.
_ − 0.7 ≟
4.3
O
4.3
_
Subtract. Write = or ≠.
The equation __ true when x = 3.6, so x = 3.6
__ a solution.
B __1a = __1; a = __3
3
4
4
1 a = __
1
__
3
4
Write the equation.
1 × ____ ≟ 1
__
__
3
4
Substitute the given value for the variable.
____
Simplify factors and multiply. Write = or ≠.
O
_1_
4
The equation _ true when a = __3, so a = 3__
4
4
_ a solution.
Example 2
The sixth-grade class president serves a term of 8 months.
Janice has already served 5 months of her term as class president. The equation
m + 5 = 8 can be used to determine the number of months m Janice has left.
Use mental math to find the solution of the equation.
Think: What number plus 5 is equal to 8?
_ plus 5 is equal to 8.
m+5= 8
Write the equation.
Substitute 3 for m.
_+5≟ 8
Add. Write = or ≠.
So, m = _ is the solution of the equation, and
_ months of Janice’s term remain.
422
_
O
8
Math
Talk
MATHEMATICAL PRACTICES 3
Apply Give an example of
an equation whose solution
is y = 7. Explain how you
know that the equation has
this solution.
© Houghton Mifflin Harcourt Publishing Company
Use substitution to check whether m = 3 is a solution.
Name
Share
hhow
Share and
and Show
Sh
MATH
M
BOARD
B
Determine whether the given value of the variable is a
solution of the equation.
1. x + 12 = 29; x = 7
2. n − 13 = 2; n = 15
3. _1_ c = 14; c = 28
2
_ + 12 ≟ 29
_
≠ 29
___
4. m + 2.5 = 4.6; m = 2.9
___
___
5. d − 8.7 = 6; d = 14.7
___
1 ; k = ___
7
6. k − 3__ = ___
5 10
10
___
___
Math
Talk
MATHEMATICAL PRACTICES 6
Explain why 2x − 6 is not an
equation.
On
On Your
Your Own
Own
Determine whether the given value of the variable is a
solution of the equation.
7. 17.9 + v = 35.8; v = 17.9
___
8. c + 35 = 57; c = 32
___
9. 18 = 2__ h; h = 12
3
___
© Houghton Mifflin Harcourt Publishing Company
10. In the equation t + 2.5 = 7, determine whether t = 4.5, t = 5,
or t = 5.5 is a solution of the equation.
11. Antonio ran a total of 9 miles in two days. The first day he
ran 5 1_4 miles. The equation 9 – d = 5 1_4 gives the distance
in miles Antonio ran the second day. Determine whether d = 4 3_4 ,
d = 4, or d = 3 3_4 is a solution of the equation, and tell what the
solution means.
Chapter 8 • Lesson 1 423
MATHEMATICAL PRACTICES .0%&-t3&"40/tM",&4&/4&
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Use the table for 12–14.
MATHEMATICAL
PRACTICE
12.
2 Connect Symbols and Words The length
of a day on Saturn is 14 hours less than a day on Mars.
The equation 24.7 − s = 14 gives the length in hours s of
a day on Saturn. Determine whether s = 9.3 or s = 10.7
is a solution of the equation, and tell what the solution
means.
13. A storm on one of the planets listed in the table lasted
for 60 hours, or 2.5 of the planet’s days. The equation
2.5h = 60 gives the length in hours h of a day on the
planet. Is the planet Earth, Mars, or Jupiter? Explain.
DEEPER
14.
A day on Pluto is 143.4 hours longer than a
day on one of the planets listed in the table. The equation
153.3 − p = 143.4 gives the length in hours p of a day on
the planet. What is the length of a storm that lasts 1_3 of a
day on this planet?
15.
What’s the Error? Jason said that
the solution of the equation 2m = 4 is m = 8. Describe
Jason’s error, and give the correct solution.
Length of Day (hours)
Earth
24.0
Mars
24.7
Jupiter
9.9
SMARTER
SMARTER
The marking period is 45 school days long. Today is the twenty-first
day of the marking period. The equation x + 21 = 45 can be used to find the number
of days left in the marking period. Using substitution, Rachel determines
20
there are
24
26
424
Planet
days left in the marking period.
© Houghton Mifflin Harcourt Publishing Company • Image Credits: ©NASA
16.
Length of Day
Practice and Homework
Name
Lesson 8.1
Solutions of Equations
COMMON CORE STANDARD—6.EE.B.5
Reason about and solve one-variable equations
and inequalities.
Determine whether the given value of the variable is
a solution of the equation.
1. x − 7 + 15; x = 8
8
2. c + 11 = 20; c = 9
3. _1h = 6; h = 2
5. 9 = 3_4e; e = 12
6. 15.5 2 y = 7.9; y = 8.4
3
2 7 0 15
fi
1
15
not a solution
4. 16.1 + d = 22; d = 6.1
Problem
Problem Solving
Solving
© Houghton Mifflin Harcourt Publishing Company
7. Terrance needs to score 25 points to win a game.
He has already scored 18 points. The equation
18 + p = 25 gives the number of points p that
Terrance still needs to score. Determine whether
p = 7 or p = 13 is a solution of the equation, and
tell what the solution means.
9.
WRITE
8. Madeline has used 50 sheets of a roll of
paper towels, which is 5_8 of the entire roll.
The equation 5_s = 50 can be used to find the
8
number of sheets s in a full roll. Determine
whether s = 32 or s = 80 is a solution of the
equation, and tell what the solution means.
Math Use mental math to find the solution to 4x = 36.
Then use substitution to check your answer.
Chapter 8
425
Lesson Check (6.EE.B.5)
1. Sheena received a gift card for $50. She has
already used it to buy a lamp for $39.99. The
equation 39.99 + x = 50 can be used to find the
amount x that is left on the gift card. What is the
solution of the equation?
TEST
PREP
2. When Pete had a fever, his temperature was
101.4°F. After taking some medicine, his
temperature was 99.2°F. The equation
101.4 – d = 99.2 gives the number of degrees
d that Pete’s temperature decreased. What is
the solution of the equation?
3. Melanie has saved $60 so far to buy a lawn
mower. This is 20% of the price of the lawn
mower. What is the full price of the lawn mower
that she wants to buy?
4. A team of scientists is digging for fossils. The
amount of soil in cubic feet that they remove is
equal to 63. How many cubic feet of soil do the
scientists remove?
5. Andrew made p picture frames. He sold 2 of
them at a craft fair. Write an expression that
could be used to find the number of picture
frames Andrew has left.
6. Write an expression that is equivalent to
4 + 3(5 + x).
FOR MORE PRACTICE
GO TO THE
426
Personal Math Trainer
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Spiral Review (6.RP.A.3c, 6.EE.A.1, 6.EE.A.4, 6.EE.B.6)
Lesson 8.2
Name
Write Equations
Expressions and Equations—
6.EE.B.7
MATHEMATICAL PRACTICES
MP2, MP3, MP4, MP6
Essential Question How do you write an equation to represent a situation?
connect You can use what you know about writing algebraic
expressions to help you write algebraic equations.
Unlock
Unlock the
the Problem
Problem
A circus recently spent $1,650 on new
trapezes. The trapezes cost $275 each.
Write an equation that could be used to
find the number of trapezes t that the
circus bought.
• Circle the information that you need to write
the equation.
• What expression could you use to represent
the cost of t trapezes?
Write an equation for the situation.
Think:
Cost per trapeze
times
number of trapezes
equals
×
t
=
© Houghton Mifflin Harcourt Publishing Company • Image Credits: (cr) ©Concept by Beytan/Alamy Images
__
total cost.
__
So, an equation that could be used to find the number of
trapezes t is ____.
Try This! Ben is making a recipe for salsa that calls for 3 1_2 cups of
tomatoes. He chops 4 tomatoes, which fill 2 1_4 cups. Write an equation that
could be used to find out how many more cups c Ben needs.
Think: Cups filled
__
plus
cups needed
equals
total cups for recipe.
+
__
=
__
So, an equation that could be used to find the number of
additional cups c is ____.
Math
Talk
MATHEMATICAL PRACTICES 4
Represent What is another
equation you could use to
model the problem?
Chapter 8
427
Example 1 Write an equation for the word sentence.
A Six fewer than a number is 46.33.
Think: Let n represent the unknown number. The phrase “fewer than” indicates
___.
is
Six fewer than a number
__ − __
=
46.33.
The expression n − 6 means
“6 fewer than n.” The expression
6 − n means “n fewer than 6.”
__
B Two-thirds of the cost of the sweater is $18.
Think: Let c represent the __ of the sweater in dollars. The word “of”
indicates ___.
Two-thirds
of
the cost of the sweater
__
×
__
Example 2
is
18.
= __
Write two word sentences for the equation.
A a + 15 = 24
B r ÷ 0.2 = 40
• The __ of a and 15 __ 24.
• The __ of r and 0.2 __ 40.
• 15 __ than a __ 24.
• r __ by 0.2 __ 40.
2.
MATHEMATICAL
PRACTICE
3 Compare Representations One student wrote
18 × d = 54 for the sentence “The product of 18 and d equals 54.”
Another student wrote d × 18 = 54 for the same sentence. Are both
students correct? Justify your answer.
428
© Houghton Mifflin Harcourt Publishing Company
1. Explain how you can rewrite the equation n + 8 = 24 so that it involves
subtraction rather than addition.
Name
Share
hhow
Share and
and Show
Sh
MATH
M
BOARD
B
1. Write an equation for the word sentence “25 is 13 more than
a number.”
What operation does the phrase “more than” indicate? ___
The equation is ___ = ___ + ___.
Write an equation for the word sentence.
2. The difference of a number and 2 is 31_.
3
3. Ten times the number of balloons is 120.
Write a word sentence for the equation.
4. x − 0.3 = 1.7
5. 25 = 1_4n
Math
Talk
On
On Your
Your Own
Own
MATHEMATICAL PRACTICES 6
Explain How does an
equation differ from an
expression?
© Houghton Mifflin Harcourt Publishing Company
Write an equation for the word sentence.
6. The quotient of a number and 20.7 is 9.
7. 24 less than the number of snakes is 35.
8. 75 is 181_ more than a number.
9. d degrees warmer than 50 degrees is
78 degrees.
2
Write a word sentence for the equation.
10. 15g = 135
11. w ÷ 3.3 = 0.6
Chapter 8 • Lesson 2 429
MATHEMATICAL PRACTICES
COMMUNICA5&t1&34E7&3&tCONSTRUCT ARGUMENTS
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OqnakdlRnkuhmf¤@ookhb`shnmr
To find out how far a car can travel on a certain amount of gas,
multiply the car’s fuel efficiency in miles per gallon by the gas
used in gallons. Use this information and the table for 12–13.
12. Write an equation that could be used to find how
many miles a hybrid SUV can travel in the city on
20 gallons of gas.
13. A sedan traveled 504 miles on the highway on a full tank of gas.
Write an equation that could be used to find the number of
gallons the tank holds.
Fuel Efficiency
Vehicle
Miles per
gallon, city
Miles per
gallon, highway
Hybrid SUV
36
31
Minivan
19
26
Sedan
20
28
SUV
22
26
WRITE
14.
MATHEMATICAL
PRACTICE
2 Connect Symbols to Words Sonya was born
Math
Show Your Work
in 1998. Carmen was born 11 years after Sonya. If you wrote
an equation to find the year in which Carmen was born, what
operation would you use in your equation?
A magazine has 110 pages. There are 23 full-page ads
and 14 half-page ads. The rest of the magazine consists of articles.
Write an equation that can be used to find the number of pages of
articles in the magazine.
16.
What’s the Error? Tony is traveling 560 miles to
visit his cousins. He travels 313 miles the first day. He says that he
can use the equation m − 313 = 560 to find the number of miles he
has left on his trip. Describe and correct Tony’s error.
17.
SMARTER
SMARTER
Jamie is making cookies for a bake sale. She
triples the recipe in order to have enough cookies to sell. Jamie
uses 12 cups of flour to make the triple batch.
Write an equation that can be used to find out how much flour f
is needed for one batch of cookies.
430
© Houghton Mifflin Harcourt Publishing Company • Image Credits: (t) ©Iain Masterton/Alamy Images
DEEPER
15.
Practice and Homework
Name
Lesson 8.2
Write Equations
COMMON CORE STANDARD—6.EE.B.7
Reason about and solve one-variable equations
and inequalities.
Write an equation for the word sentence.
1. 18 is 4.5 times a number.
2. Eight more than the number of children is 24.
18 = 4.5n
3. The difference of a number and 2_ is 3_.
3
8
4. A number divided by 0.5 is 29.
Write a word sentence for the equation.
5. x − 14 = 52
6. 2.3m = 0.46
7. 25 = k ÷ 5
8. 41_ + q = 51_
3
6
Problem
Problem Solving
Solving
© Houghton Mifflin Harcourt Publishing Company
9. An ostrich egg weighs 2.9 pounds. The difference
between the weight of this egg and the weight of
an emu egg is 1.6 pounds. Write an equation that
could be used to find the weight w, in pounds, of
the emu egg.
11.
WRITE
10. In one week, the number of bowls a potter made
was 6 times the number of plates. He made
90 bowls during the week. Write an equation that
could be used to find the number of plates p that
the potter made.
Math Explain when to use a variable when writing
a word sentence in an equation.
Chapter 8
431
TEST
PREP
Lesson Check (6.EE.B.7)
1. Three friends are sharing the cost of a bucket of
popcorn. The total cost of the popcorn is $5.70.
Write an equation that could be used to find the
amount a in dollars that each friend should pay.
2. Salimah had 42 photos on her phone. After
she deleted some of them, she had 23 photos
left. What equation could be used to find the
number of photos p that Salimah deleted?
3. A rope is 72 feet long. What is the length of the
rope in yards?
4. Julia evaluated the expression 33 + 20 ÷ 22.
What value should she get as her answer?
5. The sides of a triangle have lengths s, s + 4,
and 3s. Write an expression in simplest form
that represents the perimeter of the triangle.
6. Gary knows that p = 2 1_2 is a solution to one of
the following equations. Which one has p = 2 1_2
as its solution?
p + 21_ = 5
p − 21_ = 5
2 + p = 2_1
2
4 − p = 21_
2
2
2
FOR MORE PRACTICE
GO TO THE
432
Personal Math Trainer
© Houghton Mifflin Harcourt Publishing Company
Spiral Review (6.RP.A.3d, 6.EE.A.1, 6.EE.A.3, 6.EE.B.5)
Lesson 8.3
Name
Model and Solve Addition Equations
Essential Question How can you use models to solve addition equations?
Expressions and Equations—
6.EE.B.7
MATHEMATICAL PRACTICES
MP3, MP4, MP5
You can use algebra tiles to help you find solutions of equations.
Algebra Tiles
x tile
1 tile
Hands
On
Investigate
Investigate
Materials ■ MathBoard, algebra tiles
Thomas has $2. He wants to buy a poster that costs $7.
Model and solve the equation x + 2 = 7 to find the
amount x in dollars that Thomas needs to save in order
to buy the poster.
A. Draw 2 rectangles on your MathBoard to represent the
two sides of the equation.
B. Use algebra tiles to model the equation. Model
x + 2 in the left rectangle, and model 7 in the
right rectangle.
• What type of tiles and number of tiles did you use to
model x + 2?
_____
C. To solve the equation, get the x tile by itself on one side.
If you remove a tile from one side, you can keep the
two sides equal by removing the same type of tile from
the other side.
© Houghton Mifflin Harcourt Publishing Company
• How many 1 tiles do you need to remove from each side to
get the x tile by itself on the left side? _
• When the x tile is by itself on the left side, how many
1 tiles are on the right side? _
D. Write the solution of the equation: x = _.
So, Thomas needs to save $ _ in order to buy the poster.
Math
Talk
MATHEMATICAL PRACTICES 4
Model What operation did
you model when you removed
the tiles?
Chapter 8
433
Draw Conclusions
MATHEMATICAL
PRACTICE
1.
5 Use Appropriate Tools Describe how you could use your
model to check your solution.
2. Tell how you could use algebra tiles to model the equation x + 4 = 8.
3.
SMARTER
What would you do to solve the equation x + 9 = 12
without using a model?
Make
Make Connections
Connections
You can solve an equation by drawing a model to represent algebra tiles.
Let a rectangle represent the variable. Let a small square represent 1.
Solve the equation x + 3 = 7.
STEP 1
Draw a model of the equation.
STEP 2
Cross out _ squares on the left side and
_ squares on the right side.
STEP 3
Draw a model of the solution.
There is 1 rectangle on the left side. There are
_ squares on the right side.
So, the solution of the equation x + 3 = 7 is x = _.
434
© Houghton Mifflin Harcourt Publishing Company
Get the variable by itself on one side of the
model by doing the same thing to both sides.
Name
Share
hhow
Share and
and Show
Sh
MATH
M
BOARD
B
Model and solve the equation by using algebra tiles or iTools.
1. x + 5 = 7 __
2. 8 = x + 1 __
3. x + 2 = 5 __
4. x + 6 = 8 __
5. 5 + x = 9 __
6. 5 = 4 + x __
© Houghton Mifflin Harcourt Publishing Company
Solve the equation by drawing a model.
7. x + 1 = 5 ____
8. 3 + x = 4 ____
9. 6 = x + 4 ____
10. 8 = 2 + x ____
11.
MATHEMATICAL
PRACTICE
6 Describe a Method Describe how you would draw a model to solve
the equation x + 5 = 10.
Chapter 8 • Lesson 3 435
MATHEMATICAL PRACTICES
ANALYZEt-00,'034536$563&t13&$*4*0/
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12.
MATHEMATICAL
PRACTICE
4
Interpret a Result The table shows how long
several animals have lived at a zoo. The giraffe has lived at
the zoo 4 years longer than the mountain lion. The equation
5 = 4 + y can be used to find the number of years y the
mountain lion has lived at the zoo. Solve the equation.
Then tell what the solution means.
Zoo Animals
Animal
DEEPER
13.
Carlos walked 2 miles on Monday and 5 miles on
Saturday. The number of miles he walked on those two days is
3 miles more than the number of miles he walked on Friday.
Write and solve an addition equation to find the number of miles
Carlos walked on Friday.
14.
Sense or Nonsense? Gabriela is solving
the equation x + 1 = 6. She says that the solution must be less
than 6. Is Gabriela’s statement sense or nonsense? Explain.
Time at zoo (years)
Giraffe
5
Hippopotamus
6
Kangaroo
2
Zebra
9
SMARTER
Personal Math Trainer
SMARTER
The Hawks beat the Tigers by 5 points in
a football game. The Hawks scored a total of 12 points.
Use numbers and words to explain how this model can be used to
solve the equation x + 5 = 12.
=
436
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15.
Practice and Homework
Name
Lesson 8.3
Model and Solve Addition Equations
COMMON CORE STANDARD—6.EE.B.7
Reason about and solve one-variable equations
and inequalities.
Model and solve the equation by using algebra tiles.
1. x + 6 = 9
2. 8 + x = 10
3. 9 = x + 1
x=3
Solve the equation by drawing a model.
4. x + 4 = 7
5. x + 6 = 10
Problem
Problem Solving
Solving
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6. The temperature at 10:00 was 108F. This is
38F warmer than the temperature at 8:00. Model
and solve the equation x + 3 = 10 to find the
temperature x in degrees Fahrenheit at 8:00.
8.
WRITE
7. Jaspar has 7 more checkers left than Karen does.
Jaspar has 9 checkers left. Write and solve an
addition equation to find out how many checkers
Karen has left.
Math Explain how to use a drawing to solve an
addition equation such as x + 8 = 40.
Chapter 8
437
TEST
PREP
Lesson Check (6.EE.B.7)
1. What is the solution of the equation that is
modeled by the algebra tiles?
5
2. Alice has played soccer for 8 more years than
Sanjay has. Alice has played for 12 years. The
equation y + 8 = 12 can be used to find the
number of years y Sanjay has played. How long
has Sanjay played soccer?
3. A car’s gas tank has a capacity of 16 gallons.
What is the capacity of the tank in pints?
4. Craig scored p points in a game. Marla scored
twice as many points as Craig but 5 fewer
than Nelson scored. How many points did
Nelson score?
5. Simplify 3x + 2(4y + x).
6. The Empire State Building in New York City is
443.2 meters tall. This is 119.2 meters taller than
the Eiffel Tower in Paris. Write an equation that
can be used to find the height h in meters of the
Eiffel Tower.
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GO TO THE
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Spiral Review (6.RP.A.3d, 6.EE.A.2a, 6.EE.A.3, 6.EE.B.7)
Lesson 8.4
Name
Solve Addition and Subtraction Equations
Essential Question How do you solve addition and subtraction equations?
connect To solve an equation, you must get the variable on one side of
the equal sign by itself. You have solved equations by using models. You
can also solve equations by using Properties of Equality.
Subtraction Property of Equality
3+4=7
If you subtract the same number from
both sides of an equation, the two sides
will remain equal.
3+4−4=7−4
Expressions and Equations—
6.EE.B.7
MATHEMATICAL PRACTICES
MP2, MP8
3+0=3
3=3
Unlock
Unlock the
the Problem
Problem
The longest distance jumped on a pogo stick is 23 miles. Emilio has jumped
5 miles on a pogo stick. The equation d + 5 = 23 can be used to find the
remaining distance d in miles he must jump to match the record. Solve the
equation, and explain what the solution means.
Solve the addition equation.
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To get d by itself, you must undo the addition by 5. Operations that undo
each other are called inverse operations. Subtracting 5 is the inverse
operation of adding 5.
Write the equation.
Use the Subtraction Property of Equality.
Subtract.
Use the Identity Property of Addition.
d + 5 = 23
d + 5 − 5 = 23 − _
d+0 =_
_ = 18
Check the solution.
Write the equation.
Substitute _ for d.
The solution checks.
d + 5 = 23
_ + 5 = 23
_ = 23
So, the solution means that Emilio must jump _ more miles.
Math
Talk
MATHEMATICAL PRACTICES 8
Generalize How do you
know what number to
subtract from both sides of
the equation?
Chapter 8
439
When you solve an equation that involves subtraction, you can use
addition to get the variable by itself on one side of the equal sign.
Addition Property of Equality
7−4=3
If you add the same number to both
sides of an equation, the two sides
will remain equal.
7−4+4=3+4
7+0=7
7=7
Example
While cooking dinner, Carla pours 5_8 cup of milk from a carton. This leaves
7
_ cup of milk in the carton. Write and solve an equation to find how much
8
milk was in the carton when Carla started cooking.
STEP 1 Write an equation.
Let a represent the amount of milk in cups in the carton when Carla
started cooking.
amount
in carton
at start
minus
amount
poured
out
equals
amount
in carton
at end
a
−
____
=
____
STEP 2 Solve the equation.
Think: __5 is subtracted from a, so add 5__ to both sides to undo the subtraction.
8
__ = 7
__
a−5
Write the equation.
Use the Addition Property of Equality.
8
__ + ___ = 7
__ + ___
a−5
8
Add.
Write the fraction greater than 1 as a mixed
number, and simplify.
8
a = ______
a =_
So, there were _ cups of milk in the carton when Carla
started cooking.
440
8
Math
Talk
MATHEMATICAL PRACTICES 6
Explain How can you check
the solution of the equation?
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8
Name
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and Show
Sh
MATH
M
BOARD
B
1. Solve the equation n + 35 = 80.
n + 35 = 80
n + 35 − 35 = 80 − _
n=_
Use the
___ Property of Equality.
Subtract.
Solve the equation, and check the solution.
2. 16 + x = 42
____
5. z − 31_ = 12_
3 = __
7
4. m + __
3. y + 6.2 = 9.1
10
____
____
6. 12 = x − 24
3
____
7. 25.3 = w − 14.9
____
____
Math
Talk
On
On Your
Your Own
Own
Practice: Copy and Solve Solve the equation, and check the solution.
8. y − 3_4 = 1_
9. 75 = n + 12
© Houghton Mifflin Harcourt Publishing Company
2
11. w − 36 = 56
12. 82_ = d + 22_
5
5
10
MATHEMATICAL PRACTICES 8
Generalize What can you do
to get the variable by itself
on one side of a subtraction
equation?
10. m + 16.8 = 40
13. 8.7 = r − 1.4
14. The temperature dropped 8 degrees between 6:00 p.m. and
midnight. The temperature at midnight was 26ºF. Write and solve
an equation to find the temperature at 6:00 p.m.
15.
MATHEMATICAL
PRACTICE
2 Reason Abstractly Write an addition equation that has the solution x = 9.
Chapter 8 • Lesson 4 441
MATHEMATICAL PRACTICES .0%&-t3&"40/tM",&4&/4&
Unlock
Unlock the
the Problem
Problem
16.
DEEPER
In July, Kimberly made two
deposits into her bank account. She made no
withdrawals. At the end of July, her account
balance was $120.62. Write and solve an
equation to find Kimberly’s balance at the
beginning of July.
a. What do you need to find?
d. Solve the equation. Show your work and
describe each step.
b. What information do you need from the
bank statement?
c. Write an equation you can use to
solve the problem. Explain what the
variable represents.
WRITE
M t Show Your Work
Math
e. Write Kimberly’s balance at the beginning
of July.
If x + 6 = 35,
what is the value of x + 4?
Explain how to find the value
without solving the equation.
SMARTER
18.
SMARTER
Select the equations that
have the solution n = 23. Mark all that
apply.
A 16 + n = 39
B n − 4 = 19
C 25 = n − 2
D 12 = n − 11
442
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17.
Practice and Homework
Name
Solve Addition and Subtraction Equations
Lesson 8.4
2. x + 3 = 15
COMMON CORE STANDARD—6.EE.B.7
Reason about and solve one-variable equations
and inequalities.
3. n + _25 = _45
4. 16 = m − 14
5. w − 13.7 = 22.8
6. s + 55 = 55
7. 23 = x − 12
8. p − 14 = 14
9. m − 23_4 = 61_
Solve the equation, and check the solution.
1. y − 14 = 23
y − 14 + 14 = 23 + 14
y = 37
2
Problem
Problem Solving
Solving
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10. A recipe calls for 5 1_2 cups of flour. Lorenzo only
has 3 3_4 cups of flour. Write and solve an equation
to find the additional amount of flour Lorenzo
needs to make the recipe.
12.
WRITE
11. Jan used 22.5 gallons of water in the shower. This
amount is 7.5 gallons less than the amount she
used for washing clothes. Write and solve an
equation to find the amount of water Jan used to
wash clothes.
Math Explain how to check if your solution to an equation
is correct.
Chapter 8
443
Lesson Check (6.EE.B.7)
1. The price tag on a shirt says $21.50. The final
cost of the shirt, including sales tax, is $23.22.
The equation 21.50 + t = 23.22 can be used to
find the amount of sales tax t in dollars. What is
the sales tax?
TEST
PREP
2. The equation l – 12.5 = 48.6 can be used to find
the original length l in centimeters of a wire
before it was cut. What was the original length
of the wire?
3. How would you convert a mass in centigrams to
a mass in milligrams?
4. In the expression 4 + 3x + 5y, what is the
coefficient of x?
5. Write an expression that is equivalent to 10c.
6. Miranda bought a movie ticket and popcorn for
a total of $10. The equation 7 + x = 10 can be
used to find the cost x in dollars of the popcorn.
How much did the popcorn cost?
FOR MORE PRACTICE
GO TO THE
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Lesson 8.5
Name
Model and Solve Multiplication Equations
Essential Question How can you use models to solve
multiplication equations?
Expressions and Equations—
6.EE.B.7
MATHEMATICAL PRACTICES
MP1, MP4, MP5, MP6
You can use algebra tiles to model and solve equations that involve
multiplication.
Algebra Tiles
4x
x tile
1 tile
To model an expression involving multiplication of a variable,
you can use more than one x tile. For example, to model the
expression 4x, you can use four x tiles.
Investigate
Investigate
Materials ■ MathBoard, algebra tiles
Hands
On
Tennis balls are sold in cans of 3 tennis balls each.
Daniel needs 15 tennis balls for a tournament. Model
and solve the equation 3x = 15 to find the number
of cans x that Daniel should buy.
A. Draw 2 rectangles on your MathBoard to represent the
two sides of the equation.
B. Use algebra tiles to model the equation. Model 3x in the
left rectangle, and model 15 in the right rectangle.
C. There are three x tiles on the left side of your model. To
© Houghton Mifflin Harcourt Publishing Company
solve the equation by using the model, you need to find the
value of one x tile. To do this, divide each side of your model
into 3 equal groups.
• When the tiles on each side have been divided into
3 equal groups, how many 1 tiles are in each group on
the right side? _
D. Write the solution of the equation: x = _.
So, Daniel should buy _ cans of tennis balls.
Math
Talk
MATHEMATICAL PRACTICES 2
Reasoning What operation
did you model in Step C?
Chapter 8
445
Draw Conclusions
1. Explain how you could use your model to check your solution.
MATHEMATICAL
PRACTICE
2.
6 Describe how you could use algebra tiles to model the
equation 6x = 12.
3.
SMARTER
What would you do to solve the equation
5x = 35 without using a model?
Make
Make Connections
Connections
You can also solve multiplication equations by drawing a model
to represent algebra tiles. Let a rectangle represent x. Let a square
represent 1. Solve the equation 2x = 6.
STEP 1 Draw a model of the equation.
STEP 2 Find the value of one rectangle.
__
STEP 3 Draw a model of the solution.
There is 1 rectangle on the left side. There
are
__ squares on the right side.
So, the solution of the equation 2x = 6 is x = _ .
446
© Houghton Mifflin Harcourt Publishing Company
Divide each side of the model into
equal groups.
Name
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and Show
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MATH
M
BOARD
B
Model and solve the equation by using algebra tiles.
1. 4x = 16
2. 3x = 12
____
4. 3x = 9
3. 4 = 4x
____
5. 2x = 10
____
____
6. 15 = 5x
____
____
Solve the equation by drawing a model.
7. 4x = 8 ____
8. 3x = 18 ____
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OqnakdlRnkuhmf¤@ookhb`shnmr
9.
MATHEMATICAL
PRACTICE
5 Communicate Explain the steps you use to solve a
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multiplication equation with algebra tiles.
Chapter 8 • Lesson 5 447
MATHEMATICAL PRACTICES
COMMUNICA5&t1&34E7&3&tCONSTRUCT ARGUMENTS
The bar graph shows the number of countries that
competed in the first four modern Olympic Games.
Use the bar graph for 10–11.
10.
DEEPER
Naomi is doing a report about the
1900 and 1904 Olympic Games. Each page will contain
information about 4 of the countries that competed
each year. Write and solve an equation to find the
number of pages Naomi will need.
Olympic Games
Pose a Problem Use the
information in the bar graph to write and
solve a problem involving a multiplication
equation.
SMARTER
Countries
11.
28
24
20
16
12
8
4
0
1896 Athens
1900 Paris 1904 St. Louis 1908 London
Year and City
13.
448
SMARTER
A choir is made up of 6 vocal groups. Each group has
an equal number of singers. There are 18 singers in the choir. Solve the
equation 6p = 18 to find the number of singers in each group. Use a model.
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12. The equation 7s = 21 can be used to find the number of
snakes s in each cage at a zoo. Solve the equation. Then
tell what the solution means.
Practice and Homework
Name
Model and Solve Multiplication Equations
Lesson 8.5
COMMON CORE STANDARD—6.EE.B.7
Reason about and solve one-variable equations
and inequalities.
Model and solve the equation by using algebra tiles.
1. 2x = 8
2. 5x = 10
3. 21 = 3x
x=4
Solve the equation by drawing a model.
4. 6 = 3x
5. 4x = 12
Problem
Problem Solving
Solving
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6. A chef used 20 eggs to make 5 omelets. Model
and solve the equation 5x = 20 to find the
number of eggs x in each omelet.
8.
WRITE
7. Last month, Julio played 3 times as many video
games as Scott did. Julio played 18 video games.
Write and solve an equation to find the number
of video games Scott played.
Math Write a multiplication equation, and explain
how you can solve it by using a model.
Chapter 8
449
Lesson Check (6.EE.B.7)
1. What is the solution of the equation that is
modeled by the algebra tiles?
TEST
PREP
2. Carlos bought 5 tickets to a play for a total of
$20. The equation 5c = 20 can be used to find
the cost c in dollars of each ticket. How much
does each ticket cost?
5
3. A rectangle is 12 feet wide and 96 inches long.
What is the area of the rectangle?
4. Evaluate the algebraic expression 24 – x ÷ y for
x = 8 and y = 2.
5. Ana bought a 15.5-pound turkey at the grocery
store this month. The equation p – 15.5 = 2.5
gives the weight p, in pounds, of the turkey
she bought last month. What is the solution
of the equation?
6. A pet store usually keeps 12 birds per cage, and
there are 7 birds in the cage now. The equation
7 + x = 12 can be used to find the remaining
number of birds x that can be placed in the
cage. What is the solution of the equation?
FOR MORE PRACTICE
GO TO THE
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Spiral Review (6.RP.A.3d, 6.EE.A.2c, 6.EE.B.5, 6.EE.B.7)
Lesson 8.6
Name
Solve Multiplication and Division Equations
Essential Question How do you solve multiplication and division equations?
connect You can use Properties of Equality and inverse operations to
solve multiplication and division equations.
Division Property of Equality
Expressions and Equations—
6.EE.B.7
MATHEMATICAL PRACTICES
MP2, MP7, MP8
2 × 6 = 12
If you divide both sides of an equation by the same
nonzero number, the two sides will remain equal.
2 × 6 = 12
_____
___
2
2
1×6=6
6=6
Unlock
Unlock the
the Problem
Problem
Mei ran 14 laps around a track for a total of 4,200
meters. The equation 14d = 4,200 can be used to find
the distance d in meters she ran in each lap. Solve the
equation, and explain what the solution means.
• What operation is indicated by 14d?
Solve a multiplication equation.
To get d by itself, you must undo the multiplication by 14.
Dividing by 14 is the inverse operation of multiplying by 14.
Write the equation.
14d = 4,200
Use the Division Property of Equality.
4,200
14d = ______
____
Divide.
Use the Identity Property of
Multiplication.
1 × d = __
_ = 300
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Check the solution.
Write the equation.
Substitute ___ for d.
The solution checks.
14d = 4,200
14 × ___ = 4,200
___ = 4,200
So, the solution means that Mei ran ____ meters in each lap.
Math
Talk
MATHEMATICAL PRACTICES 8
Use Repeated Reasoning
How do you know what
number to divide both sides
of the equation by?
Chapter 8
451
Example 1 Solve the equation 2__3 n = 1__4.
Think: n is multiplied by 2__, so divide both sides by 2__ to undo the division.
3
3
2n = 1
__
__
3
4
Write the equation.
Use the
2n ÷ 2
__
__ = 1
__ ÷ ___
3
3 4
____ Property of Equality.
2
__n × 3
__ = 1
__ × ___
3
2 4
__, multiply by its reciprocal.
To divide by 2
3
n = ___
Multiply.
Multiplication Property of Equality
If you multiply both sides of an equation by the
same number, the two sides will remain equal.
12 = 3
___
4
12
___
4×
=4×3
4
1 × 12 = 12
12 = 12
Example 2
A biologist divides a water sample equally among 8 test tubes.
Each test tube contains 24.5 milliliters of water. Write and solve
an equation to find the volume of the water sample.
STEP 1 Write an equation. Let v represent the volume in milliliters.
STEP 2 Solve the equation. v is divided by 8, so multiply both sides by
8 to undo the division.
Write the equation.
Use the
______ Property of Equality.
Multiply.
So, the volume of the water sample is ____ milliliters.
452
v = 24.5
__
8
v
_ × __ = _ × 24.5
8
Math
Talk
v = __
MATHEMATICAL PRACTICES 7
Look for Structure How can
you use the Multiplication
Property of Equality to solve
Example 1?
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v = __
___
Think: The volume divided by 8 equals the volume in each test tube.
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1. Solve the equation 2.5m = 10.
2.5m = 10
10
2.5m
____ = ____
2.5
m=_
Use the _____ Property of Equality.
Divide.
Solve the equation, and check the solution.
2. 3x = 210
___
3
___
___
7. 1.3 = _4c
6. 25 = a__
1
5. _1 y = __
2
4. 1_ n = 15
3. 2.8 = 4t
5
10
___
___
___
Math
Talk
On
On Your
Your Own
Own
MATHEMATICAL PRACTICES 8
Generalize What strategy
can you use to get the
variable by itself on one side
of a division equation?
Practice: Copy and Solve Solve the equation, and check the solution.
© Houghton Mifflin Harcourt Publishing Company
8. 150 = 6m
11.
12.
9. 14.7 = b__7
10. _14 = 3_ s
5
DEEPER
There are 100 calories in 8 fluid ounces of orange juice and
140 calories in 8 fluid ounces of pineapple juice. Tia mixed 4 fluid ounces
of each juice. Write and solve an equation to find the number of calories
in each fluid ounce of Tia's juice mixture.
SMARTER
Write a division equation that has the solution x = 16.
Chapter 8 • Lesson 6 453
MATHEMATICAL PRACTICES
COMMUNICA5&t1&34E7&3&tCONSTRUCT ARGUMENTS
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What’s the Error?
13.
SMARTER
Melinda has a block of clay that weighs 14.4 ounces.
She divides the clay into 6 equal pieces. To find the weight w in ounces
of each piece, Melinda solved the equation 6w = 14.4.
Look at how Melinda solved the
equation. Find her error.
Correct the error. Solve the equation,
and explain your steps.
This is how Melinda solved the equation:
6w = 14.4
6w
___ = 6 × 14.4
6
w = 86.4
Melinda concludes that each piece of clay weighs
86.4 ounces.
So, w = _.
This means each piece of clay weighs ___.
14.
MATHEMATICAL
PRACTICE
1 Describe the error that Melinda made.
SMARTER
For numbers 14a−14d, choose Yes or No to indicate
whether the equation has the solution x = 15.
14a. 15x = 30
Yes
No
14b. 4x = 60
Yes
No
14c. _x = 3
Yes
No
14d. _x = 5
Yes
No
5
3
454
© Houghton Mifflin Harcourt Publishing Company
•
Practice and Homework
Name
Solve Multiplication and Division Equations
Lesson 8.6
COMMON CORE STANDARD—6.EE.B.7
Reason about and solve one-variable equations
and inequalities.
Solve the equation, and check the solution.
z
__
2. 16
=8
3. 3.5x = 14.7
4. 32 = 3.2c
5. 2_w = 40
a
__
6. 14
= 6.8
7. 1.6x = 1.6
8. 23.8 = 3.5b
9. _35 = 2_3t
1. 8p = 96
_ _
8p
= 96 p = 12
8
8
5
Problem
Problem Solving
Solving
© Houghton Mifflin Harcourt Publishing Company
10. Anne runs 6 laps on a track. She runs a total
of 1 mile, or 5,280 feet. Write and solve an
equation to find the distance, in feet, that she
runs in each lap.
12.
WRITE
11. In a serving of 8 fluid ounces of pomegranate
juice, there are 32.8 grams of carbohydrates.
Write and solve an equation to find the amount
of carbohydrates in each fluid ounce of the juice.
Math Write and solve a word problem that can
be solved by solving a multiplication equation.
Chapter 8
455
Lesson Check (6.EE.B.7)
1. Estella buys 1.8 pounds of walnuts for a total
of $5.04. She solves the equation 1.8p = 5.04
to find the price p in dollars of one pound of
walnuts. What does one pound of walnuts cost?
TEST
PREP
2. Gabriel wants to solve the equation 5_8 m = 25.
What step should he do to get m by itself on one
side of the equation?
3. At top speed, a coyote can run at a speed of
44 miles per hour. If a coyote could maintain its
top speed, how far could it run in 15 minutes?
4. An online store sells DVDs for $10 each. The
shipping charge for an entire order is $5.50.
Frank orders d DVDs. Write an expression that
represents the total cost of Frank’s DVDs.
5. A ring costs $27 more than a pair of earrings.
The ring costs $90. Write an equation that
can be used to find the cost c in dollars of
the earrings.
6. The equation 3s = 21 can be used to find the
number of students s in each van on a field trip.
How many students are in each van?
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Spiral Review (6.RP.A.3d, 6.EE.B.6, 6.EE.B.7)
Lesson 8.7
Name
Problem Solving • Equations with Fractions
Expressions and Equations—
6.EE.B.7
MATHEMATICAL PRACTICES
MP2, MP6, MP7, MP8
Essential Question How can you use the strategy solve a simpler problem
to solve equations involving fractions?
You can change an equation involving a fraction to an equation
involving only whole numbers. To do so, multiply both sides of the
equation by the denominator of the fraction.
Unlock
Unlock the
the Problem
Problem
On canoe trips, people sometimes carry their canoes between
bodies of water. Maps for canoeing use a unit of length called
a rod to show distances. Victoria and Mick carry their canoe 40
2
rods. The equation 40 = __
11 d gives the distance d in yards that they
carried the canoe. How many yards did they carry the canoe?
Use the graphic organizer to help you solve the problem.
Read the Problem
Solve the Problem
• Write a simpler equation.
What do I need to find?
© Houghton Mifflin Harcourt Publishing Company • Image Credits: (C) ©Radius Images/Alamy Images
I need to find
.
What information do I need to use?
I need to use ____ .
How will I use the information?
I can solve a simpler problem by changing
the equation to an equation involving
only whole numbers. Then I can solve the
simpler equation.
2d
40 = ___
Write the equation.
11
Multiply both sides by
the denominator.
2d
11 × 40 = _ × ___
Multiply.
_ = 2d
11
• Solve the simpler equation.
Write the equation.
440 = 2d
Use the Division
Property of Equality.
440 = ____
2d
____
_=d
Divide.
So, Victoria and Mick carried their canoe _ yards.
Math
Talk
MATHEMATICAL PRACTICES 1
Evaluate How can you check
that your answer to the
problem is correct?
Chapter 8
457
If an equation contains more than one fraction, you can change it to an
equation involving only whole numbers by multiplying both sides of
the equation by the product of the denominators of the fractions.
Try Another Problem
Trevor is making 2_3 of a recipe for chicken noodle soup. He adds 1_2
cup of chopped celery. The equation 2_3 c = 1_2 can be used to find the
number of cups c of chopped celery in the original recipe. How
many cups of chopped celery does the original recipe call for?
Use the graphic organizer to help you solve the problem.
Read the Problem
Solve the Problem
What do I need to find?
What information do I need to use?
How will I use the information?
So, the original recipe calls for _ cup of chopped celery.
MATHEMATICAL
PRACTICE
6 Describe a Method Describe another method that you
could use to solve the problem.
Math
Talk
458
MATHEMATICAL PRACTICES 2
Reasoning How do you
know that your answer is
reasonable?
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•
Unlock the Problem
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MATH
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B
1. Connor ran 3 kilometers in a relay race. His distance
3
represents __
10 of the total distance of the race. The
3
equation __
10 d = 3 can be used to find the total
distance d of the race in kilometers. What was the
total distance of the race?
√
√
Circle the important information.
√
Check your solution by substituting
it into the original equation.
Use the Properties of Equality when
you solve equations.
WRITE
Math t Show Your Work
First, write a simpler equation by multiplying both
sides by the denominator of the fraction.
Next, solve the simpler equation.
So, the race is ___ long.
2.
What if Connor’s distance of
2
3 kilometers represented only __
10 of the total distance of the
race. What would the total distance of the race have been?
SMARTER
3. The lightest puppy in a litter weighs 9 ounces, which
is 3_4 of the weight of the heaviest puppy. The equation
3_
4 w = 9 can be used to find the weight w in ounces of the
heaviest puppy. How much does the heaviest
puppy weigh?
© Houghton Mifflin Harcourt Publishing Company
4. Sophia took home 2_5 of the pizza that was left over from a
party. The amount she took represents 1_2 of a whole pizza.
The equation 2_5 p = 1_2 can be used to find the number of
pizzas p left over from the party. How many pizzas were
left over?
5. A city received 3_4 inch of rain on July 31. This represents
3
__
10 of the total amount of rain the city received in July. The
3
3_
equation __
10 r = 4 can be used to find the amount of rain r in
inches the city received in July. How much rain did the city
receive in July?
Chapter 8 • Lesson 7 459
MATHEMATICAL PRACTICES
ANALYZEt-00,'034536$563&t13&$*4*0/
On
On Your
Your Own
Own
6.
7.
DEEPER
Carole ordered 4 dresses for $80 each,
a $25 sweater, and a coat. The cost of the items
without sales tax was $430. What was the cost of
the coat?
SMARTER
A dog sled race is 25 miles long.
The equation _58 k = 25 can be used to estimate
the race’s length k in kilometers. Approximately
how many hours will it take a dog sled team to
finish the race if it travels at an average speed
of 30 kilometers per hour?
MATHEMATICAL
PRACTICE
6 Explain a Method Explain how you
could use the strategy solve a simpler problem to
3
solve the equation 3_4 x = __
10 .
9.
460
In a basket of fruit, 5_6 of the
pieces of fruit are apples. There are 20 apples
in the display. The equation 5_6 f = 20 can be
used to find how many pieces of fruit f are in
the basket. Use words and numbers to explain
how to solve the equation to find how many
pieces of fruit are in the basket.
SMARTER
WRITE
Math t Show Your Work
© Houghton Mifflin Harcourt Publishing Company • Image Credits: (r) ©Thinkstock/Getty Images
8.
Practice and Homework
Name
Problem Solving • Equations with Fractions
Lesson 8.7
COMMON CORE STANDARD—6.EE.B.7
Reason about and solve one-variable equations
and inequalities.
Read each problem and solve.
1. Stu is 4 feet tall. This height represents 6_ of his
7
brother’s height. The equation 6_h = 4 can be used to
7
find the height h, in feet, of Stu’s brother. How tall is
_
6
Stu’s brother?
7 3 7h = 7 3 4
6h = 28
_ _
_
_
6h
= 28
6
6
h = 42
3
2
4 feet
3
____
7
8
2. Bryce bought a bag of cashews. He served _ pound of
2
3
cashews at a party. This amount represents _ of the entire
2
3
7
8
bag. The equation _ n = _ can be used to find the number of
pounds n in a full bag. How many pounds of cashews were
in the bag that Bryce bought?
____
3. In Jaime’s math class, 9 students chose soccer as their
favorite sport. This amount represents 3_ of the entire class.
8
The equation 3_s = 9 can be used to find the total number
8
© Houghton Mifflin Harcourt Publishing Company
of students s in Jaime’s class. How many students are in
Jaime’s math class?
____
4.
Math Write a math problem for the equation.
WRITE
3
_n=5
_. Then solve a simpler problem to find the solution.
4
6
Chapter 8
461
Lesson Check (6.EE.B.7)
1. Roger served 5_8 pound of crackers, which
was 2_3 of the entire box. What was the weight of
the crackers originally in the box?
TEST
PREP
2. Bowser ate 4 1_2 pounds of dog food. That amount
is 3_4 of the entire bag of dog food. How many
pounds of dog food were originally in the bag?
3. What is the quotient 4 2_3 ÷ 4 1_5 ?
4. Miranda had 4 pounds, 6 ounces of clay. She
divided it into 10 equal parts. How heavy was
each part?
5. The amount Denise charges to repair computers
is $50 an hour plus a $25 service fee. Write an
expression to show how much she will charge
for h hours of work.
6. Luis has saved $14 for a skateboard that costs
$52. He can use the equation 14 + m = 52 to
find how much more he needs. How much
more does he need?
FOR MORE PRACTICE
GO TO THE
462
Personal Math Trainer
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Spiral Review (6.NS.A.1, 6.RP.A.3d, 6.EE.A.2a, 6.EE.B.7)
Name
Personal Math Trainer
Mid-Chapter Checkpoint
Online Assessment
and Intervention
Vocabulary
Vocabulary
Vocabulary
Choose the best term from the box to complete the sentence.
equation
inverse operations
1. A(n) ____ is a statement that two
mathematical expressions are equal. (p. 421)
solution of an equation
2. Adding 5 and subtracting 5 are ____. (p. 439)
Concepts
Concepts and
and Skills
Skills
Write an equation for the word sentence. (6.EE.B.7)
3. The sum of a number and 4.5 is 8.2.
_______
4. Three times the cost is $24.
_______
Determine whether the given value of the variable
is a solution of the equation. (6.EE.B.5)
5. x − 24 = 58; x = 82
_______
6. _1c = 3_; c = 3_4
3
8
_______
Solve the equation, and check the solution. (6.EE.B.7)
7. a + 2.4 = 7.8
© Houghton Mifflin Harcourt Publishing Company
_______
9. 3x = 27
_______
11. _4t = 16
_______
8. b − 1_4 = 31_
2
_______
10. _1s = 1_
3
5
_______
w
12. 7__
= 0.3
_______
Chapter 8
463
13. A stadium has a total of 18,000 seats. Of these, 7,500 are field seats, and
the rest are grandstand seats. Write an equation that could be used to
find the number of grandstand seats s. (6.EE.B.7)
14. Aaron wants to buy a bicycle that costs $128. So far, he has saved
$56. The equation a + 56 = 128 can be used to find the amount a
in dollars that Aaron still needs to save. What is the solution of the
equation? (6.EE.B.7)
15.
DEEPER
Ms. McNeil buys 2.4 gallons of gasoline. The total cost is $7.56.
Write and solve an equation to find the price p in dollars of one gallon of
gasoline. (6.EE.B.7)
© Houghton Mifflin Harcourt Publishing Company
16. Crystal is picking blueberries. So far, she has filled 2_3 of her basket, and
the blueberries weigh 3_4 pound. The equation 2_3 w = 3_4 can be used to
estimate the weight w in pounds of the blueberries when the basket
is full. About how much will the blueberries in Crystal’s basket weigh
when it is full? (6.EE.B.7)
464
Lesson 8.8
Name
Solutions of Inequalities
Expressions and Equations—
6.EE.B.5
MATHEMATICAL PRACTICES
MP2, MP3, MP6
Essential Question How do you determine whether a number is a
solution of an inequality?
An inequality is a mathematical sentence that compares two
expressions using the symbol <, >, ≤, ≥, or ≠. These are
examples of inequalities:
8 < 11
9>
−
4
a ≤ 50
• The symbol ≤ means “is
less than or equal to.”
x ≥ 3.2
• The symbol ≥ means “is
greater than or equal to.”
A solution of an inequality is a value of a variable that makes the
inequality true. Inequalities can have more than one solution.
Unlock
Unlock the
the Problem
Problem
A library has books from the Middle Ages. The books are more
than 650 years old. The inequality a > 650 represents the
possible ages a in years of the books. Determine whether
a = 678 or a = 634 is a solution of the inequality, and tell
what the solution means.
Use substitution to determine the solution.
© Houghton Mifflin Harcourt Publishing Company • Image Credits: (r) ©jupiterimages/Getty Images
STEP 1
Check whether a = 678 is a solution.
Write the inequality.
a > 650
Substitute 678 for a.
__ >? 650
Compare the values.
678 is __ than 650.
The inequality is true when a = 678, so a = 678 is a solution.
STEP 2
Check whether a = 634 is a solution.
Write the inequality.
a > 650
Substitute 634 for a.
__ >? 650
Compare the values.
634 __ greater than 650.
The inequality __ true when a = 634, so a = 634 __
a solution.
The solution a = 678 means that a book in the library from the
Middle Ages could be __ years old.
Math
Talk
MATHEMATICAL PRACTICES 3
Apply Give another solution
of the inequality a > 650.
Explain how you determined
the solution.
Chapter 8
465
Example 1 Determine whether the given value of the variable is a
solution of the inequality.
A b < 0.3; b = −0.2
b < 0.3
Write the inequality.
__ <? 0.3
Substitute the given value for the variable.
−
0.2 is __ than 0.3.
Compare the values.
The inequality _ true when b = −0.2, so b = −0.2 _
a solution.
B m ≥ __2; m = __3
3
5
__
m≥2
3
Write the inequality.
Substitute the given value for
the variable.
____ ? 2
__
≥3
Rewrite the fractions with a
common denominator.
Compare the values.
?
____ ≥ ______
15
9
___
15
15
10.
__ greater than or equal to ___
15
The inequality __ true when m = __3, so m = __3 __
5
5
a solution.
Example 2
An airplane can hold no more than 416 passengers. The inequality
p ≤ 416 represents the possible number of passengers p on the
airplane, where p is a whole number. Give two solutions of the
inequality, and tell what the solutions mean.
__ to 416.
• p = 200 is a solution because 200 is __ than __.
• p = __ is a solution because __ is __
than 416.
h
These solutions mean that the number of passengers on the
plane could be __ or __.
466
Mat
Talk
MATHEMATICAL PRACTICES 3
Apply Give an example of
a value of p that is not a
solution of the inequality.
How do you know it is not a
solution?
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Think: The solutions of the inequality are whole numbers __ than or
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MATH
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B
Determine whether the given value of the variable is a solution
of the inequality.
1. a ≥
−
−
6; a =
_
1; c = __1
3. c > __
4
5
2. y < 7.8; y = 8
3
? −
≥
6
___
___
___
6. t ≥ 2__; t = __3
3
4
5. d < −0.52; d = −0.51
4. x ≤ 3; x = 3
___
___
___
Math
Talk
On
On Your
Your Own
Own
MATHEMATICAL PRACTICES 6
Explain How could you use a
number line to check your
answer to Exercise 5?
Practice: Copy and Solve Determine whether s = 3_5 , s = 0,
or s = 1.75 are solutions of the inequality.
7. s > −1
8. s ≤ 1 2_3
9. s < 0.43
Give two solutions of the inequality.
© Houghton Mifflin Harcourt Publishing Company
10. e < 3
11. p > −12
___
13.
MATHEMATICAL
PRACTICE
___
12. y ≥ 5.8
___
2 Connect Symbols and Words A person must
be at least 18 years old to vote. The inequality a ≥ 18 represents
the possible ages a in years at which a person can vote. Determine
whether a = 18 , a = 1721_, and a = 91.5 are solutions of the inequality,
and tell what the solutions mean.
Chapter 8 • Lesson 8 467
MATHEMATICAL PRACTICES .0%&-t3&"40/tM",&4&/4&
OqnakdlRnkuhmf¤@ookhb`shnmr
OqnakdlRnkuhmf¤@ookhb`shnmr
The table shows ticket and popcorn prices at five
movie theater chains. Use the table for 14–15.
14.
DEEPER
The inequality p < 4.75 represents the
prices p in dollars that Paige is willing to pay for
popcorn. The inequality p < 8.00 represents the prices
p in dollars that Paige is willing to pay for a movie
ticket. At how many theaters would Paige be willing to
buy a ticket and popcorn?
Movie Theater Prices
15.
SMARTER
Sense or Nonsense? Edward
says that the inequality d ≥ 4.00 represents the
popcorn prices in the table, where d is the price
of popcorn in dollars. Is Edward’s statement
sense or nonsense? Explain.
16.
MATHEMATICAL
PRACTICE
Ticket Price ($)
Popcorn Price ($)
8.00
4.25
8.50
5.00
9.00
4.00
7.50
4.75
7.25
4.50
WRITE
Math t Show Your Work
6 Use Math Vocabulary Explain why the
Personal Math Trainer
17.
SMARTER
The minimum
wind speed for a storm to be considered a
hurricane is 74 miles per hour. The inequality
w ≥ 74 represents the possible wind speeds
of a hurricane.
Two possible solutions for the inequality w ≥ 74
71
80.
are 73 and 60.
75
40.
468
© Houghton Mifflin Harcourt Publishing Company • Image Credits: (t) ©Fuse/Getty Images
statement t > 13 is an inequality.
Practice and Homework
Name
Lesson 8.8
Solutions of Inequalities
Determine whether the given value of the variable is a solution of
the inequality.
1. s ≥ −1; s = 1
COMMON CORE STANDARD—6.EE.B.5
Reason about and solve one-variable equations
and inequalities.
2. p < 0; p = 4
3. y ≤ −3; y = −1
5. q ≥ 0.6; q = 0.23
6. b < 23_; b = _
8. z ≥ −3
9. f ≤ −5
?
1 $ 21
solution
4. u > −1_; u = 0
2
4
2
3
Give two solutions of the inequality.
7. k < 2
Problem
Problem Solving
Solving
© Houghton Mifflin Harcourt Publishing Company
10. The inequality s ≥ 92 represents the score s that
Jared must earn on his next test to get an A on his
report card. Give two possible scores that Jared
could earn to get the A.
12.
WRITE
11. The inequality m ≤ $20 represents the amount
of money that Sheila is allowed to spend on a
new hat. Give two possible money amounts that
Sheila could spend on the hat.
Math Describe a situation and write an inequality
to represent the situation. Give a number that is a solution
and another number that is not a solution of the inequality.
Chapter 8
469
TEST
PREP
Lesson Check (6.EE.B.5)
1. Three of the following are solutions of g < −1 1_2 .
Which one is not a solution?
g = −4
g = −71_
2
g=0
g = −21_
2
2. The inequality w ≥ 3.2 represents the
weight of each pumpkin, in pounds, that is
allowed to be picked to be sold. The weights of
pumpkins are listed. How many pumpkins can
be sold? Which pumpkins can be sold?
3.18 lb, 4 lb, 3.2 lb, 3.4 lb, 3.15 lb
Spiral Review (6.EE.A.1, 6.EE.A.3, 6.EE.B.7)
4. Write an expression that is equivalent to
5(3x + 2z).
5. Tina bought a t-shirt and sandals. The total cost
was $41.50. The t-shirt cost $8.95. The equation
8.95 + c = 41.50 can be used to find the cost
c in dollars of the sandals. How much did the
sandals cost?
6. Two-thirds of a number is equal to 20. What is
the number?
© Houghton Mifflin Harcourt Publishing Company
3. What is the value of 8 + (27 ÷ 9)2?
FOR MORE PRACTICE
GO TO THE
470
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Lesson 8.9
Name
Write Inequalities
Expressions and Equations—
6.EE.B.8
MATHEMATICAL PRACTICES
MP2, MP4
Essential Question How do you write an inequality to
represent a situation?
connect You can use what you know about writing equations to
help you write inequalities.
Unlock
Unlock the
the Problem
Problem
The highest temperature ever recorded at the
South Pole was 8°F. Write an inequality to show
that the temperature t in degrees Fahrenheit at the
South Pole is less than or equal to 8°F.
• Underline the words that tell you which
inequality symbol to use.
• Will you use an equal sign in your inequality?
Explain.
Write an inequality for the situation.
Think:
The temperature
is less than or equal to
t
8°F.
8
So, an inequality that describes the temperature t in
© Houghton Mifflin Harcourt Publishing Company • Image Credits: (r) ©Van Hasselt John/Corbis Sygma
degrees Fahrenheit at the South Pole is __ .
Try This! The directors of an animal shelter need to raise more than
$50,000 during a fundraiser. Write an inequality that represents the
amount of money m in dollars that the directors need to raise.
Think:
The amount of money
m
is more than
$50,000.
__
So, an inequality that describes the amount of money m in
dollars is __ .
Math
Talk
MATHEMATICAL PRACTICES 6
Explain How did you know
which inequality symbol to
use in the Try This! problem?
Chapter 8
471
Example 1 Write an inequality for the word sentence. Tell
what type of numbers the variable in the inequality can represent.
