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Contents
Domain 1 Ratios and Proportional Relationships. . . . . 4
Common Core
State Standards
Lesson 1
Computing Unit Rate . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6
7.RP.1
Lesson 2
Identifying Proportional Relationships . . . . . . . . . . . . . . . . 10
7.RP.2.a, 7.RP.2.b
Lesson 3
Representing Proportional Relationships. . . . . . . . . . . . . . 14
7.RP.2.c, 7.RP.2.d
Lesson 4
Word Problems with Ratio and Percent. . . . . . . . . . . . . . . . 18
7.RP.3
Domain 1 Review. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24
Domain 2 The Number System . . . . . . . . . . . . . . . . . . . . . . . 28
Lesson 5
Adding and Subtracting Rational Numbers. . . . . . . . . 30
Lesson 6
Applying Properties of Operations to Add and
Subtract Rational Numbers . . . . . . . . . . . . . . . . . . . . . . . . . 36
7.NS.1.a, 7.NS.1.b, 7.NS.1.c
7.NS.1.d
Lesson 7
Multiplying Rational Numbers . . . . . . . . . . . . . . . . . . . 40
7.NS.2.a, 7.NS.2.c
Lesson 8
Dividing Rational Numbers. . . . . . . . . . . . . . . . . . . . . . 46
7.NS.2.b, 7.NS.2.c
Lesson 9
Converting Rational Numbers to Decimals. . . . . . . . . . . . . 52
7.NS.2.d
Lesson 10
Problem Solving: Complex Fractions. . . . . . . . . . . . . . 56
7.NS.3
Lesson 11
Problem Solving: Rational Numbers. . . . . . . . . . . . . . . 60
7.NS.3
Domain 2 Review . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64
Lesson 12
Writing Equivalent Expressions. . . . . . . . . . . . . . . . . . . . . . 70
7.EE.1, 7.EE.2
Lesson 13
Factoring and Expanding Linear Expressions. . . . . . . . . . . 74
7.EE.1
Lesson 14
Adding and Subtracting Algebraic Expressions. . . . . . . . . 78
7.EE.1
Problem Solving: Algebraic Expressions
and Equations. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82
7.EE.3
Lesson 15
Lesson 16
Word Problems with Equations. . . . . . . . . . . . . . . 86
7.EE.4.a
Lesson 17
Word Problems with Inequalities . . . . . . . . . . . . . . . . . 92
7.EE.4.b
Domain 3 Review . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 98
2 CC12_MTH_G7_SE_FM_Final.indd 2
Problem
Solving
Fluency
Lesson
Duplicating any part of this book is prohibited by law.
Domain 3 Expressions and Equations . . . . . . . . . . . . . . . . 68
Performance
Task
15/06/12 9:31 AM
Common Core
State Standards
Domain 4 Geometry. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102
Lesson 18
Scale Drawings. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 104
7.G.1
Lesson 19
Drawing Geometric Shapes. . . . . . . . . . . . . . . . . . . . . . . . 110
7.G.2
Lesson 20 Examining Cross Sections of
Three-Dimensional Figures . . . . . . . . . . . . . . . . . . . . . . . . 114
7.G.3
Lesson 21
Area and Circumference of Circles. . . . . . . . . . . . . . . 118
7.G.4
Lesson 22
Angle Pairs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 124
7.G.5
Lesson 23
Problem Solving: Area and Surface Area
of Composite Figures. . . . . . . . . . . . . . . . . . . . . . . . . 130
7.G.6
Problem Solving: Volume of
Three-Dimensional Figures. . . . . . . . . . . . . . . . . . . . . 134
7.G.6
Lesson 24
Domain 4 Review . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 138
Domain 5 Statistics and Probability . . . . . . . . . . . . . . . . . 142
Lesson 25 Understanding Sampling. . . . . . . . . . . . . . . . . . . . . . . . . . 144
7.SP.1, 7.SP.2
Lesson 26 Using Mean and Mean Absolute Deviation . . . . . . . . . . . 150
7.SP.3, 7.SP.4
Making Comparative Inferences about
Two Populations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 156
7.SP.3, 7.SP.4
Lesson 28 Understanding Probability. . . . . . . . . . . . . . . . . . . . . . . . . 160
7.SP.5, 7.SP.6,
Lesson 27
7.SP.7.b
Lesson 29 Probabilities of Simple Events. . . . . . . . . . . . . . . . . . . . . . 164
7.SP.7.a, 7.SP.7.b
Lesson 30 Probabilities of Compound Events . . . . . . . . . . . . . . . . . . 170
7.SP.8.a, 7.SP.8.b
Lesson 31
Simulations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 176
7.SP.8.c
Duplicating any part of this book is prohibited by law.
Domain 5 Review . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 180
Glossary. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 184
Math Tools. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 187
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LE
SS
O
N
1
Computing Unit Rate
A ratio is a comparison of two quantities. A rate is a type of ratio that compares two quantities
that have different units of measure. For example, $30 for 5 pounds is a rate that compares
dollars to pounds. If a rate is a unit rate, the second quantity in the comparison is 1 unit. For example, $6 per pound is a unit rate because it compares $6 to 1 pound.
If a rate involves comparing two fractions, you can use a complex fraction to represent the rate.
A complex fraction is a fraction in which the numerator and/or the denominator is a fraction or a
mixed number.
5
1
__
EXAMPLE A Evan walks ​ __
8 ​ mile every ​ 4 ​  hour. Express his speed as a unit rate in miles per hour.
1
Write the rate as a complex fraction.
