Spectrum critical thinking for math. Grade 7. 9781483839622, 1483839621

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Spectrum critical thinking for math. Grade 7.
 9781483839622, 1483839621

Table of contents :
705119-EB.pdf
Table of Contents
Chapter 1: Adding and Subtracting Rational Numbers
Check What You Know
Lessons 1-7
Check What You Learned
Chapter 2: Multiplying and Dividing Rational Numbers
Check What You Know
Lessons 1-7
Check What You Learned
Chapter 3: Expressions, Equations, and Inequalities
Check What You Know
Lessons 1-7
Check What You Learned
Chapter 4: Ratios and Proportional Relationships
Check What You Know
Lessons 1-6
Check What You Learned
Chapters 1-4 Mid-Test
Chapter 5: Geometry
Check What You Know
Lessons 1-10
Check What You Learned
Chapter 6: Statistics
Check What You Know
Lessons 1-3
Check What You Learned
Chapter 7: Probability
Check What You Know
Lessons 1-8
Check What You Learned
Chapters 1-7 Final Test
Answer Key

Citation preview

®

Critical Thinking for Math

GRADE 7

CD-705119

Supporting your child’s educational journey every step of the way. Spectrum® provides specific support in the skills and standards that your child is learning in today’s classroom. • Comprehensive, grade-specific titles to prepare for the year ahead



• Subject-specific practice to reinforce classroom learning



• Skill-specific titles to enrich and enhance educational concepts



• Test preparation titles to support test-taking skills

Critical Thinking for Math

7

GRADE

SPECTRUM Critical Thinking for Math



®

No matter your need, Spectrum is with you every step of the way. Spectrum is available in these titles for seventh grade success:

Strategies and Activities to Extend Mathematical Understanding • Positive and negative integers • Ratios and proportions Other titles available:

Algebra

Grades 6–8

Data Analysis & Probability Grades 6–8

Geometry Grades 6–8

• Algebraic equations and inequalities

Measurement Grades 6–8

• Geometric problem-solving • Probability and statistics

U.S. $9.99

705119 CO 7.indd 1

MADE IN THE USA

EAN

carsondellosa.com

• Answer key

carsondellosa.com/spectrum 7/9/19 9:45 AM

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Critical Thinking for Math Grade 7

Published by Spectrum® an imprint of Carson-Dellosa Publishing Greensboro, NC

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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.

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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

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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

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NAME _________________________________________________________________________________ CHAPTER 1 PRETEST

Check What You Know Adding and Subtracting Rational Numbers Compare the values using , , or . 1

1

2. u214.375u ______ u13.75u

1. 227 3 _____ 27 4

Write the additive inverse of each number. 3. 242.6 _____

4. 13.25 _____

Solve the problems. 5. 13.45  0.025  5.7 5

3

1

1

6. 9 4  5 3  6 6 5

Use a number line to complete the following problems. 7. 2

3 4

1

1 (2 2 ) 5 25

0

5

25

0

5

8. 1.75  3 5

Solve the problems. Show your work. 9. 4.35 1.48 5 Spectrum Critical Thinking for Math Grade 7 

4

1

10. 1 5  (3 2 ) 5 Chapter 1 Check What You Know 5

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

1

 hane lost 5 12 pounds. Shonda lost 3 4 pounds. How much more weight did 14. S Shane lose?

Spectrum Critical Thinking for Math Grade 7 

Chapter 1 Check What You Know

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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

23 5

2

1

21 2 3 2 3

2 3

0

2 3

5

1 3

2 3

2 3

1

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.

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

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

Spectrum Critical Thinking for Math  Grade 7 

0

F

Lesson 1.1 Absolute Value & Rational Numbers 7

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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 0

25

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. 1

1

Hermine is making a skirt. She bought 5 4 yards of fabric. She used 3 2 yards to 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 Grade 7 

Lesson 1.2 Additive Inverse

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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. 3

1

Corey bought 4 5 ounces of cashews for $2.30 and 3 10 ounces of pistachios for $1.24. What is the total amount of nuts that he bought? What was the total amount of money he spent? 3

1

6

1

4 5 1 310 5 4 10 1 3 10 5 7

710 ounces of nuts

2.30  1.24 $ 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  Grade 7 

Lesson 1.3 Adding & Subtracting Rational Numbers 9

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Lesson 1.4 Adding Positive and Negative Numbers The sum of 2 positive numbers is a positive 2 2 1 23 5 25 23 number farther to the right of the first addend on the number line. 25 24 23 22 21 0 1

21355 13

2

3

4

5

2

3

4

2

3

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 number will be positive if the positive number has a greater absolute value. The sum of a positive and a negative number will be negative if the negative number has a greater absolute value.

221351 13

24 23 22 21

0

1

2 3 1 2 5 21 12

25 24 23 22 21

0

1

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 Grade 7 

Lesson 1.4 Adding Positive and Negative Numbers

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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. 3

2

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

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. 1

1

3 2  (1 2 ) 5 5

0

5

5

0

5

5

0

5

2.5 1 (0.5 ) 5

1

3

 4 1 (4 4 ) 5

Spectrum Critical Thinking for Math  Grade 7 

Lesson 1.4 Adding Positive and Negative Numbers 11

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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 Grade 7 

Lesson 1.5 Subtracting Positive and Negative Numbers

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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 2

)5417

1 2

5 11

1 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

3

250

240 230 220 210

0

10

1

Evaluate: 23 8 2 63 8 5

Evaluate: 217.56 2 13.43 5

Spectrum Critical Thinking for Math  Grade 7 

Lesson 1.5 Subtracting Positive and Negative Numbers 13

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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) Associative Property: (a 1 b) 1 c 5 a 1 (b 1 c)