A The weight is less than 31__ pounds.
2
Think: Let w represent the unknown weight in pounds.
The weight
__ pounds.
31
is less than
2
_
_,
where w is a positive number
B There must be at least 65 police officers on duty.
Think: Let p represent the number of police officers. The phrase “at least” is
equivalent to “is
__ than or equal to.”
The number of
officers
is greater than
or equal to
_
Example 2
Math
Talk
65.
_,
MATHEMATICAL PRACTICES 2
Reasoning Explain why the
value of p must be a whole
number.
where p is a __ number
Write two word sentences for the inequality.
A n ≤ 0.3
• n is __ than or __ to 0.3.
• n is no __ than 0.3.
B a > −4
• a is __ than −4.
•
472
SMARTER
Which inequality symbol would you use to show that the number
of people attending a party will be at most 14? Explain.
© Houghton Mifflin Harcourt Publishing Company
• a is __ than −4.
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Write an inequality for the word sentence. Tell what type of
numbers the variable in the inequality can represent.
1. The elevation e is greater than or equal to
15 meters.
2. A passenger’s age a must be more than
4 years.
Write a word sentence for the inequality.
3. b < 1_2
4. m ≥ 55
On
On Your
Your Own
Own
5.
MATHEMATICAL
PRACTICE
6 Compare Explain the difference
between t ≤ 4 and t < 4.
7.
SMARTER
6.
DEEPER
A children’s roller coaster is limited
to riders whose height is at least 30 inches and
at most 48 inches. Write two inequalities that
represent the height h of riders for the roller
coaster.
Match the inequality with the word sentence
© Houghton Mifflin Harcourt Publishing Company
it represents.
r > 10
Walter sold more than 10 tickets.
s ≤ 10
Fewer than 10 children are at the party.
t ≥ 10
No more than 10 people can be seated at a table.
w < 10
At least 10 people need to sign up for the class.
Chapter 8 • Lesson 9 473
Make Generalizations
The reading skill make generalizations can help you write
inequalities to represent situations. A generalization is a
statement that is true about a group of facts.
Sea otters spend almost their entire lives in the ocean.
Their thick fur helps them to stay warm in cold water. Sea
otters often float together in groups called rafts. A team of
biologists weighed the female sea otters in one raft off the
coast of Alaska. The chart shows their results.
Weights of Female
Sea Otters
Otter Number
Weight
(pounds)
First, list the weights in pounds in order from
least to greatest.
1
50
2
61
50, 51, 54, _ , _ , _ , _ , _ ,
3
62
4
69
5
71
6
54
7
68
8
62
9
58
10
51
11
61
12
66
_, _, _, _
Next, write an inequality to describe the weights by
using the least weight in the list. Let w represent the
weights of the otters in pounds.
Think: The least weight is _ pounds, so all of
the weights are greater than or equal to 50 pounds.
W
WRITE
50
Now, write an inequality to describe the weights by using
the greatest weight in the list.
Think: The greatest weight is _ pounds, so
all of the weights are
W
71
_ than or equal to
_ pounds.
So, the inequalities __ and __ represent
generalizations about the weights w in pounds of the otters.
8.
474
SMARTER
Use the chart at the right to write two
inequalities that represent generalizations about the
number of sea otter pups per raft.
Math t Show Your Work
Sea Otter Pups per
Raft
Raft Number
Number of
Pups
1
7
2
10
3
15
4
23
5
6
6
16
7
20
8
6
© Houghton Mifflin Harcourt Publishing Company • Image Credits: (t) ©Bob Gibbons/Alamy Images
Write two inequalities that represent generalizations about
the sea otter weights.
Practice and Homework
Name
Lesson 8.9
Write Inequalities
COMMON CORE STANDARD—6.EE.B.8
Reason about and solve one-variable equations
and inequalities.
Write an inequality for the word sentence. Tell what type of
numbers the variable in the inequality can represent.
1. The width w is greater than 4 centimeters.
The inequality symbol for “is greater than” is >. w > 4, where w is the width in
centimeters. w is a positive number.
2. The score s in a basketball game is greater than
or equal to 10 points.
3. The mass m is less than 5 kilograms.
4. The height h is greater than 2.5 meters.
5. The temperature t is less than or equal to −3°.
Write a word sentence for the inequality.
7. z $ 23_
6. k , −7
5
© Houghton Mifflin Harcourt Publishing Company
Problem
Problem Solving
Solving
8. Tabby’s mom says that she must read for at
least 30 minutes each night. If m represents the
number of minutes reading, what inequality can
represent this situation?
10.
WRITE
9. Phillip has a $25 gift card to his favorite
restaurant. He wants to use the gift card to
buy lunch. If c represents the cost of his lunch,
what inequality can describe all of the possible
amounts of money, in dollars, that Phillip can
spend on lunch?
Math Write a short paragraph explaining to a
new student how to write an inequality.
Chapter 8
475
TEST
PREP
Lesson Check (6.EE.B.8)
1. At the end of the first round in a quiz show,
Jeremy has at most −20 points. Write an
inequality that means “at most −20”.
2. Describes the meaning of y ≥ 7.9 in words.
3. Let y represent Jaron’s age in years. If Dawn
were 5 years older, she would be Jaron’s age.
Which expression represents Dawn’s age?
4. Simplify the expression 7 × 3g.
5. What is the solution of the equation 8 = 8f ?
6. Which of the following are solutions of the
inequality k ≤ –2?
k = 0, k = −2, k = −4, k = 1, k = −11_
2
FOR MORE PRACTICE
GO TO THE
476
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© Houghton Mifflin Harcourt Publishing Company
Spiral Review (6.EE.A.2a, 6.EE.A.4, 6.EE.B.5, 6.EE.B.7)
Lesson 8.10
Name
Graph Inequalities
Expressions and Equations—
6.EE.B.8
MATHEMATICAL PRACTICES
MP4, MP5, MP6
Essential Question How do you represent the solutions of an inequality
on a number line?
x>2
Inequalities can have an infinite number of solutions. The
solutions of the inequality x > 2, for example, include all
numbers greater than 2. You can use a number line to
represent all of the solutions of an inequality.
0
1
2
3
4
5
The empty circle at 2 shows that
2 is not a solution. The shading to
the right of 2 shows that values
greater than 2 are solutions.
The number line at right shows the solutions of the
inequality x > 2.
Unlock
Unlock the
the Problem
Problem
Forest fires are most likely to occur when the air
temperature is greater than 60°F. The inequality t > 60
represents the temperatures t in degrees Fahrenheit for
which forest fires are most likely. Graph the solutions of
the inequality on a number line.
Show the solutions of t > 60 on a number line.
Think: I need to show all solutions that are greater
© Houghton Mifflin Harcourt Publishing Company • Image Credits: (r) ©Robert McGouey/Alamy Images
than 60.
Draw an empty circle at
not a solution.
_ to show that 60 is
0
10 20 30 40 50 60 70 80 90 100
__ _ to show that
Shade to the
of
values greater than 60 are solutions.
Try This! Graph the solutions of the inequality y < 5.
Draw an empty circle at
not a solution.
_ to show that 5 is
__ _ to show that
Shade to the
of
values less than 5 are solutions.
0
•
1
MATHEMATICAL
PRACTICE
2
3
4
5
6
7
8
9
10
6
Make Connections Explain why y = 5 is not a
solution of the inequality y < 5.
Chapter 8
477
You can also use a number line to show the
solutions of an inequality that includes the
symbol ≤ or ≥.
x≥2
0
1
2
3
4
5
The filled-in circle at 2 shows that 2 is a solution.
The shading to the right of 2 shows that values
greater than 2 are also solutions.
The number line at right shows the solutions of
the inequality x ≥ 2.
Example 1 Graph the solutions of the inequality on a
number line.
A w ≤ 0.8
Draw a filled-in circle at
is a solution.
_ to show that 0.8
0
__ _
0.2
0.4
0.6
0.8
1
Shade to the
of
to show
that values less than 0.8 are also solutions.
B n ≥ −3
Draw a filled-in circle at
is a solution.
_ to show that −3
-4
__
of _ to show
−
Shade to the
that values greater than
0
2
4
3 are also solutions.
Example 2
-5
-2
-4
Write the inequality represented by the graph.
-3
-2
-1
0
1
2
3
4
5
Use x (or another letter) for the variable in the inequality.
The
__ circle at _ shows that −2
The shading to the
__ of _ shows that values
__ than −2 are solutions.
So, the inequality represented by the graph is __.
478
Math
Talk
MATHEMATICAL PRACTICES 6
Explain How do you know
whether to shade to the
right or to the left when
graphing an inequality?
© Houghton Mifflin Harcourt Publishing Company
__ a solution.
Name
hhow
Share
Share and
and Show
Sh
MATH
M
BOARD
B
Graph the inequality.
1. m < 15
_ to show that 15 is
not a solution. Shade to the __ of _ to
Draw an empty circle at
0
5
10
15 20 25 30 35 40 45
show that values less than 15 are solutions.
2. c ≥ −1.5
-3
3. b ≤ 5_
8
-2
-1
0
1
2
3
1
4
0
1
2
Math
Talk
On
On Your
Your Own
Own
3
4
1
1
14
1
12
MATHEMATICAL PRACTICES 3
Apply Why is it easier to
graph the solutions of an
inequality than it is to list
them?
Practice: Copy and Solve Graph the inequality.
5. x > −4
4. a < 2_3
6. k ≥ 0.3
7. t ≤ 6
Write the inequality represented by the graph.
8.
0
2
4
6
8
10
12
9.
__
© Houghton Mifflin Harcourt Publishing Company
10.
MATHEMATICAL
PRACTICE
20
4 Model Mathematics The
40
-8
-6
-4
-2
0
2
__
11.
inequality w ≥ 60 represents the wind speed w
in miles per hour of a tornado. Graph the
solutions of the inequality on the number line.
0
-10
60
80
100
120
DEEPER
Graph the solutions of the inequality
c < 12 ÷ 3 on the number line.
0
2
4
6
8
10
12
Chapter 8 • Lesson 10 479
MATHEMATICAL PRACTICES
COMMUNICA5&t1&34E7&3&tCONSTRUCT ARGUMENTS
OqnakdlRnkuhmf¤@ookhb`shnmr
OqnakdlRnkuhmf¤@ookhb`shnmr
The table shows the height requirements for
rides at an amusement park. Use the table for 12–16.
12. Write an inequality representing t, the heights in
inches of people who can go on Twirl & Whirl.
13. Graph your inequality from Exercise 12.
12
24
36
48
60
72
14. Write an inequality representing r, the heights in
inches of people who can go on Race Track.
15. Graph your inequality from Exercise 14.
0
16.
17.
12
24
36
48
60
72
SMARTER
Write an inequality representing b,
the heights in inches of people who can go on both
River Rapids and Mighty Mountain. Explain how
you determined your answer.
SMARTER
Alena graphed the inequality c ≤ 25.
Darius said that 25 is not part of the solution of the
inequality. Do you agree or disagree with Darius?
Use numbers and words to support your answer.
25
480
Height Requirements
Minimum
height (in.)
Ride
Mighty Mountain
44
Race Track
42
River Rapids
38
Twirl & Whirl
48
WRITE
Math t Show Your Work
© Houghton Mifflin Harcourt Publishing Company • Image Credits: ©Robert E Daemmrich/Getty Images
0
Practice and Homework
Name
Lesson 8.10
Graph Inequalities
COMMON CORE STANDARD—6.EE.B.8
Reason about and solve one-variable equations
and inequalities.
Graph the inequality.
1. h ≥ 3
Draw a filled-in circle at 3
0
to show that 3 is a
1
2 3 4 5 6 7 8 9 10
solution. Shade to the right of 3 to show
that values greater than 3 are solutions.
2. x , −4_5
3. y . −2
-1 2 -1 1 -1 - 4 - 3 - 2 - 1
5
5
5
5
5
5
0
1
5
2
5
-5 -4 -3 -2 -1 0 1
3
5
2 3 4 5
5. c ≤ −0.4
4. n ≥ 11_
2
-2
-1
0
1
2
-1
3
-0.6
-0.2
0.2
0.6
1
Write the inequality represented by the graph.
6.
7.
2 3 4 5 6 7 8 9 10
0 1
-11 -10 -9 -8 -7 -6 -5 -4 -3 -2 -1
© Houghton Mifflin Harcourt Publishing Company
Problem
Problem Solving
Solving
8. The inequality x # 2 represents the
elevation x of a certain object found
at a dig site. Graph the solutions of
the inequality on the number line.
0
10.
1
WRITE
2 3 4 5 6 7 8 9 10
9. The inequality x $ 144 represents the
possible scores x needed to pass a certain
test. Graph the solutions of the inequality
on the number line.
136
140
144
148
152
156
Math Write an inequality and graph the
solutions on a number line.
Chapter 8
481
TEST
PREP
Lesson Check (6.EE.B.8)
1. Write the inequality that is shown by the graph.
-5 -4 -3 -2 -1 0
1
2 3 4
2. Describe the graph of g < 0.6.
5
3. Write an expression that shows the product of
5 and the difference of 12 and 9.
4. What is the solution of the equation
8.7 + n = 15.1.
5. The equation 12x = 96 gives the number of egg
cartons x needed to package 96 eggs. Solve the
equation to find the number of cartons needed.
6. The lowest price on an MP3 song is $0.35. Write
an inequality that represents the cost c of an
MP3 song.
FOR MORE PRACTICE
GO TO THE
482
Personal Math Trainer
© Houghton Mifflin Harcourt Publishing Company
Spiral Review (6.EE.A.2b, 6.EE.B.5, 6.EE.B.7, 6.EE.B.8)
Name
Chapter 8 Review/Test
Personal Math Trainer
Online Assessment
and Intervention
1. For numbers 1a–1c, choose Yes or No to indicate whether the given value
of the variable is a solution of the equation.
1a.
_2 v = 10; v = 25
5
Yes
No
1b. n + 5 = 15; n = 5
Yes
No
1c. 5z = 25; z = 5
Yes
No
2. The distance from third base to home plate is 88.9 feet. Romeo was 22.1 feet
away from third base when he was tagged out. The equation 88.9 − t = 22.1
can be used to determine how far he needed to run to get to home plate.
Using substitution, the coach determines that Romeo needed
66
to run
66.8
feet to get to home plate.
111
3. There are 84 grapes in a bag. Four friends are sharing the grapes. Write an
equation that can be used to find out how many grapes g each friend will
get if each friend gets the same number of grapes.
© Houghton Mifflin Harcourt Publishing Company
4. Match each scenario with the equation that can be used to solve it.
Jane’s dog eats 3 pounds of food a week. How
many weeks will a 24-pound bag last?
3x = 39
There are 39 students in the gym, and there are
an equal number of students in each class. If
three classes are in the gym, how many students
are in each class?
4x = 24
There are 4 games at the carnival. Kevin played
all the games in 24 minutes. How many minutes
did he spend at each game if he spent an equal
amount of time at each?
3x = 24
Assessment Options
Chapter Test
Chapter 8
483
5.
DEEPER
Frank’s hockey team attempted 15 more goals than Spencer's
team. Frank’s team attempted 23 goals. Write and solve an equation that
can be used to find how many goals Spencer's team attempted.
6. Ryan solved the equation 10 + y = 17 by drawing a model. Use numbers
and words to explain how Ryan’s model can be used to find the solution.
5
5
5
8. Select the equations that have the solution m = 17. Mark all that apply.
484
A
3 + m = 21
B
m – 2 = 15
C
14 = m – 3
D
2 = m – 15
© Houghton Mifflin Harcourt Publishing Company
7. Gabriella and Max worked on their math project for a total of 6 hours.
Max worked on the project for 2 hours by himself. Solve the equation
x + 2 = 6 to find out how many hours Gabriella worked on the project.
Name
9. Describe how you could use algebra tiles to model the equation 4x = 20.
10. For numbers 10a–10d, choose Yes or No to indicate whether the
equation has the solution x = 12.
10a.
_3
4
x=9
Yes
No
10b.
3x = 36
Yes
No
10c.
5x = 70
Yes
No
10d.
_x = 4
3
Yes
No
© Houghton Mifflin Harcourt Publishing Company
11. Bryan rides the bus to and from work on the days he works at the library.
In one month, he rode the bus 24 times. Solve the equation 2x = 24 to
find the number of days Bryan worked at the library. Use a model.
Chapter 8
485
12. Betty needs 3_ of a yard of fabric to make a skirt. She bought 9 yards
4
of fabric.
Part A
Write and solve an equation to find how many skirts x she can make
from 9 yards of fabric.
Part B
Explain how you determined which operation was needed to write
the equation.
© Houghton Mifflin Harcourt Publishing Company
13. Karen is working on her math homework. She solves the equation
b
_ = 56 and says that the solution is b = 7. Do you agree or disagree
8
with Karen? Use words and numbers to support your answer. If her
answer is incorrect, find the correct answer.
486
Name
14. There are 70 historical fiction books in the school library. Historical
1 of the library’s collection. The equation __
1 b = 70
fiction books make up __
10
10
can be used to find out how many books the library has. Solve the equation
to find the total number of books in the library's collection. Use numbers
1 b = 70.
and words to explain how to solve __
10
15. Andy drove 33 miles on Monday morning. This was 3_7 of the total number
of miles he drove on Monday. Solve the equation 3_7 m = 33 to find the total
number of miles Andy drove on Monday.
Personal Math Trainer
16.
SMARTER
The maximum number of players allowed on a lacrosse
team is 23. The inequality t ≤ 23 represents the total number of players t
allowed on the team.
26.
23
Two possible solutions for the inequality are
25
© Houghton Mifflin Harcourt Publishing Company
27
and
24.
22.
17. Mr. Charles needs to have at least 10 students sign up for homework help
in order to use the computer lab. The inequality h ≥ 10 represents the
number of students h who must sign up. Select possible solutions of the
inequality. Mark all that apply.
A
7
D
10
B
8
E
11
C
9
F
12
Chapter 8
487
18. The maximum capacity of the school auditorium is 420 people. Write an
inequality for the situation. Tell what type of numbers the variable in the
inequality can represent.
19. Match the inequality to the word sentence it represents.
w < 70
The temperature did not
drop below 70 degrees.
x ≤ 70
Dane saved more than $70.
y > 70
Fewer than 70 people
attended the game.
z ≥ 70
No more than 70 people
can participate.
20. Cydney graphed the inequality d ≤ 14.
14
Part A
Dylan said that 14 is not a solution of the inequality. Do you agree
or disagree with Dylan? Use numbers and words to support
your answer.
Suppose Cydney’s graph had an empty circle at 14. Write the inequality
represented by this graph.
488
© Houghton Mifflin Harcourt Publishing Company
Part B
9
Chapter
Algebra: Relationships
Between Variables
Personal Math Trainer
Show Wha t You Know
Online Assessment
and Intervention
Check your understanding of important skills.
Name
Number Patterns Write a rule to explain the pattern.
Use the rule to find the missing numbers.
1. 127, 123, 119, ■, 111, ■
2. 5,832, ■, 648, 216, 72, ■, 8
____
____
Identify Points on a Coordinate Grid Use the
ordered pair to name the point on the grid.
y-axis
3. (4, 6 ) __
4. (8, 4 ) __
5. (2, 8 ) __
10
9
8
7
6
5
4
3
2
1
0
A
B
C
D
1 2 3 4 5 6 7 8 9 10
x-axis
Evaluate Expressions Evaluate the expression.
6. 18 + 4 − 7
__
7. 59 − 20 + 5
__
8. (40 − 15) + 30
9. 77 − (59 − 18)
__
__
© Houghton Mifflin Harcourt Publishing Company
Math in the
Terrell plotted points on the coordinate plane as shown. He
noticed that the points lie on a straight line. Write an equation
that shows the relationship between the x- and y-coordinate of
each point Terrell plotted. Then use the equation to find the
y-coordinate of a point on the line with an x-coordinate of 20.
y
10
8
(9, 6)
6
(8, 5)
4
2
0
(5, 2)
(3, 0)
2 4
6
8
10 x
Chapter 9
489
Voca bula ry Builder
Visualize It
Review Words
Use the review words to complete the bubble map.
coordinate plane
ordered pair
quadrants
x-coordinate
y-coordinate
Preview Words
dependent variable
independent variable
linear equation
Understand Vocabulary
Draw a line to match the preview word with its definition.
Preview Words
1. dependent variable
Definitions
•
2. independent variable
490
has a value that determines the value of
another quantity
•
names the point where the axes in the
coordinate plane intersect
•
has a value that depends on the value of
another quantity
•
forms a straight line when graphed
•
™Interactive Student Edition
™Multimedia eGlossary
© Houghton Mifflin Harcourt Publishing Company
3. linear equation
•
•
Chapter 9 Vocabulary
coordinate plane
dependent variable
plano cartesiano
variable dependiente
21
17
independent variable
linear equation
variable independiente
ecuación lineal
53
42
ordered pair
quadrants
par ordenado
cuadrantes
82
70
x-coordinate
y-coordinate
coordenada x
coordenada y
109
111
Example:
8
6
4
2
0
y=x−1
2
4
6
10 x
8
The four regions of the coordinate plane
separated by the x- and y-axes
y
5
4
Quadrant II 3
2
(2, 1)
1
Quadrant I
(1, 1)
-5 -4 -3 -2 -1 0 1 2 3 4 5
-1
Quadrant III - 23 Quadrant IV
(2, 2) - 4
(1, 2)
-5
x
The second number in an ordered pair; tells
the distance to move up or down from (0,0)
5
4
3
2
1
-5 -4 -3 -2 -1 0
-1
-2
-3
-4
-5
y
(3, 2)
x
1 2 3 4 5
© Houghton Mifflin Harcourt Publishing Company
© Houghton Mifflin Harcourt Publishing Company
y
10
© Houghton Mifflin Harcourt Publishing Company
An equation that, when graphed, forms a
straight line
© Houghton Mifflin Harcourt Publishing Company
© Houghton Mifflin Harcourt Publishing Company
© Houghton Mifflin Harcourt Publishing Company
© Houghton Mifflin Harcourt Publishing Company
© Houghton Mifflin Harcourt Publishing Company
A variable whose value depends on the
value of another quantity
A plane formed by a horizontal line called
the x-axis and a vertical line called the
y-axis
y
5
4
3
2
1
-5 -4 -3 -2 -1 0
-1
-2
-3
-4
-5
x
1 2 3 4 5
A variable whose value determines the
value of another quantity
A pair of numbers used to locate a point
on a grid.
5
4
3
2
1
-5 -4 -3 -2 -1 0
-1
-2
-3
-4
-5
y
(3, 2)
x
1 2 3 4 5
The first number in an ordered pair; tells
the distance to move right or left from (0,0)
5
4
3
2
1
-5 -4 -3 -2 -1 0
-1
-2
-3
-4
-5
y
(3, 2)
x
1 2 3 4 5
Going Places with
Guess
the Word
Words
XXX
For 3 to 4 players
Materials
• timer
Game
Game
Word Box
coordinate plane
dependent variable
independent
variable
linear equation
ordered pair
quadrants
x-coordinate
y-coordinate
How to Play
1. Take turns to play.
2. Choose a math term, but do not say it aloud.
3. Set the timer for 1 minute.
4. Give a one-word clue about your term. Give each player one chance
to guess the term.
5. If nobody guesses correctly, repeat Step 4 with a different clue. Repeat
until a player guesses the term or time runs out.
6. The player who guesses the term gets 1 point. If he or she can use the
word in a sentence, they get 1 more point. Then that player gets a turn.
© Houghton Mifflin Harcourt Publishing Company Image Credits: ©Elenamiv/Shutterstock
7. The first player to score 10 points wins.
XXX
XX
XXXX
Chapter 9
490A
Journal
Jo
ou
urnal
The Write Way
XXX
Reflect
Choose one idea. Write about it.
• Write and solve a word problem that includes a dependent variable and
an independent variable.
• Define linear equation in your own words.
• Tell how to determine which of the following ordered pairs makes this
linear equation true. y = x ÷ 3
(2, 6)
(6, 2)
(27, 9)
(21, 6)
(9, 3)
XXX
XX
XXXX
490B
© Houghton Mifflin Harcourt Publishing Company
Image Credits: ©Elenamiv/Shutterstock
• Write a note to a friend about something you learned in Chapter 9.
Lesson 9.1
Name
Independent and Dependent Variables
Expressions and Equations—
6.EE.C.9
MATHEMATICAL PRACTICES
MP1, MP4, MP6, MP7
Essential Question How can you write an equation to represent the
relationship between an independent variable and a dependent variable?
You can use an equation with two variables to represent a relationship
between two quantities. One variable is called the independent variable,
and the other is called the dependent variable. The value of the independent
variable determines the value of the dependent variable.
Unlock
Unlock the
the Problem
Problem
Jeri burns 5.8 calories for every minute she jogs. Identify the
independent and dependent variables in this situation. Then
write an equation to represent the relationship between the
number of minutes Jeri jogs and the total number of calories
she burns.
• Why do you need to use a variable?
• How many variables are needed to
write the equation for this problem?
Identify the independent and dependent variables.
Then use the variables to write an equation.
Let c represent the total number of __ Jeri burns.
Let m represent the number of __ Jeri jogs.
Think: The total number of calories Jeri burns depends on the number
of minutes she jogs.
_ is the dependent variable.
Math
Talk
MATHEMATICAL PRACTICES 6
Explain how you know that the
value of c is dependent on the
value of m.
_ is the independent variable.
Write an equation to represent the situation.
© Houghton Mifflin Harcourt Publishing Company
Think: The total calories burned is equal to 5.8 times the number of minutes jogged.
↓
_
↓
↓
↓
↓
=
5.8
×
_
So, the equation __ represents the number of calories c
Jeri burns if she jogs m minutes, where _ is the dependent
variable and _ is the independent variable.
Chapter 9
491
Example
Lorelei is spending the afternoon bowling with her friends. Each game she
plays costs $3.25, and there is a one-time shoe-rental fee of $2.50.
A Identify the independent and dependent variables in this
situation. Then write an equation to represent the relationship
between the number of games and the total cost.
Think: The total cost in dollars c depends on the number of games g Lorelei plays.
_ is the dependent variable.
_ is the independent variable.
Think:
The total cost is the cost of a game times the number of games plus shoe rental.
↓
↓
↓
↓
↓
↓
↓
_
=
3.25
×
_
+
_
Note that the fee for the
shoes, $2.50, is a onetime fee, and therefore
is not multiplied by the
number of games.
So, the equation ___ represents the total cost in
dollars c that Lorelei spends if she bowls g games, where _ is
the dependent variable and _ is the independent variable.
B Use your equation to find the total cost for Lorelei to play 3 games.
Think: Find the value of c when g = 3.
Write the equation.
c = 3.25g + 2.50
Substitute 3 for g.
c = 3.25( _ ) + 2.50
Follow the order of operations to solve for c.
b = _ + 2.50 = __
So, it will cost Lorelei __ to play 3 games.
2.
SMARTER
MATHEMATICAL
PRACTICE
1 Evaluate Reasonableness How can you use estimation to check that your answer
is reasonable?
492
What if there were no fee for shoe rentals? How would the equation be different?
© Houghton Mifflin Harcourt Publishing Company
1.
Name
Share
hhow
Share and
and Show
Sh
MATH
M
BOARD
B
Identify the independent and dependent variables. Then write
an equation to represent the relationship between them.
1. An online store lets customers
have their name printed on any
item they buy. The total cost c
in dollars is the price of the
item p in dollars plus $3.99 for
the name.
The __ depends on the ___.
dependent variable: _
independent variable: _
equation: _ = __
2. A raft travels downriver at a rate of 6 miles per
hour. The total distance d in miles that the raft
travels is equal to the rate times the number of
hours h.
3. Apples are on sale for $1.99 a pound. Sheila buys
p pounds of apples for a total cost of c dollars.
dependent variable: _
dependent variable: _
independent variable: _
independent variable: _
equation: __
equation: __
Math
Talk
On
On Your
Your Own
Own
Identify the independent and dependent variables. Then write
an equation to represent the relationship between them.
© Houghton Mifflin Harcourt Publishing Company
4. Sean can make 8 paper birds in an hour. The
total number of birds b is equal to the number
of birds he makes per hour times the number of
hours h.
6.
MATHEMATICAL PRACTICES 1
Analyze Relationships How
do you know which variable
in a relationship is dependent
and which is independent?
5. Billy has $25. His father is going to give him
more money. The total amount t Billy will have
is equal to the amount m his father gives him
plus the $25 Billy already has.
dependent variable: _
dependent variable: _
independent variable: _
independent variable: _
equation: __
equation: __
MATHEMATICAL
PRACTICE
2
Connect Symbols and Words
Describe a situation that can be represented by
the equation c = 12b.
7.
DEEPER
Belinda pays $4.25 for each glass she
buys. The total cost c is equal to the price per
glass times the number of glasses n plus $9.95
for shipping and handling. Write an equation
and use it to find how much it will cost Belinda
to buy 12 glasses.
Chapter 9 • Lesson 1 493
MATHEMATICAL PRACTICES
ANALYZEt-00,'034536$563&t13&$*4*0/
Unlock
Unlock the
the Problem
Problem
8. Benji decides to save $15 per week to buy a
computer program. Write an equation that
models the total amount t in dollars Benji will
have saved in w weeks.
a. What does the variable t represent?
b. Which is the dependent variable? Which is the
independent variable? How do you know?
c. How can you find the total amount saved in
w weeks?
d. Write an equation for the total amount that
Benji will have saved.
10.
DEEPER
Coach Diaz is buying hats for the baseball team. The
total cost c is equal to the number of hats n that he buys times the
sum of the price per hat h and a $2 charge per hat to have the team
name printed on it. Write an equation that can be used to find the
cost of the hats.
A steel cable that is 1_2 inch in diameter weighs 0.42 pound
per foot. The total weight in pounds w is equal to 0.42 times of the number of
feet f of steel cable. Choose the letter or equation that makes each sentence true.
SMARTER
The independent variable is
f.
The dependent variable is
w.
The equation that represents the relationship between the variables is
f.
w.
w = 0.42f.
f = 0.42w.
494
© Houghton Mifflin Harcourt Publishing Company • Image Credits: (t) ©Jaak Nilson/Alamy Images
9.
Practice and Homework
Name
Lesson 9.1
Independent and Dependent Variables
Identify the independent and dependent variables. Then write an
equation to represent the relationship between them.
1. Sandra has a coupon to save $3 off her next
purchase at a restaurant. The cost of her meal
c will be the price of the food p that she orders,
minus $3.
COMMON CORE STANDARD—6.EE.C.9
Represent and analyze quantitative
relationships between dependent and
independent variables.
cost of her meal
depends on
The _____
price
of
her
food
the _____.
c
dependent variable: _
p
independent variable: _
p23
c
5 ___
equation: _
2. An online clothing store charges $6 for shipping,
no matter the price of the items. The total cost c
in dollars is the price of the items ordered p plus
$6 for shipping.
dependent variable: _
independent variable: _
equation: _ 5 ___
3. Melinda is making necklaces. She uses 12 beads
for each necklace. The total number of beads b
depends on the number of necklaces n.
dependent variable: _
independent variable: _
equation: _ 5 ___
Problem
Problem Solving
Solving
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4. Maria earns $45 for every lawn that she mows.
Her earnings e in dollars depend on the number
of lawns n that she mows. Write an equation that
represents this situation.
6.
WRITE
5. Martin sells cars. He earns $100 per day, plus
any commission on his sales. His daily salary s in
dollars depends on the amount of commission c.
Write an equation to represent his daily salary.
Math Write a situation in which one unknown is dependent
on another unknown. Write an equation for your situation and identify
the dependent and independent variables.
Chapter 9
495
Lesson Check (6.EE.C.9)
1. There are 12 boys in a math class. The total
number of students s depends on the number
of girls in the class g. Write an equation that
represents this situation.
2. A store received a shipment of soup cans. The
clerk put an equal number of cans on each of
4 shelves. Write an equation to represent the
relationship between the total number of cans
t and the number of cans on each shelf n.
Spiral Review (6.EE.A.2a, 6.EE.B.7, 6.EE.B.8)
3. The formula F = 9_5 C + 35 gives the Fahrenheit
temperature for a Celsius temperature of
C degrees. Gwen measured a Celsius temperature
of 35 degrees. What is this temperature in
degrees Fahrenheit?
4. Write an equation to represent this sentence.
5. Drew drank 4 cups of orange juice. This is 2_5 of
the total amount of juice that was in the
container. Solve 2_5 x = 4 for x. How much juice
was in the container?
6. Graph x ≤ −4.5 on a number line.
-7
-6
-5
-4
-3
-2
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The difference of a number n and 1.8 is 2.
Lesson 9.2
Name
Equations and Tables
Expressions and Equations—
6.EE.C.9
MATHEMATICAL PRACTICES
MP2, MP3, MP4, MP7
Essential Question How can you translate between equations and tables?
When an equation describes the relationship between two
quantities, the variable x often represents the independent
variable, and y often represents the dependent variable.
Input
2
y=x+3
Output
5
A value of the independent variable is called the input
value, and a value of the dependent variable is called
the output value.
Input
4
y=x+3
Output
7
Unlock
Unlock the
the Problem
Problem
A skating rink charges $3.00 for each hour of
skating, plus $1.75 to rent skates. Write an equation
for the relationship that gives the total cost y in
dollars for skating x hours. Then make a table that
shows the cost of skating for 1, 2, 3, and 4 hours.
• What is the independent variable? What is the
Write an equation for the relationship, and use
dependent variable?
the equation to make a table.
STEP 1 Write an equation.
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Think:
The total cost
is
↓
↓
__
__
_.
for each
hour
plus
__
_.
↓
↓
↓
↓
↓
3
̣
=
__ +
1.75
So, the equation for the relationship is ___.
STEP 2 Make a table.
Input
Rule
Output
Time (hr), x
3x + 1.75
Cost ($), y
1
3 ⋅ 1 + 1.75
4.75
Replace x with each
input value, and then
evaluate the rule to find
each output value.
2
3
4
Math
Talk
MATHEMATICAL PRACTICES 5
Communicate Explain how
you could use the equation
to find the total cost of
skating for 6 hours.
Chapter 9
497
Example
Jamal downloads songs on his MP3 player. The table shows how
the time it takes him to download a song depends on the song’s
file size. Write an equation for the relationship shown in the table.
Then use the equation to find how many seconds it takes Jamal
to download a song with a file size of 7 megabytes (MB).
Download Times
File Size (MB), x
Time (s), y
4
48
5
60
6
72
7
?
8
96
STEP 1 Write an equation.
Look for a pattern between the file sizes and the download times.
File Size (MB), x
4
5
6
8
Time (s), y
48
60
72
96
Think: You can find each
download time by multiplying
12 ? 4
Think: The download time
↓
__
12 ? __
12 ? 5
is
↓
5
_
multiplied by
↓
↓
12
?
12 ? __
the file size by
__.
the file size.
↓
__
So, the equation for the relationship is __.
STEP 2 Use the equation to find the download time for a file size of
7 megabytes.
Write the equation.
y 5 12x
Replace x with 7.
y 5 12 ? __
Solve for y.
y 5 __
So, it takes Jamal __ seconds to download a 7-megabyte song.
2.
498
MATHEMATICAL
PRACTICE
Compare Representations Describe a situation in which it would
be more useful to represent a relationship between two quantities with an equation
than with a table of values.
3
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1. Explain how you can check that your equation for the relationship
is correct.
Name
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Use the equation to complete the table.
1. y = x + 3
2. y = 2x + 1
Input
Rule
Output
x
x+3
y
6
613
4
8
813
7
10
Input
Output
x
y
10
On
On Your
Your Own
Own
Write an equation for the relationship shown in the table. Then find the
unknown value in the table.
3.
5.
© Houghton Mifflin Harcourt Publishing Company
6.
x
8
9
10
11
y
16
18
?
22
4.
x
10
20
30
40
y
5
10
15
?
DEEPER
The table shows the current cost of buying apps for
a cell phone. Next month, the price of each app will double. Write an
equation you can use to find the total cost y of buying x apps
next month.
Cell Phone Apps
Number of
apps, x
Total cost ($),
y
3
9
4
12
5
15
Input
Output
Time (hr), x
Cost ($), c
1
6.00
2
7.50
3
9.00
4
10.50
SMARTER
A beach resort charges $1.50 per hour plus $4.50
to rent a bicycle. The equation c = 1.50x + 4.50 gives the total cost c
of renting a bicycle for x hours. Use numbers and words to explain
how to find the cost c of renting a bicycle for 6 hours.
Chapter 9 • Lesson 2 499
Cause and Effect
The reading skill cause and effect can help you understand how
a change in one variable may cause a change in another variable.
In karate, a person’s skill level is often shown by the color of his or
her belt. At Sara’s karate school, students must pass a test to move
from one belt level to the next. Each test costs $23. Sara hopes to
move up 3 belt levels this year. How will this affect her karate expenses?
Cause:
Sara moves to higher
belt levels.
Effect:
Sara’s karate
expenses go up.
Write an equation to show the relationship between cause and
effect. Then use the equation to solve the problem.
Let x represent the number of belt levels Sara moves up, and let y
represent the increase in dollars in her karate expenses.
Write the equation.
y 5 __ ? x
Sara plans to move up 3 levels, so replace x with __.
y 5 23 ? __
Solve for y.
y 5 __
So, if Sara moves up 3 belt levels this year, her karate expenses will
Write an equation to show the relationship between cause
and effect. Then use the equation to solve the problem.
7.
500
SMARTER
Classes at
Tony’s karate school cost $29.50
per month. This year he plans
to take 2 more months of classes
than he did last year. How will
this affect Tony’s karate expenses?
8.
MATHEMATICAL
PRACTICE
4 Write an Equation A sporting
goods store regularly sells karate uniforms
for $35.90 each. The store is putting karate
uniforms on sale for 10% off. How will this
affect the price of a karate uniform?
© Houghton Mifflin Harcourt Publishing Company • Image Credits: (t) ©Franco Vogt/Corbis
increase by $_.
Practice and Homework
Name
Lesson 9.2
Equations and Tables
COMMON CORE STANDARD—6.EE.C.9
Represent and analyze quantitative
relationships between dependent and
independent variables.
Use the equation to complete the table.
1. y = 6x
2. y = x − 7
3. y = 3x + 4
Input
Output
Input
Output
Input
Output
x
y
x
y
x
y
2
12
10
3
5
30
15
4
8
48
20
5
Write an equation for the relationship shown in the table. Then find the unknown value in the table.
4.
x
2
3
4
5
y
16
?
32
40
5.
x
18
20
22
24
y
9
10
?
12
Problem
Problem Solving
Solving
© Houghton Mifflin Harcourt Publishing Company
6. Tickets to a play cost $11 each. There is also a
service charge of $4 per order. Write an equation
for the relationship that gives the total cost y in
dollars for an order of x tickets.
8.
WRITE
7. Write an equation for the relationship shown
in the table. Then use the equation to find the
estimated number of shrimp in a 5-pound bag.
Weight of bag
(pounds), x
1
2
3
4
Estimated
number
of shrimp, y
24
48
72
96
Math Write a word problem that can be
represented by a table and equation. Solve your problem
and include the table and equation.
Chapter 9
501
Lesson Check (6.EE.C.9)
1. Write an equation that represents the
relationship shown in the table.
x
8
10
12
14
y
4
6
8
10
2. There is a one-time fee of $27 to join a gym. The
monthly cost of using the gym is $18. Write an
equation for the relationship that gives the total
cost y in dollars of joining the gym and using it
for x months.
3. Mindy wants to buy several books that each cost
$10. She has a coupon for $6 off her total cost.
Write an expression to represent her total cost
for b books.
4. When a coupon of $1.25 off is used, the cost of a
taco meal is $4.85. The equation p – 1.25 = 4.85
can be used to find the regular price p in dollars
of a taco meal. How much does a regular taco
meal cost?
5. Which of the following are solutions to the
inequality n > –7?
6. Marcus sold brownies at a bake sale. He sold
d dollars worth of brownies. He spent $5.50
on materials, so his total profit p can be found
by subtracting $5.50 from his earnings. Which
equation represents this situation?
n = –7, n = –6.9, n = –7.2, n = –61_
2
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Spiral Review (6.EE.B.5, 6.EE.B.6, 6.EE.B.7, 6.EE.C.9)
Lesson 9.3
Name
Problem Solving • Analyze Relationships
Expressions and Equations—
6.EE.C.9
MATHEMATICAL PRACTICES
MP1, MP4, MP8
Essential Question How can you use the strategy find a pattern to solve
problems involving relationships between quantities?
Unlock
Unlock the
the Problem
Problem
The table shows the amount of water pumped through a fire hose
over time. If the pattern in the table continues, how long will it take a
firefighter to spray 3,000 gallons of water on a fire using this hose?
Fire Hose Flow Rate
Time (min)
Amount of water (gal)
1
2
3
4
150
300
450
600
Use the graphic organizer to help you solve the problem.
Read the Problem
Solve the Problem
Use the table above to find the relationship
between the time and the amount of water.
What do I need to find?
I need to find _____
Think: Let t represent the time in minutes, and w
.
represent the amount of water in gallons. The amount
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of water in gallons is
minutes.
What information do I need to use?
I need to use the relationship between __
and ____.
_
_ multiplied by the time in
=
150
_
∙
Use the equation to find how long it will take to
spray 3,000 gallons.
w = 150t
Write the equation.
How will I use the information?
Substitute 3,000 for w.
3,000 = 150t
Solve for t. Divide both
sides by 150.
3,000 ______
______
= 150t
I will find a __ in the table and write an
____.
_=t
So, it will take _ minutes to spray
3,000 gallons of water.
Math
Talk
MATHEMATICAL PRACTICES 1
Evaluate How can you check
that your answer is correct?
Chapter 9
503
Try Another Problem
Dairy cows provide 90% of the world’s milk supply. The
table shows the amount of milk produced by a cow over
time. If the pattern in the table continues, how much milk
can a farmer get from a cow in 1 year (365 days)?
Cow Milk Production
Time (days), x
2
7
10
30
Amount of milk
(L), y
50
175
250
750
Read the Problem
What do I need to find?
What information do I
need to use?
How will I use the
information?
Solve the Problem
So, in 365 days, the farmer can get __ liters of milk from the cow.
• Explain how you could find the number of days it would take the
cow to produce 500 liters of milk.
504
MATHEMATICAL PRACTICES 2
Reasoning Explain how
you wrote an equation to
represent the pattern in the
table.
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Math
Talk
Name
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B
1. A soccer coach is ordering shirts for the players. The
table shows the total cost based on the number of
shirts ordered. How much will it cost the coach to
order 18 shirts?
√
√
Find a pattern in the table.
√
Check your answer.
Write an equation to represent the
pattern.
First, find a pattern and write an equation.
The cost is _ multiplied by ___.
_=_
∙
_
Next, use the equation to find the cost of 18 shirts.
Soccer Shirts
So, the cost of 18 shirts is _.
© Houghton Mifflin Harcourt Publishing Company • Image Credits: (t) ©Barry Austin/Getty Images
2.
3.
What if the coach spent $375 to purchase a
number of shirts? Could you use the same equation to find
how many shirts the coach bought? Explain.
SMARTER
Number of
Shirts, n
2
3
5
6
Cost ($), c
30
45
75
90
DEEPER
The table shows the number of miles the Carter
family drove over time. If the pattern continues, will the Carter
family have driven more than 400 miles in 8 hours? Explain.
Carter Family Trip
4.
MATHEMATICAL
PRACTICE
7 Look for a Pattern The Carter family drove a
total of 564 miles. Describe how to use the pattern in the table
to find the number of hours they spent driving.
Time (hr), x
Distance (mi), y
1
47
3
141
5
235
6
282
Chapter 9 • Lesson 3 505
MATHEMATICAL PRACTICES .0%&-t3&"40/tM",&4&/4&
On
On Your
Your Own
Own
5. A group of dancers practiced for 4 hours in March, 8 hours in April,
12 hours in May, and 16 hours in June. If the pattern continues,
how many hours will they practice in November?
6.
DEEPER
The table shows the number of hours Jacob worked
and the amount he earned each day.
Jacob’s Earnings
Time (hr), h
5
7
6
8
4
Amount earned ($), d
60
84
72
96
48
At the end of the week, he used his earnings to buy a new pair of
skis. He had $218 left over. How much did the skis cost?
7.
8.
Pose a Problem Look back at Problem 6. Use the
data in the table to write a new problem in which you could use the
strategy find a pattern. Then solve the problem.
SMARTER
MATHEMATICAL
PRACTICE
Draw Conclusions Marlon rode his bicycle 9 miles
the first week, 18 miles the second week, and 27 miles the third
week. If the pattern continues, will Marlon ride exactly 100 miles in
a week at some point? Explain how you determined your answer.
8
9.
SMARTER
A diving instructor ordered
snorkels. The table shows the cost based on the number
of snorkels ordered.
Number of
Snorkels, s
1
2
3
4
Cost ($), c
32
64
96
128
If the diving instructor spent $1,024, how many snorkels
did he order? Use numbers and words to explain your answer.
506
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Personal Math Trainer
Practice and Homework
Name
Lesson 9.3
Problem Solving • Analyze Relationships
COMMON CORE STANDARD—6.EE.C.9
Represent and analyze quantitative
relationships between dependent and
independent variables.
The table shows the number of cups of yogurt needed to make
different amounts of a fruit smoothie. Use the table for 1–3.
Batches, b
3
4
5
6
Cups of Yogurt, c
9
12
15
18
1. Write an equation to represent the relationship.
3 multiplied by the number of batches,
The number of cups needed is _
c =_
3 ×_
b .
so _
2. How much yogurt is needed for 9 batches of smoothie?
____
3. Jerry used 33 cups of yogurt to make smoothies.
How many batches did he make?
____
The table shows the relationship between Winn’s age and
his sister’s age. Use the table for 4–5.
Winn’s age, w
8
9
10
11
Winn’s sister’s age, s
12
13
14
15
© Houghton Mifflin Harcourt Publishing Company
4. Write an equation to represent the relationship.
s 5 ____
5. When Winn is 14 years old, how old will his sister be?
6.
WRITE
Math Write a problem for the table. Use
a pattern and an equation to solve your problem.
____
Hours, h
1
2
3
4
Miles, m
16
32
48
64
Chapter 9
507
Lesson Check (6.EE.C.9)
1. The table shows the total cost c in dollars of
n gift baskets. What will be the cost of 9 gift
baskets?
n
3
4
5
6
c
$36
$48
$60
$72
2. The table shows the number of minutes m
that Tara has practiced after d days. If Tara has
practiced for 70 minutes, how many days has
she practiced?
d
1
3
5
7
m
35
105
175
245
Spiral Review (6.EE.A.3, 6.EE.B.7, 6.EE.B.8, 6.EE.C.9)
5. The lowest price of an MP3 of a song in an
online store is $0.99. Write an inequality that
represents the price p of any MP3 in the store.
4. What is an equation that represents the
relationship in the table?
x
8
10
12
14
y
4
5
6
7
6. Marisol plans to make 9 mini-sandwiches for
every 2 people attending her party. Write a ratio
that is equivalent to Marisol’s ratio.
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3. Soccer shirts cost $15 each, and soccer
shorts cost $18 each. The expression
15n + 18n represents the total cost in
dollars of n uniforms. Simplify the
expression by combining like terms.
Name
Personal Math Trainer
Mid-Chapter Checkpoint
Online Assessment
and Intervention
Vocabulary
Vocabulary
Vocabulary
Choose the best term from the box to complete the sentence.
dependent variable
equation
1. A(n) ____ has a value that determines
the value of another quantity. (p. 491)
independent variable
2. A variable whose value is determined by the value of another quantity
is called a(n) ____. (p. 491)
Concepts
Concepts and
and Skills
Skills
Identify the independent and dependent variables. (6.EE.C.9)
3. Marco spends a total of d dollars on postage to
mail party invitations to each of g guests.
dependent variable: _
4. Sophie has a doll collection with 36 dolls. She
decides to sell s dolls to a museum and has r
dolls remaining.
dependent variable: _
independent variable: _
independent variable: _
Write an equation for the relationship shown in the table.
Then find the unknown value in the table. (6.EE.C.9)
5.
x
6
7
8
9
y
42
?
56
63
6.
_______
x
20
40
60
80
y
4
8
?
16
_______
© Houghton Mifflin Harcourt Publishing Company
Write an equation that describes the pattern shown in the table. (6.EE.C.9)
7. The table shows how the number of pepperoni
slices used depends on the number of pizzas
made.
Pepperonis Used
8. Brayden is training for a marathon. The table
shows how the number of miles he runs depends
on which week of training he is in.
Miles Run During Training
Pizzas, x
2
3
5
9
Week, w
3
5
8
12
Pepperoni slices, y
34
51
85
153
Miles, m
8
10
13
17
_______
_______
Chapter 9
509
9. The band has a total of 152 members. Some of the members are in the
marching band, and the rest are in the concert band. Write an equation
that models how many marching band members m there are if there are
c concert band members. (6.EE.C.9)
10. A coach is ordering baseball jerseys from a website. The jerseys cost
$15 each, and shipping is $8 per order. Write an equation that can be
used to determine the total cost y, in dollars, for x jerseys. (6.EE.C.9)
12.
510
DEEPER
Aaron wants to buy a new snowboard. The table shows the
amount that he has saved. If the pattern in the table continues, how
much will he have saved after 1 year? (6.EE.C.9)
Aaron’s Savings
Time (months)
Money saved ($)
3
135
4
180
6
270
7
315
© Houghton Mifflin Harcourt Publishing Company
11. Amy volunteers at an animal shelter. She worked 10 hours in March,
12 hours in April, 14 hours in May, and 16 hours in June. If the pattern
continues, how many hours will she work in December? (6.EE.C.9)
Lesson 9.4
Name
Graph Relationships
Essential Question How can you graph the relationship between two quantities?
connect You have learned that tables and equations are two ways
to represent the relationship between two quantities. You can also
represent a relationship between two quantities by using a graph.
Expressions and Equations—
6.EE.C.9
MATHEMATICAL PRACTICES
MP4, MP6, MP7
Unlock
Unlock the
the Problem
Problem
A cafeteria has a pancake-making machine. The table shows
the relationship between the time in hours and the number
of pancakes the machine can make. Graph the relationship
represented by the table.
Pancake Production
Time
(hours)
Use the table values to graph the relationship.
STEP 1 Write ordered pairs.
Let x represent the time in hours and y represent the number of
pancakes made. Use each row of the table to write an ordered pair.
( 1, 200 ) ( 2, __)
( 3,
__)
Pancakes
Made
1
200
2
400
3
600
4
800
5
1,000
( _, __ )
( _, __ )
STEP 2 Choose an appropriate scale for each axis of the graph.
Label the axes and give the graph a title.
Pancake Production
y
Pancakes Made
© Houghton Mifflin Harcourt Publishing Company • Image Credits: (b) ©Tom Schierlitz/Getty Images
STEP 3 Graph a point for each ordered pair.
400
200
0
1 2
Time (hours)
x
Math
Talk
MATHEMATICAL PRACTICES 7
Look for Structure What
pattern do you notice in the
set of points you graphed?
Chapter 9
511
Example
The table shows the relationship between
the number of bicycles y Shawn has left to assemble and the number
of hours x he has worked. Graph the relationship represented by the
table to find the unknown value of y.
Time (hours), x
Bicycles Left to
Assemble, y
0
10
1
8
2
?
3
4
4
2
STEP 1 Write ordered pairs.
Use each row of the table to write an ordered pair. Skip the row with
the unknown y-value.
( 0, 10 )
( 1, _ )
( 3, _ )
( _, _ )
STEP 2 Graph a point for each ordered pair on a coordinate plane.
y
10
9
8
7
6
5
4
3
2
1
0
The first value in an ordered
pair represents the independent
variable x. The second value
represents the dependent
variable y.
1 2 3 4 5 6 7 8 9 10 x
STEP 3 Find the unknown y-value.
The points on the graph appear to lie on a line. Use a ruler to draw
a dashed line through the points.
Use the line to find the y-value that corresponds to an x-value of 2.
Start at the origin, and move 2 units right. Move up until you reach
the line you drew. Then move left to find the y-value on the y-axis.
So, after 2 hours, Shawn has __ bicycles left to assemble.
•
MATHEMATICAL
PRACTICE
6 Describe another way you could find the unknown value
of y in the table.
512
MATHEMATICAL PRACTICES 2
Reasoning Describe a
situation in which it would
be more useful to represent
a function with a graph than
with a table of values.
© Houghton Mifflin Harcourt Publishing Company
When x has a value of 2, y has a value of __.
Math
Talk
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Graph the relationship represented by the table.
1.
2.
x
1
2
3
4
y
50
100
150
200
x
20
40
60
80
y
100
200
300
400
y
y
200
175
(1, 50)
150
125
(2, _)
100
75
(3, _)
50
25
(_, _)
400
350
300
250
200
150
100
50
Write ordered pairs.
Then graph.
0
1 2 3 4 5 6 7
0
x
10 20 30 40 50 60 70 80 90 100 x
Graph the relationship represented by the table to find the unknown value of y.
3.
x
4
5
6
y
9
7
5
8
4.
1
x
1
3
5
y
3
4
5
y
y
10
9
8
7
6
5
4
3
2
1
10
9
8
7
6
5
4
3
2
1
0
© Houghton Mifflin Harcourt Publishing Company
7
1 2 3 4 5 6 7 8 9 10 x
0
9
7
1 2 3 4 5 6 7 8 9 10 x
Math
Talk
On
On Your
Your Own
Own
7
Practice: Copy and Solve Graph the relationship represented
MATHEMATICAL PRACTICES 4
Use Graphs Explain how
to use a graph to find an
unknown y-value in a table.
by the table to find the unknown value of y.
5.
x
1
3
y
7
6
5
7
9
4
3
6.
x
1
2
4
y
2
3
5
6
7
8
Chapter 9 • Lesson 4 513
MATHEMATICAL PRACTICES
COMMUNICA5&t1&34E7&3&tCONSTRUCT ARGUMENTS
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The table at the right shows the typical price of a popular
brand of corn cereal over time. Use the table for 7–8.
7.
MATHEMATICAL
PRACTICE
Price of Corn Cereal
4 Use Graphs Complete the table below to show
the cost of buying 1 to 5 boxes of corn cereal in 1988. Then
graph the relationship on the coordinate plane at right.
Boxes
1
Cost in
1988 ($)
3
4
Price per box ($)
1968
0.39
1988
1.50
2008
4.50
5
Cost of Corn Cereal, 1988
1.50
DEEPER
Suppose you graphed the cost of buying 1 to 5
boxes of corn cereal using the 1968 price and the 2008 price.
Explain how those graphs would compare to the graph you
made using the 1988 price.
Cost ($)
8.
2
Year
Boxes
9.
SMARTER
A bookstore charges $4 for shipping, no matter how many
books you buy. Irena makes a graph showing the shipping cost for 1 to 5 books.
She claims that the points she graphed lie on a line. Does her statement make
sense? Explain.
Personal Math Trainer
SMARTER
Graph the relationship
represented by the table to find the
unknown value of y.
x
1
2
y
2
2.5
3
4
3.5
y
7
6
5
4
3
2
1
0
514
1 2 3 4 5 6 7
x
© Houghton Mifflin Harcourt Publishing Company
10.
Practice and Homework
Name
Lesson 9.4
Graph Relationships
COMMON CORE STANDARD—6.EE.C.9
Represent and analyze quantitative relationships
between dependent and independent variables.
Graph the relationship represented by the table.
1.
x
1
2
3
4
5
y
25
50
75
100
125
2.
x
10
20
30
40
50
y
350
300
250
200
150
y
y
175
150
125
100
75
50
25
350
300
250
200
150
100
50
0
x
1 2 3 4 5 6 7 8 9 10
0
x
5 10 15 20 25 30 35 40 45 50
Graph the relationship represented by the table to find the unknown value of y.
3.
x
3
4
y
8
7
5
6
7
5
4
4.
y
y
10
9
8
7
6
5
4
3
2
1
10
9
8
7
6
5
4
3
2
1
0
x
0
1 2 3 4 5 6 7 8 9 10
x
1
y
1
DVDs Purchased
1
2
3
4
Cost ($)
15
30
45
60
Cost ($)
© Houghton Mifflin Harcourt Publishing Company
7
9
3
4
5
1 2 3 4 5 6 7 8 9 10
Cost of DVDs
5. Graph the relationship represented by the table.
6. Use the graph to find the cost of purchasing 5 DVDs.
WRITE
5
x
Problem
Problem Solving
Solving
7.
3
Math Both tables and graphs can be used to
80
70
60
50
40
30
20
10
0
represent relationships between two variables. Explain how
tables and graphs are similar and how they are different.
_____
_
_
_
_
_
1 2 3 4 5 6 7 8 9 10
DVDs Purchased
Chapter 9
515
Lesson Check (6.EE.C.9)
1. Mei wants to graph the relationship represented
by the table. Which ordered pair is a point on
the graph of the relationship?
T-shirts purchased, x
1
2
3
4
Cost ($), y
8
16
24
32
2. An online bookstore charges $2 to ship any
book. Cole graphs the relationship that gives
the total cost y in dollars to buy and ship a book
that costs x dollars. Name an ordered pair that
is a point on the graph of the relationship.
3. Write an expression that is equivalent to 6(g + 4).
4. There are 6 girls in a music class. This
represents 3_7 of the entire class. Solve 3_7 s = 6 to
find the number of students, s, in the class.
5. Graph n > –2 on a number line.
6. Sam is ordering lunch for the people in his
office. The table shows the cost of lunch based
on the number of people. How much will lunch
cost for 35 people?
-4
-2
0
2
4
Number of people, n
5
10
15
20
Cost ($), c
40
80
120
160
FOR MORE PRACTICE
GO TO THE
516
Personal Math Trainer
© Houghton Mifflin Harcourt Publishing Company
Spiral Review (6.EE.A.3, 6.EE.B.7, 6.EE.B.8, 6.EE.C.9)
Lesson 9.5
Name
Equations and Graphs
Expressions and Equations—
6.EE.C.9
MATHEMATICAL PRACTICES
MP3, MP4, MP5
Essential Question How can you translate between equations and graphs?
The solution of an equation in two variables is an ordered pair that
makes the equation true. For example, ( 2, 5 ) is a solution of the
equation y = x + 3 because 5 = 2 + 3.
A linear equation is an equation whose solutions form a
straight line on the coordinate plane. Any point on the line is a
solution of the equation.
Unlock
Unlock the
the Problem
Problem
A blue whale is swimming at an average rate of
3 miles per hour. Write a linear equation that gives the
distance y in miles that the whale swims in x hours.
Then graph the relationship.
• What formula can you use to help you write
the equation?
Write and graph a linear equation.
STEP 1 Write an equation for the relationship.
Think:
Distance
equals
rate
multiplied by
time.
↓
↓
↓
↓
↓
_
̣
_
_
=
STEP 2 Find ordered pairs that are solutions
of the equation.
Choose several values of x and find the
corresponding values of y.
STEP 3 Graph the relationship.
Graph the ordered pairs. Draw a line through the points
to show all the solutions of the linear equation.
Distance Traveled by Blue Whale
x
3x
y
Ordered Pair
1
3?1
3
(1, 3)
2
3?