5
5
__
​ 8 ​ mi ​ __
8 ​ ____
​  __1   ​ 5 ​ __
1  ​
​ 4 ​  h
​ __
4 ​  
1
The denominator is ​ __
4 ​,  but in a unit rate,
the denominator will be 1.
2
Rewrite the complex fraction so that it
has a denominator of 1.
1
Since the denominator is ​ __
4 ​,  you
can multiply it times 4 to get a
denominator of 1.
1
1 __
4
__
​ __
4 ​  3 4 5 ​ 4 ​  3 ​ 1 ​  5 1
Multiply both the numerator and the
denominator times 4.
5
4
20
The result is a denominator of 1. So, the ratio now shows a unit rate.
20
TRY
1
5
__
Ariella jogs 3​ __
8 ​  miles in ​ 8 ​ hour. Express her speed as a unit rate in miles per hour.
6 5
___
​  8   ​ mi __
​ 2 ​ mi
1
_____
__
​  1 h   ​  5 ​ ____
1 h   ​ 5 2​ 2 ​  miles per hour
▸ Evan’s rate of speed is 2​ __21 ​  miles
per hour.
Duplicating any part of this book is prohibited by law.
5
__
​ 8 ​ 4 __
​ 8 ​ 3 ​ __1 ​  ___
​  8   ​
__
__
____
​ __1  ​ 3 ​ 4 ​ 5 ​ __1 __4  ​ 5 ​ __
1   ​
​ 4  ​
​ 4 ​  3 ​ 1 ​ 
Domain 1: Ratios and Proportional Relationships
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3
1
__
EXAMPLE B A recipe calls for 1​ __
2 ​  cups of flour for every ​ 4 ​ cup of sugar used. How many cups
of flour are needed for each cup of sugar used?
1
Write the ratio as a complex fraction.
3
2
3
__
__
​ 2 ​ c flour
​ 2 ​
__
The ratio is: ​ _______
 
 
​  
5
​ 
3
3   ​. __
​ 4 ​ c sugar ​ __
4 ​ 3
1
1 _______
1 3 2 1 1 __
__
Convert 1​ __
​  5 ​ 2 ​
2 ​  to an improper fraction: 1​ 2 ​  5 ​  2   
Use a fraction model.
3
​ __
3 __
3
2 ​ __
Since ​ __
3  ​ is the same as the quotient ​ 2 ​ 4 ​ 4 ​, you can use a
__
​ 4 ​ 3
fraction model to find the answer. Shade ​ __
2 ​ of two squares. 3
2
3
4
Then divide the squares into fourths. Move the top 3
shaded rectangles to the right of the other rectangles.
3
3
1
__
__
The diagram shows that ​ __
2 ​ (or 1​ 2 ​)  is the same as multiplying ​ 4 ​ by 2. ▸
3
4
3
​ __
2 ​ __
The quotient ​ __3  ​ 5 2, so 2 cups of flour are needed for each cup
​ 4 ​ 2
of sugar used.
3
1
__
EXAMPLE C A motorized scooter can travel 5​ __
4 ​ miles on ​ 5 ​  gallon of gasoline. How many
miles per gallon does the scooter get?
Duplicating any part of this book is prohibited by law.
1
Write the rate as a complex fraction. 3
Convert 5​ __
4 ​ to an improper fraction:
5 3 4 1 3 ___
3 ________
23
5​ __
 
​  5 ​  4   ​
4 ​ 5 ​ 
4 
23
23
___
​  4   ​ mi ___
​  4   ​
__
The rate is: ​ _____
 
 
​ 5
​ 
1
1   ​.
__
__
​ 5  ​ gal
​ 5 ​ 
2
Write the complex fraction as a unit rate.
1
5
__
Since the denominator is ​ __
5 ​,  multiply by ​ 5 ​ to get a denominator of 1.
23
23
5
115
___
​  4   ​ 5 ___
​  4   ​ 3 ​ __1 ​  ___
​  4   ​
__
_____
​  __1   ​ 3 ​ __
 
​ 5
​ 
  
​ 5 ​ ___
5
1
5
1  ​ 
__
​ 5 ​ 
​ 5 ​  3 ​ __1 ​ 
115
___
​  4   ​ mi
So, the unit rate is: ​ _____
​.
1 gal   
115
___
​  4   ​ mi
3
_____
​  1 gal   
​ 5 28​ __
4 ​ miles per gallon
C HE C K
Explain how you could use
multiplication
to check the answer OVERSET
for Example C. ▸ The scooter gets 28​ __34 ​ miles per gallon.
Lesson 1: Computing Unit Rate CC12_MTH_G7_SE_D1_Final.indd 7
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Practice
Classify each rate. Write unit rate or not a unit rate.
1.
1
3​ __
2 ​  feet per minute
$1.60 per pound
2.
3.
1
1
__
​ 4 ​  mile every ​ __
6 ​  hour
REMEMBER A unit rate, written as
a fraction, has a denominator of 1.
Find each unit rate by simplifying the given complex fraction. Show your work.
4.
3
1
__
A novelist can write 2​ __
4 ​  pages in ​ 4 ​ hour. 5.
Aaron can run 2 kilometers in
1
14​ __
2  ​ minutes. Find his rate of speed
in kilometers per minute.
2 km
_____
​ ___
  ​  5
29
​  2   ​ min
Express her writing speed as a unit rate.
9
__
​ 4 ​ pages
______
​  __3
  
​ 5
​ 4 ​ hour
pages per hour
Fill in the blanks with an appropriate word or phrase.
6.