21 1 (25) 1 (27) 1 13 1 11 {21 1 (25) 1 (27) 1 13} 1 11 (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 2

1 (23

1 3

)1

1 2

1

2 3

12

2 3

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

Spectrum Critical Thinking for Math Grade 7 

Lesson 1.6 Adding with Mathematical Properties

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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 Picked Sold Picked Sold Picked Sold Picked Sold

Weight (pounds) 4.1 19.8 21.6 17.7 18.1 22.4 12.9 15.3 23.9

Spectrum Critical Thinking for Math  Grade 7 

Lesson 1.7 Rational Numbers in the Real World 15

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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) 1

Beginning amount

1 10

Rain

4 5

Evaporation

3 10

Rain

1 2

Evaporation

1 5

Rain Evaporation

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 Grade 7 

Lesson 1.7 Rational Numbers in the Real World

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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. 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

Stock Y

1$15

2$22.60

1$18.90

2$14.25

1$7.25

CHAPTER 1 POSTTEST

Day

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 Grade 7 

Chapter 1 Check What You Learned 17

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Check What You Learned Adding and Subtracting Rational Numbers Solve the problems. Show your work.

CHAPTER 1 POSTTEST

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.

1

3

6. Jacob finds a piece of metal that is 2 5 inches thick. He files off 10 inch. He adds 2 inch thick. Then, he polishes the metal a protective covering to the metal that is 10 1 and removes an additional 10 inch. What is the final thickness of the metal?

1

7. Raquel planted an herb garden in her back yard. This year, she planted 2 of her garden with chives, 14 with oregano, and 18 with parsley. What portion of the garden is still unplanted?

Spectrum Critical Thinking for Math Grade 7

Chapter 1 Check What You Learned

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Check What You Know 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

1

1

5. 10 5 4 5 10

6. 194.75 4 10.25 5

7. 49 4 (27) 5

Spectrum Critical Thinking for Math Grade 7 

Chapter 2 Check What You Know 19

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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 6

as a decimal.

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.

1

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

Spectrum Critical Thinking for Math Grade 7

Chapter 2 Check What You Know

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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 3

2 3

)

2  4.75 5 2(4 1 0.75)

561

4 3

2  4 1 2  0.75 =

5 2(3 1

2312

2 3

1

1

5 6 1 13 5 73

8 1 1.5 5 9.5

Use the distributive property to find the product. 45

1 5

5

10  17.135 5

7  10

2 7

5

8  5.125 5

Spectrum Critical Thinking for Math  Grade 7 

Lesson 2.1 Multiplying Rational Numbers 21

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Lesson 2.1 Multiplying Rational Numbers Malachi has to pack 84 toiletry bags for the homeless shelter. He has packed the boxes. How many bags are left for him to pack? 1 4

 84 5

1 4

 (80 1 4) 5

1 4

 80 1

1 4

1 4

of

45

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?

1

Thuy has a cookie recipe that calls for 4 3 cups of flour. He wants to quadruple the recipe. How much flour should he use?

3

Rochelle walked around the block to exercise. Each lap is 1 5 miles. How far did she walk if she walked around the block 4 times?

Spectrum Critical Thinking for Math  Grade 7 

Lesson 2.1 Multiplying Rational Numbers

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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 (22)  (23) 5 6

4 5 6 7

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  Grade 7 

Lesson 2.2 Proving the Rules for Multiplying Integers 23

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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  Grade 7 

Lesson 2.2 Proving the Rules for Multiplying Integers

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Lesson 2.3 Dividing Rational Numbers To divide mixed numbers, rewrite them as improper fractions and then multiply the reciprocal of the divisor. Oscar needs 1 12 inches of twine for 1 a project. He has 5 3 inches of twine. How many projects can he make? 1

1

53 4 12 5 16 3

4

3 2 32 9

5

16 3



2 3

5

5 3 59

5

He can make 3 9 projects.

To divide decimals, multiply the divisor and dividend by a factor of 10 that will make the divisor a whole number. Cassie has a 4.35-foot piece of wood. She needs to cut it into 0.29foot pieces. How many pieces can she make? 15 29qw 435 Multiply the divisor and 2 29 dividend by 100. 145 She can make 15 pieces. 2145 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?

7

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

Spectrum Critical Thinking for Math  Grade 7 

Lesson 2.3 Dividing Rational Numbers 25

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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. 52 4 ______5 13

______ 4 (21) 5 231

______ 4 5 5 213

238 4______ 5 19

______ 4 (23) 5 27

54 4 ______ 5 227

Spectrum Critical Thinking for Math Grade 7 

Lesson 2.4 Dividing Integers

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Lesson 2.5 Multiplying and Dividing with Properties Commutative Property: The order in which numbers are multiplied does not change the product. Associative Property: The grouping of factors does not change the product. Distributive Property: The multiple of a sum is the multiple of each addend separately added together. Identity Property: The product of a factor and 1 is the factor.

ab5ba (a  b)  c 5 a  (b  c)

a  (b 1 c) 5 a  b 1 a  c a  (b 2 c) 5 a  b 2 a  c a15a

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

a050 04a50

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

1

Distributive Property: 23 2 4

1 2

5

1

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

Zero Property: (210 1 10) 4 17 5

Spectrum Critical Thinking for Math  Grade 7 

Lesson 2.5 Multiplying and Dividing with Properties 27

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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 5

as a division problem.

2 9

as a division problem.