(2, )
3
3?
( , )
4
3?
( ,
Math
Talk
MATHEMATICAL PRACTICES 2
)
Distance (mi)
© Houghton Mifflin Harcourt Publishing Company
y
20
18
16
14
12
10
8
6
4
2
0
1 2 3 4 5 6 7 8 9 10 x
Time (hr)
Reasoning Explain why
the graph does not show
negative values of x or y.
Chapter 9
517
y
beaded necklaces y that Ginger can make in x hours. Write
the linear equation for the relationship shown by the graph.
10
STEP 1 Use ordered pairs from the graph to complete the
table of values below.
STEP 2 Look for a pattern in the table.
Compare each y-value with the corresponding x-value.
0
↓
2?0
3
↓
y
1
4
↓
0
2?_
2?1
Necklaces Made
8
(4, 8)
6
(3, 6)
4
2
0
↓
x
Number of Necklaces
Example The graph shows the number of
(1, 2)
(0, 0)
2
4
6
8
10 x
2
4
6
Time (hr)
2?_
Think: Each y-value is __ times the corresponding x-value.
So, the linear equation for the relationship is y = _.
1. Explain how to graph a linear equation.
2.
MATHEMATICAL
PRACTICE
3 Compare Representations Describe a situation in which it would be
more useful to represent a relationship with an equation than with a graph.
Share
hhow
Share and
and Show
Sh
MATH
M
BOARD
B
10
2. y = 3x y
10
8
8
6
6
4
4
4
6
2
2
1. y = x − 2
y
Make a table of
values. Then graph.
x
y
2
0
8
518
0
2
4
6
8
10 x
0
x
© Houghton Mifflin Harcourt Publishing Company
Graph the linear equation.
Name
Write the linear equation for the relationship shown by the graph.
3.
4.
y
10
10
8
8
(8, 7)
6
6
(6, 5)
4
4
(4, 3)
2 (2, 1)
0
y
(4, 2)
2 (2, 1)
2
4
6
8
10 x
0
2
(6, 3)
4
6
Math
Talk
On
On Your
Your Own
Own
(8, 4)
8
10 x
MATHEMATICAL PRACTICES 6
Explain how you can tell
whether you have graphed a
linear equation correctly.
Graph the linear equation.
5. y = x + 1 y
10
8
8
6
6
4
4
2
2
2
4
6
8
10 x
0
shows the number of loaves of bread y that Kareem
bakes in x hours. Write the linear equation for the
relationship shown by the graph.
2
4
6
8
10 x
Loaves of Bread Baked
7 Identify Relationships The graph
y
Number of Loaves
© Houghton Mifflin Harcourt Publishing Company
MATHEMATICAL
PRACTICE
y
10
0
7.
6. y = 2x − 1
6
(5, 5)
(4, 4)
4
2
0
(2, 2)
(1, 1)
2
4
Time (hr)
6
x
Chapter 9 • Lesson 5 519
MATHEMATICAL PRACTICES
ANALYZEt-00,'034536$563&t13&$*4*0/
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OqnakdlRnkuhmf¤@ookhb`shnmr
The graph shows the growth of a bamboo plant.
Use the graph for 8–9.
8. Write a linear equation for the relationship shown by the
graph. Use your equation to predict the height of the bamboo
plant after 7 days.
Growth of a Bamboo Plant
y
SMARTER
The height y in centimeters of
a second bamboo plant is given by the equation
y = 30x, where x is the time in days. Describe
how the graph showing the growth of this plant
would compare to the graph showing the growth
of the first plant.
Height (centimeters)
0
10.
DEEPER
Maria graphed the linear equation y = x + 3.
Then she used her ruler to draw a vertical line through the
point (4, 0). At what point do the two lines intersect?
SMARTER
Antonio claims the linear equation
for the relationship shown by the graph is y = 1_2 x + 2.
Use numbers and words to support Antonio’s claim.
Height (cm)
11.
x
1 2 3 4 5 6
Time (days)
y
10
9
8
7
6
5
4
3
2
1
0
(8, 6)
(6, 5)
(4, 4)
(2, 3)
x
1 2 3 4 5 6 7 8 9 10
Weeks
520
© Houghton Mifflin Harcourt Publishing Company
9.
250
225
200
175
150
125
100
75
50
25
Practice and Homework
Name
Lesson 9.5
Equations and Graphs
COMMON CORE STANDARD—6.EE.C.9
Represent and analyze quantitative
relationships between dependent and
independent variables.
Graph the linear equation.
1. y 5 x 2 3
x
y
5
2
6
3
7
4
8
5
y
10
9
8
7
6
5
4
3
2
1
0
y
2. y 5 x 4 3
10
9
8
7
6
5
4
3
2
1
x
0
1 2 3 4 5 6 7 8 9 10
x
1 2 3 4 5 6 7 8 9 10
Write a linear equation for the relationship shown by the graph.
3.
y
10
9
8
7
6
5
4
3
2
1
0
4.
(7, 8)
(5, 6)
(3, 4)
(1, 2)
x
1 2 3 4 5 6 7 8 9 10
y
10
9
8
7
6
5
4
3
2
1
0
(2, 8)
(1.5, 6)
(1, 4)
x
1 2 3 4 5 6 7 8 9 10
5. Dee is driving at an average speed of
50 miles per hour. Write a linear equation
for the relationship that gives the distance y
in miles that Dee drives in x hours.
6. Graph the relationship from Exercise 5.
7.
WRITE
Math Explain how to write a linear
Dee’s Distance
Distance (miles)
© Houghton Mifflin Harcourt Publishing Company
Problem
Problem Solving
Solving
y
200
175
150
125
100
75
50
25
0
x
1
2
3
Time (hours)
4
5
equation for a line on a graph.
Chapter 9
521
Lesson Check (6.EE.C.9)
1. A balloon rises at a rate of 10 feet per second.
What is the linear equation for the relationship
that gives the height y in feet of the balloon after
x seconds?
2. Write the linear equation that is shown
by the graph.
y
10
9
8
7
6
5
4
3
2
1
0
(8, 8)
(5, 5)
(3, 3)
x
1 2 3 4 5 6 7 8 9 10
Spiral Review (6.EE.A.4, 6.EE.B.5, 6.EE.C.9)
3 + 2(9 + 2n)
7(3 + 4n)
21 + 4n
5. Red grapes cost $2.49 per pound. Write an
equation that shows the relationship between
the cost c and the number of pounds of grapes p.
4. Which of the following are solutions of j ≥ 0.6?
j = 1, j = −0.6, j = 3_, j = 0.12, j = 0.08
5
6. It costs $8 per hour to rent a bike. Niko graphs
this relationship using x for number of hours
and y for total cost. Which ordered pair is a
point on the graph of the relationship?
FOR MORE PRACTICE
GO TO THE
522
Personal Math Trainer
© Houghton Mifflin Harcourt Publishing Company
3. Of the three expressions shown, which two
are equivalent?
Name
Personal Math Trainer
Chapter 9 Review/Test
Online Assessment
and Intervention
1. A box of peanut butter crackers contains 12 individual snacks. The total
number of individual snacks s is equal to 12 times the number of boxes
of crackers b.
b.
The independent variable is
The dependent variable is
b.
s.
s.
The equation that represents the
b = 12s.
relationship between the variables is
s = 12b.
2. A stationery store charges $8 to print logos on paper purchases. The
total cost c is the price of the paper p plus $8 for printing the logo.
For numbers 2a–2d, select True or False for each statement.
2a.
True
False
2b. c is the dependent variable.
True
False
2c. p is the independent variable.
True
False
2d.
True
False
The total cost c depends on the
price of the paper.
The equation that represents the
relationship between the variables
is c = 8p.
3. An electrician charges $75 an hour for labor and an initial fee of $65.
The total cost c equals 75 times the number of hours x plus 65. Write an
equation for the relationship and use the equation to complete the table.
Time (hr), x
Cost ($), c
1
© Houghton Mifflin Harcourt Publishing Company
2
3
4
equation ___
Assessment Options
Chapter Test
Chapter 9
523
4. The community center offers classes in arts and crafts. There is a
registration fee of $125 and each class costs $79. The total cost c equals
79 times the number of classes n plus 125.
Input
Output
Number of Classes, n
Cost ($), c
1
204
2
283
3
362
4
441
For numbers 4a–4d, select True or False for each statement.
4a.
True
False
4b. n is the independent variable.
True
False
4c. c is the dependent variable.
True
False
4d. The cost for 7 classes is $678.
True
False
The registration fee is $120.
5. Ms. Walsh is buying calculators for her class. The table shows the total
cost based on the number of calculators purchased.
Number of Calculators, n
1
2
3
4
Cost ($), c
15
30
45
60
© Houghton Mifflin Harcourt Publishing Company
If Ms. Walsh spent a total of $525, how many calculators did she buy?
Use numbers and words to explain your answer.
524
Name
6. The table shows the number of cups of lemonade that can be
made from cups of lemon juice.
Lemon Juice (cups), j
2
4
5
7
Lemonade (cups), l
14
28
35
49
Mary Beth says the number of cups of lemon juice j depends on
the number of cups of lemonade l. She says the equation j = 7l
represents the relationship between the cups of lemon juice j and the
cups of lemonade l. Is Mary Beth correct? Use words and numbers to
explain why or why not.
7. For numbers 7a–7d, choose Yes or No to indicate whether the points,
when graphed, would lie on the same line.
7a. (1, 6), (2, 4), (3, 2), (4, 0)
Yes
No
7b.
Yes
No
7c. (1, 3), (2, 5), (3, 7), (4, 9)
Yes
No
7d.
Yes
No
(1, 1), (2, 4), (3, 9), (4, 16)
(1, 8), (2, 10), (3, 12), (4, 14)
Personal Math Trainer
y
SMARTER
Graph the relationship represented by
the table to find the unknown value.
Time (seconds), x
40
50
Water in Tub (gal), y
13
11.5
60
14
12
70
8.5
Water (gal)
© Houghton Mifflin Harcourt Publishing Company
8.
10
8
6
4
2
x
0
20
40
60
80
100
120
140
Time (s)
Chapter 9
525
Time (hr), x
Distance (mi), y
3
4
5
6
240
320
400
480
Distance (mi)
9. Graph the relationship represented by the table.
y
560
480
400
320
240
160
80
0
Wages ($)
10. Miranda's wages are $15 per hour. Write a linear equation
that gives the wages w in dollars that Miranda earns in
h hours.
11.
DEEPER
The table shows the number of apples a that Lucinda
uses in b batches of applesauce.
x
1 2 3 4 5 6 7 8 9 10
Time (hr)
w
80
70
60
50
40
30
20
10
0
h
1 2 3 4 5 6 7 8
Time (hr)
Batches, b
1
2
3
4
Apples, a
4
8
12
16
a
20
18
16
14
12
10
8
6
4
2
0
526
b
1 2 3 4 5 6 7 8 9 10
Batches
© Houghton Mifflin Harcourt Publishing Company
Apples
Graph the relationship between batches b and apples a. Then write the
equation that shows the relationship.
Name
12. Delonna walks 4 miles per day for exercise. The total number
of miles m she walks equals 4 times the number of days d she walks.
What is the dependent variable? __
What is the independent variable? __
Write the equation that represents the relationship between
the m and d.
13. Lacy is staying at a hotel that costs $85 per night. The total cost
of Lacy’s stay is 85 times the number of nights n she stays.
For numbers 13a–13d, select True or False for each statement.
13a. The number of nights n
is dependent on the cost c.
True
False
13b.
True
False
13c. c is the dependent variable.
True
False
13d.
True
False
n is the independent variable.
The equation that represents the
total cost is c = 85n.
14. A taxi cab company charges an initial fee of $5 and then $4 per mile
for a ride. Use the equation c = 4x + 5 to complete the table.
Input
Output
Miles (mi), x
Cost ($), c
2
4
© Houghton Mifflin Harcourt Publishing Company
6
8
Chapter 9
527
15. A grocery display of cans is arranged in the form of a pyramid with 1 can
in the top row, 3 in the second row from the top, 5 in the third row, and
7 in the fourth row. The total number of cans c equals 2 times the row r
minus 1. Use the equation c = 2r − 1 to complete the table.
Row, r
Cans, c
5
6
7
8
Words
16. The graph shows the number of words Mason read
in a given amount of minutes. If Mason continues to
read at the same rate, how many words will he have
read in 5 minutes?
w
1,000
900
800
700
600
500
400
300
200
100
0
t
1 2 3 4 5 6 7 8 9 10
Time (min)
c
200
175
150
125
100
75
50
25
0
(7, 175)
(5, 125)
(3, 75)
(1, 25)
1 2 3 4 5 6 7 8
Number of Jumps
528
j
© Houghton Mifflin Harcourt Publishing Company
Cost ($)
17. Casey claims the linear equation for the relationship
shown by the graph is c = 25j. Use numbers and
words to support Casey's claim.
Critical Area
Geometry and
Statistics
CRITICAL AREA
Solve real-world and mathematical problems involving area, surface
area, and volume.
© Houghton Mifflin Harcourt Publishing Company • Image Credits: (bg) ©Ian Shaw/Alamy
Developing understanding of statistical thinking
The San Francisco zoo in San
Francisco, California, is home to
hundreds of different animals,
including this Bengal tiger.
529
Project
This Place is a Zoo!
Planning a zoo is a difficult task. Each animal requires a special environment
with different amounts of space and different features.
WRITE
Get Started
Math
You are helping to design a new section of a zoo. The table lists some of
the new attractions planned for the zoo. Each attraction includes notes
about the type and the amount of space needed. The zoo owns a rectangle
of land that is 100 feet long and 60 feet wide. Find the dimensions of each
of the attractions and draw a sketch of the plan for the zoo.
Important Facts
Attraction
Minimum Floor
Space (sq ft)
American Alligators
400
rectangular pen with one side at least 24 feet long
Amur Tigers
750
trapezoid-shaped area with one side at least 40 feet long
Howler Monkeys
450
parallelogram-shaped cage with one side at least 30 feet long
Meerkat Village
250
square pen with glass sides
Red Foxes
350
rectangular pen with length twice as long as width
Tropical Aquarium
200
triangular bottom with base at least 20 feet long
530
Chapters 10–13
© Houghton Mifflin Harcourt Publishing Company • Image Credits: (b) ©Nicola Angella/Grand Tour/Corbis
Completed by
Notes
10
Chapter
Area
Personal Math Trainer
Show Wha t You Know
Online Assessment
and Intervention
Check your understanding of important skills.
Name
Perimeter Find the perimeter.
1.
2.
15 mm
8 mm
P = _ units
P = _ mm
17 mm
Identify Polygons Name each polygon based on the number of sides.
3.
4.
___
5.
___
___
Evaluate Algebraic Expressions Evaluate the expression.
6. 5x + 2y for x = 7
7. 6a × 3b + 4 for a = 2
8. s2 + t 2 − 23 for s = 4
and y = 9
and b = 8
and t = 6
___
___
___
© Houghton Mifflin Harcourt Publishing Company
Math in the
Ross needs to paint the white boundary lines of
one end zone on a football field. The area of the
end zone is 4,800 square feet, and one side of the
end zone measures 30 feet. One can of paint is
enough to paint 300 feet of line. Help Ross find
out if one can is enough to line the perimeter of
the end zone.
Chapter 10
531
Voca bula ry Builder
Visualize It
Complete the bubble map by using the checked words that are
types of quadrilaterals.
Review Words
acute triangle
base
height
obtuse triangle
✓ polygon
quadrilateral
quadrilateral
✓ rectangle
right triangle
✓ square
Preview Words
area
composite figure
Understand Vocabulary
Complete the sentences using the preview words.
1. The ____ of a figure is the number of square units
needed to cover it without any gaps or overlaps.
congruent
✓ parallelogram
regular polygon
✓ trapezoid
2. A polygon in which all sides are the same length and all angles
have the same measure is called a(n) ____.
3. A(n) ____ is a quadrilateral with exactly one pair
of parallel sides.
____ figures have the same size and shape.
© Houghton Mifflin Harcourt Publishing Company
4.
5. A quadrilateral with two pairs of parallel sides is called a
____.
6. A(n) ____ is made up of more than one shape.
532
™Interactive Student Edition
™Multimedia eGlossary
Chapter 10 Vocabulary
area
composite figure
área
figura compuesta
14
4
congruent
parallelogram
congruente
paralelogramo
73
15
polygon
quadrilateral
polígono
cuadrilátero
83
75
regular polygon
trapezoid
polígono regular
trapecio
90
102
Example:
A polygon with four sides and four angles
Example:
A quadrilateral with exactly one pair of
parallel sides
Examples:
© Houghton Mifflin Harcourt Publishing Company
© Houghton Mifflin Harcourt Publishing Company
A quadrilateral whose opposite sides are
parallel and congruent
© Houghton Mifflin Harcourt Publishing Company
Example:
© Houghton Mifflin Harcourt Publishing Company
© Houghton Mifflin Harcourt Publishing Company
© Houghton Mifflin Harcourt Publishing Company
© Houghton Mifflin Harcourt Publishing Company
© Houghton Mifflin Harcourt Publishing Company
A figure that is made up of two or more
simpler figures, such as triangles and
quadrilaterals
The number of square units needed to cover
a surface without any gaps or overlaps
Example:
UNITS
UNITS
Area = 21 square units
Having the same size and shape
Example:
A closed plane figure formed by three or
more line segments
Polygon
Not a polygon
A polygon in which all sides are congruent
and all angles are congruent
Example:
Going Places with
Words
Going to the
Philadelphia
Zoo
Game
Gam
G
me
ee
Gam
Word Box
area
composite figure
congruent
parallelogram
polygon
quadrilateral
regular polygon
trapezoid
For 2 players
Materials
• 1 each: red and blue connecting cubes
• 1 number cube
How to Play
1. Each player chooses a connecting cube and puts it on START.
2. Toss the number cube to take a turn. Move your connecting cube that
many spaces.
3. If you land on these spaces:
sentence. If the other player agrees that your answer is correct, jump
to the next space with the same term.
Green Space
Follow the directions printed in the space. If there
are no directions, stay where you are.
4. The first player to reach FINISH wins.
© Houghton Mifflin Harcourt Publishing Company
Image Credits: ©feraru nicolae/Fotolia
White Space Tell the meaning of the math term or use it in a
Chapter 10
532A
Game
Game
DIRECTIONS Each player chooses a connecting cube and puts it on START.
• Toss the number cube to take a turn. Move your connecting cube that
many spaces. • If you land on these spaces: White Space Tell the meaning
of the math term or use it in a sentence. If the other player agrees that
your answer is correct, jump to the next space with the same term. •
Green Space Follow the directions printed in the space. If there are no
directions, stay where you are. • The first player to reach FINISH wins.
FI N IS H
regular polygon
addends
trapezoid
quadrilateral
regular polygon
trapezoid
addends
quadrilateral
STA R T
532B
polygon
area
parallelogram
congruent
Go
back
to
polygon
area
composite figure
a
area
composite
figure
composite
figure
parallelogram
composite figure
congruent
p
polygon
parallelogram
congruent
parallelogram
© Houghton Mifflin Harcourt Publishing Company
Game
Game
area
congruent
congruent
Go
back
to
t apezoid
tr
trapezoid
polygon
parallelogram
Go back to
para
parallelogram
congruent
© Houghton Mifflin Harcourt Publishing Company
addends
regular polygon
q
quadrilateral
addends
ygon
polygon
area
composite figure
polygon
Go
back
ck
to
quadrilateral
trapezoid
area
composite figure
regular polygon
quadrilateral
addends
Image Credits: (bg) ©Digital Vision/Getty Images; (elephant) ©Elvele Images Ltd/Alamy;
(lion) ©PhotoDisc/Getty Images; (snake) ©Design Pics/Superstock
trapezoid
regular polygon
Chapter 10
532C
Journal
Jo
ou
urnal
The Write Way
Reflect
Choose one idea. Write about it.
• Explain how to find the area of a parallelogram.
• Tell how a trapezoid and a quadrilateral are related.
• Write a paragraph that uses the following words
area
congruent
regular polygon
© Houghton Mifflin Harcourt Publishing Company
Image Credits: ©Comstock/Getty Images
• Describe three real-life objects that are composite figures.
532D
ALGEBRA
Name
Lesson 10.1
Area of Parallelograms
Essential Question How can you find the area of parallelograms?
Geometry—6.G.A.1 Also
6.EE.A.2c, 6.EE.B.7
MATHEMATICAL PRACTICES
MP4, MP5, MP6, MP8
connect The area of a figure is the number of square units needed
to cover it without any gaps or overlaps. The area of a rectangle is
the product of the length and the width. The rectangle shown has an
area of 12 square units. For a rectangle with length l and width w,
A = l × w, or A = lw.
4
3
Recall that a rectangle is a special type of parallelogram.
A parallelogram is a quadrilateral with two pairs of parallel sides.
Unlock
Unlock the
the Problem
Problem
Victoria is making a quilt. She is using material in the shape
of parallelograms to form the pattern. The base of each
parallelogram measures 9 cm and the height measures
4 cm. What is the area of each parallelogram?
Activity
Use the area of a rectangle to find the area
of the parallelogram.
Materials ■ grid paper ■ scissors
height (h)
4 cm
• Draw the parallelogram on grid paper and cut it out.
base (b) 9 cm
© Houghton Mifflin Harcourt Publishing Company• Image Credits: (r) ©Roman Soumar/Corbis
• Cut along the dashed line to remove a triangle.
• Move the triangle to the right side of the figure to form
a rectangle.
width (w)
4 cm
• What is the area of the rectangle? ____
length (l) 9 cm
• What is the area of the parallelogram? ____
• base of parallelogram = ___ of rectangle
The height of a parallelogram
forms a 90º angle with the base.
height of parallelogram = ___ of rectangle
area of parallelogram = ___ of rectangle
• For a parallelogram with base b and height h, A = ___
Area of parallelogram = b × h = 9 cm × 4 cm = _ sq cm
So, the area of each parallelogram in the quilt is _ sq cm.
Math
Talk
MATHEMATICAL PRACTICES 6
Explain how you know that
the area of the parallelogram
is the same as the area of the
rectangle.
Chapter 10
533
Example 1 Use the formula A = bh to find the area
of the parallelogram.
Write the formula.
A = bh
Replace b and h with their values.
A = 6.3 ×
Multiply.
A = ___
2.1 m
__
6.3 m
So, the area of the parallelogram is _ square meters.
A square is a special rectangle in which the length and width are equal. For
a square with side length s, A = l × w = s × s = s 2, or A = s 2.
Example 2
Find the area of a square with sides
measuring 9.5 cm.
9.5 cm
Write the formula.
A=s
Substitute 9.5 for s. Simplify.
A = (__)2 = __
2
9.5 cm
So, the area of the square is ___ cm2.
Example 3
A parallelogram has an area of 98 square feet
and a base of 14 feet. What is the height of the parallelogram?
Write the formula.
A = bh
Replace A and b with their values.
__ = __ × h
Use the Division Property of Equality.
98 = ____
14h
____
Solve for h.
__ = h
?
Area 5 98 ft 2
14 ft
•
MATHEMATICAL
PRACTICE
6 Compare Explain the difference between the height of a
rectangle and the height of a parallelogram.
534
© Houghton Mifflin Harcourt Publishing Company
So, the height of the parallelogram is __ feet.
Name
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and Show
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MATH
M
BOARD
B
Find the area of the parallelogram or square.
1. A = bh
2.
1.2 m
6 ft
8.3 m
A = 8.3 × 1.2
15 ft
A = ___ m2
3. 2.5 mm
___ ft2
4.
2 ft
3
2.5 mm
___ mm2
___ ft2
3 ft
4
Find the unknown measurement for the parallelogram.
5. Area = 11 yd2
6. Area = 32 yd2
? yd
? yd
5 21 yd
___ yd
___ yd
Math
Talk
On
On Your
Your Own
Own
4 yd
MATHEMATICAL PRACTICES 2
Reasoning Explain how the
areas of some parallelograms
and rectangles are related.
Find the area of the parallelogram.
7.
8.
6.4 m
9.1 m
8 ft
21 ft
___ m2
___ ft2
Find the unknown measurement for the figure.
© Houghton Mifflin Harcourt Publishing Company
9. square
13.
10. parallelogram
11. parallelogram
12. parallelogram
A = __
A = 32 m2
A = 51 1_4 in.2
A = 121 mm2
s = 15 ft
b = __
b = 8 1_5 in.
b = 11 mm
h=8m
h = __
h = __
SMARTER
The height of a parallelogram is four times the base. The
base measures 3 1_2 ft. Find the area of the parallelogram.
Chapter 10 • Lesson 1 535
MATHEMATICAL PRACTICES .0%&-t3&"40/tM",&4&/4&
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OqnakdlRnkuhmf¤@ookhb`shnmr
14. Jane’s backyard is shaped like a parallelogram. The base of
the parallelogram is 90 feet, and the height is 25 feet. What
is the area of Jane’s backyard?
16.
17.
SMARTER
Jack made a parallelogram by putting
together two congruent triangles and a square, like
the figures shown at the right. The triangles have the
same height as the square. What is the area of Jack’s
parallelogram?
5 cm
The base of a parallelogram is 2 times the
parallelogram’s height. If the base is 12 inches, what is
the area?
MATHEMATICAL
PRACTICE
3 Verify the Reasoning of Others Li Ping
3 in.
3 in.
Find the area of
the parallelogram.
12 in.
The area is _ in2.
3 in.
3 in.
3 in.
3 in.
SMARTER
5 in.
536
8 cm
DEEPER
says that a square with 3-inch sides has a greater area than
a parallelogram that is not a square but has sides that have
the same length. Does Li Ping’s statement make sense?
Explain.
18.
5 cm
6 in.
© Houghton Mifflin Harcourt Publishing Company
15.
Practice and Homework
Name
Lesson 10.1
Area of Parallelograms
COMMON CORE STANDARD—6.G.A.1
Solve real-world and mathematical problems
involving area, surface area, and volume.
Find the area of the figure.
1.
A = bh
A = 18 3 7
A = 126 ft2
7 ft
18 ft
2.
5 cm
7 cm
__ cm2
Find the unknown measurement for the figure.
3. parallelogram
A = 9.18 m2
b = 2.7 m
h = __
4. parallelogram
5. square
6. parallelogram
A = __
A = __
A = 6.3 mm2
3m
b = 4__
s = 35 cm
b = __
10
__
h=21m
10
h = 0.9 mm
Problem
Problem Solving
Solving
© Houghton Mifflin Harcourt Publishing Company
7. Ronna has a sticker in the shape of a
parallelogram. The sticker has a base of
6.5 cm and a height of 10.1 cm. What is
the area of the sticker?
_______
9.
WRITE
8. A parallelogram-shaped tile has an area
of 48 in.2. The base of the tile measures
12 in. What is the measure of its height?
_______
Math Copy the two triangles and the square in
Exercise 15 on page 536. Show how you found the area of each piece.
Draw the parallelogram formed when the three figures are put together.
Calculate its area using the formula for the area of a parallelogram.
Chapter 10
537
Lesson Check (6.G.A.1, 6.EE.A.2c, 6.EE.B.7)
1. Cougar Park is shaped like a parallelogram and
3_
1
has an area of __
16 square mile. Its base is 8 mile.
What is its height?
2. Square County is a square-shaped county
divided into 16 equal-sized square districts. If
the side length of each district is 4 miles, what is
the area of Square County?
Spiral Review (6.EE.B.5, 6.EE.B.8, 6.EE.C.9)
y = –4
y = –6
y=0
y = –8
y=2
5. In 2 seconds, an elevator travels 40 feet. In
3 seconds, the elevator travels 60 feet. In
4 seconds, the elevator travels 80 feet. Write
an equation that gives the relationship between
the number of seconds x and the distance y the
elevator travels.
4. On a winter’s day, 9°F is the highest
temperature recorded. Write an inequality
that represents the temperature t in degrees
Fahrenheit at any time on this day.
6. The linear equation y = 4x represents the
number of bracelets y that Jolene can make in x
hours. Which ordered pair lies on the graph of
the equation?
© Houghton Mifflin Harcourt Publishing Company
3. Which of the following values of y make the
inequality y < –4 true?
FOR MORE PRACTICE
GO TO THE
538
Personal Math Trainer
Name
Explore Area of Triangles
Essential Question What is the relationship among the areas of
triangles, rectangles, and parallelograms?
Lesson 10.2
Geometry—
6.G.A.1
MATHEMATICAL PRACTICES
MP1, MP7, MP8
Hands
On
Investigate
Investigate
Materials ■ grid paper ■ tracing paper ■ ruler ■ scissors
A. On the grid, draw a rectangle with a base of
6 units and a height of 5 units.
• What is the area of the rectangle?
B. Trace the rectangle onto tracing paper. Draw
a diagonal from the top-left corner to the
lower-right corner.
• A diagonal is a line segment that connects two
nonadjacent vertices of a polygon.
C. Cut out the rectangle. Then cut along the diagonal
to divide the rectangle into two right triangles.
Compare the two triangles.
© Houghton Mifflin Harcourt Publishing Company • Image Credits: (br) ©HMH/Guy Jarvis
• Congruent figures are the same shape and size. Are
the two right triangles congruent?
• How is the area of each right triangle related to the
area of the rectangle?
• What is the area of each right triangle?
Chapter 10
539
Draw Conclusions
1. Explain how finding the area of a rectangle is like finding the area
of a right triangle. How is it different?
MATHEMATICAL
PRACTICE
2.
1 Analyze Because a rectangle is a parallelogram, its area can be found
using the formula A = b × h. Use this formula and your results from the Investigate
to write a formula for the area of a right triangle with base b and height h.
Math
Talk
Make
Make Connections
Connections
MATHEMATICAL PRACTICES 3
Apply Why did the two
triangles have to be
congruent for the formula
to make sense?
The area of any parallelogram, including a rectangle, can be found using the formula
A = b × h. You can use a parallelogram to look at more triangles.
A. Trace and cut out two copies of the acute triangle.
B. Arrange the two triangles to make a parallelogram.
• Are the triangles congruent? __
• If the area of the parallelogram is 10 square
centimeters, what is the area of each triangle?
Explain how you know.
C. Repeat Steps A and B with the obtuse triangle.
3.
MATHEMATICAL
PRACTICE
8 Generalize Can you use the formula A = _1 × b × h to find
2
the area of any triangle? Explain.
540
Obtuse triangle
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Acute triangle
Name
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MATH
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B
1. Trace the parallelogram, and cut it into two congruent
triangles. Find the areas of the parallelogram and one
triangle, using square units.
Find the area of each triangle.
3.
2.
4.
8 in.
11 yd
20 ft
10 in.
18 ft
_ ft2
_ in.2
5.
4 yd
_ yd2
6.
7.
12 cm
20 in.
33 mm
16 cm
19 in.
30 mm
_ mm2
_ in.2
_ cm2
© Houghton Mifflin Harcourt Publishing Company
OqnakdlRnkuhmf¤@ookhb`shnmr
OqnakdlRnkuhmf¤@ookhb`shnmr
8.
MATHEMATICAL
PRACTICE
5 Communicate Describe how you
can use two triangles of the same shape and size
to form a parallelogram.
9.
DEEPER
A school flag is in the shape of a right
triangle. The height of the flag is 36 inches and
the base is 3_4 of the height. What is the area of the
flag?
Chapter 10 • Lesson 2 541
MATHEMATICAL PRACTICES
SMARTER
COMMUNICA5&t1&34E7&3&tCONSTRUCT ARGUMENTS
Sense or Nonsense?
10. Cyndi and Tyson drew the models below. Each
said his or her drawing represents a triangle with
an area of 600 square inches. Whose statement
makes sense? Whose statement is nonsense?
Explain your reasoning.
Tyson’s Model
Cyndi’s Model
40 in.
40 in.
30 in.
11.
SMARTER
A flag is separated into two different colors.
Find the area of the white region. Show your work.
3 ft
5 ft
542
© Houghton Mifflin Harcourt Publishing Company • Image Credits: (tr) ©fStop/Alamy
30 in.
Practice and Homework
Name
Lesson 10.2
Explore Area of Triangles
COMMON CORE STANDARD—6.G.A.1
Solve real-world and mathematical problems
involving area, surface area, and volume.
Find the area of each triangle.
1.
2.
3.
20 mm
37 cm
10 ft
40 mm
50 cm
6 ft
30 ft2
___
4.
___
5.
30 in.
___
6.
30 cm
45 cm
12 in.
20 cm
15 cm
___
___
___
Problem
Problem Solving
Solving
© Houghton Mifflin Harcourt Publishing Company
7. Fabian is decorating a triangular pennant for
a football game. The pennant has a base of
10 inches and a height of 24 inches. What is the
total area of the pennant?
_______
9.
WRITE
8. Ryan is buying a triangular tract of land.
The triangle has a base of 100 yards and a height
of 300 yards. What is the area of the tract of land?
_______
Math Draw 3 triangles on grid paper.
Draw appropriate parallelograms to support the
formula for the area of the triangle. Tape your
drawings to this page.
Chapter 10
543
Lesson Check (6.G.A.1)
1. What is the area of a triangle with a height of
14 feet and a base of 10 feet?
2. What is the area of a triangle with a height of
40 millimeters and a base of 380 millimeters?
Spiral Review (6.EE.A.2c, 6.EE.B.7, 6.EE.B.8, 6.G.A.1)
4. Coach Herrera is buying tennis balls for his
team. He can solve the equation 4c = 92 to find
how many cans c of balls he needs. How many
cans does he need?
5. Sketch the graph of y ≤ –7 on a number line.
6. A square photograph has a perimeter of 20
inches. What is the area of the photograph?
© Houghton Mifflin Harcourt Publishing Company
3. Jack bought 3 protein bars for a total of $4.26.
Which equation could be used to find the cost c
in dollars of each protein bar?
-11 -10 -9 -8 -7 -6 -5 -4 -3
544
FOR MORE PRACTICE
GO TO THE
Personal Math Trainer
ALGEBRA
Name
Lesson 10.3
Area of Triangles
Essential Question How can you find the area of triangles?
Any parallelogram can be divided into two congruent
triangles. The area of each triangle is half the area of the
parallelogram, so the area of a triangle is half the product
of its base and its height.
Geometry—6.G.A.1
Also 6.EE.A.2c
MATHEMATICAL PRACTICES
MP1, MP5, MP8
h
b
Area of a Triangle
h
1 bh
A = __
2
where b is the base and h is the height
b
Unlock
Unlock the
the Problem
Problem
The Flatiron Building in New York is well known for its
unusual shape. The building was designed to fit the triangular
plot of land formed by 22nd Street, Broadway, and Fifth
Avenue. The diagram shows the dimensions of the triangular
foundation of the building. What is the area of the triangle?
© Houghton Mifflin Harcourt Publishing Company • Image Credits: (r) ©Frank Whitney/Getty Images
79 ft
• How can you identify the base and
the height of the triangle?
190 ft
Find the area of the triangle.
Write the formula.
__ bh
A=1
Substitute 190 for b and
79 for h.
__ ×__ × __
A=1
Multiply the base and
height.
__ ×__
A=1
Multiply by 1_2 .
A = __
2
2
2
So, the area of the triangle is ___ ft2.
Math
Talk
MATHEMATICAL PRACTICES 8
Generalize How does the
area of a triangle relate to
the area of a rectangle with
the same base and height?
Chapter 10
545
Example 1
Find the area of the triangle.
Write the formula.
__ bh
A=1
Substitute 4 1_2 for b and 3 1_2 for h.
__ × __ × __
A=1
2
Rewrite the mixed numbers as
fractions.
2
__ × ____ × ____
A=1
2
2
2
Multiply.
A = ______
Rewrite the fraction as a mixed
number.
A = __
3 21 ft
4 21 ft
8
So, the area of the triangle is __ ft2.
Example 2
Daniella is decorating a triangular pennant for
her wall. The area of the pennant is 225 in.2 and
the base measures 30 in. What is the height of the
triangular pennant?
A 5 225 in. 2
30 in.
Write the formula.
__ bh
A=1
Substitute 225 for A and 30 for b.
__ = 1__2 × __ × h
Multiply 1_2 and 30.
225 = __ × h
Use the Division Property of Equality.
225 = ________
×h
____
Simplify.
__ = h
2
So, the height of the triangular pennant is __ in.
546
© Houghton Mifflin Harcourt Publishing Company • Image Credits: (r) ©Houghton Mifflin Harcourt
? in.
Name
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MATH
M
BOARD
B
1. FInd the area of the triangle.
A = 1_ bh
2
8 cm
A = 1_ × 14 × _
2
2. The area of the triangle is
132 in.2 Find the height
of the triangle.
h
14 cm
h = __
22 in.
A = _ cm
2
Find the area of the triangle.
3.
4.
4 mm
40 mm
A = __
A = __
5.5 mm
27 mm
Math
Talk
On
On Your
Your Own
Own
SMARTER
5 in.
Explain how you can identify
the height of a triangle.
Find the unknown measurement for the figure.
Area 5 52.5 in.2
5.
MATHEMATICAL PRACTICES 6
6.
Area 5 17.2 cm2
h
h
80 mm
© Houghton Mifflin Harcourt Publishing Company
h = __
7.
MATHEMATICAL
PRACTICE
h = __
3 Verify the Reasoning of Others The height of a triangle is
twice the base. The area of the triangle is 625 in.2 Carson says the base of
the triangle is at least 50 in. Is Carson’s estimate reasonable? Explain.
Chapter 10 • Lesson 3 547
MATHEMATICAL PRACTICES
ANALYZEt-00,'034536$563&t13&$*4*0/
Unlock
Unlock the
the Problem
Problem
8.
14 in.
DEEPER
Alani is building a set of 4 shelves.
Each shelf will have 2 supports in the shape of
right isosceles triangles. Each shelf is 14 inches
deep. How many square inches of wood will she
need to make all of the supports?
14 iin.
n.
14 in.
14 in.
a. What are the base and height of each triangle?
b. What formula can you use to find the area of a
triangle?
c. Explain how you can find the area of one
triangular support.
d. How many triangular supports are needed to
build 4 shelves?
WRITE
M t Show Your Work
Math
e. How many square inches of wood will Alani
need to make all the supports?
548
SMARTER
The area of
a triangle is 97.5 cm2. The
height of the triangle is 13 cm.
Find the base of the triangle.
Explain your work.
10.
SMARTER
The area of a triangle
is 30 ft . For numbers 10a–10d, select
Yes or No to tell if the dimensions given
could be the height and base of the triangle.
2
10a. h = 3, b = 10
Yes
No
10b. h = 3, b = 20
Yes
No
10c. h = 5, b = 12
Yes
No
10d. h = 5, b = 24
Yes
No
© Houghton Mifflin Harcourt Publishing Company
9.
Practice and Homework
Name
Lesson 10.3
Area of Triangles
COMMON CORE STANDARD—6.G.A.1
Solve real-world and mathematical problems
involving area, surface area, and volume.
Find the area.
1.
6 in.
15 in.
A 5 1_2 bh
2.
3.
A 5 1_2 3 15 3 6
0.6 m
A 5 45
Area 5 45 in.2
1.2 m
___
2
3
2 ft
1
2
4 ft
___
Find the unknown measurement for the triangle.
4. A = 0.225 mi2
5. A = 4.86 yd2
b = 0.6 mi
b=
h=
h = 1.8 yd
___
___
6. A = 63 m2
7. A = 2.5 km2
b=
b = 5 km
h = 12 m
h=
___
___
© Houghton Mifflin Harcourt Publishing Company
Problem
Problem Solving
Solving
8. Bayla draws a triangle with a base of 15 cm and a
height of 8.5 cm. If she colors the space inside the
triangle, what area does she color?
_______
10.
WRITE
9. Alicia is making a triangular sign for the school
play. The area of the sign is 558 in.2. The base
of the triangle is 36 in. What is the height of
the triangle?
_______
Math Describe how you would find how much grass
seed is needed to cover a triangular plot of land.
Chapter 10
549
Lesson Check (6.G.A.1, 6.EE.A.2c)
1. A triangular flag has an area of 187.5 square
inches. The base of the flag measures 25 inches.
How tall is the triangular flag?
2. A piece of stained glass in the shape of a right
triangle has sides measuring 8 centimeters,
15 centimeters, and 17 centimeters. What is the
area of the piece?
Spiral Review (6.EE.B.7, 6.EE.C.9, 6.G.A.1)
3. Tina bought a t-shirt and sandals. The total cost
was $41.50. The t-shirt cost $8.95. The equation
8.95 + c = 41.50 can be used to find the cost
c in dollars of the sandals. How much did the
sandals cost?
4. There are 37 paper clips in a box. Carmen
places more paper clips in the box. Which
equation models the total number of paper
clips p in the box after Carmen places n more
paper clips in the box?
5. Name another ordered pair that is on the graph
of the equation represented by the table.
6. Find the area of the triangle that divides the
parallelogram in half.
1
2
3
4
Total cost of ordering
lunch special ($), y
6
12
18
24
13 cm
9 cm
FOR MORE PRACTICE
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People in group, x
Lesson 10.4
Name
Explore Area of Trapezoids
Essential Question What is the relationship between the areas of
trapezoids and parallelograms?
Geometry—
6.G.A.1
MATHEMATICAL PRACTICES
MP4, MP7, MP8
connect A trapezoid is a quadrilateral with exactly one pair of
parallel sides. The parallel sides are the bases of the trapezoid. A
line segment drawn at a 90° angle to the two bases is the height
of the trapezoid. You can use what you know about the area of a
parallelogram to find the area of a trapezoid.
base 2
height
base 1
Hands
On
Investigate
Investigate
Materials ■ grid paper ■ ruler ■ scissors
A. Draw two copies of the trapezoid on grid paper.
B. Cut out the trapezoids.
3 units
4 units
6 units
C. Arrange the trapezoids to form a parallelogram, as shown.
Examine the parallelogram.
• How can you find the length of the base of the parallelogram?
• The base of the parallelogram is _ + _ = _ units.
• The height of the parallelogram is _ units.
• The area of the parallelogram is _ × _ = _ square units.
D. Examine the trapezoids.
© Houghton Mifflin Harcourt Publishing Company
• How does the area of one trapezoid relate to the area of
the parallelogram?
• Find the area of one trapezoid. Explain how you found the area.
Chapter 10
551
Draw Conclusions
MATHEMATICAL
PRACTICE
1.
Identify Relationships Explain how knowing how to find
the area of a parallelogram helped you find the area of the trapezoid.
7
2. Use your results from the Investigate to describe how you
can find the area of any trapezoid.
MATHEMATICAL
PRACTICE
Generalize Can you use the method you described above
to find the area of a trapezoid if two copies of the trapezoid can be
arranged to form a rectangle? Explain.
3.
8
Make
Make Connections
Connections
You can use the formula for the area of a rectangle to find the area of some
types of trapezoids.
5 cm
3 cm
9 cm
A. Trace and cut out two copies of the trapezoid.
B. Arrange the two trapezoids to form a rectangle. Examine the rectangle.
• The length of the rectangle is _ + _ = _ cm.
• The area of the rectangle is _ × _ = _ cm2.
C. Examine the trapezoids.
• How does the area of each trapezoid relate to the area of the rectangle?
• The area of the given trapezoid is 1_2 × _ = _ cm2.
552
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• The width of the rectangle is _ cm.
Name
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and Show
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MATH
M
BOARD
B
1. Trace and cut out two copies of the trapezoid. Arrange the
trapezoids to form a parallelogram. Find the areas of the
parallelogram and one trapezoid using square units.
Find the area of the trapezoid.
2.
3.
6 cm
5 cm
4.
9 in.
11 ft
8 ft
8 in.
10 cm
5 ft
3 in.
__ cm2
5.
16 cm
__ in.2
6.
__ ft2
7.
8 mm
3 21 in.
6.5 mm
14 cm
22 cm
5 41 in.
14 mm
8 21 in.
__ cm2
__ mm2
__ in.2
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OqnakdlRnkuhmf¤@ookhb`shnmr
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8.
MATHEMATICAL
PRACTICE
4
Describe a Method Explain one
way to find the height of a trapezoid if you know
the area of the trapezoid and the length of both
bases.
9.
DEEPER
A patio is in the shape of a trapezoid.
The length of the longer base is 18 feet. The
length of the shorter base is two feet less than
half the longer base. The height is 8 feet. What is
the area of the patio?
Chapter 10 • Lesson 4 553
MATHEMATICAL PRACTICES .0%&-t3&"40/tM",&4&/4&
SMARTER
What’s the Error?
10. Except for a small region near its
southeast corner, the state of Nevada is
shaped like a trapezoid. The map at the
right shows the approximate dimensions
of the trapezoid. Sabrina used the map
to estimate the area of Nevada.
Look at how Sabrina solved the problem.
Find her error.
300 mi
200 mi
480 mi
400 mi
Describe the error. Find the area of the
trapezoid to estimate the area of Nevada.
Two copies of the trapezoid can be put
together to form a rectangle.
length of rectangle:
200 + 480 = 680 mi
width of rectangle: 300 mi
A = lw
= 204,000
The area of Nevada is about 204,000
square miles.
11.
SMARTER
A photo was cut in half at an angle. What is the area of
one of the cut pieces?
3 in.
6 in.
7 in.
The area is _.
554
© Houghton Mifflin Harcourt Publishing Company • Image Credits: (t) ©Elvele Images Ltd/Alamy Images
= 680 × 300
Practice and Homework
Name
Lesson 10.4
Explore Area of Trapezoids
COMMON CORE STANDARD—6.G.A.1
Solve real-world and mathematical problems
involving area, surface area, and volume.
1. Trace and cut out two copies of the trapezoid. Arrange the
trapezoids to form a parallelogram. Find the areas of the
parallelogram and the trapezoids using square units.
parallelogram: 24 square units; trapezoids:
12 square units
Find the area of the trapezoid.
2.
3.
9 in.
4.
100 yd
24 yd
11.5 ft
8 ft
48 yd
7 in.
2 in.
4.5 ft
in.2
yd2
ft2
Problem
Problem Solving
Solving
© Houghton Mifflin Harcourt Publishing Company
5. A cake is made out of two identical trapezoids.
Each trapezoid has a height of 11 inches and
bases of 9 inches and 14 inches. What is the area
of one of the trapezoid pieces?
7.
WRITE
6. A sticker is in the shape of a trapezoid. The
height is 3 centimeters, and the bases are
2.5 centimeters and 5.5 centimeters. What
is the area of the sticker?
Math Find the area of a trapezoid that has bases
that are 15 inches and 20 inches and a height of 9 inches.
Chapter 10
555
Lesson Check (6.G.A.1)
1. What is the area of figure ABEG?
A
9 yd
B
2. Maggie colors a figure in the shape of a
trapezoid. The trapezoid is 6 inches tall. The
bases are 4.5 inches and 8 inches. What is the
area of the figure that Maggie colored?
C
7 yd
G F
E
15 yd
D
Spiral Review (6.EE.A.2c, 6.EE.B.7, 6.EE.C.9, 6.G.A.1)
4. Ginger makes pies and sells them for $14 each.
Write an equation that represents the situation,
if y represents the money that Ginger earns and
x represents the number of pies sold.
5. What is the equation for the graph shown
below?
6. Cesar made a rectangular banner that is 4 feet
by 3 feet. He wants to make a triangular banner
that has the same area as the other banner. The
triangular banner will have a base of 4 feet.
What should its height be?
y
6
5
4
3
2
1
-6 -5 -4 -3 -2 -1 0
-1
-2
-3
-4
-5
-6
© Houghton Mifflin Harcourt Publishing Company
3. Cassandra wants to solve the equation 30 = 2_5 p.
What operation should she perform to isolate
the variable?
x
1 2 3 4 5 6
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ALGEBRA
Name
Lesson 10.5
Area of Trapezoids
Essential Question How can you find the area of trapezoids?
Any parallelogram can be divided into two trapezoids with the
same shape and size. The bases of the trapezoids, b 1 and b 2, form
the base of the parallelogram. The area of each trapezoid is half
the area of the parallelogram. So, the area of a trapezoid is half the
product of its height and the sum of its bases.
Geometry—6.G.A.1
Also 6.EE.A.2c
MATHEMATICAL PRACTICES
MP1, MP3, MP7
b1
b2
h
Area of a Trapezoid
b2
1 (b + b )h
A = __
1
2
b1
b1
2
where b 1 and b 2 are the two bases and h is the height
h
b2
Unlock
Unlock the
the Problem
Problem
Mr. Desmond has tables in his office with tops
shaped like trapezoids. The diagram shows the
dimensions of each tabletop. What is the area
of each tabletop?
• How can you identify the bases?
1.6 m
• How can you identify the height?
0.6 m
0.9 m
© Houghton Mifflin Harcourt Publishing Company
Find the area of the trapezoid.
Write the formula.
__(b + b )h
A=1
1
2
Substitute 1.6 for b 1, 0.9 for b 2,
and 0.6 for h.
__ × ( _ + _ ) × _
A=1
Add within the parentheses.
__ × _ × 0.6
A=1
Multiply.
__ × _ = _
A=1
2
2
2
2
So, the area of each tabletop is _ m2.
Math
Talk
MATHEMATICAL PRACTICES 1
Describe Relationships Describe the relationship
between the area of a trapezoid and the area
of a parallelogram with the same height and a
base equal to the sum of the trapezoid’s bases.
Chapter 10
557
Example 1
Find the area of the trapezoid.
Write the formula.
__ (b + b )h
A=1
1
2
Substitute 4.6 for b1,
9.4 for b2, and 3.5 for h.
A = __
× (_ + _ ) × 3.5
2
Add.
A = __
× _ × 3.5
2
Multiply.
A = _ × 3.5 = _
2
4.6 cm
1
3.5 cm
1
9.4 cm
So, the area of the trapezoid is __ cm2.
Example 2
The area of the trapezoid is 702 in.2 Find the
height of the trapezoid.
__ (b + b )h
A=1
1
2
20 in.
2
Substitute 702 for A, 20 for b 1, and 34 for b 2.
__ × (20 + _ ) × h
702 = 1
Add within the parentheses.
__ × _ × h
702 = 1
__ and 54.
Multiply 1
702 = _ × h
2
Use the Division Property of Equality.
2
2
702 =
______
_=h
Simplify.
34 in.
×h
________
Math
Talk
So, the height of the trapezoid is _ in.
•
558
MATHEMATICAL
PRACTICE
1
?
Analyze Relationships Explain why the formula for the area of a
trapezoid contains the expression b 1 + b 2.
MATHEMATICAL PRACTICES 6
Explain How can you find
the height of a trapezoid if
you know the area and the
lengths of both bases?
© Houghton Mifflin Harcourt Publishing Company
Write the formula.
Name
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MATH
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B
1. Find the area of the trapezoid.
3 cm
A = 21_(b 1 + b 2)h
4 cm
A = 21_ × ( _ + _ ) × 4
6 cm
A = 21_ × _ × 4
A = _ cm2
2. The area of the
trapezoid is 45 ft2.
Find the height of
the trapezoid.
8 ft
h
3. Find the area
of the trapezoid.
43 mm
18 mm
17 mm
10 ft
h = __
A = __
Math
Talk
On
On Your
Your Own
Own
MATHEMATICAL PRACTICES 1
Analyze Two trapezoids
have the same bases and the
same height. Are the areas
equal? Must the trapezoids
have the same shape?
Find the area of the trapezoid.
21 in.
4.
5.
2.8 m
4.2 m
14 in.
9.2 m
17 in.
© Houghton Mifflin Harcourt Publishing Company
A = __
A = __
Find the height of the trapezoid.
12.5 in.
6.
2
Area 5 500 in.
7.
3.2 cm
h
Area 5 99 cm2
h
10 cm
27.5 in.
h = __
h = __
Chapter 10 • Lesson 5 559
MATHEMATICAL PRACTICES
COMMUNICA5&t1&34E7&3&tCONSTRUCT ARGUMENTS
Home Plate
17 in.
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OqnakdlRnkuhmf¤@ookhb`shnmr
Use the diagram for 8–9.
8.
DEEPER
A baseball home plate can be divided
into two trapezoids with the dimensions shown in the
drawing. Find the area of home plate.
_______
8.5 in.
8.5 in.
17 in.
12 in.
12 in.
9. Suppose you cut home plate along the dotted line and
rearranged the pieces to form a rectangle. What would
the dimensions and the area of the rectangle be?
dimensions: ___
area: __
10.
SMARTER
A pattern used for tile floors is
shown. A side of the inner square measures 10 cm,
and a side of the outer square measures 30 cm.
What is the area of one of the yellow trapezoid tiles?
WRITE
Math
Show Your Work
11.
MATHEMATICAL
PRACTICE
3 Verify the Reasoning of Others A trapezoid
has a height of 12 cm and bases with lengths of 14 cm
and 10 cm. Tina says the area of the trapezoid is 288 cm2. Find her
error, and correct the error.
__________
12.
SMARTER
Which expression can be used to find the
area of the trapezoid? Mark all that apply.
1.5 ft
4 ft
3.5 ft
560
A
1 × (4 + 1.5) × 3.5
_
2
B
1 × (1.5 + 3.5) × 4
_
2
C
1 × (4 + 3.5) × 1.5
_
2
D
1 × (5) × 4
_
2
© Houghton Mifflin Harcourt Publishing Company • Image Credits: (t) ©Chuck Franklin/Alamy Images
_______
Practice and Homework
Name
Lesson 10.5
Area of Trapezoids
COMMON CORE STANDARD—6.G.A.1
Solve real-world and mathematical problems
involving area, surface area, and volume.
Find the area of the trapezoid.
1. A = 1_2 (b1 + b2)h
A = 1_2 × ( 11 + 17 ) × 18
11 cm
2.
5.5 ft
5 ft
18 cm
A = 1_2 × 28 × 18
A=
252
3.
cm2
6.5 ft
17 cm
0.2 cm
0.2 cm
A = ___
4.
10 in.
2 1 in.
2
5 in.
0.6 cm
A = ___
A = ___
Problem
Problem Solving
Solving
5. Sonia makes a wooden frame around a square
picture. The frame is made of 4 congruent
trapezoids. The shorter base is 9 in., the longer
base is 12 in., and the height is 1.5 in. What is
the area of the picture frame?
© Houghton Mifflin Harcourt Publishing Company
_____
7.
WRITE
6. Bryan cuts a piece of cardboard in the shape
of a trapezoid. The area of the cutout is
43.5 square centimeters. If the bases are
6 centimeters and 8.5 centimeters long,
what is the height of the trapezoid?
_____
Math Use the formula for the area of a trapezoid
to find the height of a trapezoid with bases 8 inches and 6 inches
and an area of 112 square inches.
Chapter 10
561
Lesson Check (6.G.A.1, 6.EE.A.2c)
1. Dominic is building a bench with a seat in the
shape of a trapezoid. One base is 5 feet. The
other base is 4 feet. The perpendicular distance
between the bases is 2.5 feet. What is the area
of the seat?
2. Molly is making a sign in the shape of a
trapezoid. One base is 18 inches and the other
is 30 inches. How high must she make the sign
so its area is 504 square inches?
Spiral Review (6.NS.C.6c, 6.RP.A.3d, 6.EE.A.2c)
3
3___
10
3.1
1
3__
4
5. To find the cost for a group to enter the museum,
the ticket seller uses the expression 8a + 3c
in which a represents the number of adults
and c represents the number of children in the
group. How much should she charge a group of
3 adults and 5 children?
4. Write these lengths in order from least
to greatest.
2 yards
5.5 feet
70 inches
6. Brian frosted a cake top shaped like a
parallelogram with a base of 13 inches and a
height of 9 inches. Nancy frosted a triangular
cake top with a base of 15 inches and a height
of 12 inches. Which cake’s top had the greater
area? How much greater was it?
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3. Write these numbers in order from least
to greatest.
Name
Mid-Chapter Checkpoint
Personal Math Trainer
Online Assessment
and Intervention
Vocabulary
Vocabulary
Vocabulary
area
Choose the best term from the box to complete the sentence.
congruent
1. A ___ is a quadrilateral with two pairs of
parallel sides. (p. 533)
parallelogram
trapezoid
2. The number of square units needed to cover a surface without
any gaps or overlaps is called the ___. (p. 533)
3. Figures with the same size and shape are ___. (p. 539)
Concepts
Concepts and
and Skills
Skills
Find the area. (6.G.A.1, 6.EE.A.2c)
4.
5.
3.4 cm
6 21 in.
5.7 cm
6 21 in.
________
6.
7.
8.2 mm
© Houghton Mifflin Harcourt Publishing Company
________
14 mm
18 cm
9 cm
13 cm
________
________
8. A parallelogram has an area of 276 square meters
and a base measuring 12 meters. What is the
height of the parallelogram?
9. The base of a triangle measures 8 inches and the
area is 136 square inches. What is the height of
the triangle?
________
________
Chapter 10
563
10. The height of a parallelogram is 3 times the base. The base measures
4.5 cm. What is the area of the parallelogram? (6.G.A.1)
____________________
11. A triangular window pane has a base of 30 inches and a height of
24 inches. What is the area of the window pane? (6.G.A.1)
____________________
12. The courtyard behind Jennie’s house is shaped like a trapezoid. The
bases measure 8 meters and 11 meters. The height of the trapezoid is
12 meters. What is the area of the courtyard? (6.G.A.1)
____________________
13.
DEEPER
Rugs sell for $8 per square foot. Beth bought a 9-foot-long
rectangular rug for $432. How wide was the rug? (6.G.A.1, 6.EE.A.2c)
14. A square painting has a side length of 18 inches. What is the area of the
painting? (6.G.A.1, 6.EE.A.2c)
____________________
564
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____________________
Lesson 10.6
Name
Area of Regular Polygons
Geometry—6.G.A.1
Also 6.EE.A.2C
MATHEMATICAL PRACTICES
MP7, MP8
Essential Question How can you find the area of regular polygons?
Unlock
Unlock the
the Problem
Problem
Emory is making a patch for his soccer ball. The patch he is
using is a regular polygon. A regular polygon is a polygon
in which all sides have the same length and all angles have
the same measure. Emory needs to find the area of a piece
of material shaped like a regular pentagon.
Activity
You can find the area of a regular polygon by dividing the polygon into
congruent triangles.
• Draw line segments from each vertex to the center of the pentagon to
divide it into five congruent triangles.
• You can find the area of one of the triangles if you know the side length of
the polygon and the height of the triangle.
© Houghton Mifflin Harcourt Publishing Company • Image Credits: (t) ©Houghton Mifflin Harcourt
Math
Talk
14 cm
MATHEMATICAL PRACTICES 6
Explain How do you
determine the number of
congruent triangles a regular
polygon should be divided
into in order to find the area?
20 cm
• Find the area of one triangle.
Write the formula.
Substitute 20 for b and 14 for h.
Simplify.
A = 1__ bh
2
A = 1__ × _ × _
2
A = _ cm2
• Find the area of the regular polygon by multiplying the number of
triangles by the area of one triangle.
A = _ × _ = _ cm2
So, the area of the pentagon-shaped piece is __ .
Chapter 10
565
Example
Find the area of the regular polygon.
STEP 1 Draw line segments from each vertex to the
center of the hexagon.
Into how many congruent triangles did you divide the figure? _
STEP 2 Find the area of one triangle.