A(n)
is a ratio of two quantities that have different units of measure. 7.
A(n)
is a ratio in which the second quantity in the comparison is 1 unit.
8.
A(n)
fraction has a fraction in the numerator, the denominator, or both.
9.
If you multiply the numerator and the denominator of a fraction by the same number,
the result will be a(n)
fraction.
Choose the best answer.
1
A. ​ __
10   ​ mile
1
C. __
​ 3 ​  mile
8 5
B. __
​ 18  ​ mile
3
D. 3​ __
5 ​ miles
1
1
__
11. A satellite travels 29​ __
2 ​  miles every 4​ 3 ​  
seconds. What is its unit rate of speed?
21
A.    6​ ___
26 ​  miles per second
1
B.   29​ __
2 ​  miles per second
5
C.   33​ __
6 ​ miles per second
5
D. 127​ __
6 ​ miles per second
Duplicating any part of this book is prohibited by law.
1
10. For every ​ __
6 ​  mile that a ship travels
3
north, it travels ​ __
5 ​ mile west. How many
miles does the ship travel north for
every mile it travels west?
Domain 1: Ratios and Proportional Relationships
CC12_MTH_G7_SE_D1_Final.indd 8
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Find each unit rate. Show your work.
1
12. Renting an office costs $486 per month. The office has an area of 202​ __
2 ​  square feet. What is the monthly cost per square foot to rent the office?
3
2
__
13. A recipe calls for using ​ __
4 ​ cup of brown sugar for each ​ 3 ​ cup of white sugar. How many cups
of brown sugar are used per cup of white sugar?
1
1
__
14. Lauren bikes 1​ __
3 ​  miles in ​ 10   ​ hour. What is her rate of speed in miles per hour?
1
1
__
15. Oliver reads 28​ __
2 ​  pages of a book in 1​ 6 ​  hours. Express his reading speed in pages per hour.
Solve.
16.
SHOW On mountainous terrain,
2
1
17.
2
EXPLAIN A car travels ​ __
5 ​ mile in
1
__
a semi-truck travels 2​ __
3 ​ miles on ​ 2 ​  gallon
​ __
2 ​  minute. What is the car’s speed
of fuel. How many miles can the truck
in miles per hour? Explain how you
travel per gallon of fuel? Use drawings
determined your answer.
Duplicating any part of this book is prohibited by law.
or equations to show your work.
Lesson 1: Computing Unit Rate CC12_MTH_G7_SE_D1_Final.indd 9
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Contents
Domain Assessment—Ratios and Proportional Relationships . . . . . . . . . . . . . . . . . . . . . . . . 4
Domain Assessment—The Number System . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12
Domain Assessment—Expressions and Equations. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22
Domain Assessment—Geometry. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30
Domain Assessment—Statistics and Probability. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42
Duplicating any part of this book is prohibited by law.
Summative Assessment. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54
3
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Domain Assessment • Ratios and Proportional Relationships
1.
Which of the following tables represents
a proportional relationship between x and y?
A.
B.
C.
D.
x
y
1
2
2
1
3
4
4
3
x
y
1
24
2
23
3
22
4
21
2.
Which of the following graphs represents
a proportional relationship between x and y?
A.
y
6
5
4
3
2
x
y
1
3
2
6
3
9
4
12
x
y
1
4
2
8
3
10
4
12
1
–6 –5 –4 –3 –2 –1 0
–1
–2
–3
1
2
3
4
5
6
x
1
2
3
4
5
6
x
1
2
3
4
5
6
x
1
2
3
4
5
6
x
–4
–5
–6
B.
y
6
5
4
3
2
1
–6 –5 –4 –3 –2 –1 0
–1
–2
–3
–4
–5
–6
C.
y
6
5
4
3
2
1
–6 –5 –4 –3 –2 –1 0
–1
–2
–3
D.
y
6
5
4
3
2
1
–6 –5 –4 –3 –2 –1 0
–1
–2
–3
–4
–5
–6
Go On
Duplicating any part of this book is prohibited by law.
–4
–5
–6
4
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A scientist is mixing a chemical solution for
an experiment. The solution contains 3
1
__
​ 8 ​ ounce of a chemical and ​ __
6  ​ ounce saline
solution. What is the unit rate of chemical
to saline solution?
6.
A factory produces c cans of paint in h
hours according to the equation c 5 50h.
Which of the following represents the unit
rate of cans produced per hour?
A. 50
Ratios and Proportional
Relationships
3.
B. h
1
A. ​ __
16   ​
C. c
3
B. ​ __
4 ​
h
D. ​ _c ​
9
C. __
​ 4 ​
7.
D. 4
4.
Candice rode her bike at a constant speed.
1
She traveled 36 miles in 1 ​ __
2 ​  hours. How can
she find her unit rate, in miles per hour?
1
B. add 1 ​ __
2 ​  to 36
1
D. divide 36 by 1 ​ __
2 ​ 
Duplicating any part of this book is prohibited by law.
1
Peyton reads 25 pages in ​ __
2 ​  hour.
Which equation below represents the
relationship between the number of pages
Peyton reads and how much time he
spends reading? Let p 5 number of pages
and t 5 number of hours.
1
2
8
3
12
5
20
6
24
9
36
A. They are proportional, because the
perimeter always increases by the
same amount from column to column.
1
C. multiply 36 by 1 ​ __
2 ​ 
A. p 5 50t
Side Length
Perimeter
Which of the following best explains
why these values are or are not in a
proportional relationship?
1
A. subtract 1 ​ __
2 ​  from 36
5.
