Lesson
Grade
7
Contents
Introduction................................................................... vii
Lessons
Lesson 1: Common Denominators................................. 1
Finding the Lowest Common Denominator
Lesson 2: LCDs in Mixed Numbers................................. 9
Borrowing With Mixed Numbers and LCDs
Lesson 3: Multiplying Simple Fractions........................ 17
Multiplying Whole Numbers by Fractions
Multiplying by Fractions in Word Problems
Multiplying Mixed Numbers
Canceling Fractions
Lesson 4: Dividing Fractions.......................................... 27
Dividing Whole Numbers and Fractions
Dividing Mixed Numbers
Dividing Whole Numbers and Mixed Numbers
Why You Invert and Multiply to Divide
Lesson 5: Decimal Fractions ......................................... 39
Hundredths
Thousandths
Comparing Decimals
Adding Decimals
Lesson 6: Subtracting Decimals..................................... 49
Subtracting Decimals and Whole Numbers
Multiplying Decimals
Multiplying Larger Decimals
Multiplying Decimals and Whole Numbers
iii
Lesson Contents
Grade 7 Math
Lesson 7: Dividing Decimals by Whole Numbers........ 61
Dividing With Dividends Less Than One
Dividing Decimals With Remainders
Lesson 8: Dividing Decimals By Decimals.................... 73
Dividing Whole Numbers by Decimals
Lesson 9: Exponents....................................................... 83
Square Roots
Lesson 10: Order of Operations.................................... 93
Addition and Subtraction
Parentheses
Lesson 11: Three Ways to Indicate Multiplication.... 103
Order of Operations with Multiplication and Division
Lesson 12: Multiplying Decimals by 10, 100,
and 1000................................................................... 113
Dividing Decimals by 10, 100, and 1000
Percents
Finding a Percent of a Number
Lesson 13: Converting Common Fractions
to Decimals............................................................... 125
Converting Decimals to Percents
Lesson 14: Using Percents in Word Problems............ 133
Finding Percents in Word Problems
Lesson 15: Remainders in Dividing Decimals............. 143
Repeating Decimals
Factors of Whole Numbers
Lesson 16: Finding Missing Numbers in Addition..... 151
Finding Missing Numbers in Subtraction
Missing numbers in Multiplication and Division
Lesson 17: Order of Operations With Exponents...... 161
Three Ways to Indicate Division
Fractions in Order of Operations
Lesson 18: First Semester Exam................................... 173
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Grade 7 Math
Lesson Contents
Lesson 19: Negative Numbers..................................... 181
Directions and the Number Line
Adding and Subtracting Signed Numbers
Adding Signed Numbers
Subtracting Signed Numbers
Lesson 20: Multiplying Signed Numbers.................... 193
Dividing Signed Numbers
Lesson 21: Finding the mean and the Median........... 201
Finding the Mode and Range
Lesson 22: The Metric System..................................... 211
Reading a Centimeter Ruler
Meters and Kilometers
Lesson 23: Probability.................................................. 223
Probability of a Series
Chance
Lesson 24: Addition Rule of Equations....................... 231
The Addition Rule With Negative Numbers
Lesson 25: Division Rule For Equations..................... 239
The Multiplication Rule For Equations
Lesson 26: Ratios.......................................................... 247
Proportions
Lesson 27: Using Proportions in Word Problems...... 257
Converting Units in Proportions
Lesson 28: Formulas..................................................... 265
Transforming Formulas
Lesson 29: Pi and the Circumference of a Circle........ 273
Pi and the Area of a Circle
Lesson 30: Measuring Weight With Metrics.............. 281
Measuring Liquids With Metrics
Lesson 31: Trillions in Place Value............................... 289
Prime Numbers
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v
Lesson Contents
Grade 7 Math
Lesson 32: Sequences................................................... 297
Functions
Lesson 33: Angles.......................................................... 305
Perpendicular Lines and Right Angles
Types of Angles Measuring Angles Triangles
Classifying by Angles Classifying by Sides Sum of the
Angles Area of a Triangle
Lesson 34: Using a Compass....................................... 319
Drawing an Equilateral Triangle
Polygons
Lesson 35: Geometric Solids........................................ 329
Volume of Solids
Lesson 36: Second Semester Exam............................. 337
Appendix........................................................................ 347
Answer Keys for Skill Practice and Application Practice
B-tests
vi
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Lesson
Grade 7
1
Common Denominators
When we are adding or subtracting fractions whose denominators are the
same, we simply add or subtract the numerators, and the denominator
remains the same. But in many operations with fractions, the denominators are not the same. When this is the case, we can’t just add or subtract
the numerators; we have to rename one or both of the fractions by finding
what is called a common denominator. This is a number that can be
divided evenly by both denominators in the problem.
When we’re looking for a common denominator for two fractions in a
problem, there are several ways to find this.
1. Use the largest denominator in the problem
2.Multiply the two denominators
3.Compare the multiples of both denominators and choose the lowest
multiple that both fractions have in common
To begin, we’ll look at the first two approaches:
1. Use the largest denominator in the problem
Example 1:
3 1
8 4
Step 1: U
sing the largest denominator in the problem doesn’t always work,
but this is the approach we try first, because it’s the easiest.
In this example, the largest denominator is 8 and the smallest
denominator is 4. Does 4 go into 8 evenly? Yes, 4 x 2 = 8, so 8
will be our common denominator.
