critical thinking grade 7

® GRADE

Critical Thinking 7

for Math

Strategies and Activities
to Extend Mathematical
Understanding

• Positive and negative integers
• Ratios and proportions
• Algebraic equations and inequalities
• Geometric problem-solving
• Probability and statistics
• Answer key

carsondellosa.com/spectrum



Critical Thinking for Math

Grade 7

Published by Spectrum®
an imprint of Carson-Dellosa Publishing

Greensboro, NC

Spectrum®
An imprint of Carson-Dellosa Publishing LLC
P.O. Box 35665
Greensboro, NC 27425 USA

© 2017 Carson-Dellosa Publishing LLC. Except as permitted under the United States Copyright Act, no part of this
publication may be reproduced, stored, or distributed in any form or by any means (mechanically, electronically,
recording, etc.) without the prior written consent of Carson-Dellosa Publishing LLC. Spectrum® is an imprint of
Carson-Dellosa Publishing LLC.

ISBN 978-1-4838-3962-

Table of Contents Grade 7

Chapter 1: Adding and Subtracting Rational Numbers
Check What You Know . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5
Lessons 1-7 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7-16
Check What You Learned . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17

Chapter 2: Multiplying and Dividing Rational Numbers
Check What You Know . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19
Lessons 1-7 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21-29
Check What You Learned . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30

Chapter 3: Expressions, Equations, and Inequalities
Check What You Know . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33
Lessons 1-7 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35-41
Check What You Learned . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42

Chapter 4: Ratios and Proportional Relationships
Check What You Know . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44
Lessons 1-6 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46-53
Check What You Learned . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54

Chapters 1-4 Mid-Test. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56

Chapter 5: Geometry
Check What You Know . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60
Lessons 1-10 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62-70
Check What You Learned . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71

Chapter 6: Statistics
Check What You Know . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 74
Lessons 1-3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 76-80
Check What You Learned . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81

3

Table of Contents, continued

Chapter 7: Probability
Check What You Know . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83
Lessons 1-8 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 85-94
Check What You Learned . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 95
Chapters 1-7 Final Test . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 97
Answer Key . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102-126

4

NAME _________________________________________________________________________________

Check What You Know CHAPTER 1 PRETEST

Adding and Subtracting Rational Numbers

Compare the values using , , or .

1. 227 1 _____ 27 1 2. u214.375u ______ u13.75u
3 4

Write the additive inverse of each number.

3. 242.6 _____ 4. 13.25 _____

Solve the problems. 6. 9 3  5 1  6 1 5
5. 13.45  0.025  5.7 5 4 3 6

Use a number line to complete the following problems.

7. 2 3 1 (2 1 ) 5
4 2

25 0 5
5
8. 1.75  3 5 0
25
10. 1 4  (3 1 ) 5
Solve the problems. Show your work. 5 2
9. 4.35 1.48 5

Spectrum Critical Thinking for Math Chapter 1
Grade 7 Check What You Know

5

CHAPTER 1 PRETEST NAME _________________________________________________________________________________

Check What You Know

Adding and Subtracting Rational Numbers

Solve the problems. Show your work.

11. L ucy has scored 40 points in her trivia game so far. She answers 2 more
questions correctly and scores 20 points. Then, she answers a question
incorrectly and loses 25 points. What is her final score?

12. A hiker takes a trail that increases his altitude by 26.9 feet. He switches to
another trail that will decrease his altitude by 35.6 feet. What is his overall
change in altitude?

13. The coldest record temperature in Belton is 29 degrees. The highest
temperature on record is 102 degrees. What is the difference between the two
temperatures?

14. Shane lost 5 1 pounds. Shonda lost 3 1 pounds. How much more weight did
2 4

Shane lose?

Spectrum Critical Thinking for Math Chapter 1
Grade 7 Check What You Know

6

NAME _________________________________________________________________________________

Lesson 1.1 Absolute Value and Rational Numbers

Absolute value is the distance between a number and zero on a number line.
Numbers that are opposites will have the same absolute value.

2 2 5 2 2 5 2
3 3 3 3

21 2 2 2 1 0 1 2 1
3 3 3 3

Answer the questions.

If u X uu Y u and both X and Y are negative numbers, describe the location of point X
in relation to the location of point Y on a number line.

Compare u28 1 u and u 7 2 u. Explain your thinking.
3 3

Write a statement comparing u E u and u F u. Explain your thinking.

E0 F

Spectrum Critical Thinking for Math Lesson 1.1
Grade 7 Absolute Value & Rational Numbers

7

NAME _________________________________________________________________________________

Lesson 1.2 Additive Inverse

Opposite numbers have the same absolute value. Two numbers that can be added
together to equal zero are called additive inverses.

Emelia’s mother gives her $4 to go to the store. At the store, Emelia spends $4 on
a bag of oranges. How much money does Emelia have left?

Spent $4
Given $4

25 0 1 2 3 4 5

Combine the amount of money that Emelia was given ($4) with the amount she
spent (2$4) to find how much she has left. 4 1 (24) = 0. Emelia has $0 left.