Ethan measured the side length and
perimeter of five squares, as shown in the
table below.
B. They are proportional, because the
ratio of perimeter to side length is
always 4:1.
C. They are not proportional, because
the side length increases by varying
amounts from column to column.
D. They are not proportional, because
the difference between side length
and perimeter is different for each
column.
B. p 5 ​ ___
50   ​ t
1
C. p 5 12 ​ __
2 ​  t
2
D. p 5 ​ ___
25   ​ t
Go On
5
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Leah solved 40 arithmetic problems in
1
2 ​ __
2  ​ minutes and 80 arithmetic problems
in 5 minutes. Which of the following best
represents her unit rate?
10. The line below represents the relationship
between time since tickets to a concert at
Madison Square Garden went on sale and
the number of tickets sold.
y
B. 8 problems per minute
C. 16 problems per minute
D. 40 problems per minute
9.
The line graphed below represents the
total distance traveled by a train over time.
Distance (in miles)
500
450
400
350
300
250
200
150
100
50
0
1
2
3
4
5
6
x
Time (in hours)
Which of the following represents the
constant of proportionality (unit rate) for
this line?
y
400
350
300
250
200
150
100
50
0
Number of Tickets Sold
1
A. 2 ​ __
2  ​ problems per minute
A. 40
B. 50
1
2
3
4
5
6
x
Time (in hours)
Based on this graph, which of the
following statements is true?
A. The train traveled 300 miles in 1 hour.
B. The train traveled 300 miles in 4 hours.
C. The train traveled 150 miles in 3 hours.
D. The train traveled 120 miles in 2 hours.
C. 80
D. 400
11. Toby bought a pair of jeans and a sweater.
The pair of jeans costs $30 and the sweater
costs $35. If sales tax is 6%, how much
did Toby spend in total for the jeans and
sweater?
A. $65.12
B. $66.80
C. $68.90
D. $71.00
Go On
Duplicating any part of this book is prohibited by law.
8.
6
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1
15. The graph below shows the amount of
water in a swimming pool over time.
y
Water in Pool
(in gallons)
3
A. ​ __
8 ​
2
B. __
​ 3 ​
3
C. __
​ 2 ​
8
D. __
​ 3 ​
Ratios and Proportional
Relationships
3
__
12. A park is ​ __
4 ​ mile long and ​ 2 ​  mile wide.
Which fraction represents the ratio of the
park’s length to its width?
500
450
400
350
300
250
200
150
100
50
0
1
2
3
4
5
6
7
x
Time (in minutes)
2
1
__
13. Jimena runs ​ __
3 ​ of a mile in ​ 6  ​ of an hour.
What is her unit rate, in miles per hour?
A. 2 miles per hour
B. 3 miles per hour
C. 4 miles per hour
D. 9 miles per hour
Duplicating any part of this book is prohibited by law.
14. A scale measures with a 1.5% margin of
error, which means that the measurement
given by the scale may be up to 1.5%
lower or higher than the actual weight of
the object. If Colin uses this scale to weigh
a suitcase and the scale reads 40 pounds,
what is the range of possible actual
weights for the suitcase?
What does the unit rate represent in this
relationship?
A. The pool is being filled at a rate of
25 gallons per minute.
B. The pool is being emptied at a rate of
25 gallons per minute.
C. The pool is being filled at a rate of
50 gallons per minute.
D. The pool is being emptied at a rate of
50 gallons per minute.
A. 34 pounds to 46 pounds
B. 35.5 pounds to 44.5 pounds
C. 38.5 pounds to 41.5 pounds
D. 39.4 pounds to 40.6 pounds
Go On
7
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1
1
__
16. A recipe for seafood gumbo calls for 2 ​ __
2 ​  quarts of fish stock and 1 ​ 2 ​  pounds of shrimp. What is the
unit rate of fish stock to shrimp in the gumbo?
Go On
Duplicating any part of this book is prohibited by law.
17. A corporation has agreed to donate money to a charity organization each time one of its
employees donates to that charity organization. The corporation donates $3 for every $2
donated by an employee. The total amount the corporation donates to the charity organization
is proportional to the total amount that its employees donate. Write an equation representing the
relationship between c, the amount the corporation donates, and e, the amount its employees
donate.
8
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1
60
Time (t) (in hours)
Distance (d ) (in miles)
2
120
4
240
6
360
Ratios and Proportional
Relationships
18. Naresh drove at a steady speed during a road trip. He recorded the amount of time he drove and
the distance he covered, as shown in the table below.
A. Plot these points on the graph below and connect them with a straight line.
Distance (in miles)
d
400
350
300
250
200
150
100
50
0
1
2
3
4
5
6
7
t
Time (in hours)
B. What are the coordinates of the point on the line where t 5 3?
Duplicating any part of this book is prohibited by law.
C. What do the coordinates of the point you found in part B specifically tell you about Naresh’s
progress at that point in the trip?
Go On
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19. Rachel earns 2% simple annual interest on the money she has in her savings account. Last year she
did not deposit to or withdraw from her savings account and earned $35 in interest.
A. How much money did Rachel have in her savings account at the beginning of last year?
Explain how you set up an equation to solve this.
Go On
Duplicating any part of this book is prohibited by law.
B. How much money did Rachel have in her savings account at the end of last year?
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Ratios and Proportional
Relationships
20. Last year, the attendance at the homecoming football game was 300. This year, 360 people
attended.
A. What was the percent increase from last year to this year? Show your work.
Duplicating any part of this book is prohibited by law.