Write the formula.
1 bh
A = __
2
Substitute 4.2 for b and 3.6 for h.
1×_×_
A = __
2
Simplify.
A = _ m2
3.6 m
4.2 m
STEP 3 Find the area of the hexagon.
A = _ × _ = __ m2
So, the area of the hexagon is __ m2
1.
MATHEMATICAL
PRACTICE
8 Use Repeated Reasoning Into how many congruent
2.
566
SMARTER
In an irregular polygon, the sides do not all have the
same length and the angles do not all have the same measure. Could
you find the area of an irregular polygon using the method you used in
this lesson? Explain your reasoning.
© Houghton Mifflin Harcourt Publishing Company
triangles can you divide a regular decagon by drawing line segments
from each vertex to the center of the decagon? Explain.
Name
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and Show
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MATH
M
BOARD
B
Find the area of the regular polygon.
1. number of congruent triangles inside the figure: _
6 cm
area of each triangle: _1 × _ × _ = _ cm2
2
area of octagon: _ × _ = _ cm2
2.
5 cm
3.
4m
12 mm
8 mm
6m
Math
Talk
On
On Your
Your Own
Own
MATHEMATICAL PRACTICES 1
Describe the information
you must have about a
regular polygon in order to
find its area.
Find the area of the regular polygon.
4.
5.
7 cm
43 in.
© Houghton Mifflin Harcourt Publishing Company
8 cm
6.
MATHEMATICAL
PRACTICE
28 in.
6 Explain A regular pentagon is divided into congruent triangles
by drawing a line segment from each vertex to the center. Each triangle has
an area of 24 cm2. Explain how to find the area of the pentagon.
Chapter 10 • Lesson 6 567
7.
SMARTER
Name the polygon and find its area.
Show your work.
4.8 in.
4 in.
Regular Polygons in Nature
Regular polygons are common in nature. One of the bestknown examples of regular polygons in nature is the small
hexagonal cells in honeycombs constructed by honeybees.
The cells are where bee larvae grow. Honeybees store honey
and pollen in the hexagonal cells. Scientists can measure the
health of a bee population by the size of the cells.
WRITE
Math t Show Your Work
The figure shows a typical 10-cell line of worker bee cells.
What is the width of each cell?
9.
SMARTER
The diagram shows one honeycomb cell.
Use your answer to Exercise 8 to find h, the height of the
triangle. Then find the area of the hexagonal cell.
0.3 cm
0
10.
568
h
DEEPER
A rectangular honeycomb measures
35.1 cm by 32.4 cm. Approximately how many cells
does it contain?
Honeycomb
© Houghton Mifflin Harcourt Publishing Company • Image Credits: (c) ©B.A.E. Inc./Alamy
8. Cells in a honeycomb vary in width. To find the average
width of a cell, scientists measure the combined width of
10 cells, and then divide by 10.
Practice and Homework
Name
Lesson 10.6
Area of Regular Polygons
COMMON CORE STANDARD—6.G.A.1
Solve real-world and mathematical problems
involving area, surface area, and volume.
Find the area of the regular polygon.
1.
6
number of congruent triangles inside the figure:
7 mm
8
area of each triangle: 1_2 ×
area of hexagon:
×
7
=
28
mm2
168 mm2
8 mm
2.
3.
4 in.
6.2 yd
3.3 in.
9 yd
___
___
Problem
Problem Solving
Solving
© Houghton Mifflin Harcourt Publishing Company
4. Stu is making a stained glass window in the
shape of a regular pentagon. The pentagon can
be divided into congruent triangles, each with a
base of 8.7 inches and a height of 6 inches. What
is the area of the window?
_____
6.
WRITE
5. A dinner platter is in the shape of a regular
decagon. The platter has an area of 161 square
inches and a side length of 4.6 inches. What is
the area of each triangle? What is the height of
each triangle?
_____
Math A square has sides that measure 6 inches.
Explain how to use the method in this lesson to find the area
of the square.
Chapter 10
569
Lesson Check (6.G.A.1, 6.EE.A.2c)
1. What is the area of the regular hexagon?
2. A regular 7-sided figure is divided into
7 congruent triangles, each with a base of
12 inches and a height of 12.5 inches. What is
the area of the 7-sided figure?
3m
32m
5
Spiral Review (6.EE.A.2c, 6.EE.B.5, 6.EE.C.9, 6.G.A.1)
2+b≥2
3b ≤ 14
8 – b ≤ 15
b–3≥5
5. What is the area of triangle ABC?
A
B
10 ft
D
6 ft
4. Each song that Tara downloads costs $1.25.
She graphs the relationship that gives the cost
y in dollars of downloading x songs. Name one
ordered pair that is a point on the graph of the
relationship.
6. Marcia cut a trapezoid out of a large piece of
felt. The trapezoid has a height of 9 cm and
bases of 6 cm and 11 cm. What is the area of
Marcia’s felt trapezoid?
C
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3. Which inequalities have b = 4 as one of
its solutions?
Lesson 10.7
Name
Composite Figures
Geometry—6.G.A.1
Also 6.EE.A.2c
MATHEMATICAL PRACTICES
MP1, MP2, MP5
Essential Question How can you find the area of composite figures?
A composite figure is made up of two or more simpler
figures, such as triangles and quadrilaterals.
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the Problem
Problem
The new entryway to the fun house at Happy World
Amusement Park is made from the shapes shown in
the diagram. It will be painted bright green. Juanita
needs to know the area of the entryway to determine
how much paint to buy. What is the area of the
entryway?
Find the area of the entryway.
4 ft
4 ft
10 ft
4 ft
4 ft
STEP 1 Find the area of the rectangles.
Write the formula.
A = lw
Substitute the values for l and w and evaluate.
A = 10 × _ = _
Find the total area of two rectangles.
2 × _ = _ ft2
10 ft
4 ft
STEP 2 Find the area of the triangles.
Write the formula.
__bh
A=1
Substitute the values for b and h and evaluate.
__ × 4 × _ = _
A=1
Find the total area of two triangles.
2 × _ = _ ft2
4 ft
2
4 ft
2
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STEP 3 Find the area of the square.
Write the formula.
A = s2
Substitute the value for s.
A = (_) = _ ft2
4 ft
2
4 ft
STEP 4 Find the total area of the composite figure.
Add the areas.
A = 80 ft2 + _ ft2 + _ ft2 = _ ft2
So, Juanita needs to buy enough paint to cover _ ft2.
Math
Talk
MATHEMATICAL PRACTICES 1
Describe other ways
you could divide up the
composite figure.
Chapter 10
571
Example 1
Find the area of the composite figure shown.
9 cm
6 cm
STEP 1 Find the area of the triangle, the square, and
the trapezoid.
area of triangle
1
__ × 16 × _
A = __
bh = 1
2
2
20 cm
12 cm
= _ cm2
area of square
16 cm
2
A = s2 = ( _ )
12 cm
= __ cm2
area of trapezoid
__(b + b )h = 1
__ × ( _ + _ ) × _
A=1
1
2
2
2
__ × _ × 6
=1
2
= _ cm2
STEP 2 Find the total area of the figure.
total area
A = _ cm2 + _ cm2 + _ cm2
= _ cm2
So, the area of the figure is _ cm2.
Example 2
Find the area of the shaded region.
STEP 1 Find the area of the rectangle and the square.
A = lw = _ × _
A = _ in.
area of square
1 ft
A = s2 = ( _ )2
STEP 2 Subtract the area of the square from the area of the rectangle.
A = _ in.2 − _ in.2
A = _ in.2
So, the area of the shaded region is _ in.2
572
6 in.
2
A = _ in.2
area of shaded
region
3 in.
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area of rectangle
(1 ft = 12 in.)
3 in.
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1. Find the area of the figure.
area of one rectangle
MATH
M
BOARD
B
A = lw
A = _ × _ = _ ft2
area of two rectangles
length of base of triangle
3 ft
4 ft
10 ft
A = 2 × _ = _ ft2
5 ft
b = _ ft + _ ft + _ ft
5 ft
= _ ft
A = 1_ bh
area of triangle
2
A = 1_ × _ × _ = _ ft2
2
area of composite figure
A = _ ft2 + _ ft2 = _ ft2
Find the area of the figure.
2.
8 mm
3.
13 m
5m
4 mm
13 m
6m
5m
6m
7m
11 mm
8.2 mm
12 m
11 mm
______
______
Math
Talk
MATHEMATICAL PRACTICES 6
Explain how to find the area
of a composite figure.
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On
On Your
Your Own
Own
4. Find the area
of the figure.
5.
8 in.
MATHEMATICAL
PRACTICE
6
Attend to Precision Find
the area of the shaded region.
12.75 m
10 in.
8 in.
6 in.
16 in.
______
2.5 m
8.8 m
2.5 m
4.25 m
______
Chapter 10 • Lesson 7 573
MATHEMATICAL PRACTICES
ANALYZEt-00,'034536$563&t13&$*4*0/
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Problem
6.
DEEPER
Marco made the banner
shown at the right. What is the area of the
yellow shape?
15 in.
24 in.
a. Explain how you could find the area of the
yellow shape if you knew the areas of the
green and red shapes and the area of the
entire banner.
15 in.
48 in.
c. What is the area of the red shape? What is
the area of each green shape?
d. What equation can you write to find A, the
area of the yellow shape?
b. What is the area of the entire banner? Explain
how you found it.
e. What is the area of the yellow shape?
WRIT
WR
WRITE
ITE
E
Math
M
ath tt Sh
Show
S
o Y
ow
You
Your
ourr Work
ou
Wo
ork
7. There are 6 rectangular flower gardens each measuring 18 feet by
15 feet in a rectangular city park measuring 80 feet by 150 feet. How
many square feet of the park are not used for flower gardens?
Personal Math Trainer
Sabrina wants to replace the carpet in a few rooms of
her house. Select the expression she can use to find the total area of the
floor that will be covered. Mark all that apply.
A
8 × 22 + 130 + 1_ × 10 × 9
B
2
1
_
18 × 22 − × 10 × 9
2
C
18 × 13 + 1_ × 10 × 9
D
1_ × (18 + 8) × 22
2
8 ft
2
9 ft
10 ft
13 ft
574
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8.
SMARTER
Practice and Homework
Name
Lesson 10.7
Composite Figures
COMMON CORE STANDARD—6.G.A.1
Solve real-world and mathematical problems
involving area, surface area, and volume.
Find the area of the figure.
1.
3 cm
area of square
3 cm
area of triangle
A=s×s
3
=
A = 1_2 bh
= 1_2 ×
area of trapezoid
5 cm
2
2.
2 cm
6 ft
area of composite
figure
10 ft
×
=
8
9
=
cm2
8
cm2
A = 1_2 (b1 + b2)h
= 1_2 × (
5 cm
3
×
A=
9
=
37
+
3
)×
cm2 +
8
cm2 +
5
5
=
20
20
cm2
cm2
cm2
3.
7 yd
4 yd
8 yd
9 ft
12 ft
___
11 yd
14 yd
___
Problem
Problem Solving
Solving
4. Janelle is making a poster. She cuts a triangle out
of poster board. What is the area of the poster
board that she has left?
5. Michael wants to place grass on the sides of his
lap pool. Find the area of the shaded regions that
he wants to cover with grass.
25 yd
9 in.
10 in.
© Houghton Mifflin Harcourt Publishing Company
12 yd
20 in.
_____
6.
WRITE
3 yd
12 yd
_____
Math Describe one or more situations in which
you need to subtract to find the area of a composite figure.
Chapter 10
575
Lesson Check (6.G.A.1, 6.EE.A.2c)
1. What is the area of the composite figure?
2. What is the area of the shaded region?
7m
10 m
7m
13 in.
9m
34 m
12 in.
15 in.
11 in.
21 in.
3. In Maritza’s family, everyone’s height is
greater than 60 inches. Write an inequality
that represents the height h, in inches, of any
member of Maritza’s family.
4. The linear equation y = 2x represents the cost y
for x pounds of apples. Which ordered pair lies
on the graph of the equation?
5. Two congruent triangles fit together to form
a parallelogram with a base of 14 inches and
a height of 10 inches. What is the area of each
triangle?
6. A regular hexagon has sides measuring
7 inches. If the hexagon is divided into 6
congruent triangles, each has a height of
about 6 inches. What is the approximate
area of the hexagon?
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Spiral Review (6.EE.A.2c, 6.EE.B.8, 6.EE.C.9, 6.G.A.1)
PROBLEM SOLVING
Name
Lesson 10.8
Problem Solving • Changing Dimensions
Essential Question How can you use the strategy find a pattern to show
how changing dimensions affects area?
Geometry—
6.G.A.1
MATHEMATICAL PRACTICES
MP1, MP3, MP8
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the Problem
Problem
Jason has created a 3-in. by 4-in. rectangular design to be made
into mouse pads. To manufacture the pads, the dimensions will be
multiplied by 2 or 3. How will the area of the design be affected?
3 in.
4 in.
Use the graphic organizer to help you solve the problem.
Read the Problem
What do I need to find?
I need to find how
_____
will be affected by changing
the ___.
What information do I
need to use?
How will I use the
information?
I need to use __
of the original design and
I can draw a sketch of each
rectangle and calculate
____
___ of each.
____
____.
Then I can look for
___ in my results.
Solve the Problem
Sketch
6 in.
Dimensions
Multiplier
Area
3 in. by 4 in.
1
A = 3 × 4 = 12 in.2
6 in. by 8 in.
2
A = _ × _ = _ in.2
© Houghton Mifflin Harcourt Publishing Company
8 in.
9 in.
12 in.
So, when the dimensions are multiplied by 2, the area is
multiplied by _. When the dimensions are multiplied
by 3, the area is multiplied by _.
Math
Talk
MATHEMATICAL PRACTICES 2
Reasoning What would
happen to the area of a
rectangle if the dimensions
were multiplied by 4?
Chapter 10
577
Try Another Problem
6 cm
A stained-glass designer is reducing the dimensions of an earlier design.
The dimensions of the triangle shown will be multiplied by 1_2 or 1_4 . How
will the area of the design be affected? Use the graphic organizer to help
you solve the problem.
16 cm
Read the Problem
What do I need to find?
What information do I
need to use?
How will I use the
information?
Solve the Problem
Sketch
Multiplier
1
Area
A = 1_ × 16 × _ = _ cm2
2
3 cm
1
_
2
So, when the dimensions are multiplied by 1_2 , the area is multiplied by
__. When the dimensions are multiplied by __, the area is
multiplied by __.
578
Math
Talk
MATHEMATICAL PRACTICES 8
Generalize What happens
to the area of a triangle
when the dimensions are
multiplied by a number n?
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8 cm
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B
1. The dimensions of a 2-cm by 6-cm rectangle are
multiplied by 5. How is the area of the rectangle
affected?
First, find the original area:
√
Plan your solution by deciding on the
steps you will use.
√
Find the original area and the new
area, and then compare the two.
√
Look for patterns in your results.
WRITE
Math t Show Your Work
Next, find the new area:
So, the area is multiplied by _.
2.
What if the dimensions of
the original rectangle in Exercise 1 had been
multiplied by 1_2 ? How would the area have
been affected?
SMARTER
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3. Evan bought two square rugs. The larger one
measured 12 ft square. The smaller one had an
area equal to 1_4 the area of the larger one. What
fraction of the side lengths of the larger rug were
the side lengths of the smaller one?
4.
DEEPER
On Silver Island, a palm tree, a giant
rock, and a buried treasure form a triangle with
a base of 100 yd and a height of 50 yd. On a map
of the island, the three landmarks form a triangle
with a base of 2 ft and a height of 1 ft. How many
times the area of the triangle on the map is the
area of the actual triangle?
Chapter 10 • Lesson 8 579
MATHEMATICAL PRACTICES .0%&-t3&"40/tM",&4&/4&
On
On Your
Your Own
Own
5. A square game board is divided into smaller
squares, each with sides one-ninth the length of the
sides of the board. Into how many squares is the
game board divided?
6.
7.
WRITE
Math
Show Your Work
SMARTER
Flynn County is a rectangle
measuring 9 mi by 12 mi. Gibson County is a
rectangle with an area 6 times the area of Flynn
County and a width of 16 mi. What is the length
of Gibson County?
MATHEMATICAL
PRACTICE
4 Use Diagrams Carmen left her house
and drove 10 mi north, 15 mi east, 13 mi south,
11 mi west, and 3 mi north. How far was she from
home?
8.
DEEPER
Bernie drove from his house to his
cousin’s house in 6 hours at an average rate of
52 mi per hr. He drove home at an average rate of
60 mi per hr. How long did it take him to
drive home?
Personal Math Trainer
SMARTER
Sophia wants to enlarge a 5-inch
by 7-inch rectangular photo by multiplying the
dimensions by 3.
Find the area of the original photo and the enlarged
photo. Then explain how the area of the original photo is
affected.
580
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9.
Practice and Homework
Name
Lesson 10.8
Problem Solving • Changing Dimensions
COMMON CORE STANDARD—6.G.A.1
Solve real-world and mathematical problems
involving area, surface area, and volume.
Read each problem and solve.
1. The dimensions of a 5-in. by 3-in. rectangle are multiplied by 6.
How is the area affected?
I = 6 3 5 5 30 in.
new dimensions:
W = 6 3 3 5 18 in.
540
new area = ___
_________
5 36
original area
15
original area: A = 5 3 3 5 15 in.2
new area: A = 30 3 18 5 540 in.2
The area was multiplied by
36 .
2. The dimensions of a 7-cm by 2-cm rectangle are
multiplied by 3. How is the area affected?
multiplied by ___
3. The dimensions of a 3-ft by 6-ft rectangle are
multiplied by 1_3 . How is the area affected?
multiplied by ___
4. The dimensions of a triangle with base 10 in. and
height 4.8 in. are multiplied by 4. How is the area affected?
multiplied by ___
5. The dimensions of a 1-yd by 9-yd rectangle are
multiplied by 5. How is the area affected?
multiplied by ___
6. The dimensions of a 4-in. square are multiplied
by 3. How is the area affected?
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multiplied by ___
7. The dimensions of a triangle are multiplied by 1_4 . The area
of the smaller triangle can be found by multiplying the area
of the original triangle by what number?
___
8.
WRITE
Math Write and solve a word problem that
involves changing the dimensions of a figure and finding
its area.
Chapter 10
581
Lesson Check (6.G.A.1)
1. The dimensions of Rectangle A are 6 times the
dimensions of Rectangle B. How do the areas of
the rectangles compare?
2. A model of a triangular piece of jewelry has an
area that is 1_4 the area of the jewelry. How do the
dimensions of the triangles compare?
Spiral Review (6.RP.A.3c, 6.EE.A.2c, 6.EE.B.8, 6.G.A.1)
4. Graph y > 3 on a number line.
0
5. The parallelogram below is made from two
congruent trapezoids. What is the area of the
shaded trapezoid?
50 mm
25 mm
25 mm
50 mm
1
2
3
4
5
6
7
8
6. A rectangle has a length of 24 inches and a
width of 36 inches. A square with side length
5 inches is cut from the middle and removed.
What is the area of the figure that remains?
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3. Gina made a rectangular quilt that was 5 feet
wide and 6 feet long. She used yellow fabric for
30% of the quilt. What was the area of the yellow
fabric?
35 mm
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Lesson 10.9
Name
Figures on the Coordinate Plane
Essential Question How can you plot polygons on a coordinate plane
and find their side lengths?
Geometry—6.G.A.3
Also 6.NS.C.8
MATHEMATICAL PRACTICES
MP4, MP6, MP7
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the Problem
Problem
The world’s largest book is a collection of photographs
from the Asian nation of Bhutan. A book collector models the
rectangular shape of the open book on a coordinate plane.
Each unit of the coordinate plane represents one foot. The
book collector plots the vertices of the rectangle at A(9, 3),
B(2, 3), C(2, 8), and D(9, 8). What are the dimensions of the
open book?
• What two dimensions do you need
to find?
Plot the vertices and find the dimensions of the rectangle.
10
STEP 1 Complete the rectangle on the coordinate plane.
STEP 2 Find the length of the rectangle.
8
y-axis
Plot points C(2, 8) and D(9, 8).
Connect the points to form a rectangle.
Find the distance between points A(9, 3) and B(2, 3).
The y-coordinates are the same, so the points lie on a __ line.
6
4
0
Think of the horizontal line passing through A and B as a number line.
© Houghton Mifflin Harcourt Publishing Company• Image Credits: (b) ©Toru Ysmanaka/AFP/Getty Images
B
2
A
2
4
6
x-axis
8
10
Horizontal distance of A from 0: ⎢9 ⎢ = _ ft
Horizontal distance of B from 0: ⎢2 ⎢ = _ ft
Subtract to find the distance from A to B: _ − _ = _ ft.
STEP 3 Find the width of the rectangle.
Find the distance between points C (2, 8) and B(2, 3).
The x-coordinates are the same, so the points lie on a _ line.
Think of the vertical line passing through C and B as a number line.
Vertical distance of C from 0: ⎢8 ⎢ = _ ft
Vertical distance of B from 0: ⎢3 ⎢ = _ ft
Subtract to find the distance from C to B: _ − _ = _ ft.
So, the dimensions of the open book are _ ft by _ ft.
Math
Talk
MATHEMATICAL PRACTICES 6
Explain How do you know
whether to add or subtract
the absolute values to find
the distance between the
vertices of the rectangle?
Chapter 10
583
connect You can use properties of quadrilaterals to help you
find unknown vertices. The properties can also help you graph
quadrilaterals on the coordinate plane.
Example
Find the unknown vertex, and then graph.
Three vertices of parallelogram PQRS are P(4, 2), Q(3, −3), and
R( −3, −3). Give the coordinates of vertex S and graph the
parallelogram.
The name of a polygon, such as
parallelogram PQRS, gives the
vertices in order as you move
around the polygon.
STEP 1
Plot the given points on the coordinate plane.
STEP 2
The opposite sides of a parallelogram are __.
They have the same ___.
___
Since the length of side RQ is _ units, the length of
side ___ must also be _ units.
STEP 3
Start at point P. Move horizontally _ units to the
___ to find the location of the remaining
vertex, S. Plot a point at this location.
y
5
4
3
2
1
-5 -4 -3 -2 -1 0
-1
-2
-3
-4
-5
x
1 2 3 4 5
STEP 4
Draw the parallelogram. Check that opposite sides are parallel and
congruent.
So, the coordinates of the vertex S are __.
1.
MATHEMATICAL
PRACTICE
6 Attend to Precision Explain why vertex S must be to the
2. Describe how you could find the area of parallelogram PQRS
in square units.
584
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left of vertex P rather than to the right of vertex P.
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B
1. The vertices of triangle ABC are A( −1, 3), B( −4, −2),
and C(2,___−2). Graph the triangle and find the length
of side BC .
5
4
3
2
1
Horizontal distance of B from 0: ⎢ −4⎢ = _ units
y
-5 -4 -3 -2 -1 0
-1
-2
-3
-4
-5
Horizontal distance of C from 0: ⎢2 ⎢ = _ units
The points are in different quadrants, so add to find the
distance from B to C: _ + _ = _ units.
x
1 2 3 4 5
Give the coordinates of the unknown vertex of rectangle JKLM, and graph.
2.
3.
y
y
5
4
3
2
1
10
J
8
6
2
0
K
2
4
X
6
8
10
______
x
-5 -4 -3 -2 -1 0
-1
M
-2
-3
-4
-5
4
L
K
1
2 3 4
L
5
______
On
On Your
Your Own
Own
4. Give the coordinates of the unknown vertex of
rectangle PQRS, and graph.
y
5. The vertices of pentagon PQRST are P(9, 7),
Q(9, 3), R(3, 3), S(3, 7), and T(6, 9). Graph
___ the
pentagon and find the length of side PQ.
© Houghton Mifflin Harcourt Publishing Company
10
8
y
Q
10
R
8
6
6
4
4
2
0
P
2
X
4
6
8
10
______
2
0
2
4
6
8
10
X
______
Chapter 10 • Lesson 9 585
MATHEMATICAL PRACTICES
COMMUNICA5&t1&34E7&3&tCONSTRUCT ARGUMENTS
OqnakdlRnkuhmf¤@ookhb`shnmr
OqnakdlRnkuhmf¤@ookhb`shnmr
The map shows the location of some city landmarks.
Use the map for 6–7.
6.
DEEPER
A city planner wants to locate a park
where two new roads meet. One of the new roads
will go to the mall and be parallel to Lincoln Street
which is shown in red. The other new road will go
to City Hall and be parallel to Elm Street which is
also shown in red. Give the coordinates for the
location of the park.
7. Each unit of the coordinate plane represents
2 miles. How far will the park be from City Hall?
y
9.
10.
SMARTER
___
PQ is one side of right triangle PQR.
∠P is the right angle, and the length
In the triangle,
___
of side PR is 3 units. Give all the possible coordinates
for vertex R.
MATHEMATICAL
PRACTICE
-5 -4 -3 -2 -1 0
-1
-2
-3
-4
-5
6
Use Math Vocabulary Quadrilateral
WXYZ has vertices with coordinates W( −4, 0),
X( −2, 3), Y(2, 3), and Z(2, 0). Classify the
quadrilateral using the most exact name possible
and explain your answer.
SMARTER
Kareem is drawing parallelogram
ABCD on the coordinate plane.
Find and label the coordinates of the fourth vertex,
D, of the parallelogram. Draw the parallelogram.
What is the length of side CD? How do you know?
586
P
B
5
4
3
2
1
-5 -4 -3 -2 -1 0
-1
C
-2
-3
-4
-5
Q
x
1 2 3 4 5
y
A
x
1 2 3 4 5
© Houghton Mifflin Harcourt Publishing Company
8.
5
4
3
2
1
Practice and Homework
Name
Lesson 10.9
Figures on the Coordinate Plane
COMMON CORE STANDARD—6.G.A.3
Solve real-world and mathematical problems
involving area, surface area, and volume.
1. The vertices of triangle DEF are D(−2, 3), E(3, −2),
2, −2). Graph the triangle, and find the length
and F(−___
of side DF.
3
Vertical distance of D from 0: | 3| =
Vertical distance of F from 0: | −2| =
y
D
units
2
-5 -4 -3 -2 -1 0
-1
-2
F
-3
-4
-5
units
The points are in different quadrants, so add to find the
distance from D to F:
3
2
+
5
4
3
2
1
=
5
units.
x
1 2 3 4 5
E
___
Graph the figure and find the length of side BC .
2. A(1, 4), B(1, −2), C(−3, −2), D(−3, 3)
3. A(−1, 4), B(5, 4), C(5, 1), D(−1, 1)
y
y
5
4
3
2
1
-5 -4 -3 -2 -1 0
-1
-2
-3
-4
-5
5
4
3
2
1
x
-5 -4 -3 -2 -1 0
-1
-2
-3
-4
-5
1 2 3 4 5
___
x
1 2 3 4 5
___
Length of BC = _ units
Length of BC = _ units
© Houghton Mifflin Harcourt Publishing Company
Problem
Problem Solving
Solving
4. On a map, a city block is a square with
three of its vertices at (−4, 1), (1, 1), and
(1, −4). What are the coordinates of the
remaining vertex?
5. A carpenter is making a shelf in the shape
of a parallelogram. She begins by drawing
parallelogram RSTU on a coordinate plane with
vertices R(1, 0), S(−3, 0), and T(−2, 3). What are
the coordinates of vertex U?
_____
6.
WRITE
_____
Math Explain how you would find the fourth vertex
of a rectangle with vertices at (2, 6), (−1, 4), and (−1, 6).
Chapter 10
587
Lesson Check (6.G.A.3)
1. The coordinates of points M, N, and P are
M(–2, 3), N(4, 3), and P(5, –1). What
coordinates for point Q make MNPQ a
parallelogram?
2. Dirk draws quadrilateral RSTU with vertices
R(–1, 2), S(4, 2), T(5, –1), and U(–2, –1). Which is
the best way to classify the quadrilateral?
3. Marcus needs to cut a 5-yard length of yarn
into equal pieces for his art project. Write an
equation that models the length l in yards
of each piece of yarn if Marcus cuts it into
p pieces.
4. The area of a triangular flag is 330 square
centimeters. If the base of the triangle is
30 centimeters long, what is the height of
the triangle?
5. A trapezoid is 6 1_2 feet tall. Its bases are
9.2 feet and 8 feet long. What is the area
of the trapezoid?
6. The dimensions of the rectangle below will be
multiplied by 3. How will the area be affected?
41m
2
10 m
FOR MORE PRACTICE
GO TO THE
588
Personal Math Trainer
© Houghton Mifflin Harcourt Publishing Company
Spiral Review (6.EE.A.2c, 6.EE.C.9, 6.G.A.1)
Name
Chapter 10 Review/Test
Personal Math Trainer
Online Assessment
and Intervention
1. Find the area of the parallelogram.
7.5 in.
8.5 in.
9 in.
The area is _ in.2.
2. A wall tile is two different colors. What is the area of the white part
of the tile? Explain how you found your answer.
5.5 in.
4 in.
3. The area of a triangle is 36 ft2. For numbers 3a–3d, select Yes or No to tell
if the dimensions could be the height and base of the triangle.
h = 3 ft, b = 12 ft
Yes
No
3b. h = 3 ft, b = 24 ft
Yes
No
3c. h = 4 ft, b = 18 ft
Yes
No
3d. h = 4 ft, b = 9 ft
Yes
No
© Houghton Mifflin Harcourt Publishing Company
3a.
4. Mario traced this trapezoid. Then he cut it
out and arranged the trapezoids to form a
rectangle. What is the area of the rectangle? 8 in.
_ in.2
Assessment Options
Chapter Test
4 in.
10 in.
Chapter 10
589
5. The area of the triangle is 24 ft2. Use the numbers to label the height and
base of the triangle.
2
4
6
8
10
20
ft
ft
6. A rectangle has an area of 50 cm2. The dimensions of the rectangle are
multiplied to form a new rectangle with an area of 200 cm2. By what
number were the dimensions multiplied?
7. Sami put two trapezoids with the
same dimensions together to make
a parallelogram.
8.
DEEPER
A rectangular plastic bookmark has a triangle cut out of it.
Use the diagram of the bookmark to complete the table.
Area of Rectangle
Area of Triangle
5 in.
Square Inches of
Plastic in Bookmark
2 in.
1 in.
1 in.
590
© Houghton Mifflin Harcourt Publishing Company
The formula for the area of a trapezoid is
A = 1_2 (b1 + b2) h. Explain why the bases of a
trapezoid need to be added in the formula.
Name
9. A trapezoid has an area of 32 in.2. If the lengths of the bases are 6 in. and
6.8 in., what is the height?
__ in.
10. A pillow is in the shape of a regular pentagon. The front of the pillow is
made from 5 pieces of fabric that are congruent triangles. Each triangle
has an area of 22 in.2. What is the area of the front of the pillow?
__ in.2
11. Which expressions can be used to find the area of the trapezoid? Mark
all that apply.
2 in.
5 in.
4.5 in.
A
B
1
_
2 × (5 + 2) × 4.5
1
_
2 × (2 + 4.5) × 5
C
D
1
_
2 × (5 + 4.5) × 2
1_
2 × (6.5) × 5
12. Name the polygon and find its area. Show your work.
© Houghton Mifflin Harcourt Publishing Company
6.2 in.
5 in.
polygon: __ area:__
Chapter 10
591
13. A carpenter needs to replace some flooring in a house.
7 ft
10 ft
12 ft
14 ft
Select the expression that can be used to find the total area of the
flooring to be replaced. Mark all that apply.
A
19 × 14
C
19 × 24 − 1_ × 10 × 12
B
168 + 12 × 14 + 60
D
7 × 24 + 12 × 14 + 1_ × 10 × 12
2
2
14. Ava wants to draw a parallelogram on the coordinate plane. She
plots these 3 points.
y
5
4
3
2
J
1
-5 -4 -3 -2 -1 0 1 2 3 4 5
-1
-2
I
-3 H
-4
-5
x
Part A
Find and label the coordinates of the fourth vertex, K, of the
parallelogram. Draw the parallelogram.
What is the length of side JK? How do you know?
592
© Houghton Mifflin Harcourt Publishing Company
Part B
Name
15. Joan wants to reduce the area of her posters by one-third. Draw lines to
match the original dimensions in the left column with the correct new
area in the right column. Not all dimensions will have a match.
30 in. by 12 in.
•
30 in. by 18 in.
•
20 in.2
•
•
60 in.2
12 in. by 15 in.
•
•
180 in.2
18 in. by 15 in.
•
•
360 in.2
Personal Math Trainer
16.
SMARTER
Alex wants to enlarge a 4-ft by 6-ft
vegetable garden by multiplying the dimensions of the garden by 2.
Part A
Find each area.
Area of original garden: __
Area of enlarged garden: __
Part B
Explain how the area of the original garden will be affected.
© Houghton Mifflin Harcourt Publishing Company
17. Suppose the point (3, 2) is changed to (3, 1) on this rectangle. What
other point must change so the figure remains a rectangle? What is the
area of the new rectangle?
5
4
3
2
1
-5 -4 -3 -2 -1 0
-1
-2
-3
-4
-5
y
x
1 2 3 4 5
Point :__ would change to __.
The area of the new rectangle is __ square units.
Chapter 10
593
18. Look at the figure below. The area of the parallelogram and the areas
of the two congruent triangles formed by a diagonal are related. If you
know the area of the parallelogram, how can you find the area of one
of the triangles?
19. The roof of Kamden’s house is shaped like a parallelogram. The base of
the roof is 13 m and the area is 110.5 m2. Choose a number and unit to
make a true statement.
The height of the roof is
97.5
17
8.5
m.
m2.
m3.
20. Eliana is drawing a figure on the coordinate grid. For numbers 20a–20d,
select True or False for each statement.
594
20a. The point (−1, 1) would be the
fourth vertex of a square.
True
20b. The point (1, 1) would be the
fourth vertex of a trapezoid.
True
False
20c. The point (2, −1) would be the
fourth vertex of a trapezoid.
True
False
20d. The point (−1, −1) would be the
fourth vertex of a square.
True
False
False
5
4
3
2
1
-5 -4 -3 -2 -1 0
-1
-2
-3
-4
-5
y
x
1 2 3 4 5
© Houghton Mifflin Harcourt Publishing Company
123.5
11
Chapter
Surface Area and Volume
Personal Math Trainer
Show Wha t You Know
Online Assessment
and Intervention
Check your understanding of important skills.
Name
Estimate and Find Area Multiply to find the area.
1.
2.
___
___
Area of Squares, Rectangles, and Triangles Find the area.
3.
4.
5.
8 in.
13 cm
© Houghton Mifflin Harcourt Publishing Company • Image Credits: (br) ©Andres Rodriguez/Alamy Images
13 cm
6 cm
15 in.
7 cm
A = s2
A = lw
A =1_(b × h)
Area = __
Area = __
Area = __
2
Evaluate Expressions Evaluate the expression.
6. 3 × (2 + 4)
_
7. 6 + 6 ÷ 3
_
8. 42 + 4 × 5 – 2
_
Math in the
Jerry is building an indoor beach volleyball court.
He has ordered 14,000 cubic feet of sand.
The dimensions of the court will be 30 feet by 60 feet.
Jerry needs to have a 10-foot boundary around the
court for safety. How deep will the sand be if Jerry
uses all the sand?
Chapter 11
595
Voca bula ry Builder
Visualize It
Complete the bubble map. Use the review terms
that name solid figures.
Review Words
base
cube
lateral face
polygon
polyhedron
prism
pyramid
vertex
solid figure
edge
Preview Words
net
solid figure
surface area
volume
Understand Vocabulary
Complete the sentences using the preview words.
1. A three-dimensional figure having length, width, and height is
called a(n) ___.
2. A two-dimensional pattern that can be folded into a
three-dimensional figure is called a(n) ___.
___ is the sum of the areas of all the faces,
© Houghton Mifflin Harcourt Publishing Company
3.
or surfaces, of a solid figure.
4.
___ is the measure of space a solid figure
occupies.
596
™Interactive Student Edition
™Multimedia eGlossary
Chapter 11 Vocabulary
base
lateral area
base
área lateral
47
7
net
prism
plantilla
prisma
79
65
pyramid
solid figure
pirámide
cuerpo geométrico
93
81
surface area
volume
área total
volumen
100
106
Examples:
BASES
2ECTANGULAR PRISM
4RIANGULAR PRISM
Examples:
Rectangular prism
Cone
Sphere
Pyramid
Cylinder
The measure of the space a solid figure
occupies
© Houghton Mifflin Harcourt Publishing Company
base
© Houghton Mifflin Harcourt Publishing Company
lateral
face
A two-dimensional pattern that can
be folded into a three-dimensional
polyhedron
© Houghton Mifflin Harcourt Publishing Company
© Houghton Mifflin Harcourt Publishing Company
A three-dimensional figure having length,
width, and height
Example:
In two dimensions, one side of a triangle
or parallelogram which is used to help
find the area. In three dimensions, a plane
figure, usually a polygon or circle, which
is used to partially describe a solid figure
and to help find the volume of some solid
figures.
A solid figure with a polygon base and
all other faces as triangles that meet at a
common vertex
© Houghton Mifflin Harcourt Publishing Company
© Houghton Mifflin Harcourt Publishing Company
© Houghton Mifflin Harcourt Publishing Company
A solid figure that has two congruent,
polygon-shaped bases, and other faces that
are all parallelograms
© Houghton Mifflin Harcourt Publishing Company
The sum of the areas of the lateral faces of
a solid
The sum of the areas of all the faces, or
surfaces, of a solid figure
Example:
Example:
BASE
Example:
IN
IN
IN
Surface area = 2(6 × 4) + 2(6 × 3) + 2(4 × 3)
= 180 in.2
Going Places with
Bingo
For 3–6 players
Materials
• 1 set of word cards
• 1 Bingo board for each player
• game markers
Words
Game
Game
Word Box
base
lateral area
net
prism
pyramid
solid figure
surface area
volume
How to Play
1. The caller chooses a card and reads the definition. Then the caller
puts the card in a second pile.
2. Players put a marker on the word that matches the definition each
time they find it on their Bingo boards.
3. Repeat Steps 1 and 2 until a player marks 5 boxes in a line going
down, across, or on a slant and calls “Bingo.”
4. To check the answers, the player who said “Bingo” reads the
© Houghton Mifflin Harcourt Publishing Company
Image Credits: (sky) ©Digital Vision/Getty Images
words aloud while the caller checks the definitions.
Chapter 11
596A
Journal
Jo
ou
urnal
The Write Way
Reflect
Choose one idea. Write about it.
• Define net and explain how nets can be used to understand solid figures.
• Explain and illustrate two ways to find the surface area of a cube. Use a
separate piece of paper for your drawing.
• Write two questions you have about finding a lateral area. Tell how you
would start to find the answers to those questions.
• Write a creative story about a person who needs to find
© Houghton Mifflin Harcourt Publishing Company Image Credits: (sky) ©Digital Vision/Getty Images
the volume of something.
596B
Lesson 11.1
Name
Three-Dimensional Figures and Nets
Geometry—
6.G.A.4
MATHEMATICAL PRACTICES
MP1, MP6
Essential Question How do you use nets to represent three-dimensional figures?
A solid figure is a three-dimensional figure
because it has three dimensions—length,
width, and height. Solid figures can be
identified by the shapes of their bases, the
number of bases, and the shapes of their
lateral faces.
base
vertex
lateral face
edge
Triangular Prism
Unlock
Unlock the
the Problem
Problem
A designer is working on the layout for the cereal
box shown. Identify the solid figure and draw a net
that the designer can use to show the placement of
information and artwork on the box.
• How many bases are there? __
• Are the bases congruent? __
• What shape are the bases? __
Identify the solid figure.
Recall that a prism is a solid figure with two congruent, parallel bases.
Its lateral faces are rectangles. It is named for the shape of its bases.
Is the cereal box a prism? __
What shape are the bases? __
So, the box is a ____.
Draw a net for the figure.
© Houghton Mifflin Harcourt Publishing Company
A net is a two-dimensional figure that can be folded into a solid figure.
STEP 1
STEP 2
Make a list of the shapes you will use.
Draw the net using the shapes you listed in
Step 1. One possible net is shown.
top and bottom bases: __
left and right faces: __
Corn
Twisties
front and back faces: __
Chapter 11
597
A pyramid is a solid figure with a polygon-shaped base
and triangles for lateral faces. Like prisms, pyramids are
named by the shape of their bases. A pyramid with a
rectangle for a base is called a rectangular pyramid.
lateral
face
base
Rectangular Pyramid
Example 1 Identify and draw a net for the solid figure.
Describe the base of the figure.
___________
Describe the lateral faces.
___________
The figure is a ____.
Shapes to use in the net:
Net:
base: __
lateral faces: __
Example 2 Identify and sketch the solid figure
that could be formed by the net.
The net has only _ triangles, so it cannot be a
__.
___.
•
598
MATHEMATICAL
PRACTICE
6
Compare the bases and lateral faces of prisms and pyramids.
© Houghton Mifflin Harcourt Publishing Company
The triangles must be the __ for a
Name
Share
hhow
Share and
and Show
Sh
MATH
M
BOARD
B
Identify and draw a net for the solid figure.
1.
Net:
2.
base: ___
lateral faces: ___
figure: ____
______
Identify and sketch the solid figure that could be formed by the net.
3.
4.
______
______
Math
Talk
On
On Your
Your Own
Own
MATHEMATICAL PRACTICES 1
Describe What characteristics
of a solid figure do you need
to consider when making its
net?
Identify and draw a net for the solid figure.
© Houghton Mifflin Harcourt Publishing Company
5.
6.
______
______
Chapter 11 • Lesson 1 599
MATHEMATICAL PRACTICES
ANALYZEt-00,'034536$563&t13&$*4*0/
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OqnakdlRnkuhmf¤@ookhb`shnmr
Solve.
7. The lateral faces and bases of crystals of the mineral galena
are congruent squares. Identify the shape of
a galena crystal.
8.
SMARTER
Rhianon draws the net below and
labels each square. Can Rhianon fold her net into a
cube that has letters A through G on its faces? Explain.
WRITE
Math
Show Your Work
B
A
C
E
F
G
D
10.
MATHEMATICAL
PRACTICE
1
Describe A diamond crystal
is shown. Describe the figure in terms
of the solid figures you have seen in
this lesson.
SMARTER
Sasha makes a triangular prism from paper.
rectangles.
The bases are
squares.
triangles.
rectangles.
The lateral faces are
squares.
triangles.
600
© Houghton Mifflin Harcourt Publishing Company • Image Credits: (t) ©Harry Taylor/Getty Images
9.
Practice and Homework
Name
Lesson 11.1
Three-Dimensional Figures and Nets
COMMON CORE STANDARD—6.G.A.4
Solve real-world and mathematical problems
involving area, surface area, and volume.
Identify and draw a net for the solid figure.
1.
Net
2.
rectangular prism
figure: ______
3.
Net
figure: ______
Net
figure: ______
4.
Net
figure: ______
© Houghton Mifflin Harcourt Publishing Company
Problem
Problem Solving
Solving
5. Hobie’s Candies are sold in triangular-pyramidshaped boxes. How many triangles are needed to
make one box?
_______
7.
WRITE
6. Nina used plastic rectangles to make
6 rectangular prisms. How many rectangles
did she use?
_______
Math Describe how you could draw more than one net to
represent the same three-dimensional figure. Give examples.
Chapter 11
601
Lesson Check (6.G.A.4)
1. How many vertices does a square
pyramid have?
TEST
PREP
2. Each box of Fred’s Fudge is constructed from
2 triangles and 3 rectangles. What is the shape
of each box?
Spiral Review (6.EE.B.7, 6.EE.C.9, 6.G.A.1, 6.G.A.3)
4. A hot-air balloon is at an altitude of 240 feet.
The balloon descends 30 feet per minute. What
equation gives the altitude y, in feet, of the
hot-air balloon after x minutes?
5. A regular heptagon has sides measuring
26 mm and is divided into 7 congruent
triangles. Each triangle has a height of
27 mm. What is the area of the heptagon?
6. Alexis draws quadrilateral STUV with
vertices S(1, 3), T (2, 2), U(2, –3), and
V(1, –2). What name best classifies
the quadrilateral?
© Houghton Mifflin Harcourt Publishing Company
3. Bryan jogged the same distance each day for 7
days. He ran a total of 22.4 miles. The equation
7d = 22.4 can be used to find the distance d in
miles he jogged each day. How far did Bryan jog
each day?
FOR MORE PRACTICE
GO TO THE
602
Personal Math Trainer
Lesson 11.2
Name
Explore Surface Area Using Nets
Essential Question What is the relationship between a net and the
surface area of a prism?
connect The surface area of a solid figure is the sum of the areas of
all the faces or surfaces of the figure. Surface area is measured in square
units. You can use a net to help you find the surface area of a solid figure.
Geometry—
6.G.A.4
MATHEMATICAL PRACTICES
MP1, MP2, MP3, MP4
Hands
On
Investigate
Investigate
Materials ■ centimeter grid paper, ruler, scissors
A box is shaped like a rectangular prism. The box is 8 cm long, 6 cm
wide, and 4 cm high. What is the surface area of the box?
Find the surface area of the rectangular prism.
A. Draw a net of the prism on centimeter grid paper.
B. Cut out the net.
C. Fold the net to confirm that it represents a rectangular prism
measuring 8 cm by 6 cm by 4 cm.
4 cm
D. Count the grid squares on each face of the net.
© Houghton Mifflin Harcourt Publishing Company • Image Credits: (t) ©Nikreates/Alamy
So, the surface area of the box is __ cm2.
Draw Conclusions
1. Explain how you used the net to find the surface area of the box.
2.
6 cm
8 cm
Make sure you include all
surfaces in the net of a
three-dimensional figure, not
just the surfaces you can see
in the diagram of the figure.
SMARTER
Describe how you could find the area of each face of the
prism without counting grid squares on the net.
Chapter 11
603
Make
Make Connections
Connections
You can also use the formula for the area of a rectangle to find the
surface area of the box.
Find the surface area of the box in the Investigate, which
measures 8 cm by 6 cm by 4 cm.
STEP 1 Label the rectangles in the net A through F. Then label the
dimensions.
4 cm
4 cm
8 cm
A
B
C
E
F
D
STEP 2 Find the area of each face of the prism.
Think: I can find the area of a rectangle by multiplying the rectangle’s __
times its __.
Record the areas of the faces below.
Face B: _ cm2
Face C: _ cm2
Face D: _ cm2
Face E: _ cm2
Face F: _ cm2
STEP 3 Add the areas to find the surface area of the prism.
The surface area of the prism is _ cm2.
3.
MATHEMATICAL
PRACTICE
2
Use Reasoning Identify any prism faces that have equal
areas. How could you use that fact to simplify the process of finding the
surface area of the prism?
4. Describe how you could find the surface area of a cube.
604
Math
Talk
MATHEMATICAL PRACTICES 6
Compare What do you notice
about the surface area you
found by adding the areas
of the faces and the surface
area you found by counting
grid squares? Explain.
© Houghton Mifflin Harcourt Publishing Company
Face A: 4 × 8 = 32 cm2
Name
Share
hhow
Share and
and Show
Sh
MATH
M
BOARD
B
Use the net to find the surface area of the prism.
1.
A
3 cm
B
C
D
E
4 cm
F
2 cm
Face A: _ cm2
Face D: _ cm2
Face B: _ cm2
Face E: _ cm2
Face C: _ cm2
Face F: _ cm2
Surface area: _ cm2
Find the surface area of the rectangular prism.
2.
3.
4.
10 cm
5 cm
3 cm
5 cm
15 cm
10 cm
9 cm
7 cm
10 cm
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OqnakdlRnkuhmf¤@ookhb`shnmr
5. A cereal box is shaped like a rectangular prism. The box is 20 cm long
by 5 cm wide by 30 cm high. What is the surface area of the cereal box?
© Houghton Mifflin Harcourt Publishing Company
6.
7.
MATHEMATICAL
PRACTICE
1
Darren is painting a wooden block as part of his art project.
The block is a rectangular prism that is 12 cm long by 9 cm wide by
5 cm high. Describe the rectangles that make up the net for the prism.
DEEPER
In Exercise 6, what is the surface area, in square meters,
that Darren has to paint?
Chapter 11 • Lesson 2 605
MATHEMATICAL PRACTICES .0%&-t3&"40/tM",&4&/4&
What’s the Error?
8.
SMARTER
Emilio is designing the packaging
for a new MP3 player. The box for the MP3 player is
5 cm by 3 cm by 2 cm. Emilio needs to find the surface
area of the box.
Look at how Emilio solved the problem. Find his error.
STEP 1 Draw a net.
STEP 2 Find the areas of all the
faces and add them.
3 cm
2 cm
5 cm
A
B
Face A: 3 × 2 = 6 cm2
3 cm
D
3 cm
3 cm
E
F
Face B: 3 × 5 = 15 cm2
Face C: 3 × 2 = 6 cm2
Face D: 3 × 5 = 15 cm2
C
Face E: 3 × 5 = 15 cm2
2 cm
Face F: 3 × 5 = 15 cm2
Surface area: 72 cm2
Correct the error. Find the surface area of the prism.
9.
SMARTER
For numbers 9a–9d, select True or False for each statement.
9a.
The area of face A is 10 cm2.
True
False
9b.
The area of face B is 10 cm2.
True
False
2
606
B
9c. The area of face C is 40 cm .
True
False
9d.
True
False
The surface area of the
prism is 66 cm2.
A
C
F
D
8 cm
E
5 cm
2 cm
© Houghton Mifflin Harcourt Publishing Company
So, the surface area of the prism is __.
Practice and Homework
Name
Lesson 11.2
Explore Surface Area Using Nets
Use the net to find the surface area of the
rectangular prism.
1.
A
B
2.
A: 6 squares
C
COMMON CORE STANDARD—6.G.A.4
Solve real-world and mathematical problems
involving area, surface area, and volume.
B: 8 squares
D
C: 6 squares
D: 12 squares
E
E: 8 squares
F
F: 12 squares
52 square units
_______
_______
Find the surface area of the rectangular prism.
3.
4.
5.
3 mm
7 mm
3 mm
5 in.
____
3 ft
4 in.
1 in.
____
6.5 ft
2 ft
____
© Houghton Mifflin Harcourt Publishing Company
Problem
Problem Solving
Solving
6. Jeremiah is covering a cereal box with fabric
for a school project. If the box is 6 inches long
by 2 inches wide by 14 inches high, how much
surface area does Jeremiah have to cover?
_______
8.
WRITE
7. Tia is making a case for her calculator. It is a
rectangular prism that will be 3.5 inches long
by 1 inch wide by 10 inches high. How much
material (surface area) will she need to make
the case?
_______
Math Explain in your own words how to find the surface
area of a rectangular prism.
Chapter 11
607
TEST
PREP
Lesson Check (6.G.A.4)
1. Gabriela drew a net of a rectangular prism on
centimeter grid paper. If the prism is 7 cm long
by 10 cm wide by 8 cm high, how many grid
squares does the net cover?
2. Ben bought a cell phone that came in a box
shaped like a rectangular prism. The box is
5 inches long by 3 inches wide by 2 inches high.
What is the surface area of the box?
Spiral Review (6.EE.B.5, 6.EE.C.9, 6.G.A.1, 6.G.A.4)
3. Katrin wrote the inequality x + 56 < 533. What
is the solution of the inequality?
4. The table shows the number of mixed CDs y
that Jason makes in x hours.
Mixed CDs
Hours, x
CDs, y
2
3
5
10
10
15
25
50
Which equation describes the pattern in
the table?
6. Boxes of Clancy’s Energy Bars are rectangular
prisms. How many lateral faces does each
box have?
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5. A square measuring 9 inches by 9 inches
is cut from a corner of a square measuring
15 inches by 15 inches. What is the area of the
L-shaped figure that is formed?
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Lesson 11.3
Surface Area of Prisms
Essential Question How can you find the surface area of a prism?
Geometry—6.G.A.4
Also 6.EE.A.2c
MATHEMATICAL PRACTICES
MP2, MP4, MP8
You can use a net to find the surface area of a
solid figure, such as a prism.
Unlock
Unlock the
the Problem
Problem
Alex is designing wooden boxes for his books. Each
box measures 15 in. by 12 in. by 10 in. Before he
buys wood, he needs to find the surface area of
each box. What is the surface area of each box?
• What is the shape of each face?
• What are the dimensions of each face?
Use a net to find the surface area.
12 in.
10 in.
10 in.
A
10 in.
12 in.
15 in.
B
C
D
E
15 in.
10 in.
10 in.
© Houghton Mifflin Harcourt Publishing Company
STEP 1
12 in.
F
Find the area of each lettered face.
Face A: 12 × 10 = 120 in.2
Face B: 15 × 10 = _ in.2
Face C: _ × _ = _ in.2
Face D: _ × _ = _ in.2
Face E: _ × _ = _ in.2
Face F: _ × _ = _ in.2
STEP 2
Find the sum of the areas of the faces. __
So, the surface area of each box is __.
Math
Talk
MATHEMATICAL PRACTICES 1
Describe What do you notice
about the opposite faces of
the box that could help you
find its surface area?
Chapter 11
609
12 in.
Example 1 Use a net to find the surface area of the
5 in.
triangular prism.
10 in.
The surface area equals the sum of the areas of the three rectangular
faces and two triangular bases. Note that the bases have the same area.
13 in.
__ bh = 1
__ × 12 × _ = _
area of bases A and E: A = 1
2
2
A 5 in.
area of face B: A = lw = 5 × 10 = _
area of face C: A = lw = _ × _ = _
10 in.
13 in.
D
12 in.
area of face D: A = lw = _ × _ = _
Surface area: 2 × _ + _ + _ + _ = _
So, the surface area of the triangular prism is ___.
Math
Talk
C
5 in.
B 10 in.
E
5 in.
MATHEMATICAL PRACTICES 6
Explain Why was the area
of one triangular base
multiplied by 2?
Example 2 Find the surface area of the cube.
One Way Use a net.
5 cm
STEP 1 Find the area of each face.
5 cm
All of the faces are squares with a side length of __ , so the areas of all
the squares are the same.
Area of one face: A = _ × _ = __
5 cm
5 cm
5 cm
5 cm
5 cm 5 cm
5 cm
STEP 2 Find the sum of the areas of all _ faces.
5 cm
_ + _ + _ + _ + _ + _ = __
You can also find the surface area of a cube using the formula S = 6s2, where S is
the surface area and s is the side length of the cube.
Write the formula.
S = 6s2
Replace s with 5.
S = 6 ( _ )2
Simplify.
S = 6 (_) = _
The surface area of the cube is __ .
610
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Another Way Use a formula.
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Use a net to find the surface area.
area of each face: _ × _ = _
1.
2 ft
2 ft
2 ft
2 ft
2 ft
number of faces: _
2 ft
2 ft
surface area = _ × _ = _ ft2
2 ft
2 ft
2.
10 cm
3.
8 21 in.
6 cm
16 cm
3 21 in.
8 cm
______
4 in.
______
Math
Talk
On
On Your
Your Own
Own
Use a net to find the surface area.
4.
MATHEMATICAL PRACTICES 5
Use Tools What strategy could
you use to find the surface
area of a rectangular prism
with a length of 8 ft, a width
of 2 ft, and a height of 3 ft?
5.
© Houghton Mifflin Harcourt Publishing Company
8m
3m
5m
MATHEMATICAL
PRACTICE
1
7 in.
2
1
7 in.
2
______
6.
1
7 in.
2
______
6
Attend to Precision Calculate the surface area of the
cube in Exercise 5 using the formula S = 6s2. Show your work.
Chapter 11 • Lesson 3 611
MATHEMATICAL PRACTICES
COMMUNICA5&t1&34E7&3&tCONSTRUCT ARGUMENTS
Unlock
Unlock the
the Problem
Problem
7.
SMARTER
The Vehicle
Assembly Building at Kennedy
Space Center is a rectangular
prism. It is 218 m long, 158 m
wide, and 160 m tall. There are
four 139 m tall doors in the
building, averaging 29 m in width.
What is the building’s outside surface area
when the doors are open?
a. Draw each face of the building, not including
the floor.
d. Find the building’s surface area (not including
the floor) when the doors are closed.
e. Find the area of the four doors.
WRITE
b. What are the dimensions of the 4 walls?
M t Show Your Work
Math
f. Find the building’s surface area (not including
the floor) when the doors are open.
8.
DEEPER
A rectangular prism is 11_ ft long,
2
2 ft wide, and 5_ ft high. What is the surface
_
3
6
area of the prism in square inches?
612
9.
SMARTER
A gift box is a rectangular
prism. The box measures 8 inches by 10 inches
by 3 inches. What is its surface area?
© Houghton Mifflin Harcourt Publishing Company • Image Credits: (t) ©Courtesy, NASA
c. What are the dimensions of the roof?
Practice and Homework
Name
Lesson 11.3
Surface Area of Prisms
COMMON CORE STANDARD—6.G.A.4
Solve real-world and mathematical problems
involving area, surface area, and volume.
Use a net to find the surface area.
1.
2 cm
2 cm
6 cm
5 cm A
2 cm 5 cm
5 cm
5 cm
2
Area of A and F = 2 × ( 5 × 2 ) = 20 cm
B
2
Area of B and D = 2 × ( 6 × 2 ) = 24 cm
2
Area of C and E = 2 × ( 6 × 5 ) = 60 cm
2
2
S.A.= 20 cm + 24 cm + 60 cm = 104 cm
3.
2.
D
E
6 cm
6 cm
12 cm
F
2
2
C
4.
6 in.
9 ft
3.5 in.
8 cm
9 ft
4 in.
10 cm
9 ft
____
____
____
© Houghton Mifflin Harcourt Publishing Company
Problem
Problem Solving
Solving
5. A shoe box measures 15 in. by 7 in. by 4 1_2 in.
What is the surface area of the box?
6. Vivian is working with a styrofoam cube for
art class. The length of one side is 5 inches.
How much surface area does Vivian have to
work with?
_______
7.
WRITE
_______
Math Explain why a two-dimensional net is useful for
finding the surface area of a three-dimensional figure.
Chapter 11
613
Lesson Check (6.G.A.4)
1. What is the surface area of a cubic box that
contains a baseball that has a diameter of
3 inches?
2. A piece of wood used for construction is
2 inches by 4 inches by 24 inches. What is the
surface area of the wood?
Spiral Review (6.EE.C.9, 6.G.A.1, 6.G.A.4)
4. A trapezoid with bases that measure 8 inches
and 11 inches has a height of 3 inches. What is
the area of the trapezoid?
5. City Park is a right triangle with a base of 40 yd
and a height of 25 yd. On a map, the park has a
base of 40 in. and a height of 25 in. What is the
ratio of the area of the triangle on the map to
the area of City Park?
6. What is the surface area of the prism shown by
the net?
© Houghton Mifflin Harcourt Publishing Company
3. Detergent costs $4 per box. Kendra graphs the
equation that gives the cost y of buying x boxes
of detergent. What is the equation?
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Lesson 11.4
Surface Area of Pyramids
Essential Question How can you find the surface area of a pyramid?