.2 5qw 1.0 1.0 0

= 0.2

Rewrite 2 9

1 5

= 0.2

.222 9qw 2.000 18 20 18 20

Answer the questions. Show your work. 7 8

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

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 Grade 7 

Lesson 2.6 Converting Rational Numbers Using Division

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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. 7

Five and a half feet below sea level 5 25 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 14 of a bucket of concrete mix to the sidewalk 6 times to fill the mold. The concrete was too high, so he removes 18 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  Grade 7 

Lesson 2.7 Rational Numbers in the Real World 29

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Check What You Learned Multiplying and Dividing Rational Numbers Use the distributive property to find the product.

CHAPTER 2 POSTTEST

1. 20  5

7 10

2. 19  10.25

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 Spectrum Critical Thinking for Math Grade 7

20 Chapter 2 Check What You Learned

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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 12 feet of wrapping paper. How many gifts 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 3  4  (26)  2.25 5

Spectrum Critical Thinking for Math Grade 7 

Chapter 2 Check What You Learned 31

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Check What You Learned Multiplying and Dividing Rational Numbers Solve the problems. Show your work.

CHAPTER 2 POSTTEST

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 Grade 7

Chapter 2 Check What You Learned

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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 Grade 7 

Chapter 3 Check What You Know 33

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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. 1

9. Tracy needs less than 12 3 yards of fabric to make costumes for the school play. 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 Grade 7 

Chapter 3 Check What You Know

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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 4 (x

212) 1

1 2

(x 1 8)

Spectrum Critical Thinking for Math  Grade 7

Lesson 3.1 Properties and Equivalent Expressions 35

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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 10x 1 (4 1 6x) 2 8 10x 1 (6x 1 4) 2 8 (10x 1 6x) 1 (4 2 8) x (10 1 6) 1 (4 2 8)

Distributive Property Associative Property Commutative Property Associative Property 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  Grade 7

Lesson 3.2 Creating Expressions to Solve Problems

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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 13 off?

Spectrum Critical Thinking for Math  Grade 7

Lesson 3.3 Using Variables and Expressions 37

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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 13 times as far. How far did she drive the 3rd day? Solve with equation: Solve working backward: 615  1 13 (615)  x 5 2035 Day 1: 2035  615  1420 miles remaining 1 615  820  x 5 2035 Day 2: 1 3 (615)  820 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  Grade 7

Lesson 3.4 Numeric and Algebraic Solutions

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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 x  3x  100 5 1400 4x  100 5 1400 100 5  100 51,500 4x

5 1500 4 x 5 375

4x 4

Jesse scored 375. Jerri scored 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  Grade 7

Lesson 3.5 Equations in the Real World 39

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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 3x  15  1  77 3x  14  77 14  14 3x  63 3x 63  3  3 x  21

10 is not a part of the solution. The inequality sign had to change because we had to divide by a negative number.

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 Grade 7

Lesson 3.6 Using Variables to Express Inequalities

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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 132 132 12h  132 12h  132 12 12 h  11

Niki will have to work no more than 11 hours to earn enough money.

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  Grade 7

Lesson 3.7 Inequalities in the Real World 41

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Check What You Learned Expressions, Equations, and Inequalities

CHAPTER 3 POSTTEST

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

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 12 the number of pairs of earrings that Ebony has. Together, they have 25 pairs of earrings. How many pairs of earrings does each girl have?

Spectrum Critical Thinking for Math Grade 7

Chapter 3 Check What You Learned

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

CHAPTER 3 POSTTEST

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

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 Grade 7 

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CHAPTER 4 PRETEST

NAME _________________________________________________________________________________

Check What You Know Ratios and Proportional Relationships 1. Cheyenne can type 12 of a page of her essay in can she type in 1 hour?

1 2

of an hour. How many pages

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

x y

2 3

4 6

8 12

12 11 10 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 y

40 30

80 60

120 90

Spectrum Critical Thinking for Math Grade 7

Chapter 4 Check What You Know

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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 Grade 7 

Chapter 4 Check What You Know 45

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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. 1

Carson can read 5 2 pages of his history textbook in pages can he read in 1 hour? 1 1 52  6 6 11 2  1 66  33 2

1 6

of an hour. How many

Carson can read 33 pages in 1 hour. Solve the problems. Show your work. Penny is comparing two recipes. One recipe calls for 14 stick of butter for 34 cups of 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?

1 Fran ran 4 2 miles in fastest?

2 5

of an hour. Fred ran 6 12 miles in

Spectrum Critical Thinking for Math Grade 7 

3 5

of an hour. Who ran the

Lesson 4.1 Comparing Unit Rates

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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? 2 4

4 8

6 12

Pages read

Time (minutes) Pages read

The graph is forms a straight line that goes through the origin. It is proportional.

13 12 11 10 9 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. 2 3 6 2.50 3.75 7.50

10 9 8 7 6 5 4 3 2 1 0

Georgia uses 4 pencils in 2 weeks, 7 pencils in 3 weeks, and 8 pencils in 4 weeks.

pencils

costs

Number of Pounds Cost

1 2 3 4 5 6 7 8 9 10

pounds

Spectrum Critical Thinking for Math  Grade 7 

10 9 8 7 6 5 4 3 2 1 0

1 2 3 4 5 6 7 8 9 10

weeks

Lesson 4.2 Testing Proportional Relationships 47

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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 2

,

6.75 3

proportional?

