National Glencoe Math, Course 2, © 2015
Chapter 4: Rational Numbers
Lesson 2: Compare and Order Rational Numbers
Common Core State Standards
Common Core State Standards
7.NS.2
Apply and extend previous understandings of multiplication and division and of fractions to
multiply and divide rational numbers.
7.NS.2b Understand that integers can be divided, provided that the divisor is not zero, and every
quotient of integers (with non-zero divisor) is a rational number. If p and q are integers, then
–(p/q) = (–p)/q = p/(–q). Interpret quotients of rational numbers by describing real-world
contexts.
7.EE.3
Solve multi-step real-life and mathematical problems posed with positive and negative
rational numbers in any form (whole numbers, fractions, and decimals), using tools
strategically. Apply properties of operations to calculate with numbers in any form; convert
between forms as appropriate; and assess the reasonableness of answers using mental
computation and estimation strategies. For example: If a woman making $25 an hour gets a
10% raise, she will make an additional 1/10 of her salary an hour, or $2.50, for a new salary
of $27.50. If you want to place a towel bar 9 3/4 inches long in the center of a door that is 27
1/2 inches wide, you will need to place the bar about 9 inches from each edge; this estimate
can be used as a check on the exact computation.
Mathematical Practices 1, 3, and 4 are aspects of mathematical thinking that are emphasized in every
lesson. Students are given opportunities to be persistent in their problem solving, to express their
reasoning, and apply mathematics to realworld situations.
Exercises 11 and 15 focus on Mathematical Practice 4 Model with mathematics.
Exercises 12 and 14 focus on Mathematical Practice 3 Construct viable arguments and critique the
reasoning of others.
Exercise 13 focuses on Mathematical Practice 1 Make sense of problems and persevere in solving
them.
Printed by: Nicole Sladich
Page 1 of 14
Printed on 11/04/2014 10:13 PM
National Glencoe Math, Course 2, © 2015
Chapter 4: Rational Numbers
Lesson 2: Compare and Order Rational Numbers
1 Launch the Lesson
Focus narrowing the scope
Objective Compare and order rational numbers. 7.NS.2, 7.NS.2b, 7.EE.3
Coherence connecting within and across grades
Previous
Now
Next
Students wrote rational numbers Students compare and order
Students will add and subtract
as decimals and fractions.
rational numbers by writing them fractions with like denominators.
in the same form.
Rigor pursuing concepts, fluency, and applications
See the Levels of Complexity chart in the Practice and Apply section.
Lesson Vocabulary
rational number
common denominator
least common denominator
As students enter the classroom, have them complete the Common Core Quick Check to review and
assess the skills presented in the previous lesson.
Engage Explore Explain Elaborate Evaluate
Ideas for Use
You may wish to launch the lesson using a whole group, small group, think-pair-share activity, or
independent activity.
Think-Pair-Share Give students one minute to think through their responses to the
Launch the Lesson: Vocabulary activity. Have them share their responses with a partner. Then have
each partner share their response with the whole group.
1, 6, 7
Alternate Strategy
Have students generate their own rational numbers that can be placed into each category
of the Venn diagram. For each number they generate, have them explain why that number can be
placed into that category.
1, 3, 7
Resources
Common Core Quick Check: Compare and Order Rational Numbers
Printed by: Nicole Sladich
Page 2 of 14
Printed on 11/04/2014 10:13 PM
National Glencoe Math, Course 2, © 2015
Chapter 4: Rational Numbers
Lesson 2: Compare and Order Rational Numbers
2 Teach the Concept
Focus: Mathematical Background
Rational numbers can be written as the ratio of two integers (the second integer cannot be 0).
Decimals that neither terminate nor repeat are not rational.
Fraction-Decimal Equivalents When students are memorizing common fraction-decimal equivalents,
it is useful to use coin equivalents.
Quarter:
= 0.25,
Dime:
= 0.10,
= 0.50,
= 0.30,
= 0.75
= 0.90
One dollar can be divided equally into fifths, so
= 0.20,
= 0.40, and
= 0.60.
The McGraw-Hill Professional Development Video Library provides short videos that support
Glencoe Math. The featured videos for mathematical content in this lesson are:
Compare and Order Whole Numbers, Fractions, and Percents
Rational and Irrational Numbers
Simplifying Fractions and Expressions
For classroom presentation, use the Step-by-Step Examples to demonstrate the examples from the
Student Edition.
Engage Explore Explain Elaborate Evaluate
Ask the scaffolded questions for each example to differentiate instruction.
1. Compare rational numbers.
What is the difference between the two mixed numbers? The numerators of each fraction are different.
Between which integers does each mixed number lie? -1 and -2
Explain how even though 5 > 1, the mixed number
<
. Sample answer: The mixed
numbers are negative, and as you move to the left on the number line, values decrease.