Geometry—6.G.A.4
Also 6.EE.A.2c
MATHEMATICAL PRACTICES
MP4, MP5, MP6
Most people think of Egypt when they think of pyramids, but there
are ancient pyramids throughout the world. The Pyramid of the
Sun in Mexico was built around 100 C.E. and is one of the largest
pyramids in the world.
Unlock
Unlock the
the Problem
Problem
Cara is making a model of the Pyramid of the Sun for a
history project. The base is a square with a side length
of 12 in. Each triangular face has a height of 7 in. What is
the surface area of Cara’s model?
Find the surface area of the square pyramid.
STEP 1
Label the dimensions on the net of the pyramid.
STEP 2
© Houghton Mifflin Harcourt Publishing Company • Image Credits: (t) ©David R. Frazier Photolibrary, Inc./Alamy
Find the area of the base and each triangular face.
Base:
Write the formula for the area of a square.
Substitute
_ for s and simplify.
A = s2
A = _ = _ in.2
Face:
Write the formula for the area of a triangle.
_ for b and _ for h
Substitute
and simplify.
__bh
A=1
2
__ (_)(_)
A=1
2
= _ in.2
STEP 3
Add the areas to find the surface area of the pyramid.
S = _ + 4 × _ = _ + _ = _ in.2
So, the surface area of Cara’s model is __.
Math
Talk
MATHEMATICAL PRACTICES 6
Explain Why did you
multiply the area of the
triangular face by 4 when
finding the surface area?
Chapter 11
615
Sometimes you need to find the total area of the lateral faces of a solid
figure, but you don’t need to include the area of the base. The lateral
area L of a solid figure is the sum of the areas of the lateral faces.
Example
Kwan is making a tent in the shape of a triangular
pyramid. The three sides of the tent are made of fabric, and the bottom
will be left open. The faces have a height of 10 ft and a base of 6 ft. What
is the area of the fabric Kwan needs to make the tent?
Find the lateral area of the triangular pyramid.
10 ft
6 ft
6 ft
STEP 1
Draw and label a net for the pyramid.
STEP 2
Shade the lateral area of the net.
STEP 3
Find the area of one of the lateral faces of the pyramid.
Write the formula for the area of a triangle
Substitute
_ for b and _ for h.
__bh
A=1
2
__ (_)(_)
A=1
2
A = _ ft2
Simplify
STEP 4
To find the lateral area, find the area of all three lateral faces
of the pyramid.
L = 3 × _ = _ ft2
So, the area of fabric Kwan needs is __.
MATHEMATICAL
PRACTICE
6
Compare Explain the difference between finding the
surface area and the lateral area of a three-dimensional figure.
2. Explain how you could find the amount of fabric needed if
Kwan decided to make a fabric base for the tent.
The height of the triangular base is about 5 ft.
616
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1.
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B
1. Use a net to find the surface area of the square pyramid.
Base: A = _ = _ cm2
8 cm
Face: A = 1_ ( _ )( _ ) = _ cm2
2
Surface area of pyramid: S = _ + 4 × _
5 cm
= _ + _ = _ cm2
2. A triangular pyramid has a base with an
area of 43 cm2 and lateral faces with bases
of 10 cm and heights of 8.6 cm. What is the
surface area of the pyramid?
3. A square pyramid has a base with a side length
of 3 ft and lateral faces with heights of 2 ft. What
is the lateral area of the pyramid?
Math
Talk
On
On Your
Your Own
Own
MATHEMATICAL PRACTICES 1
Describe What strategy can
you use to find the surface
area of a square pyramid if
you know the height of each
face and the perimeter of
the base?
© Houghton Mifflin Harcourt Publishing Company • Image Credits: (b) ©Terry Smith Images/Alamy
Use a net to find the surface area of the square pyramid.
4.
5.
6 cm
9 ft
6.
12.5 in.
8 in.
10 cm
8 ft
7. The Pyramid Arena is located in Memphis, Tennessee. It is in the
shape of a square pyramid, and the lateral faces are made almost
completely of glass. The base has a side length of about 600 ft and
the lateral faces have a height of about 440 ft. What is the total
area of the glass in the Pyramid Arena?
Chapter 11 • Lesson 4 617
MATHEMATICAL PRACTICES
ANALYZEt-00,'034536$563&t13&$*4*0/
OqnakdlRnkuhmf¤@ookhb`shnmr
OqnakdlRnkuhmf¤@ookhb`shnmr
Use the table for 8–9.
8. The Great Pyramids are located near Cairo, Egypt.
They are all square pyramids, and their dimensions
are shown in the table. What is the lateral area of the
Pyramid of Cheops?
Dimensions of the
Great Pyramids (in m)
9.
10.
DEEPER
What is the difference between the surface areas
of the Pyramid of Khafre and the Pyramid of Menkaure?
Name
Side Length
of Base
Height of
Lateral Faces
Cheops
230
180
Khafre
215
174
Menkaure
103
83
SMARTER
Write an expression for the
surface area of the square pyramid shown.
x ft
3 ft
11.
MATHEMATICAL
PRACTICE
3 Make Arguments A square pyramid has a
Personal Math Trainer
12.
SMARTER
Jose says the lateral area of the square
pyramid is 260 in. Do you agree or disagree with Jose? Use
numbers and words to support your answer.
2
8 in.
10 in.
618
© Houghton Mifflin Harcourt Publishing Company • Image Credits: (t) ©Peter Adams/Corbis
base with a side length of 4 cm and triangular faces with a
height of 7 cm. Esther calculated the surface area as
(4 × 4) + 4(4 × 7) = 128 cm2. Explain Esther’s error and
find the correct surface area.
Practice and Homework
Name
Lesson 11.4
Surface Area of Pyramids
COMMON CORE STANDARD—6.G.A.4
Solve real-world and mathematical problems
involving area, surface area, and volume.
Use a net to find the surface area of the square pyramid.
1.
7 mm
7 mm
2
Base: A = 5 = 25 mm2
1 (5)(7)
Face: A = _
2
= 17.5 mm2
5 mm
S.A. = 25 1 4 3 17.5
5 mm
= 25 1 70
= 95 mm2
2.
8 cm
3.
4.
4 in.
9 yd
18 cm
10 in.
2.5 yd
____
____
____
© Houghton Mifflin Harcourt Publishing Company
Problem
Problem Solving
Solving
5. Cho is building a sandcastle in the shape of
a triangular pyramid. The area of the base is
7 square feet. Each side of the base has a length
of 4 feet and the height of each face is 2 feet.
What is the surface area of the pyramid?
_______
7.
WRITE
6. The top of a skyscraper is shaped like a square
pyramid. Each side of the base has a length of
60 meters and the height of each triangle is 20
meters. What is the lateral area of the pyramid?
_______
Math Write and solve a problem finding the lateral area of
an object shaped like a square pyramid.
Chapter 11
619
Lesson Check (6.G.A.4)
1. A square pyramid has a base with a side length
of 12 in. Each face has a height of 7 in. What is
the surface area of the pyramid?
2. The faces of a triangular pyramid have a base of
5 cm and a height of 11 cm. What is the lateral
area of the pyramid?
Spiral Review (6.EE.C.9, 6.G.A.1, 6.G.A.3, 6.G.A.4)
3. What is the linear equation represented by the
graph?
y
(3, 4)
(1, 2)
(0, 1)
-5 -4 -3 -2 -1 0 1 2 3 4 5
-1
-2
(- 3, - 2) - 3
-4
-5
x
5. Carly draws quadrilateral JKLM with
vertices J( −3, 3), K(3, 3), L(2, −1), and
M( −2, −1). What is the best way to
classify the quadrilateral?
6. A rectangular prism has the dimensions 8 feet
by 3 feet by 5 feet. What is the surface area of
the prism?
© Houghton Mifflin Harcourt Publishing Company
5
4
3
2
1
4. A regular octagon has sides measuring about
4 cm. If the octagon is divided into 8 congruent
triangles, each has a height of 5 cm. What is the
area of the octagon?
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Mid-Chapter Checkpoint
Online Assessment
and Intervention
Vocabulary
Vocabulary
Vocabulary
Choose the best term from the box to complete the sentence.
1.
lateral area
net
___ is the sum of the areas of all the faces,
solid figure
or surfaces, of a solid figure. (p. 597)
surface area
2. A three-dimensional figure having length, width, and height is
called a(n) ___. (p. 603)
3. The ___ of a solid figure is the sum of the areas of
its lateral faces. (p. 616)
Concepts
Concepts and
and Skills
Skills
4. Identify and draw a net for the solid figure. (6.G.A.4)
______
5. Use a net to find the lateral area of the square pyramid.
(6.G.A.4)
12 in.
© Houghton Mifflin Harcourt Publishing Company
9 in.
______
6. Use a net to find the surface area of the prism. (6.G.A.4)
7 cm
______
10 cm
5 cm
Chapter 11
621
7. A machine cuts nets from flat pieces of cardboard. The nets can be
folded into triangular pyramids used as pieces in a board game. What
shapes appear in the net? How many of each shape are there? (6.G.A.4)
8.
DEEPER
Fran’s filing cabinet is 6 feet tall, 1 1_3 feet wide, and 3 feet
deep. She plans to paint all sides except the bottom of the cabinet.
Find the area of the sides she intends to paint. (6.G.A.4)
9. A triangular pyramid has lateral faces with bases of 6 meters and heights
of 9 meters. The area of the base of the pyramid is 15.6 square meters.
What is the surface area of the pyramid? (6.G.A.4)
11. A small refrigerator is a cube with a side length of 16 inches.
Use the formula S = 6s2 to find the surface area of the cube. (6.EE.A.2c)
622
© Houghton Mifflin Harcourt Publishing Company
10. What is the surface area of a storage box that measures 15 centimeters
by 12 centimeters by 10 centimeters? (6.G.A.4)
Lesson 11.5
Name
Fractions and Volume
Geometry—
6.G.A.2
MATHEMATICAL PRACTICES
MP5, MP6, MP7, MP8
Essential Question What is the relationship between the volume and
the edge lengths of a prism with fractional edge lengths?
connect Volume is the number of cubic units needed to occupy a given
space without gaps or overlaps. You can find the volume of a rectangular
prism by seeing how many unit cubes it takes to fill the prism. Recall that a
unit cube is a cube with a side length of 1.
Hands
On
Investigate
Investigate
Materials
net of a rectangular prism, cubes, scissors, tape
A jewelry box has a length of 3 1_2 units, a width of 1 1_2 units, and a height
of 2 units. What is the volume of the box in cubic units?
A. Each of the cubes in this activity has a side length of 1_2 unit.
How many cubes with side length 1_2 does it take to form
a unit cube? _
So, each smaller cube represents _ of a unit cube.
B. Cut out the net. Then fold and tape the net into a rectangular
© Houghton Mifflin Harcourt Publishing Company • Image Credits: (t) ©Maciej Figiel/Alamy
prism. Leave one face open so you can pack the prism with
cubes.
C. Pack the prism with cubes.
1
unit
2
How many cubes with side length 1_ does it take to fill the prism?
2
1 unit
_
D. To find the volume of the jewelry box in cubic units, determine
how many unit cubes you could make from the smaller cubes
you used to pack the prism.
Think: It takes 8 smaller cubes
to make 1 unit cube.
Divide the total number of
smaller cubes by 8. Write the
remainder as a fraction.
_÷8=_=_
So, the volume of the jewelry box is _ cubic units.
Math
Talk
MATHEMATICAL PRACTICES 2
Reasoning How did you
determine the number of
cubes with side length 1_2 it
takes to form a unit cube?
Chapter 11
623
Draw Conclusions
MATHEMATICAL
PRACTICE
1.
Draw Conclusions Could you use the method of packing
cubes to find the volume of a triangular prism? Explain.
2.
How many cubes with a side length of 1_ unit do you
2
need to form 3 unit cubes? Explain how you know.
8
SMARTER
Make
Make Connections
Connections
You can use the formula for the volume of a rectangular
prism to find the volume of the jewelry box.
STEP 2 Replace the variables using the values
you know.
STEP 3 Write the mixed numbers as fractions
greater than 1.
V=l×w×h
__ ×
V = 31
×
2
The volume of a rectangular
prism is the product of the
length, the width, and the
height: V = l × w × h.
__ × 2
V = ___ × 3
2
STEP 4 Multiply.
V = ____
STEP 5 Write the fraction as a mixed number.
V=
So, the volume of the jewelry box is _ cubic units.
624
2
__ =
4
Math
Talk
MATHEMATICAL PRACTICES 6
Compare What do you notice
about the volume you found
by using the formula and the
volume you found by packing
the prism with cubes?
© Houghton Mifflin Harcourt Publishing Company
STEP 1 Write the formula you will use.
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1. A prism is filled with 38 cubes with a side length
of 1_2 unit. What is the volume of the prism in
cubic units?
2. A prism is filled with 58 cubes with a side length
of 1_2 unit. What is the volume of the prism in
cubic units?
38 ÷ 8 = _ = _
volume = _ cubic units
Find the volume of the rectangular prism.
3.
4.
1
4 2 units
3 units
5 1 units
2 units
2
___
© Houghton Mifflin Harcourt Publishing Company • Image Credits: (b) ©intrepidina/Shutterstock
5.
6.
1
4 2 units
1
4 2 units
___
DEEPER
Theodore wants to put three flowering plants in his
window box. The window box is shaped like a rectangular prism
that is 30.5 in. long, 6 in. wide, and 6 in. deep. The three plants
need a total of 1,200 in.3 of potting soil to grow well. Is the box
large enough? Explain.
WRITE
Math Explain how use the formula V = l ×w × h
to verify that a cube with a side length of 1_ unit has a volume
2
of 1_ of a cubic unit.
8
Chapter 11 • Lesson 5 625
MATHEMATICAL PRACTICES .0%&-t3&"40/tM",&4&/4&
OqnakdlRnkuhmf¤@ookhb`shnmr
OqnakdlRnkuhmf¤@ookhb`shnmr
1
unit
2
A
1 unit
Use the diagram for 7–10.
7. Karyn is using a set of building blocks shaped like rectangular
prisms to make a model. The three types of blocks she has are
shown at right. What is the volume of an A block? (Do not include
the pegs on top.)
1 unit
1 unit
B
1 unit
8. How many A blocks would you need to take up the same amount of
space as a C block?
1 unit
1 unit
C
2 units
WRITE
9.
10.
11.
1 unit
Math
Show Your Work
DEEPER
Karyn puts a B block, two C blocks, and three A blocks
together. What is the total volume of these blocks?
SMARTER
Karyn uses the blocks to make a prism
that is 2 units long, 3 units wide, and 11_ units high. The
2
prism is made of two C blocks, two B blocks, and some
A blocks. What is the total volume of A blocks used?
MATHEMATICAL
PRACTICE
3 Verify the Reasoning of Others Jo says that
12.
626
SMARTER
A box measures 5 units by 3 units by 2 1_2 units.
For numbers 12a–12b, select True or False for the statement.
12a.
The greatest number of cubes
with a side length of 1_2 unit that
can be packed inside the box is 300.
True
False
12b.
The volume of the box is 37 1_2 cubic units.
True
False
© Houghton Mifflin Harcourt Publishing Company
you can use V = l × w × h or V = h × w × l to find the volume of a
rectangular prism. Does Jo’s statement make sense? Explain.
Practice and Homework
Name
Lesson 11.5
Fractions and Volume
COMMON CORE STANDARD—6.G.A.2
Solve real-world and mathematical problems
involving area, surface area, and volume.
Find the volume of the rectangular prism.
1.
1 unit: 54
Number of cubes with side length _
1 1 units
2
1 1 units
2
3 units
2
54 ÷ 8 = 6 with a remainder of 6
6 = 63
_
54 ÷ 8 = 6 + _
8
4
3 cubic units
Volume 5 6_
4
2.
3.
2 units
4 1 units
4.
2
5 units
1 unit
5 1 units
2
4 1 units
1 1 units
2
2
2 1 units
2
____
____
2 1 units
2
____
Problem
Problem Solving
Solving
© Houghton Mifflin Harcourt Publishing Company
5. Miguel is pouring liquid into a container that is
4 1_2 inches long by 3 1_2 inches wide by 2 inches
high. How many cubic inches of liquid will fit in
the container?
_______
7.
WRITE
6. A shipping crate is shaped like a rectangular
prism. It is 5 1_2 feet long by 3 feet wide by 3 feet
high. What is the volume of the crate?
_______
Math How many cubes with a side length of 1_4 unit would it
take to make a unit cube? Explain how you determined your answer.
Chapter 11
627
Lesson Check (6.G.A.2)
1. A rectangular prism is 4 units by 2 1_2 units by
1 1_2 units. How many cubes with a side length
of 1_2 unit will completely fill the prism?
2. A rectangular prism is filled with 196 cubes
with 1_2 -unit side lengths. What is the volume of
the prism in cubic units?
3. A parallelogram-shaped piece of stained glass
has a base measuring 2 1_2 inches and a height
of 1 1_4 inches. What is the area of the piece of
stained glass?
4. A flag for the sports club is a rectangle
measuring 20 inches by 32 inches. Within the
rectangle is a yellow square with a side length of
6 inches. What is the area of the flag that is not
part of the yellow square?
5. What is the surface area of the rectangular
prism shown by the net?
6. What is the surface area of the square pyramid?
8 cm
7 cm
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Spiral Review (6.G.A.1, 6.G.A.4)
ALGEBRA
Name
Lesson 11.6
Volume of Rectangular Prisms
Essential Question How can you find the volume of rectangular prisms
with fractional edge lengths?
Geometry— 6.G.A.2
Also 6.EE.A.2c
MATHEMATICAL PRACTICES
MP2, MP5, MP6
You can use the formula V = l × w × h to find the volume of a
rectangular prism when you know the length, width, and height
of the prism.
Unlock
Unlock the
the Problem
Problem
An obento is a single-portion meal that is common
in Japan. The meal is usually served in a box.
A small obento box is a rectangular prism that is
5 inches long, 4 inches wide, and 21_ inches high.
2
How much food fits in the box?
• Underline the sentence that tells you what you
are trying to find.
• Circle the numbers you need to use.
Find the volume of a rectangular prism
You can use the formula V = l × w × h to find the
volume of a rectangular prism when you know
the length, width, and height of the prism.
STEP 1
2 21 in.
Sketch the rectangular prism.
4 in.
© Houghton Mifflin Harcourt Publishing Company • Image Credits: (t) ©Studio Eye/Corbis
5 in.
STEP 2 Identify the value for each variable.
The length l is 5 in.
The width w is _ in.
The height h is _ in.
STEP 3 Evaluate the formula.
Write the formula.
V=l×w×h
Replace l with 5, w with
V=_×_×_
_ , and h with _.
V = _ in.3
Multiply.
So, _ in.3 of food fits in the box.
Math
Talk
MATHEMATICAL PRACTICES 2
Reasoning How do you
know what units to use for
the volume of the box?
Chapter 11
629
connect You know that the volume of a rectangular prism is
the product of its length, width, and height. Since the product of
the length and width is the area of one base, the volume is also the
product of the area of one base and the height.
Volume of a Prism
Volume = area of one base × height
V = Bh
Example 1 Find the volume of the prism.
STEP 1 Identify the value for each variable.
2 1 in.
4
The height h is _ in.
9 in.2
The area of the base B is _ in.
2
STEP 2 Evaluate the formula.
Write the formula.
V = Bh
Replace B with _ and h with _.
V=
×
V=
× ___
V=
=
Write the mixed number as a fraction
greater than 1.
Multiply and write the product as a mixed
number.
4
1
__ in.3
4
So, the volume of the prism is __.
Find the volume of the cube.
Write the formula. The area of the square
base is s2. The height of a cube is also s, so
V = Bh = s3.
V = s3
V=
Substitute _ for s.
Write the mixed number as a fraction greater
than 1. Then use repeated multiplication.
Simplify.
So, the volume of the cube is __.
630
V=
( )
( ) ( )( )( )
3
3
=
V = _____ = 42___ ft3
8
8
3 21 ft
3 21 ft
3 21 ft
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Example 2
Name
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MATH
M
BOARD
B
Find the volume.
1.
2.
V = lwh
25 in.
10 21 in.
15 in.
V=_×_×_
3
8 in.
3
in.
8
V = __ in.3
3
in.
8
____
Math
Talk
MATHEMATICAL PRACTICES 1
Describe the steps for finding
the volume of a cube.
On
On Your
Your Own
Own
Find the volume of the prism.
3.
4.
12 21 ft
8 21 ft
© Houghton Mifflin Harcourt Publishing Company
7.
5
in.
16
3 1 yd2
1 1 yd
3
5
in.
16
3
6 yd
6 21 ft
____
6.
5.
5
in.
16
____
____
DEEPER
Wayne’s gym locker is a rectangular prism with a width and height of 14 1_2 inches.
The length is 8 inches greater than the width. What is the volume of the locker?
SMARTER
Abraham has a toy box that is in the
shape of a rectangular prism.
3 feet
333_4 ft3.
The volume is
351_ ft3.
2
641_ ft3.
2
1
4 2 feet
1
2 2 feet
Chapter 11 • Lesson 6 631
Aquariums
Large public aquariums like the Tennessee Aquarium in Chattanooga
have a wide variety of freshwater and saltwater fish species from around
the world. The fish are kept in tanks of various sizes.
The table shows information about several tanks in the aquarium.
Each tank is a rectangular prism.
Find the length of Tank 1.
V = lwh
Aquarium Tanks
Length
52,500 = l × _ × _
52,500 = l × __
52,500
_______
=l
_=l
Tank 1
Width
Height
Volume
30 cm
35 cm
52,500 cm3
4m
384 m3
Tank 2
12 m
Tank 3
18 m
12 m
Tank 4
72 cm
55 cm
2,160 m3
40 cm
So, the length of Tank 1 is __.
Solve.
8. Find the width of Tank 2 and the height of Tank 3.
10.
SMARTER
To keep the fish healthy, there should be the correct
ratio of water to fish in the tank. One recommended ratio is 9 L
of water for every 2 fish. Find the volume of Tank 4. Then use the
equivalencies 1 cm3 = 1 mL and 1,000 mL = 1 L to find how many
fish can be safely kept in Tank 4.
MATHEMATICAL
PRACTICE
2 Use Reasoning Give another set of dimensions for a
tank that would have the same volume as Tank 2. Explain how you
found your answer.
632
M t Show Your Work
Math
© Houghton Mifflin Harcourt Publishing Company • Image Credits: (t) ©Comstock/Corbis
9.
WRITE
Practice and Homework
Name
Lesson 11.6
Volume of Rectangular Prisms
COMMON CORE STANDARD—6.G.A.2
Solve real-world and mathematical problems
involving area, surface area, and volume.
Find the volume.
1.
1
94m
1
3 m
5m 4
V = lwh
2.
2 in.
_ × 91
_
V = 5 × 31
4
1
5 2 in.
16
2
4 1 mm
2
____
5.
4.
6 ft
7 1 ft
2
2 1 ft
4 1 mm
1
2 2 in.
4
5
V = 150__
m3
3.
4 1 mm
2
____
6.
2 41 ft
4 1m
2
8 m2
6 ft
2
2 41 ft
____
____
____
Problem
Problem Solving
Solving
7. A cereal box is a rectangular prism that is
8 inches long and 2 1_2 inches wide. The volume of
the box is 200 in.3. What is the height of the box?
© Houghton Mifflin Harcourt Publishing Company
_______
9.
WRITE
8. A stack of paper is 8 1_2 in. long by 11 in. wide
by 4 in. high. What is the volume of the stack
of paper?
_______
Math Explain how you can find the side length of a
rectangular prism if you are given the volume and the two other
measurements. Does this process change if one of the measurements
includes a fraction?
Chapter 11
633
Lesson Check (6.G.A.2)
1. A kitchen sink is a rectangular prism with a
length of 19 7_8 inches, a width of 14 3_4 inches, and
height of 10 inches. Estimate the volume of
the sink.
TEST
PREP
2. A storage container is a rectangular prism that
is 65 centimeters long and 40 centimeters wide.
The volume of the container is 62,400 cubic
centimeters. What is the height of the
container?
3. Carrie started at the southeast corner of
Franklin Park, walked north 240 yards, turned
and walked west 80 yards, and then turned and
walked diagonally back to where she started.
What is the area of the triangle enclosed by the
path she walked?
4. The dimensions of a rectangular garage are
100 times the dimensions of a floor plan of the
garage. The area of the floor plan is 8 square
inches. What is the area of the garage?
5. Shiloh wants to create a paper-mâché box
shaped like a rectangular prism. If the box will
be 4 inches by 5 inches by 8 inches, how much
paper does she need to cover the box?
6. A box is filled with 220 cubes with a side length
of 1_2 unit. What is the volume of the box in
cubic units?
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Spiral Review (6.G.A.1, 6.G.A.2, 6.G.A.4)
PROBLEM SOLVING
Name
Lesson 11.7
Problem Solving • Geometric Measurements
Essential Question How can you use the strategy use a formula to solve
problems involving area, surface area, and volume?
Geometry—6.G.A.4 Also 6.G.A.1,
6.G.A.2
MATHEMATICAL PRACTICES
MP1, MP2
Unlock
Unlock the
the Problem
Problem
Shedd Aquarium in Chicago has one of the country’s few
full-scale animal hospitals linked to an aquarium. One tank for
sick fish is a rectangular prism measuring 75 cm long, 60 cm
wide, and 36 cm high along the outside. The glass on the tank
is 2 cm thick. How much water can the tank hold? How much
water is needed to fill the tank?
2 cm
2 cm
2 cm
36 cm
2 cm
75 cm
Use the graphic organizer to help you solve the problem.
60 cm
Read the Problem
What do I need to find?
I need to find _____and
_______.
What information do I need to use?
I need to use _____ and
_______.
How will I use the information?
© Houghton Mifflin Harcourt Publishing Company
First I will decide _____.
Solve the Problem
• Choose the measure that specifies the amount
of water that will fill a tank.
______
• Choose an appropriate formula.
______
• Subtract the width of the glass twice from the
length and width and once from the height to
find the inner dimensions.
Find the length.
75 cm − 4 cm = _ cm
Find the width.
60 cm − 4 cm = _ cm
Find the height.
36 cm − 2 cm = _ cm
Then I will choose a ___ I can
use to calculate the measure. Finally, I will
substitute the values for the ___,
and I will ___ the formula.
• Substitute and evaluate.
V = 71 × _ = _ = __ cm3
Math
Talk
So, the volume of the tank is ____.
MATHEMATICAL PRACTICES 6
Explain Why is volume the
correct measure to use to solve
the problem?
Chapter 11
635
Try Another Problem
Alexander Graham Bell, the inventor of the telephone, also
invented a kite made out of “cells” shaped like triangular pyramids.
A kite is made of triangular pyramid-shaped cells with fabric
covering one face and the base of the
pyramid. The face and base both have
20 cm
heights of 17.3 cm and side lengths of 20
17.3 cm
cm. How much fabric is needed to make
20 cm
one pyramid cell?
Read the Problem
Solve the Problem
What do I need to find?
What information do I need to use?
So, _ cm2 of fabric is needed.
• Explain how you knew which units to use for your answer.
636
Math
Talk
MATHEMATICAL PRACTICES 1
Analyze How did the strategy
of using a formula help you
solve the problem?
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How will I use the information?
Unlock the Problem
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MATH
M
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B
1. An aquarium tank in the shape of a rectangular
prism is 60 cm long, 30 cm wide, and 24 cm high.
The top of the tank is open, and the glass used to
make the tank is 1 cm thick. How much water can
the tank hold?
√
√
√
Draw a diagram.
Identify the measure needed.
Choose an appropriate formula.
First identify the measure and choose an appropriate
formula.
Next find the inner dimensions and replace the
variables with the correct values.
The Louvre Museum in Paris, France
Finally, evaluate the formula.
WRITE
Math t Show Your Work
So, the tank can hold ___ of
water.
© Houghton Mifflin Harcourt Publishing Company • Image Credits: (t) ©Cephas Picture Library/Alamy
2.
What if, to provide greater strength,
the glass bottom were increased to a thickness of 4 cm?
How much less water would the tank hold?
SMARTER
3. An aquarium tank in the shape of a rectangular prism
is 40 cm long, 26 cm wide, and 24 cm high. If the top of
the tank is open, how much tinting is needed to cover
the glass on the tank? Identify the measure you used to
solve the problem.
4. The Louvre Museum in Paris, France, has a square
pyramid made of glass in its central courtyard. The
four triangular faces of the pyramid have bases of 35
meters and heights of 27.8 meters. What is the area of
glass used for the four triangular faces of the pyramid?
Chapter 11 • Lesson 7 637
MATHEMATICAL PRACTICES
COMMUNICA5&t1&34E7&3&tCONSTRUCT ARGUMENTS
On
On Your
Your Own
Own
5.
SMARTER
A rectangular prism-shaped block of wood measures
3 m by 11_ m by 11_ m. How much of the block must a carpenter carve
2
2
1
1
away to obtain a prism that measures 2 m by _ m by _ m?
2
6.
2
DEEPER
The carpenter (Problem 5) varnished the outside of the
smaller piece of wood, all except for the bottom, which measures 1_ m
2
by 1_ m. Varnish costs $2.00 per square meter. What was the cost of
2
varnishing the wood?
7. A wax candle is in the shape of a cube with a side length of 21_ in. What
2
volume of wax is needed to make the candle?
8.
MATHEMATICAL
PRACTICE
1 Describe A rectangular prism-shaped box measures
6 cm by 5 cm by 4 cm. A cube-shaped box has a side length of 2 cm.
How many of the cube-shaped boxes will fit into the rectangular prismshaped box? Describe how you found your answer.
Personal Math Trainer
638
SMARTER
Justin is covering the outside of an open shoe box
with colorful paper for a class project. The shoe box is 30 cm long, 20 cm wide,
and 13 cm high. How many square centimeters of paper are needed to cover
the outside of the open shoe box? Explain your strategy.
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9.
Practice and Homework
Name
Problem Solving • Geometric Measurements
Lesson 11.7
COMMON CORE STANDARD—6.G.A.4
Solve real-world and mathematical problems
involving area, surface area, and volume.
Read each problem and solve.
1. The outside of an aquarium tank is 50 cm long, 50 cm wide,
and 30 cm high. It is open at the top. The glass used to
make the tank is 1 cm thick. How much water can the
tank hold?
l = 50 − 2 = 48, w = 50 − 2 = 48, h = 30 − 1 5 29
V=l×w×h
= 48 3 48 × 29
© Houghton Mifflin Harcourt Publishing Company
= 66,816
66,816 cm3
_____
2. Arnie keeps his pet snake in an open-topped glass cage. The
outside of the cage is 73 cm long, 60 cm wide, and 38 cm high.
The glass used to make the cage is 0.5 cm thick. What is the
inside volume of the cage?
_____
3. A display number cube measures 20 in. on a side. The sides
are numbered 1–6. The odd-numbered sides are covered in
blue fabric and the even-numbered sides are covered in red
fabric. How much red fabric was used?
_____
4. The caps on the tops of staircase posts are shaped like square
pyramids. The side length of the base of each cap is 4 inches.
The height of the face of each cap is 5 inches. What is the
surface area of the caps for two posts?
_____
5. A water irrigation tank is shaped like a cube and has a side
length of 2 1_2 feet. How many cubic feet of water are needed to
completely fill the tank?
6.
WRITE
_____
Math Write and solve a problem for which you use
part of the formula for the surface area of a triangular prism.
Chapter 11
639
TEST
PREP
Lesson Check (6.G.A.4)
1. Maria wants to know how much wax she
will need to fill a candle mold shaped like a
rectangular prism. What measure should
she find?
2. The outside of a closed glass display case
measures 22 inches by 15 inches by 12 inches.
The glass is 1_2 inch thick. How much air is
contained in the case?
Spiral Review (6.G.A.1, 6.G.A.2, 6.G.A.3, 6.G.A.4)
3. A trapezoid with bases that measure
5 centimeters and 7 centimeters has
a height of 4.5 centimeters. What is the
area of the trapezoid?
4. Sierra has plotted two vertices of a rectangle at
(3, 2) and (8, 2). What is the length of the side of
the rectangle?
5. What is the surface area of the
square pyramid?
6. A shipping company has a rule that all packages
must be rectangular prisms with a volume of no
more than 9 cubic feet. What is the maximum
measure for the height of the a box that has a
width of 1.5 feet and a length of 3 feet?
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11 m
4m
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Chapter 11 Review/Test
Personal Math Trainer
Online Assessment
and Intervention
1. Elaine makes a rectangular pyramid from paper.
rectangle.
rectangles.
The base is a trapezoid. The lateral faces are
triangle.
squares.
triangles.
2. Darrell paints all sides except the bottom of the box shown below.
12 cm
15 cm
20 cm
Select the expressions that show how to find the surface area
that Darrell painted. Mark all that apply.
A
240 + 240 + 180 + 180 + 300 + 300
B
2(20 × 12) + 2(15 × 12) + (20 × 15)
C
(20 × 12) + (20 × 12) + (15 × 12)+ (15 × 12)+ (20 × 15)
D
20 × 15 × 12
3. A prism is filled with 44 cubes with 1_2 -unit side lengths. What is the
volume of the prism in cubic units?
© Houghton Mifflin Harcourt Publishing Company
_ cubic units
4. A triangular pyramid has a base with an area of 11.3 square meters, and
lateral faces with bases of 5.1 meters and heights of 9 meters.
Write an expression that can be used to find the surface area of the
triangular pyramid.
Assessment Options
Chapter Test
Chapter 11
641
5. Jeremy makes a paperweight for his mother in the shape of a square
pyramid. The base of the pyramid has a side length of 4 centimeters,
and the lateral faces have heights of 5 centimeters. After he finishes, he
realizes that the paperweight is too small and decides to make another
one. To make the second pyramid, he doubles the length of the base in
the first pyramid.
For numbers 5a–5c, choose Yes or No to indicate whether the statement
is correct.
5a.
The surface area of the second
pyramid is 144 cm2.
Yes
No
5b.
The surface area doubled from the
first pyramid to the second pyramid.
Yes
No
5c. The lateral area doubled from the
first pyramid to the second pyramid.
Yes
No
6. Identify the figure shown and find its surface area. Explain how you
found your answer.
16 in.
9 in.
9 in.
2 in.
221 in.
321 in.
8 in.3
2 in.
The volume of the box is 21_ in. × 31_ in. ×
2
642
2
21_ in.
2
31_ in.
2
=
171_ in.3
2
35 in.3
© Houghton Mifflin Harcourt Publishing Company
7. Dominique has a box of sewing buttons that is in the shape of a
rectangular prism.
Name
8. Emily has a decorative box that is shaped like a cube with a height of
5 inches. What is the surface area of the box?
_ in.2
9. Albert recently purchased a fish tank for his home. Match each
question with the geometric measure that would be most appropriate
for each scenario.
How much water can
the fish tank hold?
•
•
The area of the base of
the fish tank
How much material
would it take to cover the
entire fish tank?
•
•
The surface area of the
fish tank
How much space would
the fish tank occupy on
the table?
•
•
The volume of the
fish tank
10. Select the expressions that show the volume of the rectangular prism.
Mark all that apply.
2 1 units
2
2 units
A
2(2 units × 21_ units) + 2(2 units × 1_ unit) + 2(1_ unit × 21_ units)
C
2
2(2 units × 1_ unit) + 4(2 units × 21_ units)
2
2
2 units × 1_ unit × 2_1 units
2
2
D
2.5 cubic units
B
© Houghton Mifflin Harcourt Publishing Company
1 unit
2
2
2
2
Chapter 11
643
11. For numbers 11a–11d, select True or False for the statement.
B
2 units
A
2 units
C
D
4 units
F
E
5 units
4 units
11a.
The area of face A is 8 square units.
True
False
11b.
The area of face B is 10 square units.
True
False
11c.
The area of face C is 8 square units.
True
False
11d.
The surface area of the prism
is 56 square units.
True
False
12. Stella received a package in the shape of a rectangular prism. The box
has a length of 2 1_2 feet, a width of 1 1_2 feet, and a height of 4 feet.
Part A
Stella wants to cover the box with wrapping paper. How much paper will
she need? Explain how you found your answer.
Part B
© Houghton Mifflin Harcourt Publishing Company
Can the box hold 16 cubic feet of packing peanuts? Explain how
you know.
644
Name
13. A box measures 6 units by 1_2 unit by 2 1_2 units.
For numbers 13a–13b, select True or False for the statement.
13a.
The greatest number of cubes
with a side length of 1_2 unit that
can be packed inside the box is 60.
True
False
13b.
The volume of the box is
7 1_2 cubic units.
True
False
14. Bella says the lateral area of the square pyramid is 1,224 in.2 Do you
agree or disagree with Bella? Use numbers and words to support your
answer. If you disagree with Bella, find the correct answer.
25 in.
18 in.
15.
18 in.
DEEPER
Lourdes is decorating a toy box for her sister. She will use
self-adhesive paper to cover all of the exterior sides except for the
bottom of the box. The toy box is 4 feet long, 3 feet wide, and 2 feet high.
How many square feet of adhesive paper will Lourdes use to cover
the box?
© Houghton Mifflin Harcourt Publishing Company
16. Gary wants to build a shed shaped like a rectangular prism in his
backyard. He goes to the store and looks at several different options.
The table shows the dimensions and volumes of four different sheds.
Use the formula V = l × w × h to complete the table.
Length (ft)
Shed 1
Width (ft)
10
Shed 2
18
Shed 3
12
4
Shed 4
10
12
Height (ft)
Volume (ft3)
8
960
10
2,160
288
10
Chapter 11
645
17. Tina cut open a cube-shaped microwave box to see the net. How many
square faces does this box have?
_ square faces
18. Charles is painting a treasure box in the shape of a rectangular prism.
Which nets can be used to represent Charles’ treasure box?
Mark all that apply.
A
C
B
D
Personal Math Trainer
SMARTER
Julianna is lining the inside of a basket with fabric.
The basket is in the shape of a rectangular prism that is 29 cm long,
19 cm wide, and 10 cm high. How much fabric is needed to line the
inside of the basket if the basket does not have a top? Explain your
strategy.
© Houghton Mifflin Harcourt Publishing Company
19.
646
12
Chapter
Data Displays and Measures
of Center
Personal Math Trainer
Show Wha t You Know
Online Assessment
and Intervention
Check your understanding of important skills.
Name
Read a Bar Graph Use the bar graph
Math Test Scores
to answer the questions.
2. Who has a score between 70 and 80?
3. What is the difference between the highest
Test score
1. Who has the highest test score?
and lowest scores?
100
90
80
70
60
50
40
30
20
10
0
Laura
Max Shauna
Student
Tobias
Division Find the quotient.
© Houghton Mifflin Harcourt Publishing Company • Image Credits: (b) ©Tony Garcia/Getty Images
4. 35)‾
980
5. 16)‾
352
6. 24)‾
3,456
7. 42)‾
3,276
Compare Decimals Compare. Write <, >, or =.
8. 2.48
2.53
9. 0.3
0.04
10. 4.63
4.3
11. 1.7
1.70
Math in the
Kayla scored 110 in the first game she bowled, but she can’t
remember her score from the second game. The average of the
two scores is 116. Help Kayla figure out what her second score
was.
__
Chapter 12
647
Voca bula ry Builder
Visualize It
Review Words
Sort the review words into the chart.
bar graph
line graph
For this set of information…
the temperature
of an aquarium
over a 12-hour
period
the amount of
money earned by
three different
clothing stores
Preview Words
dot plot
frequency
histogram
mean
median
mode
outlier
statistical question
…I would draw this display.
Understand Vocabulary
Complete the sentences using the preview words.
1. A(n) ___ is a bar graph that shows the frequency of
data in specific intervals.
2. The ___ is the middle value when a data set with an
odd number of values is ordered from least to greatest.
3. A(n) ___ is a value that is much less or much
greater than the other values in a data set.
© Houghton Mifflin Harcourt Publishing Company
4. A(n) ___ is a number line with dots that show
the ___ of the values in a data set.
5. You can calculate the ___ of a data set by adding
the values and then dividing the sum by the number of values.
6. The item(s) that occurs most often in a data set is called the
___ of the data.
648
™Interactive Student Edition
™Multimedia eGlossary
Chapter 12 Vocabulary
bar graph
data
gráfica de barras
datos
18
5
dot plot
frequency
diagrama de puntos
frecuencia
36
26
frequency table
histogram
tabla de frecuencia
histograma
39
37
line graph
mean
gráfica lineal
media
50
55
90
60
Minutes
Frequency
90
60
45
30
60
30
4
60
45
60
60
60
45
4
45
30
45
60
45
60
9
90
2
A type of bar graph that shows the
frequencies of data in intervals
6
5
4
3
2
1
0
0–
9
10
–1
9
20
–2
9
30
–3
9
40
–4
9
Frequency
Library Visitors on a Saturday
Example:
Age (years)
The sum of a set of data items divided by the
number of data items
Example:
The mean of the data set 85, 59, 97, 71 is
85
+ 59 + 97 + 71 = 312
________________
____ = 78.
4
4
Cars
Trucks
SUVs
Number of
Automobiles
A graph that shows frequency of data
along a number line
Example:
1
2
3
4
Miles Jogged
A table that uses numbers to record data
about how often an event occurs
Joe’s Reading Times (minutes)
Joe’s Reading Times
30
60
30
90
60
Minutes Frequency
90
60
45
30
60
60
45
60
60
60
45
30
45
60
45
30
4
45
4
60
9
90
2
A graph that uses line segments to show
how data change over time
!VERAGE 0RICE OF 'OLD
Example:
30
400
300
200
100
0
60
Example:
30
Annual Sales at an
Automobile Dealership
$OLLARS PER OUNCE
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Joe’s Reading Times
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Joe’s Reading Times (minutes)
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The number of times an event occurs
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© Houghton Mifflin Harcourt Publishing Company
© Houghton Mifflin Harcourt Publishing Company
© Houghton Mifflin Harcourt Publishing Company
© Houghton Mifflin Harcourt Publishing Company
Information collected about people
or things, often to draw conclusions
about them
A graph that uses horizontal or vertical bars
to display countable data
9EAR
Chapter 12 Vocabulary
(continued)
measure of center
median
medida de
tendencia central
mediana
59
57
mode
outlier
moda
valor atipico
61
72
relative frequency
table
statistical question
tabla de frecuencia
relativa
pregunta estadística
91
97
A value much higher or much lower than
the other values in a data set
Example:
0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18
Team Absences (days)
A question that asks about a set of data
that can vary
Example:
How many desks are in each classroom
in my school?
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8, 17, 21, 23, 26, 29, 34, 40, 45
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Example:
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© Houghton Mifflin Harcourt Publishing Company
© Houghton Mifflin Harcourt Publishing Company
The middle value when a data set is written
in order from least to greatest, or the mean
of the two middle values when there is an
even number of items
A single value used to describe the middle
of a data set
Examples: mean, median, mode
The value(s) in a data set that occurs the
most often
Example: The mode of the data set
73, 42, 55, 77, 61, 55, 68 is 55.
A table that shows the percent of
time each piece of data occurs
Macy’s Biking
Number of Miles
Frequency
Relative
Frequency
1–5
4
20%
6–10
6
30%
11–15
7
35%
16–20
3
15%
Going Places with
Picture It
For 3 to 4 players
Materials
• timer
• sketch pad
How to Play
1. Take turns to play.
2. To take a turn, choose a math term from the Word Box. Do not say
the term.
3. Set the timer for 1 minute.
4. Draw pictures and numbers on the sketch pad to give clues about
the term.
5. The first player to guess the term before time runs out gets 1 point.
Words
Game
Game
Word Box
bar graph
data
dot plot
frequency
frequency table
histogram
line graph
mean
measure of center
median
mode
outlier
relative frequency
table
statistical question
If he or she can use the word in a sentence, they get 1 more point.
Then that player gets a turn.
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Image Credits: (all) ©PhotoDisc/Getty Images
6. The first player to score 10 points wins.
Chapter 12
648A
Journal
Jo
ou
urnal
The Write Way
Reflect
Choose one idea. Write about it.
• Write a statistical question and explain why your question is a statistical one.
• Write a paragraph that uses at least three of these words or phrases.
bar graph
data
frequency table
histogram
line graph
• Work with a partner to explain and illustrate three different ways to find a
measure of center for a set of data. Use a separate piece of paper for your
drawing.
© Houghton Mifflin Harcourt Publishing Company Image Credits: (all) ©PhotoDisc/Getty Images
• Explain what is most important to understand about an outlier.
648B
Lesson 12.1
Name
Recognize Statistical Questions
Essential Question How do you identify a statistical question?
Statistics and Probability—
6.SP.A.1
MATHEMATICAL PRACTICES
MP1, MP2
If you measure the heights of your classmates, you are collecting
data. A set of data is a set of information collected about people
or things. A question that asks about a set of data that can vary is
called a statistical question.
“What are the heights of my classmates on July 1?” is a statistical
question because height usually varies in a group of people. “What is
Sasha’s height on July 1?” is not a statistical question because it asks
for only one piece of information at one time.
Unlock
Unlock the
the Problem
Problem
The New England Aquarium in Boston is home to over
80 penguins. Which of the following is a statistical question a
biologist could ask about the penguins? Explain your reasoning.
A How much does the penguin named Pip weigh this morning?
B How much does the penguin named Pip weigh each morning
on 30 different days?
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Identify the statistical question.
Question A asks for Pip’s weight at __ time(s),
so it __ ask about a set of data that varies.
Question A __ a statistical question.
Question B asks for Pip’s weight at __ time(s), and it is
likely that Pip’s weight __ vary during this period.
Question B asks about a set of data that can vary, so it __ a
statistical question.
• Another biologist asks how old the penguin named Royal Pudding is.
Is this a statistical question? Explain your reasoning.
Chapter 12
649
A statistical question can ask about an entire set of data that can
vary or a value that describes that set of data. For example, “What is
the height of the tallest person in my class?” is a statistical question
because it will tell you the greatest value in a set of data that can
vary. You will learn other ways to describe a set of data later in this
chapter.
Example
Bongos are a kind of antelope that live in central Africa.
Bongos are unusual because both males and females have
horns. Write two statistical questions a biologist could ask
about a group of bongos.
1. What is the __ in inches of the horns on the
bongo that has the __ horns in the group?
Different bongos will have different horn lengths. This
question asks about a value in a set of data that _
vary, so it _ a statistical question.
2. What is the weight of the __ bongo in the group?
about a value in a set of data that _ vary, so it _ a
statistical question.
Math
Talk
MATHEMATICAL PRACTICES 2
Reasoning Give a different
statistical question you could
ask about the heights of
students in your class.
Try This! Write a statistical question you could ask in the situations described below.
A A researcher knows the amount of
electricity used in 20 different homes on
a Monday.
650
B A museum director records the number of
students in each tour group that visits the
museum during one week.
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Different bongos will have different weights. This question asks
Name
Share
hhow
Share and
and Show
Sh
MATH
M
BOARD
B
Identify the statistical question. Explain your reasoning.
1. A. What was the low temperature in Chicago each day in March?
B. What was the low temperature in Chicago on March 7?
Question A asks for the low temperature at _ time(s),
and it is likely the temperature __.
Question B asks for the low temperature at _ time(s).
Question _ is a statistical question.
2. A. How long did it take you to get to school this morning?
B. How long did it take you to get to school each morning this week?
Write a statistical question you could ask in the situation.
3. A student recorded the number of pets in the households of 50 sixth-graders.
On
On Your
Your Own
Own
Math
Talk
Identify the statistical question. Explain your reasoning.
MATHEMATICAL PRACTICES 6
Explain How can you
determine whether a question
is a statistical question?
4. A. How many gold medals has Finland won at each of the last 10 Winter Olympics?
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B. How many gold medals did Finland win at the 2008 Winter Olympics?
Write a statistical question you could ask in the situation.
5. A wildlife biologist measured the length of time that 17 grizzly bears hibernated.
6. A doctor recorded the birth weights of 48 babies.
Chapter 12 • Lesson 1 651
MATHEMATICAL PRACTICES
ANALYZEt-00,'034536$563&t13&$*4*0/
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Use the table for 7 and 8.
7. Give a statistical question that you could ask about the
data recorded in the table.
9.
SMARTER
What statistical question
could “92 mi/hr” be the answer to?
MATHEMATICAL
PRACTICE
a new game. The manager must choose between a roleplaying game and an action game. He asks his sales staff
which of the last 10 released games sold the most copies.
Explain why this is a statistical question.
10.
11.
Name
Height (ft)
Maximum Speed
(mi/hr)
Rocket
256
83
Thunder Dolphin
281
87
Varmint
240
81
Screamer
302
92
DEEPER
Think of a topic. Record a set of data for the
topic. Write a statistical question that you could ask about
your data.
SMARTER
For numbers 11a–11d, choose Yes or No to indicate whether
the question is a statistical question.
● Yes
● No
11b. How many minutes did it take Madison to
complete her homework each night this week?
● Yes
● No
11c. How many more minutes did Andrew spend on
homework on Tuesday than on Thursday?
● Yes
● No
11d.
● Yes
● No
11a.
652
Roller Coaster Data
6 Explain A video game company will make
How many minutes did it take Ethan to complete
his homework last night?
What was the longest amount of time Abigail
spent on homework this week?
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8.
Practice and Homework
Name
Lesson 12.1
Recognize Statistical Questions
COMMON CORE STANDARD—6.SP.A.1
Develop understanding of statistical variability.
Identify the statistical question. Explain your reasoning.
1. A. How many touchdowns
did the quarterback throw
during the last game of the
season?
B. How many touchdowns did
the quarterback throw each
game of the season?
2. A. What was the
score in the first frame
of a bowling game?
3. A. How many hours of
television did you watch each
day this week?
B. What are the
scores in 10 frames
of a bowling game?
B. How many hours of
television did you watch on
Saturday?
B; the number of
touchdowns in each
game can vary.
Write a statistical question you could ask in the situation.
4. A teacher recorded the test scores of her students.
5. A car salesman knows how many of each model of a car was sold in a month.
Problem
Problem Solving
Solving
© Houghton Mifflin Harcourt Publishing Company
6. The city tracked the amount of waste that was
recycled from 2000 to 2007. Write a statistical
question about the situation.
8.
WRITE
7. The daily low temperature is recorded for
a week. Write a statistical question about
the situation.
Math Write three statistical questions that you could use to
gather data about your family. Explain why the questions are statistical.
Chapter 12
653
Lesson Check (6.SP.A.1)
1. Elise says that the question “Do you have any
siblings?” is a statistical question. Mark says
that “How many siblings do you have?” is a
statistical question. Who is correct?
2. Kate says that “What was the lowest amount
of precipitation in one month last year?” is a
statistical question. Mike says that “What is the
speed limit?” is a statistical question. Who is
correct?
Spiral Review (6.G.A.1, 6.G.A.2, 6.G.A.4)
3. A regular decagon has side lengths of
4 centimeters long. If the decagon is divided
into 10 congruent triangles, each has an
approximate height of 6.2 centimeters.
What is the approximate area of the decagon?
4. Mikki uses the net shown to make a solid figure.
5. A prism is filled with 30 cubes with 1_2 -unit side
lengths. What is the volume of the prism in
cubic units?
6. A tank in the shape of a rectangular prism has a
length of 22 inches, a width of 12 inches, and a
height of 15 inches. If the tank is filled halfway
with water, how much water is in the tank?
FOR MORE PRACTICE
GO TO THE
654
Personal Math Trainer
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What solid figure does Mikki make?
Lesson 12.2
Name
Describe Data Collection
Essential Question How can you describe how a data set was collected?
Statistics and Probability—
6.SP.B.5a, 6.SP.B.5b
MATHEMATICAL PRACTICES
MP3, MP5, MP6
Unlock
Unlock the
the Problem
Problem
One way to describe a set of data is by stating the number of observations,
or measurements, that were made. Another way is by listing the attributes
that were measured. An attribute is a property or characteristic of the
item being measured, such as its color or length.
Jeffrey’s hobby is collecting rocks and minerals. The chart gives data
on garnets he found during a recent mineral-hunting trip. Identify:
• The attribute being measured
Garnet Data
Garnet Mass (g)
Garnet Mass (g)
1
7.2
7
4.6
2
3.5
8
5.6
3
4.0
9
9.0
4
3.9
10
3.6
5
5.2
11
3.8
6
5.8
12
4.3
• The unit of measure
• The likely means by which measurements were made
• The number of observations
Describe the data set.
Think: What property or characteristic of the garnets did Jeffrey
measure?
© Houghton Mifflin Harcourt Publishing Company • Image Credits: (c) ©Greg C Grace/Alamy Images
• The attribute Jeffrey measured was the __ of the garnets.
• The unit used to measure the mass of the garnets was __ .
• To measure mass in grams, Jeffrey probably used a ___ .
• The number of observations Jeffrey made was __ .
1. Would Jeffrey likely have gotten the same data set if he had measured
a different group of garnets? Explain.
2. What other attributes of the garnets could Jeffrey have measured?
Chapter 12
655
Hands
On
Activity Collect a data set.
Materials ■ ruler
In this activity, you will work with other students to collect
data on the length of the students’ index fingers in your
group. You will present the data in a chart.
• Describe the attribute you will measure. What unit will
you use?
• Describe how you will make your measurements.
• Describe the data you will record in your chart.
• In the space at the right, make a chart of your data.
• How many observations did you make?
3.
MATHEMATICAL
PRACTICE
3 Make Arguments One of your classmates made 3
observations and another made 10 observations to answer a statistical
question. Who do you think arrived at a better answer to the statistical
question? Explain.
656
MATHEMATICAL PRACTICES 6
Explain What statistical
question does your data set
in the Activity answer?
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Math
Talk
Name
Share
hhow
Share and
and Show
Sh
MATH
M
BOARD
B
Describe the data set by listing the attribute measured, the unit
of measure, the likely means of measurement, and the number
of observations.
100-Meter Run Data
1. Greg’s 100-meter race results
Race Time (sec)
attribute: ___
unit of measure: ___
likely means by which measurements were taken: __
Race Time (sec)
1
12.8
5
13.5
2
12.5
6
13.7
3
12.9
7
12.6
4
13.4
number of observations: _
Daily Water Use (gal)
153.7 161.8 151.5 153.7 160.1
2. The Andrews family’s water use
161.9 155.5 152.3 166.7 158.3
155.8 167.5 150.8 154.6
Math
Talk
On
On Your
Your Own
Own
3. Practice: Copy and Solve Collect data on one of the topics listed
below. You may wish to work with other students. Make a chart of
your results. Then describe the data set.
• Weights of cereal boxes, soup
cans, or other items
• Numbers of family members
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4.
• Lengths of time to multiply
two 2-digit numbers
• Numbers of pets in families
SMARTER
Describe the data set by writing the
attribute measured, the unit of measure, the likely
means of measurement, and the number of
observations in the correct location on the chart.
21
yardstick
Attribute
Unit of
Measure
MATHEMATICAL PRACTICES 3
Apply Why do you think it
is important to make more
than one observation when
attempting to answer a
statistical question?
• Lengths of forearm (elbow to
fingertip)
• Numbers of pages in books
Heights of 6th Graders (in.)
50
58
56
60
58
52
50
53
54
61
48
59
48
59
55
59
62
49
57
56
61
Likely Means
of
Measurement
Number
of
Observations
inches
heights of 6th
graders
Chapter 12 • Lesson 2 657
Summarize
When you summarize a reading passage, you restate the most important
information in a shortened form. This allows you to understand more easily
what you have read. Read the followng passage:
A biologist is studying green anacondas. The green anaconda is the largest
snake in the world. Finding the length of any snake is difficult because the
snake can curl up or stretch out while being measured. Finding the length of
a green anaconda is doubly difficult because of the animal’s great size and
strength. The standard method for measuring a green anaconda is to calm
the snake, lay a piece of string along its entire length, and then measure the
length of the string. The table at the right gives data collected by the biologist
using the string method.
5.
MATHEMATICAL
PRACTICE
1 Analyze Summarize the passage in your own words.
Green Anaconda
Lengths (cm)
357.2 407.6 494.5 387.0
417.6 305.3 189.4 267.7
441.3 507.5 413.2 469.8
168.9 234.0 366.2 499.1
370.0 488.8 219.2
SMARTER
WRITE
Math t Show Your Work
Use your summary to name the attribute the biologist was
measuring. Describe how the biologist measured this attribute.
7. Give any other information that is important for describing the data set.
8.
658
DEEPER
Write the greatest green anaconda length that the biologist measured in feet.
Round your answer to the nearest foot. (Hint: 1 foot is equal to about 30 centimeters.)
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6.
Practice and Homework
Name
Lesson 12.2
Describe Data Collection
COMMON CORE STANDARDS—6.SP.B.5a,
6.SP.B.5b Summarize and describe
distributions.
Describe the data set by listing the attribute measured,
the unit of measure, the likely means of measurement,
and the number of observations.
1. Daily temperature
2. Plant heights
Daily High Temperature (°F)
Height of Plants (inches)
78
83
72
65
70
10.3
9.7
6.4
8.1
11.2
76
75
71
80
75
5.7
11.7
7.5
9.6
6.9
73
74
81
79
69
81
78
76
80
82
70
77
74
71
73
Attribute: daily temperature; unit of
measure: degrees Fahrenheit; means of
measurement: thermometer; number of
observations: 25
3. Cereal in boxes
4. Dog weights
Amount of Cereal in Boxes (cups)
Weight of Dogs (pounds)
8
7
8.5
5
5
5
6.5
6
8
8.5
7
7
9
8
8
9
22
17
34
23
19
18
20
20
© Houghton Mifflin Harcourt Publishing Company
Problem
Problem Solving
Solving
5. The table below gives the amount of time Preston
spends on homework. Name the likely means of
measurement.
7.
6. The table below shows the speed of cars on a
highway. Name the unit of measure.
Speeds of Cars (miles per hour)
Amount of Time Spent on Homework (hours)
71
55
53
65
68
61
59
62
5
70
69
57
50
56
66
67
63
3
WRITE
1
2
4
1
3
2
Math Gather data about the heights of your family
members or friends. Then describe how you collected the data set.
Chapter 12
659
Lesson Check (6.SP.B.5a, 6.SP.B.5b)
1. What is the attribute of the data set shown in
the table?
2. What is the number of observations of the data
set shown below?
Mass of Produce (grams)
Swim Times (min)
2.4
1.7
3.2
1.1
2.6
1.02
1.12
1.09
3.3
1.3
2.6
2.7
3.1
1.01
1.08
1.03
Spiral Review (6.G.A.1, 6.G.A.2, 6.G.A.4, 6.SP.A.1)
3. What is the area of the figure shown below?
7 cm
4 cm
4. Each base of a triangular prism has an area of
43 square centimeters. Each lateral face has
an area of 25 square centimeters. What is the
surface area of the prism?
7 cm
5. How much sand can this container hold?
10 in.
5 in.
6. Jay says that “How much does Rover weigh
today?” is a statistical question. Kim says
that “How long are the puppies' tails in the
pet store?” is a statistical question. Who is
NOT correct?
4 1 in.
2
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4.5 cm
Lesson 12.3
Name
Dot Plots and Frequency Tables
Statistics and Probability—
6.SP.B.4
MATHEMATICAL PRACTICES
MP4, MP5, MP6
Essential Question How can you use dot plots and frequency tables
to display data?