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) Time

2 116

3.5 238

5 340



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 Grade 7

Lesson 4.2 Testing Proportional Relationships

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Lesson 4.3 Constants of Proportionality A unit rate can also be called a constant of proportionality. The constant of proportionality, k, is the ratio of the output variable to the input variable. Days of Car Rental Cost of Car Rental

k

output input ;

3 96.75

k  96.75  32.25; 3

161.25 5

5 161.25

 32.25;

6 193.50 193.25 5

 32.25

The constant of proportionality is 32.25. The cost of renting a car is $32.25 per day. Use the information to calculate the constant of proportionality for each table. What does the constant of proportionality mean in the context of the data given? Gallons of Gas Price

3 6.54

8 17.44

15 32.70

Time (hours) Distance Biked (miles)

1.75

0.25

2.5

21

3

30



Spectrum Critical Thinking for Math  Grade 7



Lesson 4.3 Constants of Proportionality 49

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Lesson 4.4 Using Equations to Represent Proportions The constant of proportionality can be used to write an equation to represent the relationship. A recipe calls for 12 cup of sugar for every 1 13 cups of flour. Write an equation to calculate how much flour to use for each cup of sugar. How many cups of flour 1 will be used if 2 2 cups of sugar are used?

k  cups of 1

flour sugar

1



13

f  (2 23 )s f  (2 23 ) (2 2 )  ( 83 ) (

1 2 5 2 )

 

40 6

8 3

 2 23

 6 23 cups of flour

Write an equation using the constant of proportionality to represent each relationship described. It takes 1 12 gallon of gas to mow mow 2 12 acres of grass?

There are 880 feet in

1 6

1 2

acre of grass. How much gas does it take to

of a mile. How many feet are in 1 13 miles?

Spectrum Critical Thinking for Math  Grade 7

Lesson 4.4 Using Equations to Represent Proportions

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Lesson 4.5 Proportions on the Coordinate Plane A constant of proportionality can be found using the graph of a proportional relationship. Identify an ordered pair (x,y) on the line. The constant of proportionality is k  y/x.



k

0.50 5

Value ($)

Given the graph, calculate the constant of proportionality. What does the constant of proportionality represent on the graph?  0.10

0.90 0.70 0.50 0.30 0.10 0

The constant of proportionality is 0.10. It represents the value of each dime: $0.10.

1 2 3 4 5 6 7 8

# of dimes

Calculate the constant of proportionality for each graph. What does the constant of proportionality represent? 12 2.25 1.75 1.25 0.75 0.25 0

1 2 3 4 5 6 7

11 10 9 8 7 6 5 4 3 2 1

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

How much would 7 tokens be worth?

Spectrum Critical Thinking for Math Grade 7

How many blue marbles are there if there are 3 red marbles?

Lesson 4.5 Proportions on the Coordinate Plane 51

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Lesson 4.6 Proportions in the Real World Answer the questions. Show your work. Juice is sold at the grocery store in several different sizes. The prices are shown in the table. Size (fl. oz.) 16 32 64 128

Price ($) 1.89 3.49 7.59 9.99

Unit Price ($)

a. Complete the table with the unit prices of each size. Round to the nearest hundredth.

b. If you wanted 64 fluid ounces of juice, which would be the best way to purchase it?

Amount of Bill

19.00

35.00

72.00

Tip

3.42

6.30

12.96

Calculate the constant of proportionality shown in the table above. What does this constant mean within the context of the data given? How much would the tip be if the bill was $95.00?

Spectrum Critical Thinking for Math  Grade 7

Lesson 4.6 Proportions in the Real World

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Lesson 4.6 Proportions in the Real World Answer the questions. Show your work. One half of a can of paint covers 150 square feet of a wall. a. Create a table to represent this relationship.

area covered (square feet)

b. Create a graph to represent this relationship 100 900 800 700 600 500 400 300 200 100 0

1 2 3 4 5 6 7 8 9 10

cans of paint c. What is the constant of proportionality? What does it represent?

d. Write an equation to represent the relationship between cans of paint used and how much of the wall is covered. How much of the wall can be covered by 2 12 cans of paint?

Spectrum Critical Thinking for Math Grade 7

Lesson 4.6 Proportions in the Real World 53

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Check What You Learned Ratios and Proportional Relationships 3 5

CHAPTER 4 POSTTEST

1. Pool A is being filled with 23 gallon per 14 minute. Pool B is being filled with gallon per 35 minute. Which pool is being filled faster?

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

1 2

Amount Painted (square feet)

28

1

1 2

84

2 112

120 110 100 90 80 70 60 50 40 30 20 10 1 2 3 4 5 6 7 8 9 10 Spectrum Critical Thinking for Math Grade 7

Chapter 4 Check What You Learned

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Check What You Learned Ratios and Proportional Relationships 3. Use the table to find the constant of proportionality. Miles Walked

1 4

Calories Burned

25

1

3 4

3

175

1 2

350 CHAPTER 4 POSTTEST

5. Use the graph to create a table of values. Find the constant of proportionality and write an equation. What does it mean within the context of the graph? How much would the pay be after 25 hours of work?

Spectrum Critical Thinking for Math Grade 7 

pay ($)

4. Write an equation using the constant of proportionality in #3 on the previous 1 page. How many calories would you expect to burn if you walk 3 5 miles?

160 140 120 100 80 60 40 20 10 0

2

6

10

14

18

hours worked (h)

Chapter 4 Check What You Learned 55

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Mid-Test

Chapters 1–4

1. Yuri had a bank balance of $57 before he went shopping. After he used his debit card twice, his account was overdrawn by $14. What was the total amount of the debits?

2. Astrid played a board game in math class. She ended up with these cards: 3.9 4.2 3.9 4.1 6.8 8.5 10.8

CHAPTERS 1–4 MID-TEST

a. The score is the sum of the card values. What was her score? Show your work.

b. In order to win the game, the absolute value of a score must be less than 12. Is it possible for Astrid to win? Explain why or why not.