Answer the questions using the additive inverse. Show your work.

Hermine is making a skirt. She bought 5 1 yards of fabric. She used 3 1 yards to
4 2

make the skirt. How many yards can she give away if she wants to end up with no

fabric?

Juan earned $28 mowing a lawn and $15 walking a dog. The next day, he bought
a t-shirt for $20 and a jacket for $22. Was the sum of what he earned the additive
inverse of what he spent?

Spectrum Critical Thinking for Math Lesson 1.2
Grade 7 Additive Inverse

8

NAME _________________________________________________________________________________

Lesson 1.3 Adding & Subtracting Rational Numbers

When you add or subtract fractions, the denominators must be the same. When
you add or subtract decimals, the place values must be aligned.

Corey bought 4 3 ounces of cashews for $2.30 and 3 1 ounces of pistachios for
5 10

$1.24. What is the total amount of nuts that he bought? What was the total

amount of money he spent?

4 3 1 3110 5 4 6 1 3 1 5 2.30
5 10 10  1.24

7170 ounces of nuts $ 3.54 total spent

Solve the problem. Show your work.

Royal’s new smartphone has 32 gigabytes (GB) of memory. She has the following
apps and files on her phone. Does she have enough space for the operating system
upgrade that uses 9.1GB?

Royal's Phone

Program Size (GB)

Operating System 2.64

Games 1.203

Calculator App 0.08

Downloaded Files 2.75

Other Apps 2.1

Videos 5.12

Photos 4.592

Spectrum Critical Thinking for Math Lesson 1.3
Grade 7 Adding & Subtracting Rational Numbers

9

NAME _________________________________________________________________________________

Lesson 1.4 Adding Positive and Negative Numbers

The sum of 2 positive numbers is a positive 2 2 1 23 5 25 21355
23 13
number farther to the right of the
2345
first addend on the number line.
25 24 23 22 21 0 1

The sum of 2 negative numbers is a negative
number farther to the left of the first addend.

The sum of a positive and a negative 221351
number will be positive if the positive 13
number has a greater absolute value.
24 23 22 21 0 1 2 3 4

The sum of a positive and a negative 2 3 1 2 5 21
number will be negative if the negative 12
number has a greater absolute value.
25 24 23 22 21 0 1 2 3

Write always, sometimes, or never below each statement. Give an example to
show your answer.

When adding two numbers, the sum is greater than each of the addends.

When adding two negative numbers the sum is greater than each of the addends.

When adding two numbers with opposite signs, the absolute value determines the
sign of the sum.

Spectrum Critical Thinking for Math Lesson 1.4
Grade 7 Adding Positive and Negative Numbers

10

NAME _________________________________________________________________________________

Lesson 1.4 Adding Positive and Negative Numbers

Distance and direction also help to determine the sum of positive and negative
fractions and decimals on a number line.

Answer the questions.

Is an estimate of 3 for the sum of 12 3 and 15 2 reasonable? Explain your answer.
4 3

Is an estimate of 4 accurate for the sum of 24.24 and 27.8 accurate? Explain
your answer.

Use a number line to answer the questions.

3 1  (1 1 ) 5
2 2

5 0 5
0 5
2.5 1 (0.5 ) 5 0 5

 1 1 (4 3 ) 5 5
4 4 5

Spectrum Critical Thinking for Math Lesson 1.4
Grade 7 Adding Positive and Negative Numbers

11

NAME _________________________________________________________________________________

Lesson 1.5 Subtracting Positive and Negative Numbers

Subtracting a number is the same as adding the additive inverse of the second
number and applying the rules of integer addition.

4 2 7 5 4 1 (27) 5 23
4 2 (27) 5 4 1 7 5 11
Write always, sometimes, or never below each statement. Give an example to
show your answer.
When subtracting two numbers, the difference is always less than the two numbers.

Subtracting a negative number is the same as adding the absolute value of that
number.

Answer the question. Show your work.
Jamal wrote the following on his paper: 10 2 (210) 5 10 210 5 0. What was his
mistake? What is the correct answer?

Spectrum Critical Thinking for Math Lesson 1.5
Grade 7 Subtracting Positive and Negative Numbers

12

NAME _________________________________________________________________________________

Lesson 1.5 Subtracting Positive and Negative Numbers

The rules that apply to integers also apply when subtracting positive and negative
fractions and decimals.

4 2 (27.5) 5 4 1 7.5 5 11.5

4 2 (27 1 ) 5 4 1 7 1 5 11 1
2 2 2

Answer the questions.

Is an estimate of 8 as the difference between 4.7 and 23.3 reasonable? Explain.