A dot plot is a number line with marks that show the frequency of
data. Frequency is the number of times a data value occurs.
Unlock
Unlock the
the Problem
Problem
Hannah is training for a walkathon. The table shows
the number of miles she walks each day. She has one
day left in her training. How many miles is she most
likely to walk on the last day?
• What do you need to find?
Make a dot plot.
Distance Hannah Walked (mi)
STEP 1
Draw a number line with an appropriate scale.
Numbers vary from _ to _, so use a scale
from 0 to 10.
4
2
9
3
3
5
5
1
6
2
5
2
5
4
5
4
9
3
2
4
STEP 2
For each piece of data, plot a dot above the number
that corresponds to the number of miles Hannah
walked.
Complete the dot plot by making the correct number
of dots above the numbers 5 through 10.
The number of miles Hannah walked most often is the
value with the tallest stack of dots. The tallest stack in
this dot plot is for
0
1
2
3
4
5
6
7
8
9
10
Distance Walked (mi)
© Houghton Mifflin Harcourt Publishing Company
_______.
So, the number of miles Hannah is most likely to
walk on the last day of her training is
A dot plot is sometimes called
a line plot.
__.
•
MATHEMATICAL
PRACTICE
5
Communicate Explain why a dot plot is useful for solving this problem.
Chapter 12
661
A frequency table shows the number of times each data value
or range of values occurs. A relative frequency table shows the
percent of time each piece of data or group of data occurs.
Example 1
Jill’s Workout Times (minutes)
Jill kept a record of her workout times. How many of
Jill’s workouts lasted exactly 90 minutes?
30
60
30
90
60
30
60
90
60
120
30
60
90
90
Make a frequency table.
60
120
60
60
60
30
30
120
30
120
60
120
60
120
STEP 1
Jill’s Workout Times
List the workout times in the first column.
STEP 2
Minutes
Frequency
30
7
Record the frequency of each time in the Frequency
column.
60
Complete the frequency table.
90
So, _ of Jill’s workouts lasted exactly 90 minutes.
120
Example 2
Ricardo’s Lap Swimming
The table shows the number of laps Ricardo
swam each day. What percent of the days did
Ricardo swim 18 or more laps?
Make a relative frequency table.
Determine equal intervals for the data.
List the intervals in the first column.
STEP 2
Count the number of data values
in each interval. Record this in the
Frequency column.
10
15
5
12
12
5
19
3
19
16
14
17
18
13
6
17
16
11
8
Ricardo’s Lap Swimming
Number of
Laps
Frequency
Relative
Frequency
3–7
4
20%
8–12
6
30%
13–17
7
There are 20 data values.
4
___
= 0.2 = 20%
20
6
___
= 0.3 = 30%
20
18–22
3
STEP 3
Divide each frequency by the total number
of data values. Write the result as a percent
in the Relative Frequency column.
Complete the relative frequency table.
So, Ricardo swam 18 or more laps on __ of the days.
662
Math
Talk
MATHEMATICAL PRACTICES 1
Describe How could you
find the percent of days on
which Ricardo swam 13 or
more laps?
© Houghton Mifflin Harcourt Publishing Company
STEP 1
10
Name
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hhow
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and Show
Sh
MATH
M
BOARD
B
For 1−4, use the data at right.
Daily Distance Lionel Biked (km)
1. Complete the dot plot.
0
1
2
3
4
5
6
7
8
9 10 11 12
3
5
12
2
1
8
5
8
6
3
11
8
6
4
10
10
9
6
6
6
5
2
1
2
3
2. What was the most common distance Lionel biked?
How do you know?
© Houghton Mifflin Harcourt Publishing Company • Image Credits: (t) ©Ben Blankenburg/Corbis
3. Make a frequency table. Use the intervals
1−3 km, 4−6 km, 7−9 km, and 10−12 km.
4. Make a relative frequency table. Use the same
intervals as in Exercise 3.
On
On Your
Your Own
Own
Practice: Copy and Solve For 5−9, use the table.
5. Make a dot plot of the data.
Gloria’s Daily Sit-Ups
6. Make a frequency table of the data with three intervals.
13
3
14
13
12
7. Make a relative frequency table of the data with
three intervals.
12
13
4
15
12
15
13
14
3
11
13
13
12
14
15
11
14
13
15
11
8.
9.
MATHEMATICAL
PRACTICE
Describe how you decided on the
intervals for the frequency table.
1
SMARTER
Could someone use the information in the frequency
table to make a dot plot? Explain.
Chapter 12 • Lesson 3 663
MATHEMATICAL PRACTICES .0%&-t3&"40/tM",&4&/4&
Unlock
Unlock the
the Problem
Problem
10.
Fitness Center Survey
SMARTER
The manager of a fitness center
asked members to rate the fitness center. The
results of the survey are shown in the frequency
table. What percent of members in the survey rated
the center as excellent or good?
Response
a. What do you need to find?
Frequency
Excellent
18
Good
15
Fair
21
Poor
6
b. How can you use relative frequency to help you solve the problem?
c. Show the steps you use to solve the
problem.
d. Complete the sentences.
The percent of members who rated the center
as excellent is _ .
The percent of members who rated the center
as good is _ .
M t Show Your Work
Math
The percent of members who rated the center
WRITE
as excellent or good is _ .
11.
DEEPER
Use the table above. What is the difference in percent of the
members in the survey that rated the fitness center as poor versus excellent?
Personal Math Trainer
SMARTER
Julie kept a record of the number of minutes she spent
reading for 20 days. Complete the frequency table by finding the frequency
and the relative frequency (%).
Julie’s Reading Times (min)
664
Julie’s Reading Times
Minutes
Frequency
Relative Frequency (%)
15
30
15
30
30
15
5
25
30
60
15
60
45
30
15
45
30
45
15
45
60
45
30
30
30
60
© Houghton Mifflin Harcourt Publishing Company
12.
Practice and Homework
Name
Lesson 12.3
Dot Plots and Frequency Tables
COMMON CORE STANDARD—6.SP.B.4
Summarize and describe distributions.
For 1–4, use the chart.
1. The chart shows the number of pages of a
novel that Julia reads each day. Complete the
dot plot using the data in the table.
Pages Read
12
14
12
18
20
15
15
19
12
15
14
11
13
18
15
10 11 12 13 14 15 16 17 18 19 20
15
17
12
11
15
Pages Read
2. What number of pages does Julia read
most often? Explain.
______________
3. Make a frequency table in the space below. Use
the intervals 10–13, 14–17, and 18–21.
4. Make a relative frequency table in the
space below.
© Houghton Mifflin Harcourt Publishing Company
Problem
Problem Solving
Solving
5. The frequency table shows the ages of the actors
in a youth theater group. What percent of the
actors are 10 to 12 years old?
_______
6.
WRITE
Actors in a Youth Theater Group
Age
Frequency
7–9
8
10–12
22
13–15
10
Math Explain how dot plots and frequency tables are alike
and how they are different.
Chapter 12
665
Lesson Check (6.SP.B.4)
1. The dot plot shows the number of hours
Mai babysat each week. How many hours is
Mai most likely to babysit?
2. The frequency table shows the ratings that a
movie received from online reviewers. What
percent of the reviewers gave the movie a 4-star
rating?
Movie Ratings
0
1
2
3
4
5
6
7
8
9 10
Hours Babysat
Rating
Frequency
1 star
2
2 stars
5
3 stars
7
4 stars
6
3. The dimensions of a rectangular playground are
50 times the dimensions of a scale drawing of
the playground. The area of the scale drawing
is 6 square feet. What is the area of the actual
playground?
4. A square pyramid has a base side length of
8 feet. The height of each lateral face is
12 feet. What is the surface area of the pyramid?
5. A gift box is in the shape of a rectangular prism.
The box has a length of 24 centimeters, a width
of 10 centimeters, and a height of 13 centimeters.
What is the volume of the box?
6. For a science experiment, Juanita records the
height of a plant every day in centimeters. What
is the attribute measured in her experiment?
FOR MORE PRACTICE
GO TO THE
666
Personal Math Trainer
© Houghton Mifflin Harcourt Publishing Company
Spiral Review (6.G.A.1, 6.G.A.2, 6.G.A.4, 6.SP.B.5b)
Lesson 12.4
Name
Histograms
Essential Question How can you use histograms to display data?
When there is a large number of data values, it is helpful to group
the data into intervals. A histogram is a bar graph that shows the
frequency of data in intervals. Unlike a bar graph, there are no gaps
between the bars in a histogram.
Statistics and Probability—
6.SP.B.4
MATHEMATICAL PRACTICES
MP4, MP6
Unlock
Unlock the
the Problem
Problem
Ages of Best Actor Winners, 1990–2009
Interpret the histogram.
The height of each bar shows how many data
values are in the interval the bar represents.
Frequency
The histogram shows the ages of winners of
the Academy Award for Best Actor from 1990
to 2009. How many winners were under
40 years old?
How many winners were 20–29 years old?
9
8
7
6
5
4
3
2
1
0
20–29
30–39
__
40–49
50–59
60–69
Age
Which other bar represents people under 40?
__
How many winners were 30–39 years old? __
To find the total number of winners who were under 40
years old, add the frequencies for the intervals 20−29
and 30−39.
_+_=_
© Houghton Mifflin Harcourt Publishing Company
So, _ of the winners were under 40 years old.
1.
MATHEMATICAL
PRACTICE
Use Graphs Explain whether it is possible to know from
the histogram if any winner was 37 years old.
4
Chapter 12
667
Example
Ages of Best Actress Winners
The table shows the ages of winners of the Academy
Award for Best Actress from 1986 to 2009. How many of
the winners were under 40 years old?
45
21
41
26
80
42
29
33
36
45
49
39
34
26
25
33
35
35
Make a histogram.
28
30
29
61
32
33
STEP 1
Interval
Make a frequency table using
intervals of 10.
Frequency
STEP 2
0
1
12
___ axis of the graph.
10
Frequency
Write a scale for the frequencies on
7
Ages of Best Actress Winners
Set up the intervals along the
The intervals must be all the same
size. In this case, every interval
includes 10 years.
20–29 30–39 40–49 50–59 60–69 70–79 80–89
8
6
the ___ axis.
4
STEP 3
2
Graph the number of winners in
each interval.
0
20–29
30–39
40–49
50–59
Age
60–69
70–79
80–89
STEP 4
Give the graph a title and label
the axes.
Complete the histogram by drawing the bars for
the intervals 60–69, 70–79, and 80–89.
To find the number of winners who were under 40 years old,
add the frequencies for the intervals 20–29 and 30–39.
So, _ of the winners were under 40 years old.
2.
668
MATHEMATICAL
PRACTICE
Explain how you can tell from the histogram which age
group has the most winners.
6
© Houghton Mifflin Harcourt Publishing Company
_+_=_
Name
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and Show
Sh
MATH
M
BOARD
B
For 1–3, use the data at right.
Ages of People at a Health Club (yr)
1. Complete the frequency table for the age data in the
table at right.
Interval
Frequency
10–19 20–29 30–39 40–49
2
21
25
46
19
33
38
18
22
30
29
26
34
48
22
31
Ages of People at a Health Club
7
3. Use your histogram to find the number of people
at the health club who are 30 or older.
6
5
4
4.
Frequency
2. Complete the histogram for the data.
DEEPER
Use your histogram to determine the
percent of the people at the health club who are
20–29 years old.
3
2
1
0
Ages
Math
Talk
© Houghton Mifflin Harcourt Publishing Company • Image Credits: (b) ©Stockdisc/Getty Images
On
On Your
Your Own
Own
MATHEMATICAL PRACTICES 5
Use Tools Explain whether
you could use the histogram
to find the number of
people who are 25 or older.
Practice: Copy and Solve For 5–7, use the table.
5. Make a histogram of the data using the intervals 10–19,
20–29, and 30–39.
6. Make a histogram of the data using the intervals 10–14,
15–19, 20–24, 25–29, 30–34, and 35–39.
7.
MATHEMATICAL
PRACTICE
Weights of Dogs (lb)
16
20
15
24
32
33
26
30
15
21
21
12
19
21
37
10
39
21
17
35
Compare Explain how using different intervals
changed the appearance of your histogram.
6
Chapter 12 • Lesson 4 669
MATHEMATICAL PRACTICES
COMMUNICA5&t1&34E7&3&tCONSTRUCT ARGUMENTS
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OqnakdlRnkuhmf¤@ookhb`shnmr
The histogram shows the hourly salaries, to
the nearest dollar, of the employees at a small
company. Use the histogram to solve 8–11.
Hourly Salaries
14
8. How many employees make less than $20
per hour?
12
9.
DEEPER
How many employees work at
the company? Explain how you know.
Frequency
10
8
6
4
2
10.
11.
Pose a
Problem Write and solve
a new problem that uses
the histogram.
SMARTER
MATHEMATICAL
PRACTICE
0
10–14 15–19 20–24 25–29 30–34 35–39 40–44
Hourly Salary ($)
Analyze Describe the overall shape of the histogram.
What does this tell you about the salaries at the company?
1
Personal Math Trainer
The frequency table shows the TV
ratings for the show American Singer. Complete the
histogram for the data.
Frequency
TV Ratings
670
6
5
4
3
2
1
0
14.1–14.5 14.6–15.0 15.1–15.5 15.6–16.0 16.1–16.5
Rating
TV ratings
Rating
Frequency
14.1-14.5
2
14.6-15.0
6
15.1-15.5
6
15.6-16.0
5
16.1-16.5
1
© Houghton Mifflin Harcourt Publishing Company
12.
SMARTER
Practice and Homework
Name
Lesson 12.4
Histograms
COMMON CORE STANDARD—6.SP.B.4
Summarize and describe distributions.
For 1–4 use the data at right.
1. Complete the histogram for the data.
Scores on a Math Test
2. What do the numbers on the y-axis represent?
_______
85
87
69
90
82
75
74
76
84
87
99
65
75
76
83
87
91
83
92
69
3. How many students scored from 60 to 69?
Scores on a Math Test
_______
Frequency
4. Use your histogram to find the number of
students who got a score of 80 or greater.
Explain.
_______
_______
_______
9
8
7
6
5
4
3
2
1
0
60–69 70–79 80–89 90–99
Score
Problem
Problem Solving
Solving
Ages of Customers at a Restaurant
For 5–6, use the histogram.
______
© Houghton Mifflin Harcourt Publishing Company
6. How many customers are in the
restaurant? How do you know?
______
______
7.
WRITE
Frequency
5. For which two age groups are there
the same number of customers?
16
14
12
10
8
6
4
2
0
0–9
10–19 20–29 30–39 40–49 50–59
Age
Math Write a letter to another student that explains how to
make a histogram and what type of data a histogram displays.
Chapter 12
671
Lesson Check (6.SP.B.4)
Amount Spent at Museum
Gift Shop
Frequency
1. The histogram shows the amount, to the
nearest dollar, that customers spent at a
museum gift shop. How many customers
spent less than $20?
2. Use the histogram in Problem 1. How many
customers bought something at the gift shop?
9
8
7
6
5
4
3
2
1
0
0–9
10–19 20–29 30–39 40–49
Amount ($)
Spiral Review (6.G.A.2, 6.G.A.3, 6.SP.B.4)
5. DeShawn is using this frequency table to make
a relative frequency table. What percent should
he write in the Relative Frequency column for
5 to 9 push-ups?
4. A rectangular swimming pool can hold
1,408 cubic feet of water. The pool is 22 feet
long and has a depth of 4 feet. What is the
width of the pool?
DeShawn’s Daily Push-Ups
Number of Push-Ups
Frequency
0−4
3
5−9
7
10−14
8
15−19
2
FOR MORE PRACTICE
GO TO THE
672
Personal Math Trainer
© Houghton Mifflin Harcourt Publishing Company
3. Marguerite drew a rectangle with vertices
A(−2, −1), B(−2, −4), and C(1, −4). What are the
coordinates of the fourth vertex?
Name
Mid-Chapter Checkpoint
Personal Math Trainer
Vocabulary
Vocabulary
Online Assessment
and Intervention
Vocabulary
Choose the best term from the box to complete the sentence.
1. A ____ is a kind of bar graph that shows
the frequency of data grouped into intervals. (p. 667)
dot plot
histogram
statistical question
2. A question that asks about a set of data that varies is called a
____ . (p. 649)
Concepts
Concepts and
and Skills
Skills
3. A sports reporter records the number of touchdowns scored each
week during the football season. What statistical question could
the reporter ask about the data? (6.SP.A.1)
4. Flora records her pet hamster’s weight once every week for one year.
How many observations does she make? (6.SP.B.5a)
5.
DEEPER
The number of runs scored by a baseball team in 20
games is given below. Draw a dot plot of the data and use it to find the
most common number of runs scored in a game. (6.SP.B.4)
© Houghton Mifflin Harcourt Publishing Company
Runs Scored
3
1
4
3
4
2
1
7
2
3
5
3
2
9
4
3
2
1
1
4
0
1
2
3
4
5
6
7
8
9
10
Number of Runs Scored
Chapter 12
673
6. Write a statistical question you could ask about a set of data that shows
the times visitors arrived at an amusement park. (6.SP.A.1)
7. A school principal is trying to decide how long the breaks should be
between periods. He plans to time how long it takes several students to
get from one classroom to another. Name a tool he could use to collect
the data. (6.SP.B.5b)
8. The U.S. Mint uses very strict standards when making coins. On a
tour of the mint, Casey asks, “How much copper is in each penny?”
Lenny asks, “What is the value of a nickel?” Who asked a statistical
question? (6.SP.A.1)
9. Chen checks the temperature at dawn and at dusk every day for a week
for a science project. How many observations does he make? (6.SP.B.5a)
674
Song Lengths (sec)
166
157
153
194
207
150
175
168
209
206
151
201
187
162
152
209
194
168
165
156
© Houghton Mifflin Harcourt Publishing Company
10. The table shows the lengths of the songs played by a radio station
during a 90-minute period. Alicia is making a histogram of the
data. What frequency should she show for the interval 160–169
seconds? (6.SP.B.4)
Lesson 12.5
Name
Mean as Fair Share and Balance Point
Essential Question How does the mean represent a fair share and
balance point?
Statistics and Probability—
6.SP.B.5c
MATHEMATICAL PRACTICES
MP1, MP2, MP8
Hands
On
Investigate
Investigate
Materials ■ counters
On an archaeological dig, five students found 1, 5, 7, 3,
and 4 arrowheads. The students agreed to divide the
arrowheads evenly. How many arrowheads should
each student get?
A. Use counters to show how many arrowheads each of
the five students found. Use one stack of counters for
each student.
B. Remove a counter from the tallest stack and move it to
the shortest. Keep moving counters from taller stacks to
shorter stacks until each stack has the same height.
C. Count the number of counters in each stack.
The number of counters in each stack is the mean, or average, of the
data. The mean represents the number of arrowheads each student
should get if the arrowheads are shared equally.
© Houghton Mifflin Harcourt Publishing Company • Image Credits: (r) ©Houghton Mifflin Harcourt
There are 5 stacks of _ counters.
Math
Talk
So, each student should get _ arrowheads.
MATHEMATICAL PRACTICES 8
Generalize What is the
mean of the data set 3, 3, 3,
3, 3? Explain how you know.
Draw Conclusions
1. Explain what is “fair” about a fair share of a group of items.
2.
SMARTER
How could you find the fair share of arrowheads using
the total number of arrowheads and division?
Chapter 12
675
Make
Make Connections
Connections
The mean can also be seen as a kind of balance point.
Ms. Burnham’s class holds a walk-a-thon to help raise
money to update the computer lab. Five of the students
walked 1, 1, 2, 4, and 7 miles. The mean distance walked
is 3 miles.
Complete the dot plot of the data set.
0
1
2
3
4
5
6
7
8
Distance Walked (mi)
Circle the number that represents the mean.
Complete the table to find the distances of the data
points from the mean.
Values Less than the Mean
Values Greater than the Mean
Data point
1 mi
1 mi
mi
4 mi
mi
Distance from the mean
2 mi
mi
mi
mi
mi
The total distance from the mean for values less than the mean is:
2 miles + 2 miles + 1 mile = _ miles
The total distance from the mean for values greater than the mean is:
_ mile + _ miles = _ miles
The total distance of the data values less than the mean is __ the
total distance of the data values greater than the mean. The mean represents
a balance point for data values less than the mean and greater than the mean.
4.
MATHEMATICAL
PRACTICE
8 Generalize Can all of the values in a data set be greater
than the mean? Explain why or why not.
676
© Houghton Mifflin Harcourt Publishing Company
3. Explain how you found the distance of each data value from the mean.
Name
Share
hhow
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and Show
Sh
MATH
M
BOARD
B
Use counters to find the mean of the data set.
1. On the first day of a school fundraiser, five students sell 1, 1, 2, 2, and
4 gift boxes of candy.
Make _ stacks of counters with heights 1, 1, 2, 2, and 4.
Rearrange the counters so that all _ stacks have the same
height.
After rearranging, every stack has _ counters.
So, the mean of the data set is _ .
Make a dot plot for the data set and use it to check whether
the given value is a balance point for the data set.
2. Rosanna’s friends have 0, 1, 1, 2, 2, and 12 pets at home.
Rosanna says the mean of the data is 3. Is Rosanna
correct?
0
1
2
3
4
5
6
7
8
9
10 11 12
Number of Pets
The total distance from 3 for data values less than 3 is _ .
The total distance from 3 for data values greater than 3 is _ .
The mean of 3 __ a balance point.
So, Rosanna __ correct.
OqnakdlRnkuhmf¤@ookhb`shnmr
OqnakdlRnkuhmf¤@ookhb`shnmr
© Houghton Mifflin Harcourt Publishing Company
3.
DEEPER
Four people go to lunch, and the costs of their orders
are $6, $9, $10, and $11. They want to split the bill evenly. Find each
person’s fair share. Explain your work.
Chapter 12 • Lesson 5 677
MATHEMATICAL PRACTICES
ANALYZEt-00,'034536$563&t13&$*4*0/
Use the table for 4–6.
4. A grocer is preparing fruit baskets to sell as holiday
presents. If the grocer rearranges the apples in baskets A,
B, and C so that each has the same number, how many
apples will be in each basket? Use counters to find the
fair share.
5.
MATHEMATICAL
PRACTICE
3 Make Arguments Can the pears be
rearranged so that there is an equal whole number of
pears in each basket? Explain why or why not.
Fruit Baskets
7.
Oranges
Pears
A
4
2
2
B
1
2
1
C
4
2
5
SMARTER
Use counters to find the mean of the
number of pears originally in baskets B and C. Draw a
dot plot of the data set. Use your plot to explain why the
mean you found is a balance point.
SMARTER
Four friends go to breakfast and the
costs of their breakfasts are $5, $8, $9, and $10. Select
True or False for each statement.
7a.
678
Apples
The mean of the cost of the breakfasts
can be found by adding each of the
costs and dividing that total by 4.
True
False
7b. The mean cost of the four breakfasts is $10.
True
False
7c. The difference between the greatest cost
and the mean is $2.
True
False
7d. The difference between the least cost
and the mean is $2.
True
False
© Houghton Mifflin Harcourt Publishing Company • Image Credits: (t) ©Camille Moirenc/Corbis
6.
Basket
Practice and Homework
Name
Lesson 12.5
Mean as Fair Share and Balance Point
Use counters to find the mean of the data set.
1. Six students count the number of buttons on their shirts.
The students have 0, 4, 5, 2, 3, and 4 buttons.
6
Make
COMMON CORE STANDARD—6.SP.B.5c
Summarize and describe distributions.
stacks of counters with heights 0, 4, 5, 2, 3, and 4.
Rearrange the counters so that all
6
stacks have the same height.
After rearranging, every stack has
3
counters.
3
So, the mean of the data set is
.
2. Four students completed 1, 2, 2, and 3 chin-ups.
Make a dot plot for the data set and use it to check whether the given
value is a balance point for the data set.
3. Sandy’s friends ate 0, 2, 3, 4, 6, 6, and 7 pretzels.
Sandy says the mean of the data is 4. Is Sandy correct?
The total distance from 4 for
.
values less than 4 is
The total distance from 4
for values greater than 4 is
. The mean of 4
0
1
2
3
4
5
6
7
8
9
10
a balance point.
So, Sandy
correct.
Problem
Problem Solving
Solving
© Houghton Mifflin Harcourt Publishing Company
4. Three baskets contain 8, 8, and 11 soaps.
Can the soaps be rearranged so that there is an
equal whole number of soaps in each basket?
Explain why or why not.
6.
WRITE
5. Five pages contain 6, 6, 9, 10, and 11 stickers.
Can the stickers be rearranged so that there is an
equal whole number of stickers on each page?
Explain why or why not.
Math Describe how to use counters to find the mean of a
set of data. Give a data set and list the steps to find the mean.
Chapter 12
679
Lesson Check (6.SP.B.5c)
1. What is the mean of 9, 12, and 15 stamps?
2. Four friends spent $9, $11, $11, and $17 on
dinner. If they split the bill equally, how much
does each person owe?
Spiral Review (6.G.A.4, 6.SP.B.5b)
3. What figure does the net below represent?
4. Sarah paints the box below. She paints the
whole box except for the front face. What area of
the box does she paint?
front
9 cm
20 cm
Temperature at Noon
Monday
80°F
Tuesday
84°F
Wednesday
78°F
Thursday
90°F
Friday
80°F
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5. Chloe collected data and then displayed her
results in the table to the right. What is the unit
of measure of the data?
7 cm
Lesson 12.6
Name
Measures of Center
Statistics and Probability—
6.SP.B.5c Also 6.SP.A.2, 6.SP.A.3
MATHEMATICAL PRACTICES
MP3, MP6, MP7
Essential Question How can you describe a set of data using mean,
median, and mode?
A measure of center is a single value used to describe the
middle of a data set. A measure of center can be a useful way to
summarize a data set, especially when the data set is large.
Unlock
Unlock the
the Problem
Problem
Kara made a paper airplane. She flew her airplane 6 times
and recorded how long it stayed in the air during each
flight. The times in seconds for the flights are 5.8, 2.9, 6.7,
1.6, 2.9, and 4.7. What are the mean, median, and mode
of the data?
What unit of time is used in the problem?
__
How many flight times are given?
__
Find the mean, median, and mode.
The mean is the sum of the data
items divided by the number of
data items.
The median is the middle value
when the data are written in
order. If the number of data items
is even, the median is the mean
of the two middle values.
5.8 + 2.9 + 6.7 + 1.6 + 2.9 + 4.7 = _________ =
Mean = ____________________________
Order the values from least to greatest.
1.6, 2.9, 2.9, 4.7, 5.8, 6.7
The data set has an __ number of values, so the
median is the mean of the two middle values. Circle the two
middle values of the data set.
Now find the mean of the two middle values.
© Houghton Mifflin Harcourt Publishing Company
+
____________________
+ _________ =
The mode is the data value or
values that occur most often.
_ occurs twice, and all the other values occur once.
__ is the mode. Math
Talk
Try This! In 2009, an engineer named Takuo Toda set
a world record for flight time for a paper airplane. His
plane flew for 27.9 sec. If Toda’s time was included
in Kara’s set of times, what would the median be?
MATHEMATICAL PRACTICES 5
Use Tools How could you
use a dot plot and the idea
of a balance point to check
your answer for the mean?
__
Chapter 12
681
Example 1
Mrs. O’Donnell’s class has a
fundraiser for a field trip to a wildlife preservation. Five of the
donations are $15, $25, $30, $28, and $27. Find the mean, median,
and mode of the donations.
+
+
+
+
Mean = _______________________________________________
= ________ =
Order the data from least to greatest to find the median.
_, _, _, _, _
Median = _
If all of the values in a data set occur with equal frequency, then the data
set has no mode.
The data set has no repeated values, so there is no __.
Example 2
Keith surveys his classmates about how
many brothers and sisters they have. Six of the responses were
1, 3, 1, 2, 2, and 0. Find the mean, median, and mode of the data.
+
+
+
+
+
Mean = _________________________________________________________
= ________ =
Order the data from least to greatest to find the median.
The number of data values is even, so find the mean of the two middle values.
+
Median = __________________
= ________ =
The data values _ and _ appear twice in the set. If two or
more values appear in the data set the most number of times, then the
data set has two or more modes.
Modes = _ and _
682
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_ , _ , _ , _ , _, _
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1. Terrence records the number of e-mails he receives per day.
During one week, he receives 7, 3, 10, 5, 5, 6, and 6 e-mails.
What are the mean, median, and mode of the data?
Mean = __
Median = __
Mode(s) = __
2. Julie goes to several grocery stores and researches the price
of a 12 oz bottle of juice. Find the mean, median, and mode
of the prices shown.
Mean = __
Median = __
Juice Prices
$0.95
$1.09
$0.99
$1.25
$0.99
$1.99
Mode(s) = __
Math
Talk
On
On Your
Your Own
Own
3. T.J. is training for the 200-meter dash event for his school’s
track team. Find the mean, median, and mode of the
times shown in the table.
Mean = __
© Houghton Mifflin Harcourt Publishing Company
4.
5.
MATHEMATICAL
PRACTICE
Median = __
MATHEMATICAL PRACTICES 6
Explain how you can find
the median of a set of data
with an even number
of values.
T.J.’s Times (sec)
22.3
22.4
24.5
22.5
23.3
Mode(s) = __
6 Make Connections
Algebra The values of a data set can be
represented by the expressions x, 2x, 4x, and 5x. Write the data set for x = 3
and find the mean.
DEEPER
In the last six months, Sonia’s family used 456, 398, 655, 508,
1,186, and 625 minutes on their cell phone plan. To save money, Sonia’s
family wants to keep their mean cell phone usage below 600 minutes
per month. By how many minutes did they go over their goal in the
last six months?
Chapter 12 • Lesson 6 683
MATHEMATICAL PRACTICES .0%&-t3&"40/tM",&4&/4&
OqnakdlRnkuhmf¤@ookhb`shnmr
OqnakdlRnkuhmf¤@ookhb`shnmr
SMARTER
Sense or Nonsense?
6. Jeremy scored 85, 90, 72, 88, and 92 on five math tests, for a mean of 85.4.
On the sixth test he scored a 95. He calculates his mean score for all 6 tests as
shown below, but Deronda says he is incorrect. Whose answer makes sense?
Whose answer is nonsense? Explain your reasoning.
Jeremy’s Work
Deronda’s Work
The mean of my first 5 test scores was
85.4, so to find the mean of all 6 test scores,
I just need to find the mean of 85.4 and 95.
85.4 + 95 = 180.4
_____ = 90.2
Mean = _________
2
2
To find the mean of all 6 test scores, you
need to add up all 6 scores and divide by 6.
85 + 90 + 72 + 88 + 92 + 95
Mean = _________________________
6
522
____
So, my mean score for all 6 tests is 90.2.
= 6 = 87
So, Jeremy’s mean score for all 6 tests is 87.
WRITE
SMARTER
Alex took a standardized test 4 times. His test scores
were 16, 28, 24, and 32.
24.
25.
The mean of the test scores is
26.
24.
The median of the test scores is
26.
28.
16.
The mode of the test scores is
32.
no mode.
684
© Houghton Mifflin Harcourt Publishing Company
7.
M t Show Your Work
Math
Practice and Homework
Name
Lesson 12.6
Measures of Center
COMMON CORE STANDARD—6.SP.B.5c
Summarize and describe distributions.
Use the table for 1–4.
1. What is the mean of the data?
47
10 1 8 1 11 1 12 1 6
________________
5 __
5 5 9.4
5
Number of Points Blaine Scored
in Five Basketball Games
Game
Points Scored
1
10
2
8
3
11
4
12
5
6
9.4 points
_______
2. What is the median of the data?
_______
3. What is the mode(s) of the data?
_______
4. Suppose Blaine played a sixth game and scored
10 points during the game. Find the new mean,
median, and mode.
_______
_______
Problem
Problem Solving
Solving
© Houghton Mifflin Harcourt Publishing Company
5. An auto manufacturer wants their line of cars to
have a median gas mileage of 25 miles per gallon
or higher. The gas mileage for their five models
are 23, 25, 26, 29, and 19. Do their cars meet their
goal? Explain.
7.
6. A sporting goods store is featuring several new
bicycles, priced at $300, $250, $325, $780, and
$350. They advertise that the average price of
their bicycles is under $400. Is their ad correct?
Explain.
_______
_______
_______
_______
_______
_______
WRITE
Math Explain how to find the mean of a set of data.
Chapter 12
685
Lesson Check (6.SP.B.5c)
1. The prices for a video game at 5 different stores
are $39.99, $44.99, $29.99, $35.99, and $31.99.
What is the mode(s) of the data?
2. Manuel is keeping track of how long he
practices the saxophone each day. The table
gives his practice times for the past five days.
What is the mean of his practice times?
Manuel’s Practice Time
Day
Minutes Practiced
Monday
25
Tuesday
45
Wednesday
30
Thursday
65
Friday
30
Spiral Review (6.G.A.4, 6.SP.B.4, 6.SP.B.5c)
3. What is the surface area of the triangular prism
shown below?
15 cm
4. Kate records the number of miles that she bikes
each day. She displayed the number of daily
miles in the dot plot below. Each dot represents
the number of miles she biked in one day. How
many days did she bike 4–7 miles?
25 cm
9 cm
5. Six people eat breakfast together at a restaurant.
The costs of their orders are $4, $5, $9, $8, $6,
and $10. If they want to split the check evenly,
how much should each person pay?
0
1
2
3
4
5
6
7
8
9 10 11
Distance Biked
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12 cm
Lesson 12.7
Name
Effects of Outliers
Statistics and Probability—
6.SP.B.5d Also 6.SP.B.5c
MATHEMATICAL PRACTICES
MP2, MP3, MP4, MP6
Essential Question How does an outlier affect measures of center?
An outlier is a value that is much less or much greater than the other
values in a data set. An outlier may greatly affect the mean of a data
set. This may give a misleading impression of the data.
Unlock
Unlock the
the Problem
Problem
The table gives the number of days that the
24 members of the Garfield Middle School volleyball
team were absent from school last year.
• Why might a dot plot be helpful in
determining if there is an outlier?
Volleyball Team Absences (days)
4
6
7
4
5
5
3
6
6
7
3
5
8
16
5
4
5
6
5
7
6
4
5
4
Does the data set contain any outliers?
Use a dot plot to find the outlier(s).
STEP 1 Plot the data on the number line.
0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18
© Houghton Mifflin Harcourt Publishing Company • Image Credits: (t) ©Corbis
Team Absences (days)
STEP 2 Find any values that are much greater or much less than the other values.
Most of the data values are between _ and _ .
The value _ is much greater than the rest, so _ is an outlier.
1.
MATHEMATICAL
PRACTICE
6 Explain What effect do you think an outlier greater than
the other data would have on the mean of the data set? Justify your answer.
Chapter 12
687
Example
The high temperatures for the week in Foxdale,
in degrees Fahrenheit, were 43, 43, 45, 42, 26, 43, and 45. The mean of the
data is 41°F, and the median is 43°F. Identify the outlier and describe how the
mean and median are affected by it.
STEP 1 Draw a dot plot of the data and identify the outlier.
25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45
High Temperatures (°F)
The outlier is _ °F.
STEP 2 Find the mean and median of the temperatures without the outlier.
43 +
+
+
+
+
Mean = ___________________________________________________________
= _________ =
6
°F
Values ordered least to greatest: 42, _ , _ , _ , _ , _
+
_____________
Median = 43
=
2
°F
The mean with the outlier is _ °F, and the mean without the outlier is _ °F.
The outlier made the mean __.
The median with the outlier is _ °F, and the median without the outlier is _ °F.
The outlier __ affect the median.
2.
MATHEMATICAL
PRACTICE
2 Use Reasoning Explain why the mean without the outlier
3. If the outlier had been 59°F rather than 26°F, how would the mean have
been affected by the outlier? Explain your reasoning.
688
© Houghton Mifflin Harcourt Publishing Company
could be a better description of the data set than the mean with the outlier.
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1. Find the outlier by drawing a dot plot of the data.
Foul Shots Made
2
3
1
3
2
2
15
2
1
3
1
3
1
0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18
Foul Shots Made
The outlier is _.
2. The prices of the X-40 Laser Printer at five different stores are
$99, $68, $98, $105, and $90. The mean price is $92, and the
median price is $98. Identify the outlier and describe how the
mean and median are affected by it.
The outlier is _.
without the outlier: Mean = $_
Median = $_
Math
Talk
On
On Your
Your Own
Own
3. Identify the outlier in the data set of melon weights. Then
describe the effect the outlier has on the mean and median.
© Houghton Mifflin Harcourt Publishing Company
The outlier is _ oz.
4.
MATHEMATICAL
PRACTICE
MATHEMATICAL PRACTICES 2
Reasoning The mean of
a certain data set is much
greater than the median.
Explain how this can happen.
Melon Weights (oz)
47
45
48
45
49
14
45
51
46
47
47
2 Use Reasoning In a set of Joanne’s test scores, there is
an outlier. On the day of one of those tests, Joanne had the flu. Do you
think the outlier is greater or less than the rest of her scores? Explain.
Chapter 12 • Lesson 7 689
MATHEMATICAL PRACTICES
COMMUNICA5&t1&34E7&3&tCONSTRUCT ARGUMENTS
OqnakdlRnkuhmf¤@ookhb`shnmr
OqnakdlRnkuhmf¤@ookhb`shnmr
Use the table for 5–7.
5. Which player’s number of stolen bases is an outlier?
7.
DEEPER
What effect does the outlier have on the
median of the data set?
Player
Rickey Henderson
9.
690
1,406
Lou Brock
938
Billy Hamilton
914
Ty Cobb
897
Tim Raines
808
SMARTER
Miguel wrote that the mean
of the data set is 992.6. Is this the mean with or
without the outlier? Explain how you can tell
without doing a calculation.
▲ Ty Cobb steals a base.
WRITE
8.
Stolen Bases
Math t Show Your Work
SMARTER
Does an outlier have any effect on the
mode of a data set? Explain.
SMARTER
The prices of mesh athletic shorts at
five different stores are $9, $16, $18, $20, and $22. The
mean price is $17 and the median price is $18. Identify
the outlier and describe how the mean and median are
affected by it.
© Houghton Mifflin Harcourt Publishing Company • Image Credits: (r) ©MLB Photos via Getty Images
6.
Baseball All-Time
Stolen Base Leaders
Practice and Homework
Name
Lesson 12.7
Effects of Outliers
COMMON CORE STANDARD—6.SP.B.5d
Summarize and describe distributions.
1. Identify the outlier in the data set of students in each class. Then describe the effect
the outlier has on the mean and median.
Students in Each Class
30
22
26
21
24
28
23
26
28
12
12; Possible answer: The outlier decreases the mean from about 25.3 to 24. The outlier
decreases the median from 26 to 25.
2. Identify the outlier in the data set of pledge amounts. Then describe the effect the
outlier has on the mean and median.
Pledge Amounts
$100
$10
$15
$20
$17
$20
$32
$40
Problem
Problem Solving
Solving
© Houghton Mifflin Harcourt Publishing Company
3. Duke’s science quiz scores are 99, 91, 60, 94, and
95. Describe the effect of the outlier on the mean
and median.
5.
WRITE
4. The number of people who attended an art
conference for five days was 42, 27, 35, 39, and
96. Describe the effect of the outlier on the mean
and median.
Math Find or create a set of data that has an outlier. Find
the mean and median with and without the outlier. Describe the effect
of the outlier on the measures of center.
Chapter 12
691
Lesson Check (6.SP.B.5d)
1. What is the outlier for the data set?
2. The number of counties in several states is 64,
15, 42, 55, 41, 60, and 52. How does the outlier
change the median?
77, 18, 23, 29
Spiral Review (6.G.A.4, 6.SP.B.4, 6.SP.B.5b, 6.SP.B.5c)
3. Hector covers each face of the pyramid below
with construction paper. The area of the base of
the pyramid is 28 square feet. What area will he
cover with paper?
4. Mr. Stevenson measured the heights of several
students and recorded his findings in the
chart below. How many observations did
he complete?
Heights of Students (cm)
14 in.
8 in.
8 in.
160
138
148
155
159
154
155
140
135
144
142
162
170
171
8 in.
6. Sharon has 6 photo files on her computer.
The numbers below are the sizes of the files
in kilobytes. What is the median number of
kilobytes for the files?
69.7, 38.5, 106.3, 109.8, 75.6, 89.4
Lengths of Lizards (cm)
8
3
12
12
15
19
4
16
9
5
5
10
14
15
8
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5. Kendra is making a histogram for the data
in the chart. She uses the intervals 0–4, 5–9,
10–14, and 15–19. What should be the height of
the longest bar in her histogram?
Lesson 12.8
Name
Problem Solving • Data Displays
Statistics and Probability—6.SP.B.4
MATHEMATICAL PRACTICES
MP1, MP4, MP5
Essential Question How can you use the strategy draw a diagram to solve
problems involving data?
Unlock
Unlock the
the Problem
Problem
The 32 students in the History Club are researching
their family histories so they can draw family trees. The
data set at the right shows the numbers of
aunts and uncles the students have. What is the
most common number of aunts and uncles among the
students in the club?
Use the graphic organizer to help you solve the
problem.
Number of Aunts and Uncles
4
3
2
4
5
7
0
3
1
4
2
4
6
3
5
1
2
5
0
6
3
2
4
5
4
1
3
0
4
2
8
3
Read the Problem
What do I need to find?
I need to find the
___ number of
aunts and uncles among students
in the club.
What information do I
need to use?
How will I use the
information?
I need to use the number
I can draw a diagram that shows
of ___ each
student has from the table.
the __ of each value in
the data set. A good way to show the
frequency of each value in a data set
The most common number in the
is a __.
data is the __.
Solve the Problem
• Make a dot plot of the data.
© Houghton Mifflin Harcourt Publishing Company
Check: Are there the same number of dots on
the plot as there are data values?
• Use the plot to determine the mode. The mode is the data value
with the __ dots. The data value with the
most dots is _.
Math
Talk
So, the most common number of aunts and uncles is _ .
0 1 2 3 4 5 6 7 8 9 10
Number of Aunts and Uncles
MATHEMATICAL PRACTICES 3
Make Arguments Explain why displaying
the data in a dot plot is a better choice
for solving this problem than displaying
the data in a histogram.
Chapter 12
693
Try Another Problem
Attendance at 25 Pittsburgh
Pirates Games (in thousands)
The table shows the attendance for the Pittsburgh Pirates’
last 25 home games of the 2009 baseball season. What
percent of the games were attended by at least 25,000 people?
12
13
23
33
21
17
17
24
15
27
19
15
18
11
26
20
24
13
16
16
16
19
36
27
17
Read the Problem
What do I need to find?
What information do I
need to use?
How will I use the
information?
So, _ of the last 25 home games were attended by at least
25,000 people.
694
Math
Talk
MATHEMATICAL PRACTICES 1
Analyze What other type of display
might you have used to solve this
problem? Explain how you could
have used the display.
© Houghton Mifflin Harcourt Publishing Company
Solve the Problem
Name
Unlock the Problem
MATH
M
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B
1. The table shows the number of goals scored by the
Florida Panthers National Hockey League team in the
last 20 games of the 2009 season. What was the most
common number of goals the team scored?
√
Read the question carefully to be sure
you understand what you need to find.
√
Check that you plot every data value
exactly once.
√
Check that you answered the question.
Goals Scored
1
3
3
2
1
1
2
2
2
1
4
5
1
3
3
3
0
2
4
2
First, draw a dot plot of the data.
Next, use the plot to find the mode of the data: The
value _ appears _ times.
So, the most common number of goals the Panthers
0
1
2
© Houghton Mifflin Harcourt Publishing Company
3.
MATHEMATICAL
PRACTICE
4
5
Goals Scored
scored was _.
2. Draw a histogram of the hockey data. Use it to find the
percent of the games in which the Panthers scored more
than 3 goals.
3
WRITE
Math t Show Your Work
5 Use Appropriate Tools If you needed
to find the mean of a data set, which data display—dot
plot or histogram—would you choose? Explain your
reasoning.
Chapter 12 • Lesson 8 695
MATHEMATICAL PRACTICES
ANALYZEt-00,'034536$563&t13&$*4*0/
On
On Your
Your Own
Own
4.
SMARTER
Corey collected data on the ages of the parents
of his classmates. Make a data display and use it to find the percent
of parents at least 30 years old but under 50 years old.
42, 36, 35, 49, 52, 43, 41, 32, 45, 39, 50, 38, 27,
29, 37, 39
5. What is the mode of the data in Exercise 4?
MATHEMATICAL
PRACTICE
6 Explain An online retail store sold 500 electronic
FPO
devices in one week. Half of the devices were laptop computers and
20% were desktop computers. The remaining devices sold were
tablets. How many tablets were sold? Explain how you found
your answer.
7.
8.
696
DEEPER
A recipe for punch calls for apple juice and cranberry
juice. The ratio of apple juice to cranberry juice is 3:2. Tyrone
wants to make at least 20 cups of punch, but no more than 30 cups
of punch. Describe two different ways he can use apple juice and
cranberry juice to make the punch.
SMARTER
The data set shows the total points
scored by the middle school basketball team in the last
14 games. What is the most common number of points
scored in a game? Explain how to find the answer using
a dot plot.
Total Points Scored
42
36
35
49
52
43
41
32
45
39
50
38
37
39
© Houghton Mifflin Harcourt Publishing Company • Image Credits: (r) ©Brand X Pictures/Alamy
6.
Practice and Homework
Name
Lesson 12.8
Problem Solving • Data Displays
COMMON CORE STANDARD—6.SP.B.4
Summarize and describe data distributions.
Read each problem and solve.
1. Josie collected data on the number of siblings her
classmates have. Make a data display and determine
the percent of Josie’s classmates that have more than
2 siblings.
5, 1, 2, 1, 2, 4, 3, 2, 2, 6
0
1
2
3
4
5
6
7
8
9
10
40%
_______
2. The following data show the number of field goals a
kicker attempted each game. Make a data display and
tell which number of field goals is the mode.
4, 6, 2, 1, 3, 1, 2, 1, 5, 2, 2, 3
_______
3. The math exam scores for a class are shown below.
Make a data display. What percent of the scores are 90
and greater?
91, 68, 83, 75, 81, 99, 97, 80, 85, 70, 89, 92, 77, 95,
100, 64, 88, 96, 76, 88
_______
© Houghton Mifflin Harcourt Publishing Company
4. The heights of students in a class are shown below
in inches. Make a data display. What percent of the
students are taller than 62 inches?
63, 57, 60, 64, 59, 62, 65, 58, 63, 65, 58, 61, 63, 64
_______
5.
WRITE
Math Write and solve a problem for which you would
use a dot plot or histogram to answer questions about given data.
Chapter 12
697
TEST
PREP
Lesson Check (6.SP.B.4)
1. The number of student absences is shown
below. What is the mode of the absences?
2, 1, 3, 2, 1, 1, 3, 2, 2, 10, 4, 5, 1, 5, 1
2. Kelly is making a histogram of the number of
pets her classmates own. On the histogram, the
intervals of the data are 0–1, 2–3, 4–5, 6–7. What
is the range of the data?
Spiral Review (6.G.A.2, 6.SP.B.4, 6.SP.B.5c, 6.SP.B.5d)
3. The area of the base of the rectangular prism
shown below is 45 square millimeters. The
height is 51_ millimeters. What is the volume of
2
the prism?
4. The frequency table shows the number of runs
scored by the Cougars in 20 of their baseball
games. In what percent of the games did they
score 5 or fewer runs?
5 1 mm
2
45 mm2
5. There are 5 plates of bagels. The numbers
of bagels on the plates are 8, 10, 9, 10, and 8.
Shane rearranges the bagels so that each plate
has the same amount. How many bagels are
now on each plate?
698
Number of Runs
Frequency
0–2
14
3–5
3
6–8
2
9–11
1
6. By how much does the median of the data
set 12, 9, 9, 11, 14, 28 change if the outlier is
removed?
FOR MORE PRACTICE
GO TO THE
Personal Math Trainer
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Runs Scored by the Cougars
Name
Chapter 12 Review/Test
Personal Math Trainer
Online Assessment
and Intervention
1. The data set shows the total number of sandwiches sold each day for
28 days. What is the most common number of sandwiches sold in a day?
Number of sandwiches sold each day
10
14
11
12
19
13
24
12
12
18
9
17
15
20
20
21
10
13
13
16
19
21
22
18
13
15
14
10
2. Michael's teacher asks, "How many items were sold on the first day of
the fund raiser?" Explain why this is not a statistical question.
3. Describe the data set by writing the attribute measured, the unit
of measure, the likely means of measurement, and the number of
observations in the correct location on the chart.
Daily Temperature (ºF)
64
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7
53
thermometer
Attribute
Assessment Options
Chapter Test
61
39
36
degrees
Fahrenheit
Unit of Measure
43
48
daily
temperature
Likely Means of
Measurement
Number of
Observations
Chapter 12
699
4. The numbers of points scored by a football team in 7 different games are
26, 38, 33, 20, 27, 3, and 28. For numbers 4a–4c, select True or False to
indicate whether the statement is correct.
4a. The outlier in the data set is 3.
True
False
4b.
The difference between the
outlier and the median is 24.
True
False
4c.
The outlier in this set of data
affects the mean by increasing it.
True
False
5. Mr. Jones gave a quiz to his math class. The students’ scores are listed in
the table. Make a dot plot of the data.
Math Test Scores
40
100
90
40
70
70
90
80
50
70
60
90
70
60
80
100
70
50
80
90
90
80
70
80
90
70
50
60
70
80
90
100
Math Test Scores
6. Melanie scored 10, 10, 11, and 13 points in her last 4 basketball games.
10.
The mean of the test scores is
11.
10.
The median of the test scores is
10.5.
11.
10.
The mode of the test scores is
11.
no mode.
700
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13.
Name
7. The Martin family goes out for frozen yogurt to celebrate the last day
of school. The costs of their frozen yogurts are $1, $1, $2, and $4. Select
True or False for each statement.
7a. The mean cost for the frozen
yogurts can be found by adding
each cost and dividing that total by 4.
True
False
7b. The mean cost of the four frozen
yogurts is $2.
True
False
7c. The difference between the
greatest cost and the mean is $1.
True
False
7d. The difference between the least
cost and the mean is $1.
True
False
8. The histogram shows the amount of time students spent on homework
for the week. For numbers 8a–8d, choose True or False to indicate
whether the statement is correct.
Time Spent on Homework in
One Week
4
3
2
1
0
0–29
30–59
60–89
90–119 120–149
© Houghton Mifflin Harcourt Publishing Company
Minutes
8a. The number of students
that spent between 30 minutes
and 59 minutes on homework is 2.
True
False
8b. The greatest number of students
spent between 90 minutes and
119 minutes on homework.
True
False
8c. Five of the students spent less
than 60 minutes on homework for
the week.
True
False
8d. Six of the students spent 60 minutes
or more on homework for the week.
True
False
Chapter 12
701
9. The dot plot shows how many games of chess 8 different members of
the chess club played in one month. If Jackson is a new member of the
chess club, how many games of chess is he likely to play in one month?
Explain how the dot plot helped you find the answer.
0
1
2
3
4
5
6
7
8
9
10
Number of Games Played in One Month
10. Larry is training for a bicycle race. He records how far he rides each day.
Find the mode of the data.
Miles Larry Rides each Day
Monday
Tuesday
Wednesday
Thursday
Friday
Saturday
15
14
12
16
15
15
_
11.
DEEPER
The amounts of money Connor earned each week from
mowing lawns for 5 weeks are $12, $61, $71, $52, and $64. The mean
amount earned is $52 and the median amount earned is $61. Identify
the outlier and describe how the mean and median are affected by it.
Heights of Basketball Players
_
702
Inches
Frequency
60-69
3
70-79
6
80-89
3
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12. The frequency table shows the height, in inches, of 12 basketball players.
What fraction of the players are 70 inches or taller?
Name
13. A teacher surveys her students to find out how much time the students
spent eating lunch on Monday.
Monday Lunch Time (min)
hours
She uses
minutes
as the unit of measure.
seconds
15
18
18
14
15
20
16
15
15
19
15
19
14. For numbers 14a–14d, choose Yes or No to indicate whether the
question is a statistical question.
14a. What are the heights of the trees
in the park?
Yes
No
14b.
How old are the trees in the park?
Yes
No
14c. How tall is the cypress tree on
the north side of the lake this morning?
Yes
No
14d.
Yes
No
What are the diameters of the trees
in the park?
15. Five friends have 8, 6, 5, 2, and 4 baseball cards to divide equally among
themselves.
4
Each friend will get
5
cards.
6
16. The data set shows the ages of the members of the cheerleading squad.
What is the most common age of the members of the squad? Explain
how to find the answer using a dot plot.
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Ages of Cheerleaders (years)
7
8
8
11
13
12
14
12
10
11
9
11
9
10
11
12
13
14
15
Chapter 12
703
Personal Math Trainer
17.
SMARTER
The band director kept a record
of the number of concert tickets sold by 20 band
members. Complete the frequency table by
finding the frequency and the relative frequency.
Number of Concert Tickets Sold
4
6
6
7
7
8
8
9
9
9
8
11
12
11
13
15
14
18
20
19
Number of Concert Tickets Sold
Number of
Tickets Sold
Frequency
Relative
Frequency (%)
1-5
1
5
6-10
11-15
16-20
18. Gilbert is training for a marathon by running each week. The table
shows the distances, in miles, that he ran each week during the
first 7 weeks.
Week
1
2
3
4
5
6
7
Distance (miles)
8
10
9
10
15
18
21
Part A
Gilbert set a goal that the mean number of miles he runs in 7 weeks is
at least 14 miles. Did Gilbert reach his goal? Use words and numbers to
support your answer.
Suppose Gilbert had run 18 miles during week 5 and 22 miles during
week 6. Would he have reached his goal? Use words and numbers to
support your answer.
704
© Houghton Mifflin Harcourt Publishing Company
Part B
13
Chapter
Variability and
Data Distributions
Personal Math Trainer
Show Wha t You Know
Online Assessment
and Intervention
Check your understanding of important skills.
Name
Place the First Digit Tell where to place the first digit. Then divide.
1. 4qw
872
__ place
2. 8qw
256
__ place
Order of Operations Evaluate the expression.
3. 9 + 4 × 8
__
7. 23 × (22 ÷ 2)
__
4. 2 × 7 + 5
5. 6 ÷ (3 − 2)
__
6. (12 − 32) × 5
__
8. (8 − 2)2 − 9
__
9. (9 − 23) + 8
__
10. (27 + 9) ÷ 3
__
__
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Mean Find the mean for the set of data.
11. 285, 420, 345, 390 __
12. 0.2, 0.23, 0.16, 0.21, 0.2 __
13. $33, $48, $55, $52 __
14. 8.1, 7.2, 8.4 __
Math in the
Raina watched two of her friends play a game of darts.
She has to pick one of them to be her partner in a
tournament. Help Raina figure out which of her friends
is a more consistent dart player.
________
________
Dart Scores
Hector
15
5
7
19
3
19
Marin
12
10
11
11
10
14
Chapter 13
705
Voca bula ry Builder
Visualize It
Review Words
Sort the review words into the chart.
histogram
mean
Measures of Center
median
mode
Preview Words
box plot
How Do I Find It?
lower quartile
Find the sum of
all the data values
and divide the
sum by the
number of data
values.
Order the data
and find the
middle value or
the mean of the
two middle
values if the
number of
values is even.
Find the data
value(s) that
occurs most
often.
interquartile range
measure of
variability
range
upper quartile
Understand Vocabulary
Complete the sentences using the preview words.
1. The median of the upper half of a data set is the
___.
2. The ___ is the difference
between the greatest value and the least value in a data set.
© Houghton Mifflin Harcourt Publishing Company
3. A(n) ___ is a graph that shows the median,
quartiles, least value, and greatest value of a data set.
4. A data set’s ____ is the difference between
its upper and lower quartiles.
5. You can describe how spread out a set of data is using a(n)
____.
706
™Interactive Student Edition
™Multimedia eGlossary
Chapter 13 Vocabulary
box plot
interquartile range
diagrama de caja
rango intercuartil
45
8
lower quartile
mean absolute
deviation
primer cuartil
desviación absoluta
respecto a la media
54
measure of variability
median
medida de dispersión
mediana
59
58
range
upper quartile
rango
tercer cuartil
85
56
107
35
42
47
54
60
75
29
57
lower quartile
upper quartile
The interquartile range is 57 – 29 = 28.
The mean of the distances from each data
value in a set to the mean of the set
The middle value when a data set is written in
order from least to greatest, or the mean of
the two middle values when there is an even
number of items
Example:
8, 17, 21, 23, 26, 29, 34, 40, 45
The median of the upper half of a data set
Example:
24
26
32
35
42
47
54
60
75
29
57
lower quartile
upper quartile
© Houghton Mifflin Harcourt Publishing Company
32
© Houghton Mifflin Harcourt Publishing Company
26
© Houghton Mifflin Harcourt Publishing Company
24
© Houghton Mifflin Harcourt Publishing Company
© Houghton Mifflin Harcourt Publishing Company
© Houghton Mifflin Harcourt Publishing Company
© Houghton Mifflin Harcourt Publishing Company
© Houghton Mifflin Harcourt Publishing Company
The difference between the upper and
lower quartiles of a data set
A graph that shows how data are
distributed using the median, quartiles,
least value, and greatest value
Example:
8
0
19
10
26
20
37
30
45
40
50
The median of the lower half of a data set
Example:
24
26
32
35
42
47
54
60
75
29
57
lower quartile
upper quartile
A single value used to describe how the
values in a data set are spread out
Examples: range, interquartile range, mean
absolute deviation
The difference between the greatest and
least numbers in a data set
Example: The range of the data set
60, 35, 22, 46, 81, 39 is 81 − 22 = 59.
Going Places with
Words
Concentration
For 2–3 players
Materials
• 1 set of word cards
How to Play
1. Place the cards face-down on a table in
even rows. Take turns to play.
Game
Game
Word Box
box plot
interquartile range
lower quartile
mean absolute
deviation
measure of
variability
median
range
upper quartile
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2. Choose two cards and turn them face-up.
• If the cards show a word and its meaning, it’s a match. Keep the
pair and take another turn.
• If the cards do not match, turn them over again.
3. The game is over when all cards have been matched. The players
count their pairs. The player with the most pairs wins.
Chapter 13
706A
Journal
Jo
ou
urnal
The Write Way
Reflect
Choose one idea. Write about it.
• Summarize the information represented in a box plot.
• Explain in your own words what the mean absolute deviation is.
• Joe says that the median for a set of data is a measure of variability. Yossi
says that the range is. Identify who is correct and explain why.
© Houghton Mifflin Harcourt Publishing Company
Image Credits: (bg) ©Comstock/Getty Images; (t) ©PhotoDisc/Getty Images
• Describe how to calculate the interquartile range for a set of data.
706B
Lesson 13.1
Name
Patterns in Data
Essential Question How can you describe overall patterns in a data set?
connect Seeing data sets in graphs, such as dot plots and
histograms, can help you find and understand patterns in
the data.