B. If the increase in the number of people from this year’s to next year’s homecoming football
game is the same as from last year to this year, is the percent of increase the same? Explain
your answer.
STOP
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Contents
Instructional Overview. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4
Common Core State Standards Correlation Chart . . . . . . . . . . . . . . . . . 12
Domain 1 Ratios and Proportional Relationships. . . . . . . 16
Common Core
State Standards
Lesson 1
Computing Unit Rate . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18
7.RP.1
Lesson 2
Identifying Proportional Relationships . . . . . . . . . . . . . . . . . . 20
7.RP.2.a, 7.RP.2.b
Lesson 3
Representing Proportional Relationships. . . . . . . . . . . . . . . . 22
7.RP.2.c, 7.RP.2.d
Lesson 4
Word Problems with Ratio and Percent. . . . . . . . . . . . . . . . . . 24
7.RP.3
Domain 2 The Number System . . . . . . . . . . . . . . . . . . . . . . . . . . 26
Lesson 5
Adding and Subtracting Rational Numbers . . . . . . . . . . . . . . 28
7.NS.1.a, 7.NS.1.b,
7.NS.1.c
Applying Properties of Operations to
Add and Subtract Rational Numbers. . . . . . . . . . . . . . . . . . . . 30
7.NS.1.d
Lesson 7
Multiplying Rational Numbers. . . . . . . . . . . . . . . . . . . . . . . . . 32
7.NS.2.a, 7.NS.2.c
Lesson 8
Dividing Rational Numbers . . . . . . . . . . . . . . . . . . . . . . . . . . . 34
7.NS.2.b, 7.NS.2.c
Lesson 9
Converting Rational Numbers to Decimals. . . . . . . . . . . . . . . 36
7.NS.2.d
Lesson 6
Lesson 10
Problem Solving: Complex Fractions. . . . . . . . . . . . . . . . 38
7.NS.3
Lesson 11
Problem Solving: Rational Numbers. . . . . . . . . . . . . . . . . 40
7.NS.3
Lesson 12
Writing Equivalent Expressions. . . . . . . . . . . . . . . . . . . . . . . . 44
7.EE.1, 7.EE.2
Lesson 13
Factoring and Expanding Linear Expressions. . . . . . . . . . . . . 46
7.EE.1
Lesson 14
Adding and Subtracting Algebraic Expressions. . . . . . . . . . . 48
7.EE.1
Lesson 15
Problem Solving: Algebraic
Expressions and Equations. . . . . . . . . . . . . . . . . . . . . . . . 50
7.EE.3
Lesson 16
Word Problems with Equations. . . . . . . . . . . . . . . . . . . . . 52
7.EE.4.a
Lesson 17
Word Problems with Inequalities. . . . . . . . . . . . . . . . . . . . . . . 54
7.EE.4.b
Domain 4 Geometry. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56
Scale Drawings. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58
7.G.1
Lesson 19 Drawing Geometric Shapes. . . . . . . . . . . . . . . . . . . . . . . . . . . 60
7.G.2
Lesson 18
Problem
Solving
Fluency
Lesson
Duplicating any part of this book is prohibited by law.
Domain 3 Expressions and Equations . . . . . . . . . . . . . . . . . . . 42
Performance
Task
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Common Core
State Standards
Lesson 20 Examining Cross Sections of
Three-Dimensional Figures . . . . . . . . . . . . . . . . . . . . . . . . . . . 62
7.G.3
Area and Circumference of Circles . . . . . . . . . . . . . . . . . . . . . 64
7.G.4
Lesson 22 Angle Pairs. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66
7.G.5
Problem Solving: Area and Surface
Area of Composite Figures . . . . . . . . . . . . . . . . . . . . . . . . 68
7.G.6
Problem Solving: Volume of
Three-Dimensional Figures. . . . . . . . . . . . . . . . . . . . . . . . 70
7.G.6
Lesson 21
Lesson 23
Lesson 24
Domain 5 Statistics and Probability . . . . . . . . . . . . . . . . . . . . . 72
Lesson 25 Understanding Sampling. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 74
7.SP.1, 7.SP.2
Lesson 26 Using Mean and Mean Absolute Deviation . . . . . . . . . . . . . . 76
7.SP.3, 7.SP.4
Making Comparative Inferences about
Two Populations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 78
7.SP.3, 7.SP.4
Lesson 28 Understanding Probability. . . . . . . . . . . . . . . . . . . . . . . . . . . . 80
7.SP.5, 7.SP.6,
Lesson 27
7.SP.7.b
Lesson 29 Probabilities of Simple Events. . . . . . . . . . . . . . . . . . . . . . . . . 82
7.SP.7.a, 7.SP.7.b
Lesson 30 Probabilities of Compound Events . . . . . . . . . . . . . . . . . . . . . 84
7.SP.8.a, 7.SP.8.b
Lesson 31
Simulations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 86
7.SP.8.c
Answer Key. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 88
Math Tools. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 130
Appendix A: Fluency Practice. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . A
Duplicating any part of this book is prohibited by law.
Appendix B: Standards for Mathematical Practice. . . . . . . . . . . . . . . . . . B
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1
Computing Unit Rate
Learning Objective
• Students will set up and simplify complex fractions to find
unit rates.