25 24 23 22 21 0 1 2 3 4 5

On a number line, what is the difference between 261.5 and 223.4?
270 260 250 240 230 220 210 0 10

Evaluate: 23 3 2 63 1 5
8 8

Evaluate: 217.56 2 13.43 5

Spectrum Critical Thinking for Math Lesson 1.5
Grade 7 Subtracting Positive and Negative Numbers

13

NAME _________________________________________________________________________________

Lesson 1.6 Adding with Mathematical Properties

Mathematical properties can be used to add rational numbers quickly.
21 1 11 1 13 1 (25) 1 (27)

Commutative Property: (a 1 b 5 b 1 a) 21 1 (25) 1 (27) 1 13 1 11

Associative Property: {21 1 (25) 1 (27) 1 13} 1 11
(a 1 b) 1 c 5 a 1 (b 1 c) (213 1 13) 1 11

Identity Property of Addition: a 1 0 5 a 0 1 11 5 11

Simplify the expressions using mathematical properties.
23.25 1 4.2 1 3.2 1 (22.1) 1 0.05

2 1 1 (23 1 ) 1 1 1 2 1 2 2
2 3 2 3 3

4 1 18 1 7 1 (29) 13 1 (29)

Spectrum Critical Thinking for Math Lesson 1.6
Grade 7 Adding with Mathematical Properties

14

NAME _________________________________________________________________________________

Lesson 1.7 Rational Numbers in the Real World

Solve the problems. Show your work.

Lynn begins with a bank balance of $64. After three checks are written, the account
now has a balance of 2$13. What was the total amount of the checks?

Aaron picks peaches from his family’s orchard and sells them at a farmers market.
Each morning, he picks more peaches and adds them to the unsold peaches from
the previous day. Last week he picked and sold the amounts shown in the table. How
many pounds did he have at the end of the week?

Beginning amount Weight (pounds)
Picked 4.1
Sold 19.8
Picked 21.6
Sold 17.7
Picked 18.1
Sold 22.4
Picked 12.9
Sold 15.3
23.9

Spectrum Critical Thinking for Math Lesson 1.7
Grade 7 Rational Numbers in the Real World

15

NAME _________________________________________________________________________________

Lesson 1.7 Rational Numbers in the Real World

Solve the problems. Show your work.

Monica is monitoring the amount of water in a container in her backyard. During
the rainy season, it rains each day. Then, the sun comes out and evaporates some
of the water in the container. The table below tracks the amount of rain added and
evaporated each week. What is the final height of the water?

Height (cm)

Beginning amount 1 1
Rain 10

Evaporation 4
Rain 5

Evaporation 3
Rain 10

Evaporation 1
2

1
5

2 2
5

7
10

Mrs. McCoy’s original loan balance was $3,467. She made her regular payment of
$291 as well as an extra payment of $79. She was charged $23 in interest. Write
and simplify an expression to find Mrs. McCoy’s new balance.

Spectrum Critical Thinking for Math Lesson 1.7
Grade 7 Rational Numbers in the Real World

16

NAME _________________________________________________________________________________

Check What You Learned

Adding and Subtracting Rational Numbers

The table shows stock market gains and losses that were recorded over a 5-day
period. Use the table to answer the questions. Show your work.

Day 1 2 3 4 5
Stock X 1$6.75 2$12.50 1$21 2$9.20 1$4.35

Day 1 2 3 4 5 CHAPTER 1 POSTTEST
Stock Y 1$15 2$22.60 1$18.90 2$14.25 1$7.25

1. On which day did Stock X have the biggest change (gain or loss)?

2. On which day did Stock Y have the biggest change?

3. How much was lost in total on day 4 for Stock X and Stock Y?

4. For Stock Y, what was the difference between day 3 and day 2?

Spectrum Critical Thinking for Math Chapter 1
Grade 7 Check What You Learned

17

NAME _________________________________________________________________________________

Check What You Learned

Adding and Subtracting Rational Numbers
Solve the problems. Show your work.

5. The goal of Tarik’s card game is to have a score of 0. Find two more cards he
could pick to win if he is holding cards with the following values: 27, 3, 4, 29.

CHAPTER 1 POSTTEST 6. Jacob finds a piece of metal that is 2 1 inches thick. He files off 3 inch. He adds
5 10
2
a protective covering to the metal that is 10 inch thick. Then, he polishes the metal

and removes an additional 1 inch. What is the final thickness of the metal?
10

7. Raquel planted an herb garden in her back yard. This year, she planted 1 of her
2
1 1
garden with chives, 4 with oregano, and 8 with parsley. What portion of the

garden is still unplanted?

Spectrum Critical Thinking for Math Chapter 1
Grade 7 Check What You Learned

18

NAME _________________________________________________________________________________

Check What You Know CHAPTER 2 PRETEST

Multiplying and Dividing Rational Numbers

Solve the problems. Show your work.

1. 6  3 1
3

2. 5  4.75

3. 10  (22)  (23) 5

4. 22  (211)  (25) 5

5. 10 1 4 5 1
5 10

6. 194.75 4 10.25 5

7. 49 4 (27) 5

Spectrum Critical Thinking for Math Chapter 2
Grade 7 Check What You Know

19

CHAPTER 2 PRETEST NAME _________________________________________________________________________________

Check What You Know

Multiplying and Dividing Rational Numbers
8. True or false: 25  (217)  25 5 25  (25)  (217)

9. Write 5 as a decimal.
6

10. Katie is digging a hole to plant a tree. She digs 1 foot deep each day for 3
days. Write and evaluate a numeric expression to represent this situation.