3. A number j is positive and another number k is negative. Based on this information, can you determine whether j  k is positive or negative? Explain.

Spectrum Critical Thinking for Math Grade 7

Chapters 1–4 Mid-Test

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Mid-Test

Chapters 1–4

4. A mini-shelf in the food pantry can hold 3 34 pounds. If a can weighs 38 of a pound, how many cans can it hold? If you add more support so the shelf can hold 5 14 pounds, how many cans can the shelf hold now? Show your work.

5. Mary Ellen says that the expression 5  7  8(4  2) simplifies to a negative number because if you multiply three negative numbers, the final answer will be negative. Is she correct? Show why or why not.

Spectrum Critical Thinking for Math Grade 7

CHAPTERS 1–4 MID-TEST

6. A number is multiplied by  34 , divided by 12 , and then divided by  78 . The resulting number is 96. Work backward to get the original number by performing opposite operations.

Chapters 1–4 Mid-Test 57

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Mid-Test

Chapters 1–4

7. Determine whether the expression 2(x  3) is equal to (4x  Identify the properties you used in your solution steps.

1 2

)  (8x  5

1 2

).

CHAPTERS 1–4 MID-TEST

8. A plumber charges $110 for a service call and $65 for each hour of work after the first hour. Let h represent the hours the electrician works on a service call. Write an expression to represent the cost. How many hours did it take the plumber to complete a job if the total cost is $240?

9. Candace has $65.25. She spent $31.50 on a new throw rug for her bedroom. She wants to buy some matching throw pillows. How many pillows can she buy if the pillows cost $11.25 each?

Spectrum Critical Thinking for Math Grade 7

Chapters 1–4 Mid-Test

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Mid-Test

Chapters 1–4

10. A museum is keeping track of the number of visitors per day.



15,000 14,000 13,000 12,000 11,000 10,000 9,000 8,000 7,000 6,000 5,000 4,000 3,000 2,000 1,000 0

1 2 3 4 5 6 7 8 9 1011121314

a. Create a table that represent the proportional relationship.

c. Write an equation to represent the relationship.

CHAPTERS 1–4 MID-TEST

b. What is the constant of proportionality to the nearest hundredth? What does it represent on the graph?

d. Predict how many people will have visited the museum after 33 days.

Spectrum Critical Thinking for Math Grade 7

Chapters 1–4 Mid-Test 59

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CHAPTER 5 PRETEST

NAME _________________________________________________________________________________

Check What You Know Geometry 1. Find the length of the missing side for the pair of similar triangles. 12 ft .

14 ft .

18 ft .

18 ft . 27 ft .

2. Can these lengths form a triangle? Side 1: 9 cm Side 2: 5 cm Side 3: 11 cm

3. Name the shape that is created by each cross section.

Spectrum Critical Thinking for Math Grade 7

Chapter 5 Check What You Know

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Check What You Know Geometry 4. Find the circumference and area of the circle. Use 3.14 for . Round answers to the nearest thousandth, if necessary. 4.3 m

A  _______ square feet C 5 _______ feet

5. If AGB is 120 degrees, what is the measure of HGE?

A H

B G



C E F

HGE  _______ degrees

D

6. What is the volume of a rectangular prism with a length of 10mm, a width of 8mm, and a height of 5mm?

7. What is the combined area of a rectangle with a length of 13.2 in. and a width of 4.1 in., and a triangle with a base of 13.2 in. and a height of 6.5 in.?

Spectrum Critical Thinking for Math Grade 7 

Chapter 5 Check What You Know 61

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Lesson 5.1 Scale Drawings Scale drawings are used to represent an object. Scale drawings can be smaller, larger, or the same size as the original object. The scale factor shows the proportional relationship between the original object and the scale drawing. Benita makes a scale drawing so she can rearrange her room. Her actual room is 12 feet by 14 feet. Her drawing is 6 inches by 7 inches. What is the scale factor? She wants to draw her bed in a new spot. If her bed is 4 feet by 8 feet, what size should she draw it on her diagram? 6 in. 0.5 in. scale factor  inches feet  12 f t .  1 ft . The scale factor can also be written as 0.5 in.:1 ft.

4 ft. 

0.5 in. 1 f t.

 2 inches

8 ft. 

0.5 in. 1 f t.

 4 inches

7 in.

6 in.

The bed should be 2 inches by 4 inches on the diagram. Solve the problem. Show your work. The scale in the drawing is 2 cm:5 m (2 cm  5 m). Find the length and width of the actual room. What is the area? What is the perimeter? 20 cm

5 cm

Spectrum Critical Thinking for Math  Grade 7

Lesson 5.1 Scale Drawings

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Lesson 5.2 Forming Triangles The sum of the lengths of two sides of a triangle must be greater than the length of the third side. Jorge is planning to build a plant box in the shape of a triangle. He has 3 planks of wood that are 4 feet, 6 feet, and 3 feet long. Will he be able to build a rectangular plant box? 463 634 346 Jorge will be able to build a triangular box using these three planks because each sum of two sides is greater than the length of the remaining side. Don has 3 straws with lengths of 3 cm, 4 cm, and 9 cm. He is trying to make a triangle with the straws. Will he be successful? If not, which straw should he replace? What is the minimum length of the replacement?

There are 3 line segments with lengths x  2, x 2 3,and x  1. What is the minimum value of x that allows these line segments to form a triangle? Assume that x is an integer.