Vocabulary
complex fraction a fraction in which the numerator and/or
denominator is a fraction
rate a ratio that compares two quantities that have different
units of measure
a
ratio a comparison of two numbers, written as __
​ b ​,  a:b, or a to b
unit rate a rate in which the second quantity in the
comparison is 1 unit
Common Core State Standard
7.RP.1 Compute unit rates
associated with ratios of fractions,
including ratios of lengths, areas
and other quantities measured
in like or different units. For
1
example, if a person walks __
​ 2  ​mile
1
in each ​ __
4 ​ hour, compute the unit
rate as the complex fraction
Before the Lesson
Review the fact that a ratio compares two quantities and that a rate
compares two quantities that compare different units of measure.
Pose this problem to the class: Suppose you will bring your favorite
beverage to a family picnic. You can ask students to identify their
favorite beverages, if you wish. Then ask: Suppose a 2-liter bottle
of that beverage costs $4 and a 3-liter bottle of that beverage costs
$4.50. Which is the better buy? Give students a chance to determine
their answers and then discuss. Students may have used proportional
reasoning. For example, they may reason that buying three 2-liter
bottles means buying 6 liters for $12 and buying two 3-liter bottles
means buying 6 liters for $9, so the 3-liter bottle is the better buy.
Other students may have found the unit price per liter for each bottle:
1
__
​ 2  ​
​ __
1  ​ miles per hour, equivalently
__
​ 4  ​
2 miles per hour.
So, the 3-liter bottle is the better buy.
However students determine the answer, be sure to mention that one
way is to determine the unit price for each bottle and compare them.
Explain that a unit price is an example of a unit rate and that one way
to determine a unit rate is to write a fraction to represent the rate and
then simplify it. Segue into the lesson, which shows how to write
complex fractions to determine unit rates.
Duplicating any part of this book is prohibited by law.
$4
_____
   ​ 5 $4 4 2L 5 $2.00 per liter
​ 2 liters
$4.50
​ _____
 ​ 5 $4.50 4 3 L 5 $1.50 per liter
3 liters 
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EXAMPLE A This example introduces how to write
and simplify a complex fraction to find a unit rate.
Use Step 1 to show how to set up the complex
fraction. Then use Step 2 to show that in order to
get a denominator of 1, students must multiply
both the numerator and the denominator by 4. Ask
some questions, such as: If the denominator were
1
__
​ 3 ​ instead, by what number would you multiply? Be
3
1
sure that students see that since __
​ 3  ​3 __
​ 1 ​ 5 1, in that
case, they would need to multiply by 3 in order
5
__
​ 2 ​mi
to get a denominator of 1. Finally, review how ____
​  1 h   ​
5
1
shows a unit rate of __
​ 2 ​miles per hour or 2 ​ __
2 ​ miles
5
per hour. If necessary, review how to convert __
​ 2 ​to
5
1
__
a mixed number: ​ __
2 ​5 5 4 2 5 2R1 5 2 ​ 2 ​. 
TRY MP1 Explain that this problem is a little more
1
complicated. Students must convert 3 ​ __
8 ​ to an
25
improper fraction, ​ ___
8   ​. Since the denominator is
not a unit fraction, encourage students to use the
EXAMPLE C This example is similar to Example B,
since it involves writing a complex fraction with
a mixed number as the numerator and a proper
fraction as the denominator. The difference is
that this problem is solved using a computational
method, not a fraction model.
CHECK MP3 To reinforce the relationship
between multiplication and division, have students
use multiplication to check their work.
Explanations may vary. Possible explanation: The
23
___
​  4   ​
3
23 __
3
1
___
__
​  __1   ​5 28 ​ __
rate is __
4 ​. This is the same as ​  4   ​4 ​ 5 ​ 5 28 ​ 4 ​.
​ 5  ​
So, to check the answer, I can multiply:
3
1
115
1
115
23
__ ___ __ ___ ___
28 ​ __
4 ​3 ​ 5  ​5 ​  4   ​3 ​ 5 ​ 5 ​ 20  ​5 ​  4   ​
fact that a fraction represents division to find the
Practice
as ___
​  8   ​ 4 __
​ 8 ​.
5 miles per hour
25
1
___
Work may vary. Possible work: 3 ​ __
8 ​ 5 ​  8   ​;
As students are working, pay special attention to
problems 12–15, which require students not only
to find unit rates but also to show their work. When
reviewing answers, be sure to review the work that
students did to find those answers.
For answers, see page 88.
25
___
​  8   ​
quotient. They can rewrite the complex fraction __
​  __5  ​
​ 8 ​
25
5
25
Duplicating any part of this book is prohibited by law.
First, draw 2 squares and divide them in halves
3
by drawing vertical line segments. Shade __
​ 2 ​. Then
show how those halves can be divided into fourths
by drawing horizontal line segments. Note that
dividing into halves and then into fourths actually
results in squares divided into eighths. Explain
3
that together, all the shaded squares show __
​ 2 ​, so
students can move the top 3 shaded squares to the
right of the other rectangles and still have the model
3
show __
​ 2 ​. Demonstrate that this now shows that
3 __
3
3
3
__
__
2 3 ​ __
4 ​5 ​ 2 ​, so ​ 2 ​ 4 ​ 4 ​5 2.
___
​  8   ​ x
__
​  __5  ​5 __
​ 1  ​;
​ 8 ​
25
5 ___
25 __
5
8 __
__
​ ___
8   ​ 4 ​ 8 ​5 ​  8   ​3 ​ 5 ​5 ​ 1 ​ 
EXAMPLE B This example shows how to write
a complex fraction with a mixed number as
the numerator and a proper fraction as the
denominator. Then it illustrates how to use a fraction
model to divide the numerator by the denominator
and find the unit rate. Use the fraction model to
3
3
​ 4 ​.