11. A tree fell in Kyle’s back yard during a storm. Each day, he cuts a 2 1 foot
1 3
3
section off of the tree. If the tree was 23 feet tall, how many days will it take

him to cut up the entire tree?

Spectrum Critical Thinking for Math Chapter 2
Grade 7 Check What You Know

20

NAME _________________________________________________________________________________

Lesson 2.1 Multiplying Rational Numbers

The distributive property can be used to multiply a whole number and a rational
number.

Distributive Property: a  (b 1 c) 5 a  b 1 a  c

2  3 2 5 2(3 1 2 ) 2  4.75 5 2(4 1 0.75)
3 3

2  3 1 2  2 5 6 1 4 2  4 1 2  0.75 =
3 3

5 6 1 1 1 5 7 1 8 1 1.5 5
3 3

9.5

Use the distributive property to find the product.

4  5 1 5
5

10  17.135 5

7  10 2 5
7

8  5.125 5

Spectrum Critical Thinking for Math Lesson 2.1
Grade 7 Multiplying Rational Numbers

21

NAME _________________________________________________________________________________

Lesson 2.1 Multiplying Rational Numbers

Malachi has to pack 84 toiletry bags for the homeless shelter. He has packed 1 of
4
the boxes. How many bags are left for him to pack?

1  84 5 1  (80 1 4) 5 1  80 1 1 45
4 4 4 4

20 1 1 5 21
Malachi has already packed 21 bags, so he has 63 more bags to pack.

Solve the problems. Show your work.

Leanne bought 20 pounds of potting soil for her new plants. The soil costs $3.17 per
pound. How much did Leanne spend on potting soil?

Thuy has a cookie recipe that calls for 4 1 cups of flour. He wants to quadruple the
3

recipe. How much flour should he use?

Rochelle walked around the block to exercise. Each lap is 1 3 miles. How far did she
5

walk if she walked around the block 4 times?

Spectrum Critical Thinking for Math Lesson 2.1
Grade 7 Multiplying Rational Numbers

22

NAME _________________________________________________________________________________

Lesson 2.2 Proving the Rules for Multiplying Integers

Multiplication is repeated addition. A number line can be used to model integer
multiplication.
2  3 can be modeled by moving 2 units to the right 3 times. (22)  3 can be
modeled by moving 2 units to the left 3 times.

22 22 22 12 12 12

27 26 25 24 23 22 21 0 1 2 3 4 5 6 7
2  3 5 6; –2  (3) = –6

2  (23) can be modeled by graphing the opposite of 2 units to the right 3 times.

12 12 12

27 26 25 24 23 22 21 0 1 2 3 4 5 6 7
23  2 5 6; 2  (–3) = –6

(22)  (23) can be modeled by graphing the opposite of 2 units to the left
3 times.

27 26 25 24 23 22 21 0 1 2 3 4 5 6 7
(22)  (23) 5 6

Answer the questions based on the models above.

If the product of 2 integers is positive, what must be true about the sign of each
factor?

If the product of 2 integers is negative, what must be true about the sign of each
factor?

Spectrum Critical Thinking for Math Lesson 2.2
Grade 7 Proving the Rules for Multiplying Integers

23

NAME _________________________________________________________________________________

Lesson 2.2 Proving the Rules for Multiplying Integers

When multiplying more than 2 factors, the same rules apply. Multiply 2 factors at
a time.

28  1  (22)  4 5 28  (22)  4 5
16  4 5 64

Find the product.
23  2  4 5

23  (22)  (24) 5

21  (22)  (23)  (24) 5

22  (23)  (24)  (25)  (22) 5

Given a numeric expression in the form a  b  c  d  e  …, how can you predict if
the product will be positive or negative before you begin your calculations? Give an
example.

Spectrum Critical Thinking for Math Lesson 2.2
Grade 7 Proving the Rules for Multiplying Integers

24

NAME _________________________________________________________________________________

Lesson 2.3 Dividing Rational Numbers

To divide mixed numbers, rewrite To divide decimals, multiply the
them as improper fractions and then divisor and dividend by a factor of
multiply the reciprocal of the divisor. 10 that will make the divisor a whole
number.

Oscar needs 1h21asin5ch31 eisncohfetswoinfetwfoinre. Cassie has a 4.35-foot piece of
a project. He

How many projects can he make? wood. She needs to cut it into 0.29-

5 1 4 1 1 5 foot pieces. How many pieces can she
3 2
make? 15
16 4 3 5 16  2 5 29qw435
3 2 3 3 Multiply the divisor and
dividend by 100. 2 29
32 5 3 5 145
9 9

He can make 3 5 projects. She can make 15 pieces. 2145
9 0

Solve the problems. Show your work.