Spectrum Critical Thinking for Math Grade 7

Lesson 5.2 Forming Triangles 63

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Lesson 5.3 Cross Sections of 3-Dimensional Figures A cross section is the intersection of a 3-dimensional figure and a plane. Here are some examples:

Intersect each 3-D figure with the given 2-D shape. A rectangle

A square

A quadrilateral

A triangle

Spectrum Critical Thinking for Math  Grade 7

Lesson 5.3 Cross Sections of 3-Dimensional Figures

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Lesson 5.4 Circles: Circumference The perimeter of a circle is called the circumference.

C  2r (r  radius) or C  d (d  diameter), where   3.14 A plate has a diameter of 10 inches. What is the circumference of the plate?

C  3.14(10)  31.4 inches Answer the questions. Show your work. Use 3.14 for p. In college basketball rules, the ball can have a maximum circumference of 30 inches. What is the maximum diameter of a basketball to the nearest hundredth?

A round stained-glass window has a circumference of 195 inches. What is the radius of the window to the nearest inch?

Spectrum Critical Thinking for Math Grade 7

Lesson 5.4 Circles: Circumference 65

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Lesson 5.5 Circles: Area The area of a circle is A  r2, where   3.14. A circular pool has a radius of 10 feet. What is the area of the pool? A  3.14(10)2 A  314 square feet Answer the questions. Show your work. Use 3.14 for p. A playground area is circular with a diameter of 32 feet. What is the area of the playground? Round your answer to the nearest tenth.

A frying pan has a diameter of 11 inches. What is the area to the nearest square inch of the smallest cover that will fit on top of the frying pan?

Justin just got his driver’s license. His parents are giving him permission to drive within a 25-mile radius of his home. What is the area Justin is restricted to when driving? Round your answer to the nearest tenth.

Spectrum Critical Thinking for Math  Grade 7

Lesson 5.5 Circles: Area

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Lesson 5.6 Angle Relationships When two lines intersect, they form angles that have special relationships. • Vertical angles have the same measure. • Supplementary angle are two angles with the sum of 180 degrees • Complimentary angles are two angles with the sum of 90 degrees. 828

p

What is the value of p? The angles are supplementary, so the sum is 180.



p  82  180 82 82 p  98 degrees

If 4 is a right angle and 5 measures 40 degrees, find the measures of the remaining angles.

G L

5

4I

H

3 1

2

J

K

Spectrum Critical Thinking for Math Grade 7

Lesson 5.6 Angle Relationships 67

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Lesson 5.7 Area of Composite Shapes Shapes that are composed of other shapes are called composite shapes The area of a composite shape is equal to the sum of each shape it is made of. Find the area of the composite shape:

3m

11.4 m

Area = area of rectangle + area of semicircle area of rectangle  lw  (11.4)(3)  34.2 m2 Area of semicircle  12 r2  34.2  3.53  37.73 m2

1 2

(3.14) ( 32 )2  3.53 m2

Find the area of the composite shapes. Show your work. Use 3.14 for p. Round answers to the nearest hundredth.

5 ft.

3 ft.

3.1 cm 7.4 cm

4.2 cm

Spectrum Critical Thinking for Math  Grade 7

Lesson 5.7 Area of Composite Shapes

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Lesson 5.8 Volume of Rectangular Prisms Volume is the amount of space an object occupies. The volume of a rectangular prism can be calculated using the formula V  bh, where b  area of the base and h  height. The area of the base is b  lw, where l  length and w  width. The volume of a rectangular prism is 210 cm3. If it has a length of 5 cm and a width of 6 cm, what is the height?

V  210 cm3 210  (5)(6)h 210  30h h  70 cm Answer the questions. Show your work. Penny is using colored sand to fill a jar that is shaped like a rectangular prism. The bag of sand contains 150 cubic inches. The base of the prism is 6.5 inches by 7.4 inches. The height of the box is 2.2 inches. Will all the sand fit in the jar?

A rectangular prism has a volume of 966 ft3. The prism’s height is 4 feet, and its length is 14 feet. What is its width?

Spectrum Critical Thinking for Math Grade 7

Lesson 5.8 Volume of Rectangular Prisms 69

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Lesson 5.9 Volume of Triangular Prisms A triangular prism is a prism whose base is a triangle. The volume of a triangular prism is the product of the area of the base and the height of the prism. Volume  bh, where b  12 bh The triangular base has a height of 3 cm and a base of 8 cm. The height of the prism is 12 cm.



b  12 (8)(3)  12 (24)  12 cm2 V  (12)(12)  144 cm3

3 cm

12 cm

8 cm

Find the volume of each figure. Show your work.

6 cm

12

cm

4 cm

4 cm

4 cm 12.5 cm 3 cm

Spectrum Critical Thinking for Math Grade 7

Lesson 5.9 Volume of Triangular Prisms

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Check What You Learned Geometry 1. At the photo lab, a customer brings in a photograph that is 4 inches wide by 6 inches high. The customer wants the photograph enlarged to 20 inches wide by 25 inches high. Can this be done? Explain your reasoning.

CHAPTER 5 POSTTEST

2. If a triangle XYZ has two sides with lengths of 5 cm and 8 cm, what is the maximum and minimum length of the third side? Explain your answer. Assume that the length of the third side is an integer.

3. What are two possible shapes that can be formed by a cross section of this shape? Describe the angle of the cross section.

Spectrum Critical Thinking for Math Grade 7 

Chapter 5 Check What You Learned 71

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Check What You Learned Geometry

CHAPTER 5 POSTTEST

4. A mini pancake has a circumference of 3 centimeters. A regular pancake has a circumference of 6 centimeters. Is the area of the regular pancake twice the area of the mini pancake? Use 3.14 for .