Statistics and Probability—6.SP.B.5c
Also 6.SP.A.2
MATHEMATICAL PRACTICES
MP5, MP7, MP8
Unlock
Unlock the
the Problem
Problem
Many lakes and ponds contain freshwater fish species such
as bass, pike, bluegills, and trout. Jacob and his friends went
fishing at a nearby lake. The dot plot shows the sizes of the fish
that the friends caught. What patterns do you see in the data?
Fish Caught
• Circle any spaces with no data.
• Place a box around any groups
5
6
7
8
9
10
11
of data.
12 13 14
Length (inches)
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Analyze the dot plot.
A gap is an interval that contains no data.
A cluster is a group of data points that lie within a
small interval.
Does the dot plot contain any gaps?
There is a cluster from _ to _ and
If so, where? ____
another cluster from _ to _.
So, there were no fish from _ to _ inches long,
and there were two clusters of fish measuring from _
to _ inches long and from _ to _ inches long.
1. Summarize the information shown in the dot plot.
2.
MATHEMATICAL
PRACTICE
Math
Talk
MATHEMATICAL PRACTICES 7
Look for Structure What is the
mode(s) of the data? Explain how
you know.
8 Draw Conclusions What conclusion can you draw
about why the data might have this pattern?
Chapter 13
707
You can also analyze patterns in data that are displayed in histograms.
Some data sets have symmetry about a peak, while others do not.
Example
Analyze a histogram.
Erica made this histogram to show the weights of the
pumpkins grown at her father’s farm in October. What
patterns do you see in the data?
Pumpkin Weights
Number Grown
250
200
150
100
50
0
0–10
11–20 21–30 31–40
Weight (pounds)
41–50
STEP 1 Identify any peaks in the data.
The histogram has _ peak(s).
The interval representing the greatest number of pumpkins is for
weights between _ and _ pounds.
The number of pumpkins increases from 0 to _ pounds
and __ from 30 to 50 pounds.
STEP 3 Describe any symmetry the graph has.
If I draw a vertical line through the interval for _ to
_ pounds, the left and right parts of the histogram are very
close to being mirror images. The histogram __ line
symmetry.
So, the data values increase to one peak in the interval for _ to
_ pounds and then decrease. The data set __ line
symmetry about the peak.
708
A geometric figure has line
symmetry if you can draw a line
through it so that the two parts
are mirror images of each other.
© Houghton Mifflin Harcourt Publishing Company • Image Credits: (r) ©Siaukia/Alamy Images
STEP 2 Describe how the data changes across the intervals.
Name
Share
hhow
Share and
and Show
Sh
MATH
M
BOARD
B
For 1–3, use the dot plot.
1. The dot plot shows the number of paintings students in the art club
displayed at the art show. Does the dot plot contain any gaps?
If so, where? _______
1
2. Identify any clusters in the data.
2
3
4
5
6
7
Number of Paintings
_________
3. Summarize the information in the dot plot.
On
On Your
Your Own
Own
Monday Zoo Visitors
What patterns do you see in the
histogram data?
Number of Visitors
4.
DEEPER
______
______
______
225
200
175
150
125
100
75
50
25
0
0–9
______
10–19 20–29 30–39 40–49 50–59 60–69
Age (years)
______
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5.
SMARTER
The dot plot shows the number
of errors made by a baseball team in the first 16 games
of the season. For numbers 7a-7e, choose Yes or No to
indicate whether the statement is correct.
7a.
There is a gap from 4 to 5.
Yes
No
7b.
There is a peak at 0.
Yes
No
7c. The dot plot has line symmetry.
Yes
No
7d.
There are two modes.
Yes
No
7e.
There is one cluster.
Yes
No
0
1
2
3
4
5
6
7
Errors per Game
Chapter 13 • Lesson 1
709
Big Cats
There are 41 species of cats living in the world today. Wild
cats live in places as different as deserts and the cold forests
of Siberia, and they come in many sizes. Siberian tigers may
be as long as 9 feet and weigh over 2,000 pounds, while
bobcats are often just 2 to 3 feet long and weigh between 15
and 30 pounds.
You can find bobcats in many zoos in the United States. The histogram below shows the
weights of several bobcats. The weights are rounded to the nearest pound.
Bobcat Weights
Number
Num
mber of Bobcats
10
9
8
7
6
5
4
3
2
1
0
12–14
12
14
15
15–17
17
18
18–20
20 21
21–23
23 24
24–26
26 27
27–29
29
WRITE
Math
Weights (pounds)
t Show Your Work
6.
7.
710
MATHEMATICAL
PRACTICE
7 Look for a Pattern Describe the overall shape of the histogram.
Sense or Nonsense? Sunny says that the graph might
have a different shape if it was redrawn as a bar graph with one bar for each
number of pounds. Is Sunny’s statement sense or nonsense? Explain.
SMARTER
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Use the histogram for 6 and 7.
Practice and Homework
Name
Lesson 13.1
Patterns in Data
COMMON CORE STANDARD—6.SP.B.5c
Summarize and describe distributions.
For 1–2, use the dot plot.
1. The dot plot shows the number of omelets ordered at Paul’s
Restaurant each day. Does the dot plot contain any gaps?
Yes; from 12 to 13, and at 17
10
2. Identify any clusters in the data.
11
12
13
14 15 16 17 18 19
Omelets Ordered Per Day
For 3–4, use the histogram.
Store Visitors Per Day
3. The histogram shows the number of people that
visited a local shop each day in January. How many
peaks does the histogram have?
12
Frequency
4. Describe how the data values change across the intervals.
14
10
8
6
4
2
0
0–9
10–19
20–29
Number of Visitors
30–39
Problem
Problem Solving
Solving
5. Look at the dot plot at the right. Does the graph
have line symmetry? Explain.
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5
10 15 20 25 30 35 40 45 50
Gift Cards Purchased This Week
6.
WRITE
Math A histogram that shows the ages of students at a
library has intervals 1–5, 6–10, 11–15, 16–20, and 21–25. There is a peak
at 11–15 years and the graph is symmetric. Sketch what the histogram
could look like and describe the patterns you see in the data.
Chapter 13
711
Lesson Check (6.SP.B.5c)
1. What interval in the histogram has the greatest
frequency?
7
6
Frequency
5
4
3
2
1
0
0–4
5–9
10–14
15–19
20–24
2. Meg makes a dot plot for the data 9, 9, 4, 5, 5, 3,
4, 5, 3, 8, 8, 5. Where does a gap occur?
Spiral Review (6.G.A.2, 6.SP.B.4, 6.SP.B.5c)
3. A rectangular fish tank is 20 inches long,
12 inches wide, and 20 inches tall. If the tank is
filled halfway with water, how much water is in
the tank?
4. Look at the histogram below. How many
students scored an 81 or higher on the
math test?
Math Test Scores
14
10
8
6
4
2
0
61–70
71–80
81–90 91–100
Number of Students
FOR MORE PRACTICE
GO TO THE
712
Personal Math Trainer
© Houghton Mifflin Harcourt Publishing Company
5. The Little League coach uses a radar gun to
measure the speed of several of Kyle’s baseball
pitches. The speeds, in miles per hour, are
52, 48, 63, 47, 47. What is the median of Kyle’s
pitch speeds?
Frequency
12
Lesson 13.2
Name
Box Plots
Statistics and Probability—
6.SP.B.4
MATHEMATICAL PRACTICES
MP3, MP4, MP6
Essential Question How can you use box plots to display data?
The median is the middle value, or the mean of the two middle
values, when data is written in order. The lower quartile is the
median of the lower half of a data set, and the upper quartile is
the median of the upper half of a data set.
Unlock
Unlock the
the Problem
Problem
In 1885, a pair of jeans cost $1.50. Today, the cost of
jeans varies greatly. The chart lists the prices of jeans
at several different stores. What are the median, lower
quartile, and upper quartile of the data?
Prices of Jeans
$35 $28 $42 $50 $24 $75 $47 $32 $60
Find the median, lower quartile, and upper quartile.
STEP 1 Order the numbers from least to
greatest.
$24
$28
$32
$35
$42
STEP 2 Circle the middle number, the
median.
The median is $ _.
$47
$50
$60
$75
STEP 3 Calculate the upper and lower quartiles.
© Houghton Mifflin Harcourt Publishing Company • Image Credits: (r) ©Corbis
Find the median of each half of the data set.
When a data set has an odd
number of values, do not include
the median when finding the
lower and upper quartiles.
Think: If a data set has an even number of values, the
median is the mean of the two middle values.
$24
$28
$32
$35
$42
$47
lower quartile
2
$60
$75
upper quartile
$28 + $32 _______
_________
= $
=$
2
$50
_
$
+$
_______________
_______ = $
=$
2
2
_
So, the median is $_ , the lower quartile is $_, and the
upper quartile is $_.
Chapter 13
713
A box plot is a type of graph that shows how data are distributed by using
the least value, the lower quartile, the median, the upper quartile, and
the greatest value. Below is a box plot showing the data for jean prices
from the previous page.
20 25 30 35 40 45 50 55 60 65 70 75
Prices of Jeans (in dollars)
Example
Make a box plot.
The data set below represents the ages of the top ten finishers in a
5K race. Use the data to make a box plot.
Ages of Top 10 Runners (in years)
33
18
21
23
35
19
38
30
23
25
STEP 1 Order the data from least to greatest. Then find the median
and the lower and upper quartiles.
18, _ , _ , _ , _ , _ , _ , _ , _ , _
+
Median = _____________
= _ years
Lower quartile = _ years
The lower quartile is the median of the lower half of the
data set, which goes from 18 to 23.
Upper quartile = _ years
The upper quartile is the median of the upper half of the
data set, which goes from 25 to 38.
STEP 2 Draw a number line. Above the number line, plot a point
for the least value, the lower quartile, the median, the
upper quartile, and the greatest value.
15
20
25
30
35
40
Ages of Top Ten Runners
STEP 3 Draw a box from the lower to upper quartile. Inside the
box, draw a vertical line segment through the median.
Then draw line segments from the box to the least and
greatest values.
•
MATHEMATICAL
PRACTICE
6 Explain Would the box plot change if the data point for
38 years were replaced with 40 years? Explain.
714
Math
Talk
MATHEMATICAL PRACTICES 6
Describe the steps for making a box
plot.
© Houghton Mifflin Harcourt Publishing Company • Image Credits: (t) ©Image Source/Getty Images
2
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Find the median, lower quartile, and upper quartile of the data.
1. the scores of 11 students on a geography quiz:
87, 72, 80, 95, 86, 80, 78, 92, 88, 76, 90
72, 76, 78, 80, 80, 86, 87, 88, 90, 92, 95
Order the data from least to greatest.
median: _
lower quartile: _
upper quartile: _
2. the lengths, in seconds, of 9 videos posted online:
50, 46, 51, 60, 62, 50, 65, 48, 53
median: _
lower quartile: _
upper quartile: _
3. Make a box plot to display the data set in Exercise 2.
45
50
55
60
65
70
Lengths of Online Videos (seconds)
Math
Talk
On
On Your
Your Own
Own
Compare How are box plots and dot
plots similar? How are they different?
Find the median, lower quartile, and upper quartile of the data.
4. 13, 24, 37, 25, 56, 49, 43, 20, 24
MATHEMATICAL PRACTICES 6
5. 61, 23, 49, 60, 83, 56, 51, 64, 84, 27
median: _
median: _
lower quartile: _
lower quartile: _
upper quartile: _
upper quartile: _
6. The chart shows the height of trees in a park.
Display the data in a box plot.
5
© Houghton Mifflin Harcourt Publishing Company
Tree Heights (feet)
8
7.
12
MATHEMATICAL
PRACTICE
20
30
25
18
18
8
10
28
26
10
15
20
25
30
35
30
35
Tree Heights (feet)
29
1 Analyze Eric made this box plot
for the data set below. Explain his error.
5
Number of Books Read
5
13
22
8
31
37
25
24
10
10
15
20
25
Number of Books Read
Chapter 13 • Lesson 2 715
MATHEMATICAL PRACTICES .0%&-t3&"40/tM",&4&/4&
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SMARTER
Pose a Problem
8. The box plots show the number of flights delayed per day
for two different airlines. Which data set is more spread out?
Find the distance between the least and greatest values for
each data set.
Airline A: greatest value − least value =
_−_=_
0
Airline B: greatest value − least value =
5
10
15
20
Flights Delayed: Airline A
_−_=_
So, the data for __ is more spread out.
0
Write a new problem that can be solved using the
data in the box plots.
5
15
20
Flights Delayed: Airline B
Pose a Problem
9.
10
Solve Your Problem
SMARTER
The data set shows the cost of the dinner specials at a
restaurant on Friday night.
30
24
24
16
24
25
19
28
19.
The median is
24.
25.
716
18
19
26
26.
16.
The lower quartile is
18.
19.
The upper quartile is
28.
30.
© Houghton Mifflin Harcourt Publishing Company
Cost of Dinner Specials ($)
Practice and Homework
Name
Lesson 13.2
Box Plots
COMMON CORE STANDARD—6.SP.B.4
Summarize and describe distributions.
Find the median, lower quartile, and upper quartile of the data.
1. the amounts of juice in 12 glasses, in fluid ounces:
11, 8, 4, 9, 12, 14, 9, 16, 15, 11, 10, 7
Order the data from least to greatest: 4, 7, 8, 9, 9, 10, 11, 11, 12, 14, 15, 16
median:
10.5
8.5
lower quartile:
upper quartile:
13
2. the lengths of 10 pencils, in centimeters:
18, 15, 4, 9, 14, 17, 16, 6, 8, 10
median:
lower quartile:
upper quartile:
3. Make a box plot to display the data set in Exercise 2.
0
2
4
6
8
10 12 14 16 18 20
Lengths of Pencils (centimeters)
4. The numbers of students on several teams are 9, 4, 5, 10, 11, 9, 8, and 6.
Make a box plot for the data.
0
1
2
3
4
5
6 7 8 9 10 11 12
Number of Students on a Team
© Houghton Mifflin Harcourt Publishing Company
Problem
Problem Solving
Solving
5. The amounts spent at a gift shop today are
$19, $30, $28, $22, $20, $26, and $26. What is
the median? What is the lower quartile?
7.
WRITE
6. The weights of six puppies in ounces are 8,
5, 7, 5, 6, and 9. What is the upper quartile
of the data?
Math Draw a box plot to display this
data: 81, 22, 34, 55, 76, 20, 56.
Chapter 13
717
Lesson Check (6.SP.B.4)
1. The values in a data set are 15, 7, 11, 12, 6, 3, 10,
and 6. Where would you draw the box in a box
plot for the data?
2. What is the lower quartile of the following
data set?
22, 27, 14, 21, 22, 26, 18
Spiral Review (6.SP.A.1, 6.SP.B.5c, 6.SP.B.5d)
3. Jenn says that “What is the average number of
school lunches bought per day?” is a statistical
question. Lisa says that “How many lunches did
Mark buy this week?” is a statistical question.
Who is NOT correct?
4. The prices of several chairs are $89, $76, $81,
$91, $88, and $70. What is the mean of the
chair prices?
5. By how much does the mean of the following
data set change if the outlier is removed?
6. Where in the dot plot does a cluster occur?
50
51
52
53
54
55
56
57
58
FOR MORE PRACTICE
GO TO THE
718
Personal Math Trainer
© Houghton Mifflin Harcourt Publishing Company
13, 19, 16, 40, 12
Lesson 13.3
Name
Mean Absolute Deviation
Statistics and Probability—
6.SP.B.5c
MATHEMATICAL PRACTICES
MP2, MP4, MP6, MP8
Essential Question How do you calculate the mean absolute
deviation of a data set?
One way to describe a set of data is with the mean. However, two
data sets may have the same mean but look very different when
graphed. When interpreting data sets, it is important to consider how
far away the data values are from the mean.
Hands
On
Investigate
Investigate
Materials ■ counters, large number line from 0–10
The number of magazine subscriptions sold by two teams
of students for a drama club fundraiser is shown below.
The mean number of subscriptions for each team is 4.
Team A
3
3
4
Team B
5
5
0
1
4
7
8
A. Make a dot plot of each data set using counters for the
dots. Draw a vertical line through the mean.
B. Count to find the distance between each counter and
the mean. Write the distance underneath each counter.
© Houghton Mifflin Harcourt Publishing Company • Image Credits: (r) ©Houghton Mifflin Harcourt
Team A
Distance from
mean = 1
1
0
1
2
3
4
5
6
7
8
9
10
C. Find the mean of the distances for each data set.
Team A
1+
+
+
+
___________________________
5
= ____
=
5
Team B
+
+
+
+
______________________________
= ______ =
Chapter 13
719
Draw Conclusions
1.
SMARTER
Which data set, Team A or B, looks more spread
out in your dot plots? Which data set had a greater average distance
from the mean? Explain how these two facts are connected.
MATHEMATICAL
PRACTICE
2.
2 Reason Quantitatively The table shows
the average distance from the mean for the heights of
players on two basketball teams. Tell which set of heights
is more spread out. Explain how you know.
Heights of Players
Average Distance from
Mean (in.)
Team
Chargers
2.8
Wolverines
1.5
Make
Make Connections
Connections
The mean of the distances of data values from the mean of the data set
is called the mean absolute deviation. As you learned in the Investigation,
mean absolute deviation is a way of describing how spread out a data set is.
The dot plot shows the ages of gymnasts registered for the school team.
The mean of the ages is 10. Find the mean absolute deviation of the data.
Math
Talk
STEP 1 Label each dot with its distance from the mean.
Age of Gymnasts
MATHEMATICAL PRACTICES 3
8
9
10
11
12
13
14
Age (years)
STEP 2 Find the mean of the distances.
+
+
+
+
+
+
+
+
+
+
+
_________________________________________________________________________
So, the mean absolute deviation of the data is _ years.
720
= ______ =
© Houghton Mifflin Harcourt Publishing Company
Apply Is it possible for the mean
absolute deviation of a data set to
be zero? Explain.
Name
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Use counters, a dot plot, or iTools to find the mean absolute deviation of the data.
1. Find the mean absolute deviation for both data sets. Explain
which data set is more spread out.
the number of laps Shawna swam on 5 different
days:
the number of laps Lara swam on 5
different days:
5, 6, 6, 8, 10
1, 3, 7, 11, 13
mean = 7
mean = 7
2_________________________
+
+
+
+
= ____ =
mean absolute deviation = _ laps
mean absolute deviation = _ laps
The data set of __ laps is more spread out because the mean
absolute deviation of her data is __.
Use the dot plot to find the mean absolute deviation of the data.
2. mean = 7 books
3. mean = 29 pounds
Books Read Each Semester
Packages Shipped on Tuesday
26 27 28 29 30 31 32 33 34 35
4
5
6
7
8
9
10
11
Weight (pounds)
Number of Books
mean absolute deviation = __
© Houghton Mifflin Harcourt Publishing Company
4.
5.
mean absolute deviation = __
WRITE Math The mean absolute deviation of the number of
daily visits to Scott’s website for February is 167.7. In March, the
absolute mean deviation is 235.9. In which month did the number
of visits to Scott’s website vary more? Explain how you know.
MATHEMATICAL
PRACTICE
Write an Inequality Algebra In April, the data for Scott’s
website visits are less spread out than they were in February. Use a to
represent the mean absolute deviation for April. Write an inequality to
describe the possible values of a.
4
Chapter 13 • Lesson 3 721
MATHEMATICAL PRACTICES
COMMUNICA5&t1&34E7&3&tCONSTRUCT ARGUMENTS
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OqnakdlRnkuhmf¤@ookhb`shnmr
DEEPER
6.
Use the table.
Days of Precipitation
Jan
Feb
Mar
Apr
May
Jun
Jul
Aug
Sep
Oct
Nov
Dec
10
12
13
18
10
8
7
6
16
14
8
10
The mean of the data is 11. What is the mean absolute deviation of
the data?
7.
Suppose all of the players on a basketball team had
the same height. Explain how you could use reasoning to find the
mean absolute deviation of the players’ heights.
8.
MATHEMATICAL
PRACTICE
Explain Tell how an outlier that is much greater than
the mean would affect the mean absolute deviation of the data set.
Explain your reasoning.
6
SMARTER
The data set shows the number
of soccer goals scored by players in 3 games.
For numbers 9a–9c, choose Yes or No to indicate
whether the statement is correct.
9a.
722
Number of Goals Scored
Player A
1
2
1
Player B
2
2
2
Player C
3
2
1
The mean absolute deviation of Player A is 1.
Yes
No
9b. The mean absolute deviation of Player B is 0.
Yes
No
9c. The mean absolute deviation of Player C is greater
than the mean absolute deviation of Player A.
Yes
No
© Houghton Mifflin Harcourt Publishing Company • Image Credits: (t) ©Johner Images/Alamy Images
9.
SMARTER
Practice and Homework
Name
Lesson 13.3
Mean Absolute Deviation
COMMON CORE STANDARD—6.SP.B.5c
Summarize and describe distributions.
Use counters and a dot plot to find the mean
absolute deviation of the data.
1. the number of hours Maggie spent practicing
soccer for 4 different weeks:
2. the heights of 7 people in inches:
60, 64, 58, 60, 70, 71, 65
9, 6, 6, 7
mean 5 64 inches
mean 5 7 hours
2111110
_____________
4
4
__
5451
mean absolute deviation 5
1 hour
mean absolute deviation 5
Use the dot plot to find the mean absolute
deviation of the data.
3. mean 5 10
4
4. mean 5 8
6
8
10
12
14
2
Ages of Students in Dance Class
4
6
8
10
12
Weekly Hours Spent Doing Homework
mean absolute deviation 5
mean absolute deviation 5
Problem
Problem Solving
Solving
© Houghton Mifflin Harcourt Publishing Company
5. In science class, Troy found the mass, in grams,
of 6 samples to be 10, 12, 7, 8, 5, and 6. What is
the mean absolute deviation?
7.
WRITE
6. Five recorded temperatures are 71°F, 64°F,
72°F, 81°F, and 67°F. What is the mean
absolute deviation?
Math Make a dot plot of the following data:
10, 10, 11, 12, 12, 13, 13, 15. Use the dot plot to find the
mean absolute deviation.
Chapter 13
723
Lesson Check (6.SP.B.5c)
1. Six test grades are 86, 88, 92, 90, 82, and 84.
The mean of the data is 87. What is the mean
absolute deviation?
2. Eight heights in inches are 42, 36, 44, 46, 48, 42,
48, and 46. The mean of the data is 44. What is
the mean absolute deviation?
Spiral Review (6.G.A.2, 6.SP.B.4)
3. What is the volume of a rectangular prism with
dimensions 4 meters, 1 1_2 meters, and 5 meters?
4. Carrie is making a frequency table showing
the number of miles she walked each day
during the 30 days of September. What value
should she write in the Frequency column for
9 to 11 miles?
Carrie’s Daily Walks
Frequency
0–2
17
3–5
8
6–8
4
9–11
?
6. What is the upper quartile of the following data?
43, 48, 55, 50, 58, 49, 38, 42, 50
© Houghton Mifflin Harcourt Publishing Company
5. The following data shows the number of laps
each student completed. What number of laps
is the mode?
Number of Miles
9, 6, 7, 8, 5, 1, 8, 10
FOR MORE PRACTICE
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Lesson 13.4
Name
Measures of Variability
Statistics and Probability— 6.SP.B.5c
Also 6.SP.A.2, 6.SP.A.3
MATHEMATICAL PRACTICES
MP1, MP7, MP8
Essential Question How can you summarize a data set by using range,
interquartile range, and mean absolute deviation?
connect A measure of variability is a single value used to
describe how spread out a set of data values are. The mean
absolute deviation is a measure of variability.
Unlock
Unlock the
the Problem
Problem
In gym class, the students recorded how far they could jump.
The data set below gives the distances in inches that Manuel
jumped. What is the mean absolute deviation of the data set?
Manuel's Jumps (in inches)
54
58
56
59
60
55
Find the mean absolute deviation.
STEP 1 Find the mean of the data set.
Add the data values and divide the sum
by the number of data values.
54
+
+
+
+
+
_______________________________________
= ______ =
The mean of the data set is _ inches.
© Houghton Mifflin Harcourt Publishing Company • Image Credits: (tr) ©Jim Lane/Alamy Images
STEP 2 Find the distance of each data
value from the mean.
Subtract the lesser value from
the greater value.
STEP 3 Add the distances.
STEP 4 Find the mean of the
distances.
Data Value
Subtract (Mean = 57)
Distance between
data value and the mean
54
57 − 54 =
3
58
58 − 57 =
56
57 − 56 =
59
59 − 57 =
60
60 − 57 =
55
57 − 55 =
Total of distances from the mean:
_÷6=_
Divide the sum of the
distances by the number
of data values.
So, the mean absolute deviation of the data is _ inches.
Math
Talk
MATHEMATICAL PRACTICES 7
Look for Structure Give an example
of a data set that has a small mean
absolute deviation. Explain how
you know that the mean absolute
deviation is small without doing any
calculations.
Chapter 13
725
Range is the difference between the greatest value and the least
value in a data set. Interquartile range is the difference between
the upper quartile and the lower quartile of a data set. Range and
interquartile range are also measures of variability.
Example
Use the range and interquartile
range to compare the data sets.
The box plots show the price in dollars of the handheld game
players at two different electronic stores. Find the range
and interquartile range for each data set. Then compare the
variability of the prices of the handheld game players at the
two stores.
Store A
24
20
48
40
52
Store B
150
72
60
80
100
120
140
30
160
20
42
40
100
68
60
80
100
120
120
140
160
Costs of Handheld Game Players (in dollars)
STORE A
STORE B
150 − 24 = _
120 − _ = _
The range for Store A is _.
The range for Store B is _.
Calculate the interquartile
range.
72 − 48 = _
100 − _ = _
Find the difference between the
upper quartile and lower quartile.
The interquartile range for
The interquartile range for
Store A is _.
Store B is _.
Find the difference between the
greatest and least values.
So, Store A has a greater __, but
Store B has a greater ___.
726
Math
Talk
MATHEMATICAL PRACTICES 6
Compare Explain how range
and interquartile range
are alike and how they are
different.
© Houghton Mifflin Harcourt Publishing Company • Image Credits: (tr) ©Randy Faris/Corbis
Calculate the range.
Name
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1. Find the range and interquartile range of the data in the box plot.
0
5
10
15
20
Cost of T-shirts (in dollars)
For the range, find the difference between the
greatest and least values.
For the interquartile range, find the difference
between the upper and lower quartiles.
_−_=_
_−_=_
range: $ _
interquartile range: $ _
Practice: Copy and Solve Find the mean absolute deviation for the data set.
2. heights in inches of several tomato plants:
3. times in seconds for students to run one lap:
16, 18, 18, 20, 17, 20, 18, 17
68, 60, 52, 40, 64, 40
mean absolute deviation: __
mean absolute deviation: __
Math
Talk
On
On Your
Your Own
Own
MATHEMATICAL PRACTICES 6
Explain how you can find the mean
absolute deviation of a data set.
Use the box plot for 4 and 5.
4. What is the range of the data? _
5. What is the interquartile range of the data?
30
35
40
45
50
55
60
Price of Pottery Sold (in dollars)
Practice: Copy and Solve Find the mean absolute deviation for the data set.
© Houghton Mifflin Harcourt Publishing Company
6. times in minutes spent on a history quiz
7. number of excused absences for one semester:
35, 35, 32, 34, 34, 32, 34, 36
1, 2, 1, 10, 9, 9, 10, 6, 1, 1
mean absolute deviation: __
mean absolute deviation: __
8. The chart shows the price of different varieties of dog
food at a pet store. Find the range, interquartile range,
and the mean absolute deviation of the data set.
Cost of Bag of Dog Food ($)
18
24
20
26
24
20
32
20
16
20
Chapter 13 • Lesson 4 727
MATHEMATICAL PRACTICES
ANALYZEt-00,'034536$563&t13&$*4*0/
OqnakdlRnkuhmf¤@ookhb`shnmr
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9.
10.
DEEPER
Hyato’s family began a walking program.
They walked 30, 45, 25, 35, 40, 30, and 40 minutes each
day during one week. At the right, make a box plot of
the data. Then find the interquartile range.
MATHEMATICAL
PRACTICE
6 Compare Jack recorded the number of
minutes his family walked each day for a month. The
range of the data is 15. How does this compare to the
data for Hyato’s family?
20
25
30
35
40
45
50
55
60
Time Spent Walking (in minutes)
12.
Sense or Nonsense? Nathan claims that the interquartile
range of a data set can never be greater than its range. Is Nathan’s claim sense
or nonsense? Explain.
SMARTER
SMARTER
The box plot shows the heights
of corn stalks from two different farms.
Farm A
Farm B
54
56
58
60
62
64
66
Heights (in.)
the same as
The range of Farm A's heights is
less than
greater than
728
the range of Farm B's heights.
68
70
72
74
© Houghton Mifflin Harcourt Publishing Company • Image Credits: (t) ©Houghton Mifflin Harcourt
11.
Practice and Homework
Name
Lesson 13.4
Measures of Variability
COMMON CORE STANDARD—6.SP.B.5c
Summarize and describe distributions.
1. Find the range and interquartile range of the data in the box plot.
0
2
4
6
8
10 12 14 16 18 20
Miles Walked
For the range, find the difference between the
greatest and least values.
17 2
range:
1
For the interquartile range, find the difference
between the upper and lower quartiles.
5 16
16 miles
12 2
4
8
5
8 miles
interquartile range:
Use the box plot for 2 and 3.
2. What is the range of the data?
50
3. What is the interquartile range of the data?
60
70
80
90
100
Quiz Scores
Find the mean absolute deviation for the set.
4. heights in centimeters of several flowers:
5. ages of several children:
14, 7, 6, 5, 13
mean absolute deviation:
5, 7, 4, 6, 3, 5, 3, 7
mean absolute deviation:
© Houghton Mifflin Harcourt Publishing Company
Problem
Problem Solving
Solving
6. The following data set gives the amount of time,
in minutes, it took five people to cook a recipe.
What is the mean absolute deviation for the data?
33, 38, 31, 36, 37
8.
WRITE
7. The prices of six food processors are $63, $59,
$72, $68, $61, and $67. What is the range,
interquartile range, and mean absolute deviation
for the data?
Math Find the range, interquartile range, and mean
absolute deviation for this data set: 41, 45, 60, 61, 61, 72, 80.
Chapter 13
729
Lesson Check (6.SP.B.5c)
1. Daily high temperatures recorded in a certain
city are 65°F, 66°F, 70°F, 58°F, and 61°F. What is
the mean absolute deviation for the data?
2. Eight different cereals have 120, 160, 135, 144,
153, 122, 118, and 134 calories per serving.
What is the interquartile range for the data?
Spiral Review (6.SP.B.4, 6.SP.B.5c)
3. Look at the histogram. How many days did the
restaurant sell more than 59 pizzas?
Pizzas Sold Per Day
35
Frequency
30
25
20
15
10
5
5. What is the mode of the data set?
14, 14, 18, 20
0
0–19
20–39
40–59
Number of Pizzas
60–79
6. The data set below lists the ages of people on a
soccer team. The mean of the data is 23. What is
the mean absolute deviation?
24, 22, 19, 19, 23, 23, 26, 27, 24
FOR MORE PRACTICE
GO TO THE
730
Personal Math Trainer
© Houghton Mifflin Harcourt Publishing Company
4. Look at the histogram. Where does a peak in the
data occur?
Name
Mid-Chapter Checkpoint
Vocabulary
Vocabulary
Personal Math Trainer
Online Assessment
and Intervention
Vocabulary
Choose the best term from the box to complete the sentence.
1. The ___ is the difference between the
upper quartile and the lower quartile of a data set. (p. 726)
box plot
interquartile range
mean absolute
deviation
2. A graph that shows the median, quartiles, and least and greatest
values of a data set is called a(n) ___. (p. 714)
measure of variability
range
3. The difference between the greatest value and the least value in a
data set is the ___. (p. 726)
4. The ____ is the mean of the distances
between the values of a data set and the mean of the data set. (p. 720)
Concepts
Concepts and
and Skills
Skills
5. Make a box plot for this data set: 73, 65, 68, 72, 70, 74. (6.SP.B.4)
Find the mean absolute deviation of the data. (6.SP.B.5c)
© Houghton Mifflin Harcourt Publishing Company
6. 43, 46, 48, 40, 38
____
7. 26, 20, 25, 21, 24, 27, 26, 23
____
8. 99, 70, 78, 85, 76, 81
____
Find the range and interquartile range of the data. (6.SP.B.5c)
9. 2, 4, 8, 3, 2
____
10. 84, 82, 86, 87, 88, 83, 84
____
11. 39, 22, 33, 45, 42, 40, 28
____
Chapter 13
731
12. Yasmine keeps track of the number of hockey goals scored by her
school’s team at each game. The dot plot shows her data.
0
1
2
3
4
Goals Scored
Where is there a gap in the data? (6.SP.B.5c)
13. What is the interquartile range of the data shown in the dot plot
with Question 12? (6.SP.B.5c)
14.
DEEPER
Randall’s teacher added up the class scores for the
quarter and used a histogram to display the data. How many
peaks does the histogram have? Explain how you know. (6.SP.B.5c)
Number of Students
Class Scores This Quarter
9
8
7
6
5
4
3
2
1
0
15. In a box plot of the data below, where would the box be drawn?
(6.SP.B.4)
55, 37, 41, 62, 50, 49, 64
732
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201–300 301–400 401–500 501–600 601–700 701–800
Points
Lesson 13.5
Name
Choose Appropriate Measures of Center
and Variability
Statistics and Probability—
6.SP.B.5d
MATHEMATICAL PRACTICES
MP1, MP2, MP3
Essential Question How can you choose appropriate measures of center
and variability to describe a data set?
Outliers, gaps, and clusters in a set of data can affect both the
measures of center and variability. Some measures of center and
variability may describe a particular set of data better than others.
Unlock
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the Problem
Problem
Thomas is writing an article for the school newsletter about
a paper airplane competition. In the distance category,
Kara’s airplanes flew 17 ft, 16 ft, 18 ft, 15 ft, and 2 ft. Should
Thomas use the mean, median, or mode to best describe
Kara’s results? Explain your reasoning.
• Do you need to order the numbers?
Find the mean, median, and mode and compare them.
Mean =
=
+
+
+
+
=
Order the data from least to greatest to find the median.
_, _, _, _, _
Median = _
The measures of center for some data
sets may be very close together. If that
is the case, you can list more than one
measure as the best way to describe
the data.
The data set has no repeated values so there is no __.
The mean is __ than 4 of the 5 values, so it is not a good
© Houghton Mifflin Harcourt Publishing Company
description of the center of the data. The __ is closer to
most of the values, so it is the best way to describe Kara’s results.
So, Thomas should use the __ to describe Kara’s results.
1. Explain why the two modes may be a better description than the mean
or median of the data set 2, 2, 2, 2, 7, 7, 7, 7.
Chapter 13
733
Example Mr. Tobin is buying a book online. He compares
prices of the book at several different sites. The table shows his results.
Make a box plot of the data. Then use the plot to find the range and
interquartile range. Which measure better describes the data? Explain
your reasoning.
STEP 1 Make a box plot.
Write the data in order from least to
greatest.
_, _, _, _,
_, _, _
Find the median of the data.
median = _
Find the lower quartile—the median
of the lower half of the data.
lower quartile = _
Find the upper quartile—the median
of the upper half of the data.
upper quartile = _
Prices of Book
Site
Price ($)
1
15
2
35
3
17
4
18
5
5
6
16
7
17
Make the plot.
Math
Talk
3 5 7 9 11 13 15 17 19 21 23 25 27 29 31 33 35 37 39
Prices of Books (in dollars)
MATHEMATICAL PRACTICES 2
Reasoning Describe a data
set for which the range is a
better description than the
interquartile range.
STEP 2 Use the box plot to find the range and the interquartile range.
range = _ − _ = _
interquartile range = _ − _ = _
__ of the seven prices are within the ___.
The other two prices are much higher or lower.
So, the ___ better describes the data because the
they actually do.
2.
734
SMARTER
How can you tell from the box plot how varied the
data are? Explain.
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___ makes it appear that the data values vary more than
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1. The distances in miles students travel to get to
school are 7, 1, 6, 8, 9, and 8. Decide which
measure(s) of center best describes the data set.
Explain your reasoning.
mean = __
median = __
mode = __
The __ is less than 4 of the 6 data points, and the __ describes only 2 of
the data points. So, the __ best describes the data.
2.
MATHEMATICAL
PRACTICE
4 Use Graphs The numbers of
different brands of orange juice carried in
several stores are 2, 1, 3, 1, 12, 1, 2, 2, and 5.
Make a box plot of the data and find the range
and interquartile range. Decide which measure
better describes the data set and explain your
reasoning.
1
2
3
4
5
6
7
8
9
10
11
12
13
Number of Juice Brands
range = _
interquartile range = _
Math
Talk
On
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Your Own
Own
3.
MATHEMATICAL
PRACTICE
2 Use Reasoning The ages of
© Houghton Mifflin Harcourt Publishing Company
students in a computer class are 14, 13, 14, 15, 14,
35, 14. Decide which measure of center(s) best
describes the data set. Explain your reasoning.
4.
MATHEMATICAL PRACTICES 6
Explain How does an
outlier affect the range of
a data set?
mean = __
median = __
mode = __
DEEPER
Mateo scored 98, 85, 84, 80, 81, and 82 on six math tests.
When a seventh math test score is added, the measure of center that best
describes his scores is the median. What could the seventh test score be?
Explain your reasoning.
Chapter 13 • Lesson 5 735
MATHEMATICAL PRACTICES .0%&-t3&"40/tM",&4&/4&
Unlock
Unlock the
the Problem
Problem
5.
Swim Team Results
SMARTER
Jaime is on the community
swim team. The table shows the team’s results
in the last 8 swim meets. Jaime believes they
can place in the top 3 at the next swim meet.
Which measure of center should Jaime use to
persuade her team that she is correct?
Explain.
a. What do you need to find?
Meet
Place
Meet 1
1
Meet 2
2
Meet 3
3
Meet 4
18
Meet 5
1
Meet 6
2
Meet 7
3
Meet 8
2
b. What information do you need to solve the
problem?
c. What are the measures of center?
d. Which measure of center should Jaime use? Explain.
M t Show Your Work
Math
Personal Math Trainer
6.
736
SMARTER
The numbers of sit-ups students completed in one
minute are 10, 42, 46, 50, 43, and 49. The mean of the data values is 40
and the median is 44.5. Which measure of center better describes the
data, the mean or median? Use words and numbers to support
your answer.
© Houghton Mifflin Harcourt Publishing Company • Image Credits: (r) ©Digital Vision/Getty Images
WRITE
Practice and Homework
Name
Lesson 13.5
Choose Appropriate Measures
of Center and Variability
1. The distances, in miles, that 6 people travel to get
to work are 14, 12, 2, 16, 16, and 18. Decide which
measure(s) of center best describes the data set.
Explain your reasoning.
is less than 4 of the data
The
describes only
points, and the
best
2 of the data points. So, the
describes the data.
2. The numbers of pets that several children have
are 2, 1, 2, 3, 4, 3, 10, 0, 1, and 0. Make a box plot
of the data and find the range and interquartile
range. Decide which measure better describes
the data set and explain your reasoning.
COMMON CORE STANDARD—6.SP.B.5d
Summarize and describe distributions.
13 miles
mean 5
15 miles
median 5
16 miles
mode 5
0
1
2
3
range 5
4
5
6 7 8 9 10 11 12
interquartile range 5
______________
______________
______________
Problem
Problem Solving
Solving
3. Brett’s history quiz scores are 84, 78, 92, 90,
85, 91, and 0. Decide which measure(s) of
center best describes the data set. Explain
your reasoning.
© Houghton Mifflin Harcourt Publishing Company
mean 5
5.
4. Eight students were absent the following number
of days in a year: 4, 8, 0, 1, 7, 2, 6, and 3. Decide if
the range or interquartile range better describes
the data set, and explain your reasoning.
median 5
interquartile range 5
mode 5
range 5
______
______
______
______
______
______
WRITE
Math Create two sets of data that would be best described
by two different measures of center.
Chapter 13
737
Lesson Check (6.SP.B.5d)
1. Chloe used two box plots to display some data.
The box in the plot for the first data set is wider
than the box for the second data set. What does
this say about the data?
2. Hector recorded the temperature at noon
for 7 days in a row. The temperatures are
20°F, 20°F, 20°F, 23°F, 23°F, 23°F, and 55°F.
Which measure of center would best describe
the data?
Spiral Review (6.SP.B.4, 6.SP.B.5c, 6.SP.B.5d)
3. By how much does the median of the following
data set change if the outlier is removed?
13, 20, 15, 19, 22, 26, 42
4. What percent of the people surveyed spent at
least an hour watching television?
Daily Minutes Spent Watching TV
14
Number of People
12
10
8
6
4
2
0
5. What is the lower quartile of the following data?
12, 9, 10, 8, 7, 12
20–39
40–59
Number of Minutes
60–79
6. What is the interquartile range of the data
shown in the box plot?
6
8
10
12
14
16
18
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0–19
Lesson 13.6
Name
Apply Measures of Center and
Variability
Statistics and Probability— 6.SP.A.3
Also 6.SP.A.2
MATHEMATICAL PRACTICES
MP4, MP6, MP7
Essential Question What do measures of center and variability indicate
about a data set?
Unlock
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the Problem
Problem
Julia is collecting data on her favorite sports teams
for a report. The table shows the median and
interquartile range of the heights of the players on
her favorite baseball and basketball teams. How do
the heights of the two teams compare?
Compare the medians and interquartile ranges of the
two teams.
Sports Team Data
Median
Interquartile
Range
Baseball
Team Heights
70 in.
6 in.
Basketball
Team Heights
78 in.
4 in.
Median
The median of the __ players’ heights is _ inches
greater than the median of the __ players’ heights.
Interquartile Range
The interquartile range of the baseball team is __ the
interquartile range of the basketball team, so the heights
of the baseball players vary __ the heights of the basketball team.
So, the players on the __ team are typically taller than the
players on the __ team, and the heights of the __
© Houghton Mifflin Harcourt Publishing Company
team vary more than the those of the __ team.
Math
Talk
1. Julia randomly picks one player from the basketball team and one
player from the baseball team. Given data in the table, can you say that
the basketball player will definitely be taller than the baseball player?
Explain your reasoning.
MATHEMATICAL PRACTICES 2
Reasoning What if the mean
of the heights of players on
the baseball team is 75 in.?
Explain what this could tell
you about the data.
Chapter 13
739
Example Compare the means and ranges of the two data sets.
Kamira and Joey sold T-shirts during lunch to raise money for a
charity. The table shows the number of T-shirts each student sold
each day for two weeks. Find the mean and range of each data set,
and use these measures to compare the data.
T-Shirts Sold
Kamira
5, 1, 2, 1, 3, 3, 1, 4, 5, 5
Joey
0, 1, 2, 13, 2, 1, 3, 4, 4, 0
STEP 1 Find the mean of each data set.
Kamira:
+
+
+
+
+
+
+
+
+
Mean = ___________________________________________________________
= ______ =
Joey:
+
+
+
+
+
+
+
+
+
Mean = _____________________________________________________________
Make sure you include
zeroes when you count
the total number of data
values.
= ______ =
STEP 2 Find the range of each data set.
Kamira:
Range =
Joey:
−
=
Range =
−
=
The mean of Joey’s sales is ___ the mean of Kamira’s sales.
The range of Joey’s sales is ___ the range of Kamira’s sales.
So, the typical number of shirts Joey sold each day was ___ the
typical number of shirts Kamira sold. However, since the range of Joey’s
data was __ than Kamira’s, the number of shirts Joey sold
varied __ from day to day than the number of shirts Kamira sold.
2.
740
MATHEMATICAL
PRACTICE
6 Explain Which measure of center would better describe Joey’s data set? Explain.
© Houghton Mifflin Harcourt Publishing Company • Image Credits: (tr) ©Ocean/Corbis
STEP 3 Compare the mean and range.
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1. Zoe collected data on the number of points her favorite basketball
players scored in several games. Use the information in the table to
compare the data.
Points Scored
The mean of Player 1’s points is __ the mean of Player 2’s
points.
Mean
Interquartile
Range
Player 1
24
8
Player 2
33
16
The interquartile range of Player 1’s points is __ the
interquartile range of Player 2’s points.
So, Player 2 typically scores __ points than Player 1, but
Player 2’s scores typically vary __ Player 1’s scores.
2. Mark collected data on the weights of puppies at two animal shelters.
Find the median and range of each data set, and use these measures to
compare the data.
Puppy Weight, in pounds
Shelter A:
7, 10, 5, 12, 15, 7, 7
Shelter B:
4, 11, 5, 11, 15, 5, 13
© Houghton Mifflin Harcourt Publishing Company • Image Credits: (br) ©Stockbyte/Getty Images
On
On Your
Your Own
Own
Kwan analyzed data about the number of hours musicians in her band
practice each week. The table shows her results. Use the table for
Exercises 3–5.
3. Which two students typically practiced the same amount each week,
with about the same variation in practice times?
4. Which two students typically practiced the same number of hours, but
had very different variations in their practice times?
5. Which two students had the same variation in practice times, but
typically practiced a different number of hours per week?
Hours of Practice per Week
Mean
Range
Sally
5
2
Matthew
9
12
Tim
5
12
Jennifer
5
3
Chapter 13 • Lesson 6 741
MATHEMATICAL PRACTICES
COMMUNICA5&t1&34E7&3&tCONSTRUCT ARGUMENTS
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OqnakdlRnkuhmf¤@ookhb`shnmr
6.
MATHEMATICAL
PRACTICE
Miles Run
6 Compare The table shows the number of
Week 1
2, 1, 5, 2, 3, 3, 4
miles Johnny ran each day for two weeks. Find the median
and the interquartile range of each data set, and use these
measures to compare the data sets.
7.
Week 2
3, 8, 1, 8, 1, 3, 1
Sense or Nonsense? Yashi made the
box plots at right to show the data he collected on plant
growth. He thinks that the variation in bean plant growth
was about the same as the variation in tomato plant
growth. Does Yashi’s conclusion make sense? Why or
why not?
SMARTER
1
2 3 4 5 6 7 8 9 10 11 12 13 14 15
Bean Plant Growth (inches)
1
2 3 4 5 6 7 8 9 10 11 12 13 14 15
Tomato Plant Growth (inches)
Personal Math Trainer
8.
SMARTER
Kylie’s teacher collected data
on the heights of boys and girls in a sixth-grade class.
Use the information in the table to compare the data.
Heights (in.)
Girls
55
60
56
51
60
63
65
Boys
72
68
70
56
58
62
64
The mean of the boys’ heights is
less than
the mean of the girls’ heights.
greater than
the same as
The range of the boys’ heights is
less than
greater than
742
the range of the girls’ heights.
© Houghton Mifflin Harcourt Publishing Company
the same as
Practice and Homework
Name
Lesson 13.6
Apply Measures of Center and Variability
COMMON CORE STANDARD—6.SP.A.3
Develop understanding of statistical variability.
Solve.
1. The table shows temperature data for two cities. Use the
information in the table to compare the data.
The mean of City 1’s temperatures is
less than
the
mean of City 2’s temperatures.
Daily High Temperatures (°F)
Mean
Interquartile
Range
City 1
60
7
City 2
70
15
interquartile range
The
of City 1’s temperatures is
interquartile
range
less than the
of City 2’s temperatures.
So, City 2 is typically
more than
vary
warmer than
City 1, but City 2’s temperatures
City 1’s temperatures.
2. The table shows weights of fish that were caught in two
different lakes. Find the median and range of each data
set, and use these measures to compare the data.
Fish Weight (pounds)
Lake A: 7, 9, 10, 4, 6, 12
Lake B: 6, 7, 4, 5, 6, 4
Problem
Problem Solving
Solving
© Houghton Mifflin Harcourt Publishing Company
3. Mrs. Mack measured the heights of her students
in two classes. Class 1 has a median height
of 130 cm and an interquartile range of 5 cm.
Class 2 has a median height of 134 cm and an
interquartile range of 8 cm. Write a statement
that compares the data.
5.
WRITE
4. Richard’s science test scores are 76, 80, 78, 84,
and 80. His math test scores are 100, 80, 73, 94,
and 71. Compare the medians and interquartile
ranges.
Math Write a short paragraph to a new student that
explains how you can compare data sets by examining the mean and the
interquartile range.
Chapter 13
743
Lesson Check (6.SP.A.3)
1. Team A has a mean of 35 points and a range
of 8 points. Team B has a mean of 30 points
and a range of 7 points. Write a statement that
compares the data.
2. Jean’s test scores have a mean of 83 and an
interquartile range of 4. Ben’s test scores have
a mean of 87 and an interquartile range of 9.
Compare the students’ scores.
Spiral Review (6.SP.A.3, 6.SP.B.4, 6.SP.B.5d)
3. Look at the box plots below. What is the
difference between the medians for the two
groups of data?
4. The distances in miles that 6 people drive to
get to work are 10, 11, 9, 12, 9, and 27. What
measure of center best describes the data set?
School A
20
22
24
26
28
30
Number of Students in a Class
School B
18
20
22
24
26
5. Which two teams typically practice the same
number of hours, but have very different
variations in their practice times?
Hours of Practice Per Week
28
Number of Students in a Class
30
Team
Mean
Range
A
7
1.5
B
10.5
1.5
C
7.5
5
D
10
2
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18
Lesson 13.7
Name
Describe Distributions
Essential Question How can you describe the distribution of a data set
collected to answer a statistical question?
Statistics and Probability—
6.SP.A.2 Also 6.SP.B.5d
MATHEMATICAL PRACTICES
MP1, MP3, MP6
Activity
Ask at least 20 students in your school how many pets they
have. Record your results in a frequency table like the one
shown.
Pet Survey
Number of Pets
Frequency
0
1
2
3
4
• What statistical question could you use your data to
answer?
Unlock
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the Problem
Problem
You can graph your data set to see the center,
spread, and overall shape of the data.
• What type of graph will you use?
Make a dot plot or a histogram of your data.
© Houghton Mifflin Harcourt Publishing Company
• How will you label your graph?
Math
Talk
MATHEMATICAL PRACTICES 5
Use Tools Explain why you
chose the display you used.
Chapter 13
745
Think about the overall distribution of your data.
• Are there any clusters?
• Are there peaks in the data?
• Are there gaps in the data?
• Does the graph have symmetry?
1.
MATHEMATICAL
PRACTICE
6 Use Math Vocabulary Describe the overall distribution
of the data. Include information about clusters, gaps, peaks, and symmetry.
Example
Find the mean, median, mode, interquartile
range, and range of the data you collected.
STEP 1 Find the mean, median, and mode.
Mean: ____ Median: ____
Model:
____
STEP 2 Draw a box plot of your data and use it to find the interquartile
range and range.
Interquartile range: ____ Range: ____
2. Which measure of center do you think best describes your data? Why?
4. What is the answer to the statistical question you wrote on the
previous page?
746
Math
Talk
MATHEMATICAL PRACTICES 6
Describe Compare your data
set to the data set of one of
your classmates. How are the
data sets similar and how
they are different?
© Houghton Mifflin Harcourt Publishing Company
3. Does the interquartile range or range best describe your data? Why?
Name
MATH
M
BOARD
B
Connie asked people their ages as they
entered the food court at the mall. Use
the histogram of the data she collected
for 1–5.
1. What statistical question could
Connie ask about her data?
Ages of People at the Food Court
Number of People
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8
7
6
5
4
3
2
1
0
1–10
11–20
21–30
31–40 41–50
Age (years)
51–60
61–70
2. Describe any peak or gap in the data.
© Houghton Mifflin Harcourt Publishing Company • Image Credits: (cr) ©Lee Foster/Alamy Images
3. Does the graph have symmetry? Explain your reasoning.
On
On Your
Your Own
Own
4. The lower quartile of the data set is 16.5 years, and the upper
quartile is 51.5 years. Find the interquartile range. Is it a better
description of the data than the range? Explain your reasoning.
5.
MATHEMATICAL
PRACTICE
Math
Talk
MATHEMATICAL PRACTICES 1
Analyze Explain what, if any,
information you would need
to answer the statistical
question you wrote in
Exercise 1 and what
calculations you would
need to do.
3 Make Arguments The mode of the data is 16 years old.
Is the mode a good description of the center of the data? Explain.
Chapter 13 • Lesson 7 747
MATHEMATICAL PRACTICES
ANALYZEt-00,'034536$563&t13&$*4*0/
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OqnakdlRnkuhmf¤@ookhb`shnmr
Use the dot plot for 6–8.
6.
MATHEMATICAL
PRACTICE
3 Make Arguments Jason collected data
about the number of songs his classmates bought
online over the past 3 weeks. Does the data set have
symmetry? Why or why not?
7.
8.
9.
1
2
3
4
5
6
7
8
9
10
11
12
13
Number of Songs Bought Online
DEEPER
Jason claims that the median is a good description of
his data set, but the mode is not. Does his statement make sense?
Explain.
SMARTER
Trinni surveyed her classmates about how many
siblings they have. A dot plot of her data increases from 0 siblings
to a peak at 1 sibling, and then decreases steadily as the graph
goes to 6 siblings. How is Trinnis dot plot similar to Jason’s? How is
it different?
SMARTER
Diego collected data on the number of movies
seen last month by a random group of students.
Number of Movies Seen Last Month
1
3
2
1
0
5
12
2
3
2
2
3
Draw a box plot of the data and use it to find the interquartile range
and range.
Interquartile range _
Range _
0
1
2
3
4
5
6
7
8
9 10 11 12 13 14
Number of Movies Seen Last Month
748
© Houghton Mifflin Harcourt Publishing Company
0
Practice and Homework
Name
Lesson 13.7
Describe Distributions
COMMON CORE STANDARD—6.SP.A.2
Develop understanding of statistical variability.
Chase asked people how many songs they have bought
online in the past month. Use the histogram of the data he
collected for 1–4.
1. What statistical question could Chase ask about
the data?
Possible answer: What is the median number
Number of Songs Purchased
Online in a Month
14
of songs purchased?
Frequency
2. Describe any peaks in the data.
12
10
8
6
4
2
3. Describe any gaps in the data.
0
0–4
5–9
10–14
15–19
Number of Songs
20–24
4. Does the graph have symmetry? Explain your reasoning.
Problem
Problem Solving
Solving
© Houghton Mifflin Harcourt Publishing Company
5. Mr. Carpenter teaches five classes each day.
For several days in a row, he kept track of the
number of students who were late to class and
displayed the results in a dot plot. Describe
the data.
0
2
4
6
8
10
Number of Students
Late to Class Each Day
6.
WRITE
Math Describe how a graph of a data set can be used to
understand the distribution of the data.
Chapter 13
749
Lesson Check (6.SP.A.2)
1. The ages of people in a restaurant are 28, 10, 44,
25, 18, 8, 47, and 30. What is the median age of
the people in the restaurant?
2. What is the median in the dot plot?
8
9
10
11
12
13
14 15 16
Cost (in dollars) of Dinners
Ordered on Friday
Spiral Review (6.SP.A.2, 6.SP.A.3, 6.SP.B.5c, 6.SP.B.5d)
3. Look at the dot plot. Where does a gap occur in
the data?
30
5. Which two teams had similar variations in
points earned, but typically earned a different
number of points per game?
Points Earned Per Game
Team
Mean
Range
Red
20
8
Blue
28
8
Green
29
4
Orange
28
4
34
36
38
40
42
Number of Movies Ordered Per Day
6. Manny’s monthly electric bills for the past
6 months are $140, $165, $145, $32, $125, and
$135. What measure of center best represents
the data?
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4. Look at the dot plot. Where does a peak occur
in the data?
32
PROBLEM SOLVING
Name
Lesson 13.8
Problem Solving • Misleading Statistics
Essential Question How can you use the strategy work backward to
draw conclusions about a data set?
Statistics and Probability—
6.SP.A.2 Also 6.SP.B.5c
MATHEMATICAL PRACTICES
MP1, MP3, MP6
Unlock
Unlock the
the Problem
Problem
Mr. Owen wants to move to a town where the daily high
temperature is in the 70s most days. A real estate agent
tells him that the mean daily high temperature in a certain
town is 72°. Other statistics about the town are given in the
table. Does this location match what Mr. Owen wants?
Why or why not?
Town Statistics for the Past Year
(Daily High Temperature)
Minimum
62º
Maximum
95º
Median
69º
Mean
72º
Use the graphic organizer to help you solve the problem.
Read the Problem
What do I need to find?
I need to decide if the daily high
temperature in the town
____.
What information do I
need?
How will I use the
information?
I need the __ in the
table.
I will work backward from the
statistics to draw conclusions
about the __ of data.
© Houghton Mifflin Harcourt Publishing Company
Solve the Problem
The minimum high temperature is _.
Think: The high temperature is sometimes __
The maximum high temperature is _.
Think: The high temperature is sometimes __
The median of the data set is _.
Think: The median is the middle value in the data set.
than 70°.
than 80°.
Because the median is 69°, at least half of the days must have high
temperatures less than or equal to 69°.
So, the location does not match what Mr. Owen wants. The median
indicates that most days __ have a high temperature in
the 70s.
Math
Talk
MATHEMATICAL PRACTICES 6
Explain Why is the mean
temperature misleading in
this example?
Chapter 13
751
Try Another Problem
Ms. Garcia is buying a new car. She would like to visit a
dealership that has a wide variety of cars for sale at many
different price ranges. The table gives statistics about one
dealership in her town. Does the dealership match Ms.
Garcia’s requirements? Explain your reasoning.
Statistics for New Car Prices
Lowest Price
$12,000
Highest Price
$65,000
Lower Quartile Price
$50,000
Median Price
$55,000
Upper Quartile Price
$60,000
Read the Problem
What do I need to find?
What information do I
need?
How will I use the
information?
Solve the Problem
15 20 25 30 35 40 45 50 55 60 65
New Car Prices (in thousands of dollars)
• What would the box plot look like for a dealership that does meet
Ms. Garcia’s requirements?
N
752
© Houghton Mifflin Harcourt Publishing Company
10
Name
Share
hhow
Share and
and Show
Sh
MATH
M
BOARD
B
1. Josh is playing a game at the carnival. If his arrow lands
on a section marked 25 or higher, he gets a prize. Josh
will only play if most of the players win a prize. The
carnival worker says that the average (mean) score is
28. The box plot shows other statistics about the game.
Should Josh play the game? Explain your reasoning.
0
5
10
Unlock the Problem
√
√
√
√
Circle important facts.
Organize the information.
Choose a strategy.
Check to make sure you answered the
question.
15 20 25 30 35 40 45
Points Scored
First, look at the median. The median is _ points.
Next, work backward from the statistics.
The median is the __ value of the data.
So, at least __ of the values are scores
less than or equal to _.
Finally, use the statistics to draw a conclusion.
© Houghton Mifflin Harcourt Publishing Company
2.
3.
What if a score of 15 or greater resulted in a prize? How would that
affect Josh’s decision? Explain.
SMARTER
DEEPER
A store collects data on the sales of DVD players
each week for 3 months. The manager determines that the data has a
range of 62 players and decides that the weekly sales were
very inconsistent. Use the statistics in the table to decide if
the manager is correct. Explain your answer.