Anderson spent $11.76 on some vegetables. How many pounds did he buy if the
cost was $1.47 per pound?

Jen buys a piece of fabric that is 6 7 yards long. She wants to make pillow covers
1 8
2
that require 1 yards of fabric each. How many pillow cases can she make?

Spectrum Critical Thinking for Math Lesson 2.3
Grade 7 Dividing Rational Numbers

25

NAME _________________________________________________________________________________

Lesson 2.4 Dividing Integers

Dividing is the opposite of multiplying.
Rewrite 235 4 7 as 7  __  235.

We know that 25 will finish the equation, because a negative factor times a
positive factor is a negative product.

The rules of integer multiplication also apply to integer division. The quotient
of two integers with the same sign is positive. The quotient of two integers with
different signs is negative.

2144 4 24  26
2144 4 (224)  6

Answer the questions.

A number with an absolute value of 42 was divided by a number with an absolute
value of 7. The quotient is -6. Write two possible numeric equations.

A number with an absolute value of 63 was divided by a number with an absolute
value of 9. The quotient is 7. Write two possible numeric equations.

Complete the equations. ______ 4 (21) 5 231 ______ 4 5 5 213
52 4 ______5 13

238 4______ 5 19 ______ 4 (23) 5 27 54 4 ______ 5 227

Spectrum Critical Thinking for Math Lesson 2.4
Grade 7 Dividing Integers

26

NAME _________________________________________________________________________________

Lesson 2.5 Multiplying and Dividing with Properties

Commutative Property: The order in ab5ba
which numbers are multiplied does not change
the product. (a  b)  c 5 a  (b  c)
Associative Property: The grouping of
factors does not change the product. a  (b 1 c) 5 a  b 1 a  c
a  (b 2 c) 5 a  b 2 a  c
Distributive Property: The multiple of a
sum is the multiple of each addend separately a15a
added together.
Identity Property: The product of a factor a050
and 1 is the factor. 04a50

Properties of Zero: The product of a factor
and 0 is 0. The quotient of the dividend 0 and
any divisor is 0.

Use the given property to evaluate the expression.
Commutative Property: 21.4  5  (210) 5

Distributive Property: 23 1 4 1 5
2 2

Associative & Identity Properties: 2 1  (23  4)  (22) 5
3

Zero Property: (210 1 10) 4 17 5

Spectrum Critical Thinking for Math Lesson 2.5
Grade 7 Multiplying and Dividing with Properties

27

NAME _________________________________________________________________________________

Lesson 2.6 Converting Rational Numbers Using Division

Fractions can be converted to decimals using long division. If a decimal in the
answer is repeating, draw a line over the digits that repeat.

Rewrite 1 as a division problem. .2
5 5qw1.0

1 = 0.2 1.0
5 0

Rewrite 2 as a division problem. .222
9 9qw2.000

2 = 0.2 18
9 20
18
20

Answer the questions. Show your work.

7 of Kiara’s homework is done. Write this as a decimal.
8

Li invited 4 of the 11 people in her math group to the study session. Write the
fraction of students who were not invited as a decimal.

Spectrum Critical Thinking for Math Lesson 2.6
Grade 7 Converting Rational Numbers Using Division

28

NAME _________________________________________________________________________________

Lesson 2.7 Rational Numbers in the Real World

Negative and positive numbers are used to represent how far something is above
or below a point of reference. The point of reference, such as sea level, zero
balance, or target amount, is considered to be zero.

Five and a half feet below sea level 5 25 7
8

$100.25 balance in the bank 5 1100.25

Write and evaluate a numeric expression to represent each situation.

Marcy is tracking the depth of a baby shark in the ocean. The baby shark swims
3 more feet below sea level each day. At this rate, how deep will the shark be
swimming after 5 days?

The construction worker adds 1 of a bucket of concrete mix to the sidewalk 6 times to
4
1
fill the mold. The concrete was too high, so he removes 8 of a bucket 2 times to level

it out. What is the overall amount of concrete used to make the sidewalk?

Spectrum Critical Thinking for Math Lesson 2.7
Grade 7 Rational Numbers in the Real World

29

NAME _________________________________________________________________________________

Check What You Learned

Multiplying and Dividing Rational Numbers

Use the distributive property to find the product.

1. 20  5 7 2. 19  10.25
10

CHAPTER 2 POSTTEST Solve the problem. Show your work.

3. The Apps 'R' Us store charges $1.09 per app. Susan downloaded 2 apps on
Friday and 3 apps on Saturday. How much did she spend?

4. Without multiplying, find the sign of the product. Explain your thinking.
227  14  (210)  (272)  45

5. Use a number line to find the product: 5  (23) =

220 20

Spectrum Critical Thinking for Math Chapter 2
Grade 7 Check What You Learned
30

NAME _________________________________________________________________________________

Check What You Learned

Multiplying and Dividing Rational Numbers

Solve the problems. Show your work.

6. The school club collected toys to donate to young children. Club members

wrapped each coat individually. They used 8 rolls of wrapping paper. Each roll
1
was 47 2 feet long. Each gift box used 9 1 feet of wrapping paper. How many gifts
2

were wrapped?