Explain how you could compare the numbers without locating them on a number line. Sample answer:
Both mixed numbers have the same whole number part. By comparing the numerators,
farther left than
is
.
Need Another Example?
Printed by: Nicole Sladich
Page 3 of 14
Printed on 11/04/2014 10:13 PM
National Glencoe Math, Course 2, © 2015
Chapter 4: Rational Numbers
Lesson 2: Compare and Order Rational Numbers
Fill in the
with <, >, or = to make
a true sentence. >
2. Compare rational numbers.
Can we simply compare the numerators 7 and 8 to determine which fraction is greater? Explain. no;
The denominators are different.
What is the LCD of 12 and 18? 36
Explain how to rewrite each fraction using the LCD. Multiply
How do you know that
is equivalent to
by
; multiply
? We multiplied by
by
.
, which is the same as
multiplying by 1.
Would your result be different if you simplified
prior to comparing the numbers? Explain. no;
Sample answer: The result is the same because of equivalent fractions.
Need Another Example?
Fill in the
with <, >, or = to make
a true sentence. <
3. Compare rational numbers.
What are you trying to determine? the class that owns more roller shoes
What are the two numbers we need to compare? 20% and 5 out of 29
What will you have to do to compare these values? Sample answer: Express each number as a
decimal.
How do you determine that 5/29 0.1724? Divide the numerator by the denominator.
How could you use estimation to compare?
=
and
>
Need Another Example?
In a football game, the quarterback completed 11 out of 19 passes. The next week, he completed
60% of his passes. During which game did he complete the greater fraction of his passes? the
second game
4. Order a set of rational numbers.
In what form could we compare these numbers? Sample answer: decimal form
Printed by: Nicole Sladich
Page 4 of 14
Printed on 11/04/2014 10:13 PM
National Glencoe Math, Course 2, © 2015
Chapter 4: Rational Numbers
Lesson 2: Compare and Order Rational Numbers
Will annexing zeros change the value of a decimal? No, you can annex as many zeros as needed.
Why should you extend each decimal place to the thousandths? because each pair has the same
tenths and hundredths values
How do you know just by looking at the numbers 3.44 and 3.4 which one is greater? Sample answer:
3.4 repeats the digit 4 forever. Even if it just repeated the digit to the thousandths place, 3.444 > 3.44.
Is a terminating decimal, repeating decimal, or neither? Explain. Neither; is a non-terminating,
non-repeating decimal.
Need Another Example?
Order the set
from least to greatest.
5. Order a set of rational numbers.
In what form can we compare these numbers? Sample answer: decimal form
Could you use another form to compare? If so, which form? yes; Sample answer: percent
What are the fractional values for each number? decimal values? percentage values?
,
,
,
; 0.67, 0.64, 0.60, 0.69; 67%, 64%, 60%, 69%
Which form do you prefer to use to compare the numbers? See students’ preferences.
Need Another Example?
Maddie is a guard on the basketball team. She made 70% of her field goals in the first game. She
made 0.62,
, and 66% of her field goals in the next three games. List Maddie’s made field goal
statistics from least to greatest. 0.62, 66%, 70%,
Alternate Teaching Strategy
If students have trouble finding the LCD of the fractions,
Then use one of these reteach options:
1. Quick Review Math Handbook, pp. 153-155
2. Problem-Based Learning Book 2, Unit B: Rational Numbers, Investigation 2: Order and Compare
Rational Numbers
3. Have students write equivalent fractions for each fraction until they find fractions with a common
Printed by: Nicole Sladich
Page 5 of 14
Printed on 11/04/2014 10:13 PM
National Glencoe Math, Course 2, © 2015
3.
Chapter 4: Rational Numbers
Lesson 2: Compare and Order Rational Numbers
denominator. For example, to compare
and
,
,
,
the comparison.
. Since
and
and
and
, students would write
,
,
,
,
have a common denominator, they can be used for
.
, so
.
Formative Assessment Use these exercises to assess students' understanding of the concepts in
this lesson.
If some of your students are not ready for assignments, use the differentiated activities
below.
Pairs Discussion Have students work together to complete Exercises 1-4. Give them
blank number lines for Exercises 3 and 4. Students should place values on a decimal or fraction
number line.
1, 5
Numbered Heads Together Provide a list of varying fraction, decimal, and percentage
values. Have students work in groups generate equivalent values of remaining forms for each given
value. Call on one group member to share the group’s responses.
1, 7
Tips for New Teachers
Rational Numbers Help students remember the definition of rational numbers by pointing out that
rational numbers can be written as a ratio of integers (provided the second integer is not 0). You may
also wish to remind students that terminating decimals can be written as fractions.