Weekly DVD Player Sales
Minimum
16
Maximum
78
Lower quartile
58
Upper quartile
72
Chapter 13 • Lesson 8 753
MATHEMATICAL PRACTICES .0%&-t3&"40/tM",&4&/4&
On
On Your
Your Own
Own
4.
5.
6.
DEEPER
Gerard is fencing in a yard that is 21 feet by
18 feet. How many yards of fencing material does Gerard
need? Explain how you found your answer.
SMARTER
Susanna wants to buy a fish that grows
to be about 4 in. long. Mark suggests she buys the same
type of fish he has. He has five of these fish with lengths of
1 in., 1 in., 6 in., 6 in., and 6 in., with a mean length of 4 in.
Should Susanna buy the type of fish that Mark suggests?
Explain.
MATHEMATICAL
PRACTICE
7 Look for a Pattern The graph shows the
Number of Stamps
number of stamps that Luciano collected over several
weeks. If the pattern continues, how many
stamps will Luciano collect in Week 8? Explain.
Stamps Collected
6
5
4
3
2
1
0
1
2
3
4
5
6
7.
SMARTER
The data set shows the number of hours Luke plays the piano
each week. Luke says he usually plays the piano 3 hours per week. Why is Luke’s
statement misleading?
Hours Playing the Piano
1
754
2
1
3
2
10
2
© Houghton Mifflin Harcourt Publishing Company
Week
Practice and Homework
Name
Lesson 13.8
Problem Solving • Misleading Statistics
COMMON CORE STANDARD—6.SP.A.2
Develop understanding of statistical variability.
Mr. Jackson wants to make dinner reservations at a
restaurant that has most meals costing less than $16.
The Waterside Inn advertises that they have meals that
average $15. The table shows the menu items.
Menu Items
Meal
Price
Potato Soup
$6
Chicken
$16
Steak
$18
Pasta
$16
Shrimp
$18
Crab Cake
$19
1. What is the minimum price and maximum price?
min 5
$6
max 5
$19
2. What is the mean of the prices?
____
3. Construct a box plot for the data.
6
8
10
12
14
16
18
20
22
4. What is the range of the prices?
____
5. What is the interquartile range of the prices?
____
© Houghton Mifflin Harcourt Publishing Company
6. Does the menu match Mr. Jackson’s requirements? Explain your reasoning.
7.
WRITE
Math Give an example of a misleading statistic. Explain
why it is misleading.
Chapter 13
755
Lesson Check (6.SP.A.2)
1. Mary’s science test scores are 66, 94, 73, 81, 70,
84, and 88. What is the range of Mary’s science
test scores?
2. The heights in inches of students on a team
are 64, 66, 60, 68, 69, 59, 60, and 70. What is the
interquartile range?
Spiral Review (6.SP.B.4, 6.SP.B.5c, 6.SP.B.5d)
3. By how much does the median of the following
data set change if the outlier is removed?
4. Look at the box plot. What is the interquartile
range of the data?
26, 21, 25, 18, 0, 28
40
44
46
48
50
52
Trivia Game Results
Game
Score
Game 1
20
Game 2
20
Game 3
18
Game 4
19
Game 5
23
Game 6
40
Game 7
22
Game 8
19
FOR MORE PRACTICE
GO TO THE
756
Personal Math Trainer
© Houghton Mifflin Harcourt Publishing Company
5. Erin is on the school trivia team. The table
shows the team’s scores in the last 8 games.
Erin wants to build confidence in her team so
that they will do well in the last game. What
measure of center would be best for Erin to use
to motivate her teammates?
42
Name
Personal Math Trainer
Chapter 13 Review/Test
Online Assessment
and Intervention
1. The dot plot shows the number of chin-ups done by a gym class.
0
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
Number of Chin-ups
For numbers 1a–1e, choose Yes or No to indicate whether the
statement is correct.
1a. There are two peaks.
Yes
No
1b.
Yes
No
1c. There is a gap between 6 and 8.
Yes
No
1d.
The most chin-ups anyone did
was 15.
Yes
No
1e.
The modes are 3, 4, and 9.
Yes
No
There are no clusters.
2. The histogram shows the high temperatures in degrees Fahrenheit of
various cities for one day in March.
High Temperatures
Frequency
15
10
5
0
© Houghton Mifflin Harcourt Publishing Company
21–30
31–40
41–50
51–60
Temperatures ºF
61–70
Select the best word to complete each sentence.
zero
has
The histogram has
one
peak(s). The histogram
line symmetry.
does not have
two
Assessment Options
Chapter Test
Chapter 13
757
3. The data set shows the scores of the players on the winning team of a
basketball game.
Scores of Players on Winning Team
0
17
47
13
4
1
22
0
5
6.
9
1
30
0.
9.
The median is
6
The lower quartile is
1.
13.
4.
15
The upper quartile is
19.5
26
4. The data set shows the number of desks in 12 different classrooms.
Classroom Desks
24
21
18
17
21
19
17
20
21
22
20
16
Find the values of the points on the box plot.
A
A=
B
B=
C D
E
C=
D=
E=
5. The box plot shows the number of boxes sold at an office supply store
each day for a week.
2
4
6
8
10
12
14
16
18
20
22
For numbers 5a–5d, select True or False for each statement.
758
5a.
The median is 18.
True
False
5b.
The range is 15.
True
False
5c. The interquartile range is 9.
True
False
5d.
True
False
The upper quartile is 18.
© Houghton Mifflin Harcourt Publishing Company
Boxes of Paper Sold
Name
6. The data set shows the number of glasses of water Dalia drinks each day for
a week.
Glasses of Water
6
7
9
9
8
7
10
Part A
What is the mean number of glasses of water Dalia drinks each day?
__
Part B
What is the mean absolute deviation of the number of glasses of water Dalia
drinks each day? Round your answer to the nearest tenth. Use words and
numbers to support your answer.
7. The numbers of emails Megan received each hour are 9, 10, 9, 8, 7, and 2. The
mean of the data values is 7.5 and the median is 8.5. Which measure of center
better describes the data, the mean or median? Use words and numbers to
support your answer.
© Houghton Mifflin Harcourt Publishing Company
8. The number of miles Madelyn drove between stops was 182, 180, 181,
184, 228, and 185. Which measure of center best describes the data?
A
mean
B
median
C
mode
Chapter 13
759
9. The histogram shows the weekly earnings of part-time workers. What
interval(s) represents the most common weekly earnings?
Weekly Earnings
9
8
Frequency
7
6
5
4
3
2
1
0
291–300 301–310 311–320 321–330 331–340 341–350
Earnings ($)
10. Jordan surveyed a group of randomly selected smartphone users and
asked them how many applications they have downloaded onto their
phones. The dot plot shows the results of Jordan's survey. Select the
statements that describe patterns in the data. Mark all that apply.
35 36 37 38 39 40 41 42 43 44 45
760
A
The modes are 37 and 42.
B
There is a gap from 38 to 40.
C
There is a cluster from 41 to 44.
D
There is a cluster from 35 to 36.
© Houghton Mifflin Harcourt Publishing Company
Number of Applications
Name
11. Mrs. Gutierrez made a histogram of the birth month of the students in
her class. Describe the patterns in the histogram by completing
the chart.
Birth Month of Students
4.5
4
3.5
Frequency
3
2.5
2
1.5
1
0.5
0
Jan
Feb
Mar
DEEPER
May
Jun
Jul
Month
Identify any increases
across the intervals.
Identify any peaks.
12.
Apr
Aug
Sep
Oct
Nov
Dec
Identify any decreases
across the intervals.
Ian collected data on the number of children in 13 different
families.
Number of Children
© Houghton Mifflin Harcourt Publishing Company
1
2
4
3
2
1
0
8
1
1
0
2
3
Draw a box plot of the data and use it to find the interquartile range
and range.
0
1
2
3
4
5
6
Interquartile range: _
7
8
9
10
Range: _
Chapter 13
761
Personal Math Trainer
SMARTER
Gavin wants to move to a county where it rains about
13.
5 inches every month. The data set shows the monthly rainfall in inches
for a county. The mean of the data is 5 and the median is 4.35. After
analyzing the data, Gavin says that this county would be a good place to
move. Do you agree or disagree with Gavin? Use words and numbers to
support your answer.
Monthly Rainfall (in.)
4.4
3.7
6
2.9
4.3
5.4
6.1
14.1
4.3
0.5
4.5
3.8
14. The data set shows the number of books Peyton reads each month.
Peyton says she usually reads 4 books per month. Why is Peyton's
statement misleading?
Books Read
2
3
2
4
3
11
3
15. The data set shows the scores of three players for a board game.
Player A
90
90
90
Player B
110
100
90
Player C
95
100
95
For numbers 15a–15d, choose Yes or No to indicate whether the
statement is correct.
762
15a. The mean absolute deviation
of Player B’s scores is 0.
Yes
No
15b. The mean absolute deviation
of Player A’s scores is 0.
Yes
No
15c. The mean absolute deviation of
Player B’s scores is greater than the mean
absolute deviation of Player C’s scores.
Yes
No
© Houghton Mifflin Harcourt Publishing Company
Board Game Scores
Pronunciation Key
a
ā
â(r)
ä
b
ch
d
e
ē
add, map
ace, rate
care, air
palm,
father
bat, rub
check, catch
dog, rod
end, pet
equal, tree
f
g
h
i
ī
j
k
l
m
fit, half
go, log
hope, hate
it, give
ice, write
joy, ledge
cool, take
look, rule
move, seem
n
ng
o
ō
ô
oi
ou
nice, tin
ring, song
odd, hot
open, so
order, jaw
oil, boy
pout, now
took, full
pool, food
ə the schwa, an unstressed vowel
representing the sound spelled a in
above, e in sicken, i in possible, o in
melon, u in circus
absolute value [absə•lt valy] valor absoluto
The distance of an integer from zero on a
number line (p. 165)
acute angle [ə•kyt anggəl]
ángulo agudo An angle that has a measure
less than a right angle (less than 90º and
greater than 0º)
Example:
© Houghton Mifflin Harcourt Publishing Company
acute triangle [ə•kyt trīang•gəl]
triángulo acutángulo A triangle that has
three acute angles
p
r
s
sh
t
th
th
u
u̇
pit, stop
run, poor
see, pass
sure, rush
talk, sit
thin, both
this, bathe
up, done
pull book
û(r) burn, term
y fuse, few
v vain, eve
w win, away
y yet, yearn
z zest, muse
zh vision,
pleasure
Other symbols:
•
separates words into syllables
indicates stress on a syllable
Addition Property of Equality [ə•dishən
präpər•tē əv ē•kwôlə•tē] propiedad de suma
de la igualdad The property that states that
if you add the same number to both sides of
an equation, the sides remain equal
additive inverse [adə•tiv inv ûrs] inverso aditivo
The number which, when added to the given
number, equals zero
algebraic expression [al•jə•brāik ek•spreshən]
expresión algebraica An expression that
includes at least one variable (p. 369)
Examples: x + 5, 3a − 4
angle [anggəl] ángulo A shape formed by two
rays that share the same endpoint
Example:
addend [adend] sumando A number that is
added to another in an addition problem
addition [ə•dishən] suma The process of finding
the total number of items when two or
more groups of items are joined; the inverse
operation of subtraction
area [ârē•ə] área The number of square units
needed to cover a surface without any gaps
or overlaps (p. 533)
Glossary
H1
array [ə•rā] matriz An arrangement of objects in
rows and columns
Example:
base [bās] (geometry) base In two dimensions,
one side of a triangle or parallelogram
which is used to help find the area. In three
dimensions, a plane figure, usually a polygon
or circle, which is used to partially describe a
solid figure and to help find the volume of
some solid figures. See also height.
Examples:
height
Associative Property of Addition [ə•sōshē•ə•āt•iv
präpər•tē əv ə•dishən] propiedad asociativa de
la suma The property that states that when
the grouping of addends is changed, the sum
is the same
Example: (5 + 8) + 4 = 5 + (8 + 4)
Associative Property of Multiplication
[ə•sōshē•ə•tiv präpər•tē əv mul•tə•pli•kāshən]
propiedad asociativa de la multiplicación The
property that states that when the grouping
of factors is changed, the product is the same
Example: (2 × 3) × 4 = 2 × (3 × 4)
bar graph [bär graf] gráfica de barras A graph
that uses horizontal or vertical bars to display
countable data
Example:
base
base
base
base
benchmark [benchmärk] punto de referencia A
familiar number used as a point of reference
billion [bilyən] millardo 1,000 millions; written as
1,000,000,000
box plot [bäks plät] diagrama de caja A graph
that shows how data are distributed using the
median, quartiles, least value, and greatest
value (p. 714)
Example:
20 25 30 35 40 45 50 55 60 65 70 75
Prices of Jeans (in dollars)
capacity [kə•pasi•tē] capacidad The amount a
container can hold (p. 321)
Examples: 1_2 gallon, 2 quarts
Sports
base [bās] (arithmetic) base A number used as a
repeated factor (p. 357)
Example: 83 = 8 × 8 × 8. The base is 8.
H2
Glossary
Celsius (°C) [selsē•əs] Celsius (°C) A metric scale
for measuring temperature
© Houghton Mifflin Harcourt Publishing Company
Number of
Students
FAVORITE SPORT
12
10
8
6
4
2
0
closed figure [klōzd figyər] figura cerrada A figure
that begins and ends at the same point
coefficient [kō•ə•fishənt] coeficiente A number
that is multiplied by a variable (p. 376)
Example: 6 is the coefficient of x in 6x
common denominator [kämən dē•nämə•nāt•ər]
denominador común A common multiple of
two or more denominators
Example: Some common denominators for 1_4
and 5_6 are 12, 24, and 36.
common factor [kämən faktər] factor común
A number that is a factor of two or more
numbers (p. 23)
common multiple [kämən multə•pəl] múltiplo
común A number that is a multiple of two or
more numbers
composite number [kəm•päzit numbər] número
compuesto A number having more than two
factors
Example: 6 is a composite number, since its
factors are 1, 2, 3, and 6.
cone [kōn] cono A solid figure that has a flat,
circular base and one vertex
Example:
congruent [kən•grənt] congruente Having the
same size and shape (p. 539)
Example:
Commutative Property of Addition
[kə•myt ə•tiv präpər•tē əv ə•dishən] propiedad
conmutativa de la suma The property that
states that when the order of two addends is
changed, the sum is the same
Example: 4 + 5 = 5 + 4
Commutative Property of Multiplication
[kə•mytə•tiv präpər•tē əv mul•tə•pli•kāshən]
propiedad conmutativa de la multiplicación
The property that states that when the order of
two factors is changed, the product is the same
Example: 4 × 5 = 5 × 4
conversion factor [kən•vûrzhən faktər] factor de
conversión A rate in which two quantities are
equal, but use different units (p. 315)
coordinate plane [kō•ôrdn•it plān] plano cartesiano
A plane formed by a horizontal line called the
x-axis and a vertical line called the y-axis (p. 177)
Example:
compatible numbers [kəm•patə•bəl numbərz]
números compatibles Numbers that are easy to
compute with mentally
© Houghton Mifflin Harcourt Publishing Company
composite figure [kəm•päzit figyər] figura
compuesta A figure that is made up of two
or more simpler figures, such as triangles and
quadrilaterals (p. 571)
Example:
8 in.
10 in.
8 in.
6 in.
16 in.
Glossary
H3
cube [kyb] cubo A solid figure with six
congruent square faces
Example:
cubic unit [kybik ynit] unidad cúbica A unit
used to measure volume such as cubic foot (ft3),
cubic meter (m3), and so on
degree Celsius (°C) [di•grē selsē•əs] grado Celsius
A metric unit for measuring temperature
degree Fahrenheit (°F) [di•grē fârən•hīt] grado
Fahrenheit A customary unit for measuring
temperature
denominator [de•nämə•nāt•ər] denominador The
number below the bar in a fraction that tells
how many equal parts are in the whole or in
the group
3
Example: __
4
denominator
dependent variable [de•pendənt vârē•ə•bəl]
variable dependiente A variable whose value
depends on the value of another quantity
(p. 491)
data [dātə] datos Information collected about
people or things, often to draw conclusions
about them (p. 649)
decagon [dekə•gän] decágono A polygon with
10 sides and 10 angles
Examples:
difference [difər•əns] diferencia The answer to a
subtraction problem
digit [dijit] dígito Any one of the ten symbols 0, 1,
2, 3, 4, 5, 6, 7, 8, 9 used to write numbers
dimension [də•menshən] dimensión A measure in
one direction
distribution [dis•tri•byshən] distribución The
overall shape of a data set
decimal [desə•məl] decimal A number with one or
more digits to the right of the decimal point
decimal point [desə•məl point] punto decimal A
symbol used to separate dollars from cents in
money, and the ones place from the tenths
place in decimal numbers
divide [də•vīd] dividir To separate into equal
groups; the inverse operation of multiplication
dividend [divə•dend] dividendo The number that
is to be divided in a division problem
Example: 36 4 6; 6qw
36 The dividend is 36.
© Houghton Mifflin Harcourt Publishing Company
degree (°) [di•grē] grado (°) A unit for measuring
angles or for measuring temperature
Distributive Property [di•striby•tiv präpər•tē]
propiedad distributiva The property that states
that multiplying a sum by a number is the same
as multiplying each addend in the sum by the
number and then adding the products (p. 24)
Example: 3 3 (4 1 2) 5 (3 3 4) 1 (3 3 2)
3 3 6 5 12 1 6
18 5 18
H4
Glossary
divisible [də•vizə•bəl] divisible A number is divisible
by another number if the quotient is a counting
number and the remainder is zero
Example: 18 is divisible by 3.
division [də•vizhən] división The process of sharing
a number of items to find how many groups can
be made or how many items will be in a group;
the operation that is the inverse of multiplication
equation [i•kwāzhən] ecuación An algebraic
or numerical sentence that shows that two
quantities are equal (p. 421)
equilateral triangle [ē•kwi•latər•əl trīang•gəl]
triángulo equilátero A triangle with three
congruent sides
Example:
Division Property of Equality [də•vizhən präpər•tē
əv ē•kwôlə•tē] propiedad de división de la
igualdad The property that states that if you
divide both sides of an equation by the same
nonzero number, the sides remain equal
divisor [də•vīzər] divisor The number that
divides the dividend
Example: 15 4 3; 3qw
15 The divisor is 3.
equivalent [ē•kwivə•lənt] equivalente Having
the same value
dot plot [dot plät] diagrama de puntos A graph
that shows frequency of data along a number
line (p. 661)
Example:
equivalent decimals [ē•kwivə•lənt desə•məlz]
decimales equivalentes Decimals that name
the same number or amount
Example: 0.4 5 0.40 5 0.400
™
™
™
™
1
2
™
™
™
™
3
4
5
equivalent expressions [ē•kwivə•lənt ek•spreshənz]
expresiones equivalentes Expressions that are
equal to each other for any values of their
variables (p. 401)
Example: 2x + 4x = 6x
6
7
Miles Jogged
equivalent fractions [ē•kwivə•lənt frakshənz]
fracciones equivalentes Fractions that name
the same amount or part
3 6
Example: __ 5 __
4 8
equivalent ratios [ē•kwivə•lənt rāshē•ōz] razones
equivalentes Ratios that name the same
comparison (p. 223)
edge [ej] arista The line where two faces
of a solid figure meet
Example:
estimate [estə•māt] verb estimar (v) To find a
number that is close to an exact amount
© Houghton Mifflin Harcourt Publishing Company
edge
estimate [estə•mit] noun estimación (s) A number
close to an exact amount
Glossary
H5
evaluate [ē•valy•āt] evaluar To find the value of
a numerical or algebraic expression (p. 363)
factor [faktər] factor A number multiplied by
another number to find a product
even [ēvən] par A whole number that has a 0, 2,
4, 6, or 8 in the ones place
factor tree [faktər trē] árbol de factores A
diagram that shows the prime factors of a
number
Example:
expanded form [ek•spandid fôrm] forma
desarrollada A way to write numbers by
showing the value of each digit
Example: 832 5 800 1 30 1 2
exponent [ekspōn•ənt] exponente A number
that shows how many times the base is used
as a factor (p. 357)
Example: 103 5 10 3 10 3 10;
3 is the exponent.
5
X
X
2
6
X
3
expression [ek•spreshən] expresión A
mathematical phrase or the part of a number
sentence that combines numbers, operation
signs, and sometimes variables, but does not
have an equal or inequality sign
face [fās] cara A polygon that is a flat surface of
a solid figure
Example:
face
fact family [fakt famə•lē] familia de operaciones
A set of related multiplication and division, or
addition and subtraction, equations
7 3 8 5 56; 8 3 7 5 56;
Example:
56 4 7 5 8; 56 4 8 5 7
formula [fôrmy•lə] fórmula A set of symbols
that expresses a mathematical rule
Example: A 5 b 3 h
fraction [frakshən] fracción A number that names
a part of a whole or a part of a group
frequency [frēkwən•sē] frecuencia The number of
times an event occurs (p. 661)
frequency table [frēkwən•sē tābəl] tabla de
frecuencia A table that uses numbers to record
data about how often an event occurs (p. 662)
greatest common factor (GCF) [grātest kämən
faktər] máximo común divisor (MCD) The
greatest factor that two or more numbers
have in common (p. 23)
Example: 6 is the GCF of 18 and 30.
grid [grid] cuadrícula Evenly divided and equally
spaced squares on a figure or flat surface
© Houghton Mifflin Harcourt Publishing Company
Exponent comes from the combination of
the Latin roots ex (“out of”) 1 ponere (“to
place”). In the 17th century, mathematicians
began to use complicated quantities. The
idea of positioning a number by raising it
“out of place” is traced to René Descartes.
Glossary
5
Fahrenheit (°F) [fârən•hīt] Fahrenheit (°F) A
customary scale for measuring temperature
Word History
H6
30
horizontal [hôr•i•zäntəl] horizontal Extending left
and right
height [hīt] altura The length of a perpendicular
from the base to the top of a plane figure or
solid figure
Example:
hundredth [hundrədth] centésimo One of one
hundred equal parts
56
Examples: 0.56, ___
, fifty-six hundredths
100
height
hexagon [heksə•gän] hexágono A polygon with six
sides and six angles
Examples:
histogram [histə•gram] histograma A type of bar
graph that shows the frequencies of data in
intervals. (p. 667)
Example:
Heights of 6th Grade Students
9
Identity Property of Multiplication [ī•dentə•tē
präpər•tē əv mul•tə•pli•kāshən] propiedad de
identidad de la multiplicación The property
that states that the product of any number and
1 is that number
independent variable [in•dē•pendənt vârē•ə•bəl]
variable independiente A variable whose value
determines the value of another quantity (p. 491)
inequality [in•ē•kwôlə•tē] desigualdad A
mathematical sentence that contains the
symbol ,, ., #, $, or Þ (p. 491)
integers [intə•jərz] enteros The set of whole
numbers and their opposites (p. 139)
8
7
interquartile range [intûr•kwôrtīl rānj] rango
intercuartil The difference between the upper
and lower quartiles of a data set (p. 726)
6
Frequency
Identity Property of Addition [ī•dentə•tē präpər•tē
əv ə•dishən] propiedad de identidad de la suma
The property that states that when you add zero
to a number, the result is that number
5
intersecting lines [in•tər•sekting līnz] líneas secantes
Lines that cross each other at exactly one point
Example:
4
3
2
1
0
48–52
53–57
58–62
© Houghton Mifflin Harcourt Publishing Company
Height (inches)
63–67
inverse operations [invûrs äp•pə•rāshənz]
operaciones inversas Opposite operations,
or operations that undo each other, such as
addition and subtraction or multiplication and
division (p. 439)
Glossary
H7
key [kē] clave The part of a map or graph that
explains the symbols
kite [kīt] cometa A quadrilateral with exactly two
pairs of congruent sides that are next to each
other; no two sides are parallel
Example:
line [līn] línea A straight path in a plane,
extending in both directions with no
endpoints
Example:
line graph [līn graf] gráfica lineal A graph that
uses line segments to show how data change
over time
line segment [līn segmənt] segmento A part of a
line that includes two points called endpoints
and all the points between them
Example:
line of symmetry [līn əv sim ə•trē] eje de simetría
A line that divides a figure into two halves
that are reflections of each other (p. 184)
line symmetry [līn sim ə•trē] simetría axial A figure
has line symmetry if it can be folded about a
line so that its two parts match exactly. (p. 184)
linear equation [linē•ər ē•kwāzhən] ecuación lineal
An equation that, when graphed, forms a
straight line (p. 517)
ladder diagram [lad ər dīə•gram] diagrama de
escalera A diagram that shows the steps of
repeatedly dividing by a prime number until
the quotient is 1 (p. 12)
lateral area [latər•əl ârē•ə] área lateral The sum
of the areas of the lateral faces of a solid
linear unit [linē•ər ynit] unidad lineal A
measure of length, width, height, or distance
lower quartile [lōər kwôrtīl] primer cuartil The
median of the lower half of a data set (p. 713)
lateral face [latər•əl fās] cara lateral Any surface
of a polyhedron other than a base
least common multiple (LCM) [lēst kämən
mínimo común múltiplo (m.c.m.) The least
number that is a common multiple of two or
more numbers (p. 17)
multə•pəl]
like terms [līk tûrmz] términos semejantes
Expressions that have the same variable with
the same exponent (p. 395)
H8
Glossary
mean [mēn] media The sum of a set of data items
divided by the number of data items
(p. 681)
mean absolute deviation [mēn absə•lt
dē•vē•āshən] desviación absoluta respecto a la
media The mean of the distances from each
data value in a set to the mean of the set
(p. 720)
© Houghton Mifflin Harcourt Publishing Company
least common denominator (LCD) [lēst kämən
dē•nämə•nāt•ər] mínimo común denominador
(m.c.d.) The least common multiple of two or
more denominators
Example: The LCD for 1_4 and 5_6 is 12.
measure of center [mezhər əv sentər] medida
de tendencia central A single value used to
describe the middle of a data set (p. 681)
Examples: mean, median, mode
measure of variability [mezhər əv vârē•ə•bilə•tē]
medida de dispersión A single value used to
describe how the values in a data set are spread
out (p. 725)
Examples: range, interquartile range, mean
absolute deviation
median [mēdēən] mediana The middle value when
a data set is written in order from least to
greatest, or the mean of the two middle values
when there is an even number of items
(p. 681)
midpoint [midpoint] punto medio A point on a
line segment that is equally distant from either
endpoint
multiply [multə•plī] multiplicar When you combine
equal groups, you can multiply to find how
many in all; the inverse operation of division
negative integer [negə•tiv intə•jər] entero
negativo Any integer less than zero
Examples: −4, −5, and −6 are
negative integers.
net [net] plantilla A two-dimensional pattern
that can be folded into a three-dimensional
polyhedron (p. 597)
Example:
million [milyən] millón 1,000 thousands; written as
1,000,000
mixed number [mikst numbər] número mixto A
number that is made up of a whole number
and a fraction
Example: 1 5_8
mode [mōd] moda The value(s) in a data set that
occurs the most often (p. 681)
multiple [multə•pəl] múltiplo The product of two
counting numbers is a multiple of each of those
numbers
multiplication [mul•tə•pli•kāshən] multiplicación
A process to find the total number of items made
up of equal-sized groups, or to find the total
number of items in a given number of groups; It
is the inverse operation of division.
Multiplication Property of Equality
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[mul•tə•pli•kāshən präpər•tē əv ē•kwôlə•tē]
propiedad de multiplicación de la igualdad
The property that states that if you multiply
both sides of an equation by the same
number, the sides remain equal
not equal to (≠) [not ēkwəl t] no igual a
A symbol that indicates one quantity is not
equal to another
number line [numbər līn] recta numérica A line on
which numbers can be located
Example:
numerator [nmər•āt•ər] numerador The number
above the bar in a fraction that tells how many
equal parts of the whole are being considered
3
numerator
Example: __
4
numerical expression [n•meri•kəl ek•spreshən]
expresión numérica A mathematical phrase
that uses only numbers and operation
signs (p. 363)
multiplicative inverse [multə•pli•kāt•iv invûrs] inverso
multiplicativo A reciprocal of a number that is
multiplied by that number resulting in a product
of 1 (p. 108)
Glossary
H9
obtuse triangle [äb•ts trīang•gəl] triángulo
obtusángulo A triangle that has one obtuse
angle
octagon [äktə•gän] octágono A polygon with
eight sides and eight angles
Examples:
odd [od] impar A whole number that has a 1, 3,
5, 7, or 9 in the ones place
open figure [ōpən figyər] figura abierta A figure
that does not begin and end at the same point
opposites [äpə•zits] opuestos Two numbers
that are the same distance, but in opposite
directions, from zero on a number line (p. 139)
order of operations [ôrdər əv äp•ə•rāshənz]
orden de las operaciones A special set of rules
which gives the order in which calculations are
done in an expression (p. 363)
ordered pair [ôrdərd pâr] par ordenado A pair of
numbers used to locate a point on a grid. The
first number tells the left-right position and
the second number tells the up-down position.
(p. 177)
origin [ôrə•jin] origen The point where the two
axes of a coordinate plane intersect; (0,0)
(p. 177)
outlier [outlī•ər] valor atípico A value much
higher or much lower than the other values in
a data set (p. 687)
overestimate [ōvər•es•tə•mit] sobrestimar
An estimate that is greater than the exact
answer
H10
Glossary
parallel lines [pârə•lel līnz] líneas paralelas Lines
in the same plane that never intersect and are
always the same distance apart
Example:
parallelogram [pâr•ə•lelə•gram] paralelogramo
A quadrilateral whose opposite sides are parallel
and congruent (p. 533)
Example:
parentheses [pə•renthə•sēz] paréntesis The
symbols used to show which operation or
operations in an expression should be done
first
partial product [pärshəl prädəkt] producto parcial
A method of multiplying in which the ones,
tens, hundreds, and so on are multiplied
separately and then the products are added
together
pattern [patərn] patrón An ordered set of
numbers or objects; the order helps you
predict what will come next
Examples: 2, 4, 6, 8, 10
pentagon [pentə•gän] pentágono A polygon
with five sides and five angles
Examples:
© Houghton Mifflin Harcourt Publishing Company
obtuse angle [äb•ts anggəl] ángulo obtuso
An angle whose measure is greater than 90°
and less than 180°
Example:
percent [pər•sent] porcentaje The comparison
of a number to 100; percent means “per
hundred” (p. 269)
perimeter [pə•rimə•tər] perímetro The distance
around a closed plane figure
point [point] punto An exact location in space
polygon [päli•gän] polígono A closed plane figure
formed by three or more line segments
Examples:
period [pir ē•əd] período Each group of three
digits separated by commas in a multidigit
number
Example: 85,643,900 has three periods.
perpendicular lines [pər•pən•diky•lər līnz] líneas
perpendiculares Two lines that intersect to
form four right angles
Example:
Polygons
Not Polygons
polyhedron [päl•i•hēdrən] poliedro A solid
figure with faces that are polygons (p. 596)
Examples:
positive integer [päzə•tiv intə•jər] entero
positivo Any integer greater than zero
pictograph [piktə•graf] pictografía A graph
that displays countable data with symbols
or pictures
Example:
HOW WE GET TO SCHOOL
Walk
Ride a Bus
Ride in a Car
= 10 students
place value [plās valy] valor posicional The value
of each digit in a number based on the location
of the digit
plane [plān] plano A flat surface that extends
without end in all directions
Example:
© Houghton Mifflin Harcourt Publishing Company
prime factorization [prīm fak•tə•rə•zāshən]
descomposición en factores primos A number
written as the product of all its prime factors
(p. 11)
prime number [prīm numbər] número primo
A number that has exactly two factors: 1 and
itself
Examples: 2, 3, 5, 7, 11, 13, 17, and 19 are prime
numbers. 1 is not a prime number.
Ride a Bike
Key: Each
prime factor [prīm fak tər] factor primo A factor
that is a prime number
prism [prizəm] prisma A solid figure that has two
congruent, polygon-shaped bases, and other
faces that are all parallelograms
Examples:
rectangular prism
triangular prism
product [prädəkt] producto The answer to a
multiplication problem
plane figure [plān figyər] figura plana A figure
that lies in a plane; a figure having length and
width
Glossary
H11
pyramid [pirə•mid] pirámide A solid figure with
a polygon base and all other faces as triangles
that meet at a common vertex
Example:
ratio [rāshē•ō] razón A comparison of two
numbers, a and b, that can be written as a
fraction _ba (p. 211)
rational number [rash•ən•əl numbər] número
racional Any number that can be written as a
ratio _ba where a and b are integers and b ≠ 0.
(p. 151)
Word History
A fire is sometimes in the shape of a
pyramid, with a point at the top and a
wider base. This may be how pyramid got
its name. The Greek word for fire was
pura, which may have been combined with
the Egyptian word mer.
ray [rā] semirrecta A part of a line; it has one
endpoint and continues without end in one
direction
Example:
reciprocal [ri•siprə•kəl] recíproco Two numbers
are reciprocals of each other if their product
equals 1. (p. 108)
rectangle [rektang•gəl] rectángulo A
parallelogram with four right angles
Example:
quadrilateral [kwä•dri•latər•əl] cuadrilátero
A polygon with four sides and four angles
Example:
quotient [kwōshənt] cociente The number that
results from dividing
Example: 8 4 4 5 2. The quotient is 2.
range [rānj] rango The difference between the
greatest and least numbers in a data set
(p. 726)
rate [rāt] tasa A ratio that compares two
quantities having different units of measure
(p. 218)
H12
Glossary
rectangular prism [rek•tanggyə•lər prizəm]
prisma rectangular A solid figure in which all
six faces are rectangles
Example:
reflection [ri•flek shən] reflexión A movement of a
figure to a new position by flipping it over
a line; a flip
Example:
regroup [rē•grp ] reagrupar To exchange
amounts of equal value to rename a number
Example: 5 1 8 5 13 ones or 1 ten 3 ones
© Houghton Mifflin Harcourt Publishing Company
quadrants [kwädrənts] cuadrantes The four
regions of the coordinate plane separated by
the x- and y-axes (p. 183)
regular polygon [regyə•lər päli•gän] polígono
regular A polygon in which all sides are
congruent and all angles are congruent
(p. 565)
relative frequency table [relə•tiv frēkwən•sē tābəl]
tabla de frecuencia relativa A table that shows
the percent of time each piece of data occurs
(p. 662)
remainder [ri•māndər] residuo The amount
left over when a number cannot be divided
equally
rhombus [rämbəs] rombo A parallelogram with
four congruent sides
Example:
Word History
Rhombus is almost identical to its Greek
origin, rhombos. The original meaning
was “spinning top” or “magic wheel,”
which is easy to imagine when you look at
a rhombus, an equilateral parallelogram.
right triangle [rīt trīang•gəl] triángulo rectángulo
A triangle that has a right angle
Example:
simplest form [simpləst fôrm] mínima expresión
A fraction is in simplest form when the
numerator and denominator have only 1 as
a common factor
simplify [simplə•fī] simplificar The process of
dividing the numerator and denominator of a
fraction or ratio by a common factor
solid figure [sälid figyər] cuerpo geométrico
A three-dimensional figure having length,
width, and height (p. 597)
solution of an equation [sə•lshən əv an
ē•kwāzhən] solución de una ecuación A value
that, when substituted for the variable, makes
an equation true (p. 421)
solution of an inequality [sə•lshən əv an
in•ē•kwôlə•tē] solución de una desigualdad A
value that, when substituted for the variable,
makes an inequality true (p. 465)
square [skwâr] cuadrado A polygon with four
equal, or congruent, sides and four right
angles
square pyramid [skwâr pirə•mid] pirámide cuadrada
A solid figure with a square base and with four
triangular faces that have a common vertex
Example:
square unit [skwâr ynit] unidad cuadrada A unit
used to measure area such as square foot (ft2),
square meter (m2), and so on
© Houghton Mifflin Harcourt Publishing Company
round [round] redondear To replace a number with
one that is simpler and is approximately the
same size as the original number
Example: 114.6 rounded to the nearest ten is
110 and to the nearest unit is 115.
sequence [sēkwəns] secuencia An ordered set of
numbers
Glossary
H13
standard form [standərd fôrm] forma normal A
way to write numbers by using the digits 0–9,
with each digit having a place value
Example: 456
standard form
statistical question [stə•tisti•kəl kweschən]
pregunta estadística A question that asks
about a set of data that can vary (p. 649)
Example: How many desks are in each
classroom in my school?
Substitution Property of Equality [sub•stə•tshən
präpər•tē əv ē•kwôlə•tē] propiedad de
sustitución de la igualdad The property that
states that if you have one quantity equal to
another, you can substitute that quantity for
the other in an equation
subtraction [səb•trakshən] resta The process of
finding how many are left when a number
of items are taken away from a group of
items; the process of finding the difference
when two groups are compared; the inverse
operation of addition
Subtraction Property of Equality [səb•trakshən
präpər•tē əv ē•kwôlə•tē] propiedad de resta de
la igualdad The property that states that if
you subtract the same number from both sides
of an equation, the sides remain equal
sum [sum] suma o total The answer to an
addition problem
terms [tûrmz] términos The parts of an expression
that are separated by an addition or
subtraction sign (p. 376)
thousandth [thouzəndth] milésimo One of one
thousand equal parts
Example: 0.006 5 six thousandths
three-dimensional [thrē də•menshə•nəl]
tridimensional Measured in three directions,
such as length, width, and height
three-dimensional solid [thrē də•menshə•nəl sälid]
figura tridimensional See solid figure
trapezoid [trapi•zoid] trapecio A quadrilateral
with exactly one pair of parallel sides (p. 551)
Examples:
tree diagram [trē dīə•gram] diagrama de árbol
A branching diagram that shows all possible
outcomes of an event
trend [trend] tendencia A pattern over time, in all
or part of a graph, where the data increase,
decrease, or stay the same
triangle [trīang•gəl] triángulo A polygon with
three sides and three angles
Examples:
surface area [sûrfis ârē•ə] área total The sum
of the areas of all the faces, or surfaces, of a
solid figure (p. 603)
tally table [talē tābəl] tabla de conteo A table
that uses tally marks to record data
two-dimensional [t də•menshə•nəl]
bidimensional Measured in two directions,
such as length and width
tenth [tenth] décimo One of ten equal parts
Example: 0.7 5 seven tenths
two-dimensional figure [t də•menshə•nəl figyər]
figura bidimensional See plane figure
H14
Glossary
© Houghton Mifflin Harcourt Publishing Company
triangular prism [trī•anggyə•lər prizəm] prisma
triangular A solid figure that has two
triangular bases and three rectangular faces
underestimate [un•dər•estə•mit] subestimar An
estimate that is less than the exact answer
unit fraction [ynit frakshən] fraccion unitaria A
fraction that has 1 as a numerator
unit rate [ynit rāt] tasa por unidad A rate
expressed so that the second term in the ratio
is one unit (p. 218)
Example: 55 mi per hr
unlike fractions [unlīk frakshənz] fracciones
no semejantes Fractions with different
denominators
upper quartile [upər kwôrtīl] tercer cuartil The
median of the upper half of a data set (p. 713)
variable [vârē•ə•bəl] variable A letter or symbol
that stands for an unknown number or
numbers (p. 369)
Venn diagram [ven dīə•gram] diagrama de Venn
A diagram that shows relationships among sets
of things
Example:
vertex [vûrteks] vértice The point where two
or more rays meet; the point of intersection
of two sides of a polygon; the point of
intersection of three (or more) edges of a
solid figure; the top point of a cone; the
plural of vertex is vertices
Examples:
vertex
vertex
vertical [vûrti•kəl] vertical Extending up and down
volume [välym] volumen The measure of the
space a solid figure occupies (p. 623)
weight [wāt] peso How heavy an object is
whole number [hōl numbər] número entero
One of the numbers 0, 1, 2, 3, 4, . . . ; the set
of whole numbers goes on without end
x-axis [eks aksis] eje de la x The horizontal
number line on a coordinate plane (p. 177)
© Houghton Mifflin Harcourt Publishing Company
x-coordinate [eks kō•ôrdn•it] coordenada x
The first number in an ordered pair; tells the
distance to move right or left from (0,0) (p. 177)
Glossary
H15
y-axis [wī aksis] eje de la y The vertical number
line on a coordinate plane (p. 177)
y-coordinate [wī kō•ôrd•n•it] coordenada y The
second number in an ordered pair; tells the
distance to move up or down from (0,0) (p. 177)
© Houghton Mifflin Harcourt Publishing Company
Zero Property of Multiplication [zērō präpər•tē
əv mul•tə•pli•kāshən] propiedad del cero de la
multiplicación The property that states that
when you multiply by zero, the product is zero
H16
Glossary
COMMON CORE STATE STANDARDS
Standards You Will Learn
Student Edition Lessons
Mathematical Practices
MP1
Make sense of problems and
persevere in solving them.
Lessons 1.1, 2.6, 2.9, 6.1, 6.3, 6.5,
12.8, 13.4, 13.7
MP2
Reason abstractly and quantitatively.
Lessons 1.3, 1.8, 1.9, 7.3, 7.4, 7.6,
12.5, 12.7, 13.5
MP3
Construct viable arguments and
critique the reasoning of others.
Lessonss 1.7, 2.4, 3.5, 6.4, 7.8, 8.3,
13.5, 13.7, 13.8
MP4
Model with mathematics.
Lessons 1.4, 2.5, 2.8, 5.6, 6.2, 7.6,
12.8, 13.2, 13.3
MP5
Use appropriate tools strategically.
Lessons 2.8, 3.1, 3.2, 6.4, 7.5, 7.7,
12.3, 12.8, 13.1
MP6
Attend to precision.
Lessons 1.6, 2.9, 3.7, 7.4, 7.9, 8.1,
13.6, 13.7, 13.8
MP7
Look for and make use of structure.
Lessons 1.2, 2.7, 3.1, 5.2, 6.5, 8.7,
13.1, 13.4, 13.6
MP8
Look for and express regularity in
repeated reasoning.
Lessons 1.9, 2.7, 3.2, 4.5, 5.2, 6.2,
12.5, 13.1, 13.3
Domain: Ratios and Proportional Relationships
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Understand ratio concepts and use ratio reasoning to solve problems.
6.RP.A.1
Understand the concept of a ratio
and use ratio language to describe
a ratio relationship between two
quantities. For example, “The ratio
of wings to beaks in the bird house
at the zoo was 2:1, because for every
2 wings there was 1 beak.”“For every
vote candidate A received, candidate
C received nearly three votes.”
Lessons 4.1, 4.2
6.RP.A.2
Understand the concept of a unit
rate a/b associated with a ratio a:b
with b & 0, and use rate language in
the context of a ratio relationship.
For example, “This recipe has a ratio
of 3 cups of flour to 4 cups of sugar,
so there is 3/4 cup of flour for each
cup of sugar.” “We paid $75 for
15 hamburgers, which is a rate of
$5 per hamburger.”
Lessons 4.2, 4.6
Correlations
H17
Standards You Will Learn
Student Edition Lessons
Domain: Ratios and Proportional Relationships (Continued)
Use ratio and rate reasoning to
solve real-world and mathematical
problems, e.g., by reasoning about
tables of equivalent ratios, tape
diagrams, double number line
diagrams, or equations.
a. Make tables of equivalent ratios
relating quantities with wholenumber measurements, find
missing values in the tables, and
plot the pairs of values on the
coordinate plane. Use tables to
compare ratios.
Lessons 4.3, 4.4, 4.5, 4.8
b. Solve unit rate problems including
those involving unit pricing and
constant speed. For example, if it
took 7 hours to mow 4 lawns, then
at that rate, how many lawns could
be mowed in 35 hours? At what
rate were lawns being mowed?
Lessons 4.6, 4.7
c. Find a percent of a quantity as
a rate per 100 (e.g., 30% of a
quantity means 30/100 times the
quantity); solve problems involving
finding the whole, given a part and
the percent.
Lessons 5.1, 5.2, 5.3, 5.4, 5.5, 5.6
d. Use ratio reasoning to convert
measurement units; manipulate
and transform units appropriately
when multiplying or dividing
quantities.
Lessons 6.1, 6.2, 6.3, 6.4, 6.5
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6.RP.A.3
H18
Correlations
Standards You Will Learn
Student Edition Lessons
Domain: The Number System
Apply and extend previous understandings of multiplications and division to divide
fractions by fractions.
6.NS.A.1
Interpret and compute quotients of
fractions, and solve word problems
involving division of fractions by
fractions, e.g., by using visual fraction
models and equations to represent
the problem. For example, create
a story context for (2/3) ÷ (3/4) and
use a visual fraction model to show
the quotient; use the relationship
between multiplication and division
to explain that (2/3) ÷ (3/4) = 8/9
because 3/4 of 8/9 is 2/3. (In general,
(a/b) ÷ (c/d) = ad/bc.) How much
chocolate will each person get if
3 people share 1/2 lb of chocolate
equally? How many 3/4-cup servings
are in 2/3 of a cup of yogurt? How
wide is a rectangular strip of land
with length 3/4 mi and area 1/2
square mi?
Lessons 2.5, 2.6, 2.7, 2.8, 2.9, 2.10
© Houghton Mifflin Harcourt Publishing Company
Compute fluently with multi-digit numbers and find common factors and multiples.
6.NS.B.2
Fluently divide multi-digit numbers
using the standard algorithm.
Lesson 1.1
6.NS.B.3
Fluently add, subtract, multiply, and
divide multi-digit decimals using
the standard algorithm for each
operation.
Lessons 1.6, 1.7, 1.8, 1.9
6.NS.B.4
Find the greatest common factor
of two whole numbers less than or
equal to 100 and the least common
multiple of two whole numbers
less than or equal to 12. Use the
distributive property to express a sum
of two whole numbers 1–100 with
a common factor as a multiple of a
sum of two whole numbers with no
common factor. For example, express
36 + 8 as 4 (9 + 2).
Lessons 1.2, 1.3, 1.4, 1.5, 2.3, 2.4
Correlations
H19
Standards You Will Learn
Student Edition Lessons
Apply and extend previous understandings of numbers to the system of rational numbers.
Understand that positive and
negative numbers are used together
to describe quantities having opposite
directions or values (e.g., temperature
above/below zero, elevation above/
below sea level, credits/ debits,
positive/negative electric charge);
use positive and negative numbers
to represent quantities in real-world
contexts, explaining the meaning of 0
in each situation.
6.NS.C.6
Understand a rational number as a
point on the number line. Extend
number line diagrams and coordinate
axes familiar from previous grades
to represent points on the line and
in the plane with negative number
coordinates.
H20
Lesson 3.1, 3.3
a. Recognize opposite signs of
numbers as indicating locations on
opposite sides of 0 on the number
line; recognize that the opposite
of the opposite of a number is the
number itself, e.g., −(–3) = 3, and
that 0 is its own opposite.
Lessons 3.1, 3.3
b. Understand signs of numbers in
ordered pairs as indicating locations
in quadrants of the coordinate
plane; recognize that when two
ordered pairs differ only by signs, the
locations of the points are related by
reflections across one or both axes.
Lesson 3.8
c. Find and position integers and
other rational numbers on a
horizontal or vertical number line
diagram; find and position pairs of
integers and other rational numbers
on a coordinate plane.
Lessons 2.1, 3.1, 3.3, 3.7
Correlations
© Houghton Mifflin Harcourt Publishing Company
6.NS.C.5
Standards You Will Learn
Student Edition Lessons
Apply and extend previous understandings of numbers to the system of
rational numbers. (Continued)
6.NS.C.7
© Houghton Mifflin Harcourt Publishing Company
6.NS.C.8
Understand ordering and absolute
value of rational numbers.
a. Interpret statements of inequality
as statements about the relative
position of two numbers on a
number line diagram. For example,
interpret –3 > –7 as a statement
that –3 is located to the right of
–7 on a number line oriented from
left to right.
Lessons 2.2, 3.2, 3.4
b. Write, interpret, and explain
statements of order for rational
numbers in real-world contexts.
For example, write –3°C > –7°C
to express the fact that –3°C is
warmer than –7°C.
Lessons 3.2, 3.4
c. Understand the absolute value of a
rational number as its distance from
0 on the number line; interpret
absolute value as magnitude for a
positive or negative quantity in a
real-world situation. For example,
for an account balance of –30
dollars, write | –30| = 30 to describe
the size of the debt in dollars.
Lesson 3.5
d. Distinguish comparisons of
absolute value from statements
about order. For example,
recognize that an account balance
less than –30 dollars represents a
debt greater than 30 dollars.
Lesson 3.6
Solve real-world and mathematical
problems by graphing points in all
four quadrants of the coordinate
plane. Include use of coordinates
and absolute value to find distances
between points with the same first
coordinate or the same second
coordinate.
Lessons 3.9, 3.10
Correlations
H21
Standards You Will Learn
Student Edition Lessons
Domain: Expressions and Equations
Apply and extend previous understandings of arithmetic to algebraic expressions.
Write and evaluate numerical
expressions involving whole-number
exponents.
6.EE.A.2
Write, read, and evaluate expressions
in which letters stand for numbers.
6.EE.A.3
H22
Lessons 7.1, 7.2
a. Write expressions that record
operations with numbers and with
letters standing for numbers. For
example, express the calculation
“Subtract y from 5” as 5 − y.
Lesson 7.3
b. Identify parts of an expression
using mathematical terms (sum,
term, product, factor, quotient,
coefficient); view one or more
parts of an expression as a single
entity. For example, describe the
expression 2(8 + 7) as a product
of two factors; view (8 + 7) as both
a single entity and a sum of two
terms.
Lesson 7.4
c. Evaluate expressions at specific
values of their variables. Include
expressions that arise from
formulas used in real-world
problems. Perform arithmetic
operations, including those
involving whole-number exponents,
in the conventional order when
there are no parentheses to
specify a particular order (Order of
Operations). For example, use the
formulas V = s³ and A = 6s² to find
the volume and surface area of a
cube with sides of length s = 1/2.
Lessons 7.5, 10.1, 10.5, 10.6,
10.7, 11.3, 11.4, 11.6
Apply the properties of operations
to generate equivalent expressions.
For example, apply the distributive
property to the expression 3(2 + x)
to produce the equivalent expression
6 + 3x; apply the distributive property
to the expression 24x + 18y to
produce the equivalent expression
6(4x + 3y ); apply properties of
operations to y + y + y to produce the
equivalent expression 3y.
Lessons 7.7, 7.8
Correlations
© Houghton Mifflin Harcourt Publishing Company
6.EE.A.1
Standards You Will Learn
Student Edition Lessons
Apply and extend previous understandings of arithmetic to algebraic expressions.
(Continued)
6.EE.A.4
Identify when two expressions
are equivalent (i.e., when the
two expressions name the same
number regardless of which value is
substituted into them). For example,
the expressions y + y + y and 3y are
equivalent because they name the
same number regardless of which
number y stands for.
Lesson 7.9
© Houghton Mifflin Harcourt Publishing Company
Reason about and solve one-variable equations and inequalities.
6.EE.B.5
Understand solving an equation or
inequality as a process of answering
a question: which values from
a specified set, if any, make the
equation or inequality true? Use
substitution to determine whether a
given number in a specified set makes
an equation or inequality true.
Lessons 8.1, 8.8
6.EE.B.6
Use variables to represent numbers
and write expressions when solving a
real-world or mathematical problem;
understand that a variable can
represent an unknown number, or,
depending on the purpose at hand,
any number in a specified set.
Lesson 7.6
6.EE.B.7
Solve real-world and mathematical
problems by writing and solving
equations of the form x + p = q and
px = q for cases in which p, q and x
are all nonnegative rational numbers.
Lessons 8.2, 8.3, 8.4, 8.5, 8.6,
8.7, 10.1
6.EE.B.8
Write an inequality of the form x > c
or x < c to represent a constraint
or condition in a real-world or
mathematical problem. Recognize
that inequalities of the form x > c
or x < c have infinitely many
solutions; represent solutions of such
inequalities on number line diagrams.
Lessons 8.9, 8.10
Correlations
H23
Standards You Will Learn
Student Edition Lessons
Represent and analyze quantitative relationships between dependent and
independent variables.
6.EE.C.9
Use variables to represent two
quantities in a real-world problem
that change in relationship to one
another; write an equation to express
one quantity, thought of as the
dependent variable, in terms of the
other quantity, thought of as the
independent variable. Analyze the
relationship between the dependent
and independent variables using
graphs and tables, and relate these
to the equation. For example, in a
problem involving motion at constant
speed, list and graph ordered pairs
of distances and times, and write the
equation d = 65t to represent the
relationship between distance and
time.
Lessons 9.1, 9.2, 9.3, 9.4, 9.5
Domain: Geometry
6.G.A.1
Find the area of right triangles, other
triangles, special quadrilaterals,
and polygons by composing into
rectangles or decomposing into
triangles and other shapes; apply
these techniques in the context of
solving real-world and mathematical
problems.
Lessons 10.1, 10.2, 10.3, 10.4, 10.5,
10.6, 10.7, 10.8, 11.7
6.G.A.2
Find the volume of a right
rectangular prism with fractional
edge lengths by packing it with
unit cubes of the appropriate unit
fraction edge lengths, and show that
the volume is the same as would
be found by multiplying the edge
lengths of the prism. Apply the
formulas V = lwh and V = bh to find
volumes of right rectangular prisms
with fractional edge lengths in the
context of solving real-world and
mathematical problems.
Lessons 11.5, 11.6, 11.7
H24
Correlations
© Houghton Mifflin Harcourt Publishing Company
Solve real-world and mathematical problems involving area, surface area, and volume.
Standards You Will Learn
Student Edition Lessons
Solve real-world and mathematical problems involving area, surface area, and volume.
(Continued)
6.G.A.3
Draw polygons in the coordinate
plane given coordinates for the
vertices; use coordinates to find
the length of a side joining points
with the same first coordinate or
the same second coordinate. Apply
these techniques in the context of
solving real-world and mathematical
problems.
Lesson 10.9
6.G.A.4
Represent three-dimensional figures
using nets made up of rectangles and
triangles, and use the nets to find the
surface area of these figures. Apply
these techniques in the context of
solving real-world and mathematical
problems.
Lessons 11.1, 11.2, 11.3, 11.4, 11.7
Domain: Statistics and Probability
© Houghton Mifflin Harcourt Publishing Company
Develop understanding of statistical variability.
6.SP.A.1
Recognize a statistical question as
one that anticipates variability in
the data related to the question and
accounts for it in the answers. For
example, “How old am I?” is not a
statistical question, but “How old
are the students in my school?” is
a statistical question because one
anticipates variability in students’
ages.
Lesson 12.1
6.SP.A.2
Understand that a set of data
collected to answer a statistical
question has a distribution which can
be described by its center, spread,
and overall shape.
Lessons 12.6, 13.1, 13.4, 13.6,
13.7, 13.8
6.SP.A.3
Recognize that a measure of center
for a numerical data set summarizes
all of its values with a single number,
while a measure of variation
describes how its values vary with a
single number.
Lessons 12.6, 13.4, 13.6
Correlations
H25
Standards You Will Learn
Student Edition Lessons
Summarize and describe distributions.
6.SP.B.4
Display numerical data in plots on
a number line, including dot plots,
histograms, and box plots.
6.SP.B.5
Summarize numerical data sets in
relation to their context, such as by:
Lessons 12.3, 12.4, 12.8, 13.2
Lesson 12.2
b. Describing the nature of the
attribute under investigation,
including how it was measured and
its units of measurement.
Lesson 12.2
c. Giving quantitative measures of
center (median and/or mean) and
variability (interquartile range
and/or mean absolute deviation),
as well as describing any overall
pattern and any striking deviations
from the overall pattern with
reference to the context in which
the data were gathered.
Lessons 12.5, 12.6, 12.7, 13.1, 13.3,
13.4, 13.8
d. Relating the choice of measures of
center and variability to the shape
of the data distribution and the
context in which the data were
gathered.
Lessons 12.7, 13.5, 13.7
© Houghton Mifflin Harcourt Publishing Company
a. Reporting the number of
observations.
H26
Correlations
© Houghton Mifflin Harcourt Publishing Company
Absolute value, 165–168
compare, 171–174
defined, 138, 165
Activity, 95, 108, 113, 171, 184, 211, 533, 565,
656, 745
Addition
Addition Property of Equality, 420, 440
Associative Property of, 401
Commutative Property of, 401
decimals, 37–40
Distributive Property, 401–404
equations
model and solve, 433–436
solution, 421–424
Identity Property of, 401
order of operations, 38–39, 363
properties of, 401
Addition Properties, 355
Addition Property of Equality, 420, 440
Algebra
algebraic expressions
combine like terms, 395–398
defined, 369
equivalent
generate, 401–404
identifying, 407–410
evaluating, 381–384, 419, 531
exponents, 357–360
identifying, 375–378
simplifying, 395–398
terms of, 376
translating between tables and, 378–381
translating between words and, 369–372
use variables to solve problems, 389–392
variable in, 369, 389–392
writing, 369–372
area
composite figures, 571–574
defined, 532
parallelograms, 533–536
rectangles, 533–536
regular polygons, 565–568
squares, 533–536
surface area, 603–606, 609–612, 615–618
trapezoids, 551–554, 557–560
triangles, 539–542, 545–548
distance, rate, and time formulas, 341–344
equations
addition, 433–442
defined, 420
division, 451–454
with fractions, 457–460
multiplication, 445–448, 451–454
solution of, 420, 421–424
subtraction, 439–442
write from word sentence, 428–429
writing, 427–430
equivalent ratios
graph to represent, 255–258
to solve problems, 229–232
evaluate, 121
integers
absolute value, 165–168
compare and order, 145–148
defined, 138, 139, 163
opposites, 138
order of operations, 38, 39, 44, 45, 51, 57,
82, 83, 109, 121, 363, 364, 381, 705
inverse operations, 7
least common multiple, 19
patterns
divide mixing patterns, 121
proportions, equivalent ratios to solve, 235–238
reasoning, 191
finding least common multiple, 19
finding the missing number, 167
surface area, 603–606, 609–612, 615–618
unit rates to solve problems, 249–252
volume, 629–632
Algebraic expressions
combine like terms, 395–398
defined, 369
equivalent
generate, 401–404
identifying, 407–410
evaluating, 381–384, 419, 531
exponents, 357–360
identifying parts, 375–378
like terms, 395–398
simplifying, 395–396
terms of, 376
translating between tables and, 378–381
translating between words and, 370–371
use variables to solve problems, 389–392
variables in, 389–392
writing, 369–372
Area
composite figures, 571–574
defined, 532
finding, 595
of parallelograms, 533–536
of rectangles, 595
Index
H27
Balance point
mean as, 675–678
Bar graph
reading, 647
Bar model, 250–251, 289, 295, 297
Base
of a number, 356
of solid figure, 597–600
Basic fact, 11
Benchmark, 68, 82
Benchmark fractions, 67
Box plots, 713–716
defined, 706, 713
for finding interquartile range, 726–727, 734
for finding range, 726–727, 734
Bubble map, 68, 210, 268, 356, 490, 532, 596
Capacity
converting units of, 321–324
customary units of, 321
defined, 314
metric units of, 322
Cause and effect, 500
Centigrams, 328, 3