CHAPTER 2 POSTTEST

7. Anna went to the store and bought 3 items that cost $10.75, $8.90, and $5.10.
What was the average cost of the items?

8. How can the commutative and associative properties be used to make this problem
easier to solve?

2 2  4  (26)  2.25 5
3

Spectrum Critical Thinking for Math Chapter 2
Grade 7 Check What You Learned

31

CHAPTER 2 POSTTEST NAME _________________________________________________________________________________

Check What You Learned

Multiplying and Dividing Rational Numbers
Solve the problems. Show your work.
9. Jayvon collected 9 of the 12 hidden treasures in his video game. Write as a

decimal.

10. The high temperature in Alaska was recorded for 5 straight days in the winter.
The recorded temperatures were 25.5, 22, 1, 0, and 24 degrees. What was
the average temperature during this period of time?

11. During the first quarter of a game, a football team made 3 plays that each
resulted in a loss of 3 yards. The team also made 4 plays that each resulted in a
gain of 2 yards. What was the team’s net gain or loss at the end of the quarter?

Spectrum Critical Thinking for Math Chapter 2
Grade 7 Check What You Learned

32

NAME _________________________________________________________________________________ CHAPTER 3 PRETEST

Check What You Know

Expressions, Equations, and Inequalities
Name the property represented (associative, commutative, or distributive).
1. 3(x 1 y) 5 3x 1 3y _______________
2. 4x 1 2y 5 2y 1 4x _______________
3. (2x 1 y) 1 z 5 2x 1 (y 1 z) _______________

4. Sheldon bought 6 pieces of gum for $0.35 each, 10 pieces of licorice for $0.15
each, and 2 candy bars for $1.25 each. Write and evaluate an expression for the
total amount Sheldon spent on candy.

5. Solve for x: 7x 1 9 5 25

6. Naomi put the same amount of money in the bank each week for 9 weeks. She
took $50 out to go to the fair. She had $143.50 left in the account. How much
was she putting into the account each week?

Spectrum Critical Thinking for Math Chapter 3
Grade 7 Check What You Know

33

CHAPTER 3 PRETEST NAME _________________________________________________________________________________

Check What You Know

Expressions, Equations, and Inequalities
7. What is the difference between solving an equation and an inequality?

8. Solve the inequality: 5x  3  38

Solve the problems. Show your work.

9. Tracy needs less than 12 1 yards of fabric to make costumes for the school play.
3
1
She already has 4 6 yards of fabric. How much more fabric can she buy?

10. Eric bought wants a new computer that costs more than $1,275. His
grandmother gives him $475. He makes $40 for each lawn that he mows. How
many lawns will he have to mow? Write and solve an inequality.

Spectrum Critical Thinking for Math Chapter 3
Grade 7 Check What You Know

34

NAME _________________________________________________________________________________

Lesson 3.1 Properties and Equivalent Expressions

The Commutative, Associative, and Distributive properties can be used to create
equivalent expressions.

7x 1 2 1 5x Original expression
7x 1 (2 1 5x) Associative Property
7x 1 (5x 1 2) Commutative Property
(7x 1 5x) 1 2 Associative Property
x (7 1 5) 1 2 Distributive Property
12x 1 2 Commutative Property

Use the Commutative, Associative, and Distributive Properties to simplify the
expressions.
17x 1 6 1 13x 23

1 (x 212) 1 1 (x 1 8)
4 2

Spectrum Critical Thinking for Math Lesson 3.1
Grade 7 Properties and Equivalent Expressions

35

NAME _________________________________________________________________________________

Lesson 3.2 Creating Expressions to Solve Problems

A rectangle has a length of 5x 1 2 and a width of 3x – 4. What is the perimeter
of the rectangle? Use the properties to simplify the expression.

P 5 2l 1 2w 5 2(5x 1 2) 1 2(3x 2 4)

10x 1 4 1 6x 2 8 Distributive Property
10x 1 (4 1 6x) 2 8 Associative Property
10x 1 (6x 1 4) 2 8 Commutative Property
(10x 1 6x) 1 (4 2 8) Associative Property
x (10 1 6) 1 (4 2 8) Distributive Property

16x 1 (24) 516x 2 4
The perimeter of the rectangle is 16x 24.

Solve the problems.

A jewelry store is having a sale. All necklaces are 25% off. Using number properties,
write two equivalent expressions that can be used to calculate the sales price of any
necklace at the store.

Joan pays her daughter $10 a week plus $5 per chore she completes. She pays
her younger son $7 a week plus $3 for each chore he completes. Using the number
properties, write two equivalent expressions. Assume that each child does the same
number of chores.

Spectrum Critical Thinking for Math Lesson 3.2
Grade 7 Creating Expressions to Solve Problems

36

NAME _________________________________________________________________________________

Lesson 3.3 Using Variables and Expressions

A problem can be solved by writing an expression that is equal to the unknown
variable.