Teach with Tech
Interactive WhiteBoard Draw a number line on the IWB to compare and order rational numbers. Have
students use the highlighter tool to show the placement of various rational numbers as suggested by
other students in the class. Have students decide on the order of value of the rational numbers. Repeat
with different sets of rational numbers.
Podcast Have students write a song, rap, or poem about rational numbers, integers, whole numbers,
and natural numbers. Included should be what type of number is in each set and how the different sets
of numbers relate to each other. Have students record the result using software and share it by way of
a podcast.
Resources
Virtual Manipulatives
Fractions as Decimals
Fractions as Decimals
Mixed Numbers as Decimals
Write a Fraction as a Decimal Using Division
Write a Fraction as a Decimal Using Division
Write a Fraction as a Repeating Decimal
Real-World Example: Write a Decimal as a Fraction
Compare Rational Numbers
Compare Rational Numbers
Order Rational Numbers
Printed by: Nicole Sladich
Page 6 of 14
Printed on 11/04/2014 10:13 PM
National Glencoe Math, Course 2, © 2015
Chapter 4: Rational Numbers
Lesson 2: Compare and Order Rational Numbers
Real-World Example: Order Rational Numbers
eToolkit
Problem-Based Learning Book 2, Unit B, Student Edition: Rational Numbers
Problem-Based Learning Book 2, Unit B, Teacher Edition: Rational Numbers
Hot Topic 2-9: Fraction, Decimal, and Percent Relationships
Compare and Order Whole Numbers, Fractions, and Percents
Simplifying Fractions and Expressions
Manipulative Masters: Number Lines
Compare with Percents
Printed by: Nicole Sladich
Page 7 of 14
Printed on 11/04/2014 10:13 PM
National Glencoe Math, Course 2, © 2015
Chapter 4: Rational Numbers
Lesson 2: Compare and Order Rational Numbers
3 Practice & Apply
Engage Explore Explain Elaborate Evaluate
Independent Practice and Extra Practice
The Independent Practice pages are meant to be used as the homework assignment. The Extra
Practice page can be used for additional reinforcement or as a second-day assignment.
Levels of Complexity
The levels of the exercises progress from 1 to 3, with Level 1 indicating the lowest level of complexity.
Suggested Assignments
You can use the table below that includes exercises of all complexity levels to select appropriate
exercises for your students' needs.
Differentiated Homework Options
Approaching Level
1–9, 11, 12, 14, 15, 24, 25
On Level
1–7 odd, 9–12, 14, 15, 24, 25
Beyond Level
9–15, 24, 25
can provide additional remediation support with personalized instruction and
practice.
Create Your Own Homework Online
eSolutionsManual™ can be used to create worksheets for the suggested assignments above, or to
create your own worksheets for differentiated homework or review.
The following worksheets are also available for additional homework practice:
The Skills Practice editable worksheet provides students with additional practice in the skills taught in
this lesson.
The Homework Practice editable worksheet helps students practice skills in this lesson and use those
skills to solve problems.
The Problem-Solving Practice editable worksheet helps students become better problem solvers by
providing frequent opportunities to solve real-world problems.
The Extra Practice worksheet provides additional practice for the skills taught in this lesson.
The Step-by-Step Solutions provide a worked out solution for selected exercises from the
Independent Practice in the Student Edition.
Common Error Students may have trouble comparing fractions with unlike denominators because they
Printed by: Nicole Sladich
Page 8 of 14
Printed on 11/04/2014 10:13 PM
National Glencoe Math, Course 2, © 2015
Chapter 4: Rational Numbers
Lesson 2: Compare and Order Rational Numbers
may only compare the numerators. Have students circle the denominators and remind them to rewrite
the fractions using the LCD. Then compare the numerators.
Resources
Step by Step Solutions: Compare and Order Rational Numbers
Skills Practice: Compare and Order Rational Numbers
Skills Practice Answers: Compare and Order Rational Numbers
Homework Practice: Compare and Order Rational Numbers
Homework Practice Answers: Compare and Order Rational Numbers
Problem-Solving Practice: Compare and Order Rational Numbers
Problem-Solving Practice Answers: Compare and Order Rational Numbers
Extra Practice: Compare and Order Rational Numbers
Extra Practice Answers, Chapter 4
Homework Practice: Compare and Order Rational Numbers
Problem-Solving Practice: Compare and Order Rational Numbers
Skills Practice: Compare and Order Rational Numbers
Extra Practice: Compare and Order Rational Numbers
eSolutions, Glencoe Math Course 2
Printed by: Nicole Sladich
Page 9 of 14
Printed on 11/04/2014 10:13 PM
National Glencoe Math, Course 2, © 2015
Chapter 4: Rational Numbers
Lesson 2: Compare and Order Rational Numbers
Additional Activities for Differentiated Instruction
Differentiated Instruction activities for Approaching Level, On Level, and Beyond Level Learners can
also be found at point-of-use in the Launch the Lesson and Teach the Concept sections. The following
are additional activities that can be used with this lesson.