1
Lesson 1
Grade 7 Math
Common
Denominators
(continued)
Step 2: Write the fractions in vertical format and add the equal sign
and the common denominator. Since the top fraction isn’t
changing, write that as is:
3
3
=
8
8
1
=
4
8
Step 3: Look at the denominator in the bottom fraction. Say to yourself, “4 goes into 8 two times.” Look at the numerator in
the bottom fraction and say, “Two times 1 is 2.” Write the 2
2
above the 8 in the bottom fraction to create the fraction 8 ,
then subtract as usual:
3
3
=
8
8
1
2
=
4
8
1
8
That approach will work well with some problems, and it’s the one
you should always try first because it’s the easiest. But if you find
a problem that won’t work with this approach, then try the next
approach:
2.Multiply the two denominators
Example 2:
2 3
+
3 4
Step 1: In this problem we can’t use the largest denominator, because
3 won’t divide evenly into 4, so we’ll try the second approach:
we’ll multiply the two denominators. In this example, we would
multiply 3 by 4 and get 12. This is the common denominator.
2
Oak Meadow
Grade 7 Math
Lesson 1
As before, write the problem in a vertical format and add the
equal sign and a common denominator:
2
=
3 12
3
+
=
4 12
Common
Denominators
(continued)
Step 2: Look at the denominator in the top fraction and say, “How
many times does 3 go into 12?” The answer is 4, so you
then say, “4 times 2 is 8” and you put the 8 above the 12,
8
to create 12 .
2
8
=
3 12
3
+
=
4 12
Step 3: Look at the denominator in the bottom fraction and say,
“How many times does 4 go into 12?” The answer is 3, so you
then say, “3 times 3 is 9” and you put the 9 above the 12, to
9
make 12 . Then add as usual, and reduce the fraction to
lowest terms:
12
2
8
=
3 12
3
9
+
=
4 12
17
5
= 1
12
12
Practice these two approaches to finding common denominators in the
following Skill Practice. When you’re finished, we’ll review the third approach to finding common denominators.
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3
Lesson 1
Grade 7 Math
Common
Denominators
(continued)
Finding the Lowest Common Denominator
Multiplying the two denominators will always give you a common
denominator, but often this denominator will be quite large and must
be reduced to lowest terms at the end of the problem. To avoid having to
reduce fractions at the end of the problem, you always look for the lowest
common denominator, or LCD. Sometimes, the first two approaches will give
us the LCD. If not, we use a third approach:
3.Compare the multiples of both denominators and choose the lowest
multiple that both fractions have in common
Example 1:
1 1
+
4 6
Step 1: A
s usual, we first look to see if we can use the largest denominator in the problem as a common denominator, but we find
that this doesn’t work. We can’t divide 4 into 6 evenly. So
we try the second approach: we multiply the two denominators. We can do this, but we end up with 24 for a denominator. Since the numbers are small, we think there may be a
smaller denominator, so we compare the multiples of the two
denominators:
Multiples of 4:4
8 12 16 20 24 28
Multiples of 6:6 12 18 24 30 36 42
We find that 24 is a multiple of both denominators, but 12 is also a
multiple, and since 12 is a lower number, we choose that as the lowest common denominator.
Step 2: Once we’ve found the LCD, we complete the problem as usual:
1
3
=
4 12
1
2
+
=
6 12
5
12
4
Oak Meadow
Grade 7 Math
Math 7
Skill Practice A
Lesson 1 - 4
Lesson 1
SKILL PRACTICE A
Reduce answers to the lowest terms.
Reduce answers to lowest terms
1.
4.
7.
10.
2
3
−
−
−
− 41
2.
+
7
8
1
2
3.
3
4
− 21
2
4
− 31
3
4
2
3
5.
3
5
+ 21
6.
4
5
1
2
8.
2
5
+ 130
9.
3
4
1
3
11.
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+
3
4
5
8
12.
+
1
2
3
8
4
16
+
1
4
5
Lesson 1
Math 7
Lesson 1 - 6
Skill Practice B
Grade 7 Math
SKILL PRACTICE B
Find the lowest
common
denominator
anddenominator
solve.
Find
the lowest
common
and solve
1.
4.
7.
10.
6
3
8
−
+
−
− 61
2.
+
1
6
1
9
3.
3
4
+ 130
5
6
− 34
5
6
1
8
5.
3
8
+ 152
6.
1
6
1
8
8.
5
8
+ 172
9.
3
8
1
12
11.
−
1
4
1
10
12.
−
3
4
5
6
2
9
+ 56
Oak Meadow
Grade 7 Math
Math 7
Lesson 1 - 7
Lesson 1
REVIEW 1
Lesson 1 Review
Reduce all fractions
in answers
to lowest
Reduce
all fractions
in terms.
answers to lowest terms
1.
−
3
5
1
3
2. 5,0 8 9
× 375
3.
6. 1 2 4 1 5 2
4.
1
5
+ 170
5.
7.
1
3
+ 34
8. 7 1 1 0 6
10.
307
×294
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11.
−
−
3
4
1
2
2
4
1
6
9.
12.
1
8
2
3
−
+ 125
+ 59
5
6
1
9
7
Lesson 1
Grade 7 Math
Lesson 1 Review continued
13. Simon orders books for The Book Barn. He ordered 175 copies of a new bestseller at a cost
of $12 per copy. What was the total cost of the 175 books he ordered?
14. The Ferndale Kiwanis Club bought 700 Christmas trees to sell at 5 locations throughout
Ferndale. If they distribute the trees equally to each location, how many trees will be available at each location?
15. Ali is buying furniture, and the salesman says she will have to pay $35 a month for 24
months to buy what she wants. When she completes all her monthly payments, how much
will she have paid for the furniture?
16. The Spaghetti Factory is giving its employees a year-end bonus, and $7,800 has been set
aside to distribute equally to all employees. If there are 24 employees, how much will each
employee receive?
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