Darryl bought 2 pairs of pants, 3 shirts, and 1 pair of shoes. How much did he
spend if the pants cost $31.25 each, the shirts cost $17.50 each, and the shoes
were $50.75?

s  amount spent
s  2(31.25)  3(17.50)  50.75
s  62.50  52.50  50.75  165.75

Darryl spent $165.75.

Write and simplify expressions to solve the following problems.

Bruno and Mark were shopping for school supplies. Mark bought 3 packs of pencils,
4 packs of paper, and 2 notebooks. Bruno bought 2 packs of pencils, 3 packs
of paper, and 1 notebook. Packs of pencils cost $3.20, paper costs $0.75, and
notebooks cost $4.50. How much did the supplies cost altogether?

How would the amount that Bruno and Mark spent change if the pencils were 25%

off and the paper was 1 off?
3

Spectrum Critical Thinking for Math Lesson 3.3
Grade 7 Using Variables and Expressions

37

NAME _________________________________________________________________________________

Lesson 3.4 Numeric and Algebraic Solutions

Chelsea is driving across the country. The trip is 2,035 miles. She takes 3 days to

drive. The first day, she drove 615 miles. The second day she drove 1 1 times as
3
far. How far did she drive the 3rd day?

Solve working backward: Solve with equation:

Day 1: 2035  615  1420 miles remaining 615  1 1 (615)  x 5 2035
3

Day 2: 1 1 (615)  820 615  820  x 5 2035
3
1435  x 5 2035
1420  820  600 miles
21435 21435
Chelsea drove 600 miles the 3rd day.
x 5 600

Solve each problem working backward. Then, solve with an equation.

A chef adds 2 more cups of cheese to the original amount in a recipe. She doubles
the total amount to 6 cups. What was the original amount given in the recipe?

Five less than 3 times a number is 25. What is the number?

Spectrum Critical Thinking for Math Lesson 3.4
Grade 7 Numeric and Algebraic Solutions

38

NAME _________________________________________________________________________________

Lesson 3.5 Equations in the Real World

Jerri is teaching Jesse how to play a new video game. They play a round against
each other. Jerri’s score is 100 less than 3 times Jessie’s score. Their scores add up
to 1400. Write and solve an equation to find out each of their scores.

x 5 Jesse’s score; 3x 100 5 Jerri’s score 4x 5 1500
x  3x  100 5 1400 4 4
x 5 375
4x  100 5 1400
100 5  100 Jesse scored 375. Jerri scored
4x 51,500 3(375)  100  1,025.

Write and solve an equation for each problem.

Ruth paid $50.25 for a dress. The original price was $67. What was the discount on
the dress?

Sayeed sold magazine subscriptions for the school fundraiser and raised $21.25 in
donations. Robyn sold three-quarters of the number of magazine subscriptions that
Sayeed sold and raised $15.50 in donations. Together, they raised $127.75. How
much did each student make in magazine subscriptions?

Spectrum Critical Thinking for Math Lesson 3.5
Grade 7 Equations in the Real World

39

NAME _________________________________________________________________________________

Lesson 3.6 Using Variables to Express Inequalities

Inequalities have more than one number as a part of the solution. Inequalities can
be solved the same way that equations are solved. If you multiply or divide by a
negative number to solve, the inequality sign needs to be reversed.

Solve the inequality: 3(x  5) 2 1  77. Is 10 a part of the solution?

3(x  5)  1  77 10 is not a part of the solution. The
inequality sign had to change because we
3x  15  1  77 had to divide by a negative number.

3x  14  77

14  14

3x  63

 3x  63
3 3

x  21

Solve each inequality. Show your work.

Is 20 a part of the solution?
3(p  3)  5p  23

Is 100 a part of the solution?
1.25(x  16)  140.75

Spectrum Critical Thinking for Math Lesson 3.6
Grade 7 Using Variables to Express Inequalities

40

NAME _________________________________________________________________________________

Lesson 3.7 Inequalities in the Real World

Niki has saved $132. She earns $12 an hour babysitting. She wants to buy a
tablet that costs no more than $264. How many hours does she have to babysit
to earn enough money?

132  12h  264 Niki will have to work no more than 11
hours to earn enough money.
132 132

12h  132

12h  132
12 12
h  11

Write and solve an inequality for the scenario.

A laser tag arena offers two payment plans for laser tag games. Plan A charges $6
per game plus a one-time membership fee of $35. Plan B offers unlimited games
for a year for a one-time membership fee of $149. What is the minimum number of
games you would have to play in order for the unlimited plan to be the best deal?

Spectrum Critical Thinking for Math Lesson 3.7
Grade 7 Inequalities in the Real World

41

NAME _________________________________________________________________________________

Check What You Learned

Expressions, Equations, and Inequalities

1. Write an equivalent expression using the Commutative, Associative, and
Distributive properties.
8x  2y  4  3y  3

CHAPTER 3 POSTTEST 2. Write two equivalent expressions to represent the perimeter of a triangle that has
2 sides with length 3x  1 and 1 side with length 2x  2.