​ 2 ​by __
review the conceptual meaning of dividing __
Domain 1
Examples
Common Errors
When the complex fraction that represents a unit
rate includes a mixed number, some students may
not rewrite the complex fraction as an improper
fraction before computing an answer. Be sure that
students understand the importance of this step.
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Identifying Proportional Relationships
Learning Objectives
• Students will show that ratios are equivalent and are therefore
proportional and will identify the constant of proportionality for
quantities in a proportional relationship.
• Students will represent ratios as points on a coordinate grid to
show that the ratios are equivalent and will demonstrate that an
equation represents a directly proportional relationship.
Vocabulary
constant of proportionality the constant ratio by which
two quantities co-vary in a proportion; also called the unit
rate or constant ratio
origin the point named by (0, 0) on a coordinate grid, where
the axes intersect
proportion an equation that shows that two ratios are
equivalent
Before the Lesson
Review the fact that a ratio compares two quantities.
Pose this problem: You are in a bike shop looking at
bicycles. The number of tires you see in the showroom
depends on the number of bicycles in the showroom.
How many tires does 1 bicycle have? Draw a table on
the board, labeling one row Bicycles and the second
row Tires. In the bicycles row, write 2, 3, 4, and 5.
Then ask: What is the total number of tires if there are
2 bicycles? 3 bicycles? 4? 5? Fill in the table as shown.
Bicycles
1
2
3
4
5
Tires
2
4
6
8
10
Understand
Common Core State Standards
7.RP.2 Recognize and represent
proportional relationships
between quantities.
a. Decide whether two
quantities are in a proportional
relationship, e.g., by testing
for equivalent ratios in a table
or graphing on a coordinate
plane and observing whether
the graph is a straight line
through the origin.
b.Identify the constant of
proportionality (unit rate) in
tables, graphs, equations,
diagrams, and verbal
descriptions of proportional
relationships.
Explain that each column of the table shows a
ratio that compares the total number of bicycles
to the total number of tires. Ask: If more bicycles
are added to the showroom, will the number of
tires in the showroom also increase? Use this to
help students understand that when ratios are
in a proportional relationship, as one quantity
increases, the other quantity will also increase.
In the case of the above example, as the number
of bicycles increases by 1, the number of tires
increases by 2. Explain that students can show
that quantities in a table show a proportional
relationship. Segue into the lesson.
Connect
Tables can be used to help students recognize
proportional relationships. To help develop
conceptual understanding, show that the pairs of
values in each column of the table showing Tina’s
Earnings can be written as a ratio. Use step 1 to
1
show that all of these ratios simplify to __
​ 12   ​, so they
are all equivalent ratios. Explain that since the
ratios are equivalent, the table shows a directly
proportional relationship.
Explain that Tina’s hourly wage shows the number
of dollars she earns if she works 1 hour and that
this is also the constant of proportionality.
Explain that the constant of proportionality is the
Duplicating any part of this book is prohibited by law.
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Duplicating any part of this book is prohibited by law.
Connect students’ understanding of the constant of
proportionality to this graph. On page 10, students
recognize that the constant of proportionality
showed Tina’s hourly wage, $12 per hour. Explain
that her hourly wage is a unit rate, because it is the
$12
ratio ___
​ 1 h  ​. Connect this to the graph by showing
that the point (1, 12) shows that the constant of
proportionality is 12. Remind students that if a graph
shows a directly proportional relationship, the point
(1, k) shows k, the constant of proportionality.
TRY MP2 Use this to illustrate that directly
proportional relationships can be represented as
equations, as well as in tables and graphs.
Answers may vary. Possible answer:
Each pair of values makes the equation true.
(1, 12): 12 5 12(1) is true.
(2, 24): 24 5 12(2) is true.
(3, 36): 36 5 12(3) is true.
(4, 48): 48 5 12(4) is true.
(5, 60): 60 5 12(5) is true.
(6, 72): 72 5 12(6) is true.
The equation y 5 12x is in the form y 5 kx, so it
represents a directly proportional relationship.
Since k 5 12, the constant of proportionality is 12.
Practice
As students are working, pay special attention
to problems 2, 5, and 6, which provide an
opportunity for students to recognize that not all
tables, graphs, and equations show proportional
relationships. It is critical that students understand
why these representations do not show
proportional relationships.
Also pay careful attention to questions 9–12,
since these are the first times students are asked
to determine if verbal statements or mapping
diagrams show proportional relationships. Again,
be sure that students understand why problems
10 and 11 do not show proportional relationships.
For answers, see pages 88 and 89.
Domain 1
amount by which each x-value must be multiplied
to get each y-value. So, if Tina works for 1 hour,
she earns 1 3 12, or 12, dollars. If she works for
2 hours, she earns 2 3 12, or 24, dollars. The
amount by which each x-value is multiplied is
always the same, 12. Ask: Does it make sense
that the constant is 12 given that this is an hourly
wage? Have students discuss the fact that the
amount Tina earned can be found by multiplying
$12 by the number of hours she works. So, it
makes sense that the constant of proportionality
is also her hourly wage.
To extend students’ understanding of the concept
that values in a table can represent a directly
proportional relationship, show that the pairs of
values in the table on page 10 can be plotted as
points. Use steps 1 and 2 to illustrate that the points
can be connected to form a straight line that passes
through the origin, and that this means they show a
directly proportional relationship.