The Reteach Worksheet provides additional examples and practice for students who may have
difficulty in grasping the math concepts in this lesson.
The Enrich Worksheet provides students with valuable opportunities for extending this lesson.
Resources
Reteach: Compare and Order Rational Numbers
Reteach Answers: Compare and Order Rational Numbers
Enrich: Intersection and Union of Sets
Enrich Answers: Intersection and Union of Sets
Reteach: Compare and Order Rational Numbers
Enrich: Intersection and Union of Sets
Printed by: Nicole Sladich
Page 10 of 14
Printed on 11/04/2014 10:13 PM
National Glencoe Math, Course 2, © 2015
Chapter 4: Rational Numbers
Lesson 2: Compare and Order Rational Numbers
Additional Activities for English Learners
Differentiated Instruction activities for English Learners can also be found at point-of-use in the Launch
the Lesson and Teach the Concept sections. The following are additional activities that can be used
with this lesson.
Option 1: Use Chapter 4, Lesson 2: Compare and Order Rational Numbers from the
Interactive Guide for English Learners.
Option 2: Kinesthetic Learners Place students in groups of four. Give each group twelve cards,
each of which shows a fraction or a decimal.
Each member of the group draws one card. Then have students order the cards from least to
greatest. Repeat the activity several times.
Option 3: Verbal/Linguistic Learners Create a word wall with examples of GCF, LCM, and
LCD.
Explain what each term is, how it is used, and whether the terms are related in any way. Be sure
to give examples of all three.
Word Wall Printable English/Spanish Word Wall cards are available for the following vocabulary
words in this lesson:
common denominator
least common denominator
rational number
Resources
Word Wall Cards
Interactive Guide for English Learners, Student Edition, Course 2, Lesson 4-2: Compare and Order Rational
Numbers
Interactive Guide for English Learners, Teacher Edition, Course 2, Lesson 4-2: Compare and Order Rational
Numbers
Interactive Guide for English Learners, Course 2, Teacher Edition
Interactive Guide for English Learners, Course 2, Student Edition
Visual Kinesthetic Vocabulary, Course 2, Student Edition
Visual Kinesthetic Vocabulary, Course 2, Teacher Edition
Printed by: Nicole Sladich
Page 11 of 14
Printed on 11/04/2014 10:13 PM
National Glencoe Math, Course 2, © 2015
Chapter 4: Rational Numbers
Lesson 2: Compare and Order Rational Numbers
Assessment
Formative Assessment
Use this activity as a closing formative assessment before dismissing students from your class.
Write the following on the board:
and
; and
and
. Have each
student explain out loud to a partner how they would compare each pair of rational numbers. Then have
them summarize the process by writing it on a piece of paper to hand in at the end of class. See
students' work.
The online Self-Check Quiz for this lesson allows students to review concepts from each lesson.
Exercises 24 and 25 prepare students for more rigorous thinking needed when taking the CCSS
assessment.
24.
This test item requires students to explain and apply mathematical concepts and
solve problems with precision, while making use of structure.
Depth of Knowledge
DOK2
CCSS Content Standard
7.EE.3
Mathematical Practices
MP1
Online Experience
Students will be required to type their
responses in the boxes.
Scoring Rubric
2 points
Students correctly order 4 counties, list the sales
tax rate for each, and determine the county with
the lowest sales tax.
1 point
Students correctly order 4 counties, list the sales
tax for each, but fail to determine the county with
the lowest sales tax OR students correctly order
the counties, but fail to list the correct sales tax
rate for each and may or may not identify the
county with the lowest sales tax.
25.
This test item requires students to reason abstractly and quantitatively when problem
solving.
Depth of Knowledge
DOK1
CCSS Content Standard
7.EE.3
Mathematical Practices
MP1
Online Experience
Students will be required to click on the
bubbles to select all answers that are
correct.
Printed by: Nicole Sladich
Page 12 of 14
Printed on 11/04/2014 10:13 PM
National Glencoe Math, Course 2, © 2015
Chapter 4: Rational Numbers
Lesson 2: Compare and Order Rational Numbers
Scoring Rubric
1 point
Students correctly answer each part of the
question.
Quick Check
Are students continuing to struggle with comparing and ordering rational numbers?
If Yes
Reteach Worksheet
If No
Skills Practice Worksheet
Enrich Worksheet
Resources
Self-Check Quiz: Compare and Order Rational Numbers
Printed by: Nicole Sladich
Page 13 of 14
Printed on 11/04/2014 10:13 PM
National Glencoe Math, Course 2, © 2015
Chapter 4: Rational Numbers
Lesson 2: Compare and Order Rational Numbers
Printed by: Nicole Sladich
Page 14 of 14
Printed on 11/04/2014 10:13 PM