3. Tia has 4 more than 1 the number of pairs of earrings that Ebony has. Together,
2

they have 25 pairs of earrings. How many pairs of earrings does each girl have?

Spectrum Critical Thinking for Math Chapter 3
Grade 7 Check What You Learned

42

NAME _________________________________________________________________________________

Check What You Learned

Expressions, Equations, and Inequalities
4. Solve for x: 6x  1.5  8.7

5. Is 0 a part of the solution?  1 (x  27)  17  1 CHAPTER 3 POSTTEST
3

6. Rhonda is buying a video game system that costs $325. She also wants to buy
an equal number of strategy games and action games. Strategy games cost $20
each, and action games cost $35 each. How many games can she buy if she
spends no more than $435?

Spectrum Critical Thinking for Math Chapter 3
Grade 7 Check What You Learned

43

NAME _________________________________________________________________________________

CHAPTER 4 PRETEST Check What You Know

Ratios and Proportional Relationships

1. Cheyenne can type 1 of a page of her essay in 1 of an hour. How many pages
2 2

can she type in 1 hour?

2. Graph the values in the table to see if they represent a proportional relationship.

12
11
x248 10
y 3 6 12
9
8
7
6
5
4
3
2
1

0 1 2 3 4 5 6 7 8 9 10 11 12

3. Use the table to find the constant of proportionality.



x 40 80 120
y 30 60 90

Spectrum Critical Thinking for Math Chapter 4
Grade 7 Check What You Know

44

NAME _________________________________________________________________________________ CHAPTER 4 PRETEST

Check What You Know

Ratios and Proportional Relationships
4. Wayne takes 5 steps every time that Jade takes 7 steps. What is the constant of

proportionality? Use it to write an equation.

5. Given the graph, what is the constant of proportionality?

10
9
8
7
6
5
4
3
2
1
0 1 2 3 4 5 6 7 8 9 10

6. Write an equation using the constant of proportionality from #5.

7. Use the equation in #6 to predict the value of y when x  50.

Spectrum Critical Thinking for Math Chapter 4
Grade 7 Check What You Know

45

NAME _________________________________________________________________________________

Lesson 4.1 Comparing Unit Rates

A rate is a special ratio of two values with different units. When one of the values
is 1, it is a unit rate. The two values can be divided to calculate the unit rate.

Carson can read 5 1 pages of his history textbook in 1 of an hour. How many
2 6

pages can he read in 1 hour? 1 1
2 6
5 

11  6
2 1

66  33
2

Carson can read 33 pages in 1 hour.

Solve the problems. Show your work.

Penny is comparing two recipes. One recipe calls for 1 stick of butter for 3 cups of
4 4
1 2
milk. The other recipe calls for 2 stick of butter for 1 3 cups of milk. Which recipe has

more butter per 1 cup of milk?

Fran ran 4 1 miles in 2 of an hour. Fred ran 6 1 miles in 3 of an hour. Who ran the
2 5 2 5

fastest?

Spectrum Critical Thinking for Math Lesson 4.1
Grade 7 Comparing Unit Rates

46

NAME _________________________________________________________________________________

Lesson 4.2 Testing Proportional Relationships

When two quantities have a proportional relationship, this means the ratio of one
quantity to the other quantity is constant. When graphed on a coordinate plane,
the proportional relationship will form a straight line through the origin.

Does this represent a proportional relationship? 13
12
Time (minutes) 2 46 Pages read 11
Pages read 4 8 12 10

The graph is forms a straight line that 9
goes through the origin. It is proportional. 8
7
6
5
4
3
2
1

0 1 2 3 4 5 6 7 8 9 10

Time (minutes)

Graph these relationships to determine if they are proportional.

Number of Pounds 2 3 6 Georgia uses 4 pencils in 2 weeks, 7
pencils in 3 weeks, and 8 pencils in
Cost 2.50 3.75 7.50 4 weeks.

costs 10 10pencils
9 9
8 8
7 7
6 6
5 5
4 4
3 3
2 2
1 1

0 1 2 3 4 5 6 7 8 9 10 0 1 2 3 4 5 6 7 8 9 10

pounds weeks

Spectrum Critical Thinking for Math Lesson 4.2
Grade 7 Testing Proportional Relationships

47

NAME _________________________________________________________________________________

Lesson 4.2 Testing Proportional Relationships

You can also test proportionality by cross-multiplying. If a relationship is
proportional, the cross products will be equal. Use cross products to check the
proportionality.

Is 4.5 , 6.75 proportional?
2 3

4.5  3  2  6.75?
13.5  13.5?

Yes, the relationship is proportional.

Cross-multiply to determine if each relationship is proportional.

Time (hours) 2 3.5 5
Time 116 238 340



Mike is trying to choose a data plan for his phone. 3GB costs $27, 4GB costs $36,
and 7GB costs $63.

Spectrum Critical Thinking for Math Lesson 4.2
Grade 7 Testing Proportional Relationships

48