Common Errors
Some students may mistakenly think that the
constant of proportionality, k, must be a whole
number. If so, they may think that the equation in
1
problem 8, y 5 __
​ 5  ​ k, does not show a proportional
relationship. Be sure to point out that k does not
need to be a whole number. For example, if a child
earned $0.20 for each cookie she sold, then one
could multiply the number of cookies sold by $0.20
1
​ 5  ​ , this could
to find the total earned. Since 0.20 5 __
1
be represented as y 5 0.2x or y 5 __
​ 5  ​ x. You could
also generate points for that situation and graph
them to show that they form a straight line.
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Representing Proportional
Relationships
Learning Objectives
• Students will write equations to represent proportional
relationships.
• Students will examine graphs of proportional relationships,
interpreting the meanings of points and identifying unit rates.
Before the Lesson
Explain that one way to represent a word problem is using an
expression or equation. Then remind students that a directly
proportional relationship can be represented in the form y 5 kx,
where k is the constant of proportionality. Because this lesson focuses
on directly proportional relationships, the equations should all
involve multiplication. Ask: What key words, when they appear in a
problem, signal multiplication? Which key words signal an equal sign?
Make a chart showing the words students suggest on the board.
Include examples of how these words may be used. The chart may
look similar to this.
Key Word(s)
In Words
In Symbols
times
5 times x
5 3 x or 5x
twice
twice as much as a number
2 3 n or 2n
is equal to
Cara's age, c, is equal to
twice Justin's age, j
c 5 2j
is, are, was,
were
Sam's height, s, is 3 times
Betty's height, b
s 5 3b
Common Core State Standards
7.RP.2 Recognize and represent
proportional relationships
between quantities.
c. Represent proportional
relationships by equations.
For example, if total cost t is
proportional to the number
n of items purchased at
a constant price p, the
relationship between the
total cost and the number
of items can be expressed
as t 5 pn.
d.Explain what a point (x, y) on
the graph of a proportional
relationship means in terms
of the situation, with
special attention to the
points (0, 0) and (1, r) where
r is the unit rate.
Understand
Connect
If a relationship is directly proportional, all the (x, y)
pairs are related in the same way: y 5 kx, and the
constant of proportionality, k, holds for each ordered
pair in the relationship. To help develop conceptual
understanding, emphasize that although a directly
proportional relationship could be displayed in
different ways, the equation that represents this
relationship is the same. For example, use the
given verbal description about charges for a
baby-sitting job. Ask: How would the same
relationship be displayed in a table of x and y values?
Ask a volunteer to express that relationship by
making a table of values in which the Hours Worked
column increases from 1 to 4 in increments of 1 and
the Total Charges column increases from 10 to 40 in
increments of 10. Similarly, ask a volunteer to create
a graph by plotting the ordered pairs (1, 10), (2, 20),
(3, 30), and (4, 40) to express this relationship.
Duplicating any part of this book is prohibited by law.
Continue to list as many key words and examples as students can
think of. Explain that because proportional relationships are always
in the form y 5 kx, the key words that indicate multiplication (3) and
equality (5) will be useful in this lesson.
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  
  
As students are working, pay special attention to
problems 6 and 7, which provide an opportunity
for students to interpret a point on the graph in the
context of the given situation and to use the graph
to identify the unit rate for this situation.
For answers, see pages 89 and 90.
Common Errors
Some students may make errors when writing
equations to represent quantities in a table if they
do not pay careful attention to how the x-values and
y-values are increasing. For example, in problem 11,
5
if students only recognize that each y is __
​ 2 ​more than
the previous one, they may mistakenly choose D,
5
y 5 ​ __
2 ​ x, as the answer. Be sure to point out that
1
since the x-values are increasing by __
​ 2  ​ s, students
cannot simply look at how the y-values are
increasing to determine the answer. Ask: How
could you determine the relationship in this case?
Students may suggest rewriting the table so it only
lists values when x is 1, 2, or 3.
x
y
1
5
2
10
3
15
This shows that as the x-values increase by 1s, the
y-values increase by 5s. So, C, y 5 5x, is the correct
answer.
Duplicating any part of this book is prohibited by law.
  
Practice
Domain 1
Although the relationship is displayed in three
different ways, the same equation is used for each
representation: y 5 kx, where k 5 10, the constant
of proportionality.
To connect the concept to procedural
understanding, stress the importance of
determining the meaning of the (x, y) relationship in
terms of the context of the situation. Use Step 1 on
the Connect page to show that each point on the
graph can be interpreted in the context of the given
problem. For example, (4, 8) indicates that after 4
minutes, there will be 8 gallons of water in the tank.
To test for understanding, ask additional questions:
What does the point (2, 4) on the graph represent?
Then use Step 2 to show that whatever point on the
graph has a first coordinate of 1 will have a second
coordinate that shows the unit rate. Connect the
knowledge that (1, r) shows the unit rate, r, to the
fact that students know that a point (1, k) on the
graph of a proportional relationship shows the
constant of proportionality, k. Remind students that
the unit rate is also the constant of proportionality.
TRY MP4 MP6 Use this problem to highlight the
fact that the point (1, r) shows a unit rate, but a point
(x, 1) does not. Make sure that students understand
1
​ 2  ​, 1  ​is
that although one of the coordinates of ​ __
a 1, since the first coordinate is not 1, the second
coordinate does not show a unit rate.
Also use this problem to illustrate that every point
on the line graph has a meaning in the context
of the problem situation. Even though the graph
1
​ 2  ​, 1  ​, it is on the line, so it has
shows no dot ​ __
meaning.
1
1
​ 2  ​, 1  ​shows that after __
​ 2  ​minute, there is
The point ​ __
1 gallon of water in the tub.
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