5. If 4 is a right angle and 5  40°, find the measure of the remaining angles. C

5

A

1

B

4 2 3

D

E

F

Spectrum Critical Thinking for Math Grade 7 

Chapter 5 Check What You Learned

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Check What You Learned Geometry 6. Find the volume of the figure. 2 mm

CHAPTER 5 POSTTEST

10 mm

5 mm 2 mm 2 mm

9 mm

7. Bill wants to fill this triangular prism need?

2 3

full of water. How much water does he

20 m 16 m 12 m

Spectrum Critical Thinking for Math Grade 7 

Chapter 5 Check What You Learned 73

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CHAPTER 6 PRETEST

NAME _________________________________________________________________________________

Check What You Know Statistics 1. Are the following samples biased or random? Explain your answer. a. Wendy wants to find out the favorite sports of the students at her school. She asks 25 students who were at the basketball team tryouts.

b. Jake wants to know how many students are interested in buying a yearbook this year. He used a random number generator to randomly select 25 students from each grade level.

2. The graph represents a sample of football players’ heights. If there were 100 players, how many players could be expected to be 70 inches tall?  

Number of Players

Football Players’ Heights (in.) 3

2

1

0

62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78

Heights (in.)

Spectrum Critical Thinking for Math Grade 7

Chapter 6 Check What You Know

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Check What You Know Statistics 3. A sample of people were asked how far they drive to work. What percentage of people drive 6 miles to work? Round your answer to the nearest tenth of a percent.

1

2

3

4

5

6

7

8

9

10

Distance to Work from Home

4. What can you infer from this data collected about the number of apps on a sample of smart phones?

1

2 3 4

5 6 7

8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25

5. A factory produces 92,000 tubes of toothpaste each day. The quality manager claims that fewer than 750 defective tubes are produced each day. In a random sample of 420 tubes of toothpaste, 3 are defective. Is the quality manager’s claim correct? Explain your answer.

Spectrum Critical Thinking for Math Grade 7 

Chapter 6 Check What You Know 75

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Lesson 6.1 Sampling and Drawing Inferences When a population has a large number of data points, a sample can be taken to help summarize information and make inferences about the entire population. A random sample has individuals who are chosen by chance, and each member of the population has an equal chance of being included. In a biased sample, some members of the population are less likely to be chosen. Samples that are random are better predictors of trends for the bigger population. Rosewood Middle School has 714 students. Susan surveys a random sample of 34 students and finds that 9 of them play a sport outside of school. How many students at the school are likely to play a sport outside of school? 9  s ; 34s  (9) (714)  6426 34 714 34s  6426; s  189 34 34 189 students are likely to play sports outside of the school. Answer the questions. Show your work. A high-tech company makes 3,500 widgets a day. The quality department chooses a random sample of 50 widgets and finds that 3 are defective. How many high tech widgets per day are likely to be defective?

Grace hears that the average gas price has risen to $2.89 during the gas shortage. She checks gas prices at stations near her school, and finds that the average is $3.20. Why are the averages different?

Spectrum Critical Thinking for Math  Grade 7

Lesson 6.1 Sampling and Drawing Inferences

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Lesson 6.2 Comparing Similar Data Sets What can you infer from the two histograms? In class 1, no students were shorter than 34 inches or taller than 46 inches. In class 2, the range of heights is 25 inches, but the range in class 1 is just 12 inches. The median for both classes is 42 inches. 50% of the students in class 1 are between 42 and 46 inches. 12-

10

84

40-

6-

6

62-

8-

0 30 34

0 38

42

46

6

5

4

4

4-

Frequency

Frequency

10-

1

20-

50

30

35

40

45

50

55

Heights of Class #2

Heights of Class #1

5-

Frequency

3-

4

4 3

8-

3

210-

7

6-

0

0

150 200 250 300 350 400 450

Weights of Chickens: Soybean Diet (dkg)

Frequency

4-

4-

2

20-

0

1

1

1

150 200 250 300 350 400 450

Weights of Chickens: Sunflower Diet (dkg)

In which range will the median occur for each diet?

What percentage of the chickens are between 300 dkg and 350 dkg for each type of feed? Round your answers to the nearest percent.

Spectrum Critical Thinking for Math Grade 7

Lesson 6.2 Comparing Similar Data Sets 77

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Lesson 6.2 Comparing Similar Data Sets Class 1: Number of People in Household Class 2: Number of People in Household

8, 2, 5, 5, 3, 1, 6, 2, 4, 4 3, 5, 4, 4, 5, 4, 3, 2, 4, 4

Find the mean, median, and mode of each set of data. How do the data sets compare? Class 1 Mean: 4; Median: 4 Mode: 2, 4, 5; Range: 7 This data is spread out fairly evenly between 1 and 8. There are a variety of household sizes in this class.

Class 2 Mean: 3.8; Median: 4 Mode: 4; Range: 3 This data is more compact and closer to the center. There is less variety in sizes. Four is the most common size.

Find the mean, median, and mode of each set of data. How do the data sets compare? Class 1: Teacher Donations to Charity Fund Class 2: Student Donations to Charity Fund

Spectrum Critical Thinking for Math  Grade 7

2 3 7 8 10 11 12 14 15 20 17 20 14 12 11 12 20 20 20 20 1 2 5 7 8 9 1 11 14 15 17 19 19 17 11 8 2 2 11

Lesson 6.2 Comparing Similar Data Sets

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Lesson 6.2 Comparing Similar Data Sets Box-and-whisker plots can help you interpret the distribution of data. Each section of a box and whisker plot contains 25% of the data points. Active time data is collected from a group of high school students and a group of elementary students. High School Students Elementary Students 50

52

54

56

58

60

62

64

66

68

70

72

74

76

78

The double box and whisker plot shows that the high school students are overall less active with a median of 59 minutes a day. The middle 50% of students sampled are active between 54 and 63 minutes a day. The elementary students are more active. The median is 64 minutes a day. The middle 50% exercise between 56 and 72 minutes. This double box-and-whisker plot displays the test scores of students who studied alone and the scores of students who studied with a study group. Use it to compare the data sets. w/o study group with study group 10

15

20

25

30

35

40

45

Spectrum Critical Thinking for Math Grade 7

50

55

60

65

70

75

80

85

90

95

100

Lesson 6.2 Comparing Similar Data Sets 79

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Lesson 6.3 Data in the Real World Tameka is planning a party for her brother. She invites 180 of the people in her brother’s class. When she sent out invitations, she listed the wrong phone number for RSVP, so she will not be getting any responses. She is trying to figure out how many people are planning to come to the party. a. T ameka decides to ask the 20 students who live in her neighborhood. 12 of them say that they will be able to come to the party. What is the population in this event? Is this a random sample? Could this sample be biased? Are these results too low or too high? Explain.

b. T ameka decides to look at the graduation program and call every 10th person on the list of graduates to see if they plan to come. She calls 18 people and 8 of them say that they will be able to come. How many people can she expect to come to the party?

c. I s this a random sample? Could this sample be biased? Compare these results to the results from the first sample.

Spectrum Critical Thinking for Math  Grade 7

Lesson 6.3 Data in the Real World

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Check What You Learned Statistics 1. Will wants to survey a sample of students at his school to find out how many play musical instruments. He surveys students coming out of band class. Is Will’s sample biased or random? Why?

2. The graph shows a sample of heights of sixth graders and eighth graders. Compare the data. What can you infer? 6

5

5

4

4

3

3

2

2

1

1

0

50-59 60-69 Heights of 6th Graders

0

CHAPTER 6 POSTTEST

6

50-59 60-69 70-79 Heights of 8th Graders

3. A factory produces 74,000 sets of headphones each day. The quality manager claims that fewer than 600 defective tubes are produced each day. In a random sample of 310 sets of headphones, 3 are defective. Is the quality manager’s claim correct? Explain your answer.

Spectrum Critical Thinking for Math Grade 7

Chapter 6 Check What You Learned 81

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Check What You Learned Statistics 4. The graph shows a sample of heights of plants that were grown with no fertilizer and plants that were grown with fertilizer. What can you infer from the box and whisker plots? (no fertilizer) (w/fertilizer) CHAPTER 6 POSTTEST

1

2

3

4

5

6

7

8

9

10

11 12

5. The graph shows a sample group of girls and a sample group of boys, and the number of books they read during the school year. If there are 200 boys and 200 girls at the school, how many girls and boys read 10 books?

0

1

2

3

4

5

6

7

8

9

10

11 12 13 14 15 16 17

18 19 20

# of books read during school year (boys)

0

1

2

3

4

5

6

7

8

9

10

11 12 13 14 15 16 17

18 19 20

# of books read during school year (girls)

Spectrum Critical Thinking for Math  Grade 7

Chapter 6 Check What You Learned

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Check What You Know Probability 1. Of the 50 U.S. states, 13 were the original colonies. If you select 1 state randomly, how likely is it to be one of the original colonies?

2. David takes 20 shots and scores 6 goals at soccer practice. What is the experimental probability that he will miss his next shot?

3. Evan hits 6 out of 14 pitches during practice. What does an experimental probability of 47 describe?

4. At Luvski Ski Resort, there are two chair lifts to the top of the mountain. There are five ski trails to the bottom of the mountain. What is the probability of riding on Chair 1 and skiing on Trail 3?

Spectrum Critical Thinking for Math Grade 7

Chapter 7 Check What You Know 83

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CHAPTER 7 PRETEST

NAME _________________________________________________________________________________

Check What You Know Probability 5. The school picnic is a two-day weekend event. It has been scheduled for May. The area routinely gets 16 rainy days in May. What is the probability that the weekend will be dry? Round your answer to the nearest percent.

6. In basketball, Alan makes 1 out of every 4 free throws he tries. What is the probability that Alan will make his next 3 free throws? Round your answer to the nearest tenth of a percent.

7. Gregg has 12 cards. Half are black, and half are red. He picks 2 cards out of the deck. What is the probability that both cards are red?

8. Lucy places 5 cards face down on the table and mixes them up. The cards are numbered 1 through 6. What is the likelihood that her friend Harry will draw an even-numbered card?

Spectrum Critical Thinking for Math Grade 7

Chapter 7 Check What You Know

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Lesson 7.1 Understanding Probability The probability of an event measures the likelihood that the event will occur.

sible

s Impo 0

0%

ely

Unlik 1 4

0.25 25%

lly

Equa

likely

/Un Likely

1 2

0.50 50%

in

Likely

Certa

3 4

1

0.75 75%

100%

The complement of an event is the set of all outcomes not included in the event. Answer the questions. What is the sum of the probabilities of an event and its complement?

Students in Ms. Baldwin’s class are picking numbers out of a hat. The hat has 8 pieces of paper. Four pieces of the paper are black, and the other pieces are white. Where does the probability of picking a white piece of paper out of the hat fall on the number line above?

Where would the probability of picking a white piece of paper fall on the number line if there were 6 pieces of white paper and 2 pieces of black paper in the hat?

Spectrum Critical Thinking for Math Grade 7

Lesson 7.1 Understanding Probability 85

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Lesson 7.2 Frequency Tables The experimental probability of an event is found by comparing the number of times the event occurs to the total number of trials. A frequency table is used to keep track of the trials. Marvin has a bag of marbles. He removes a marble, records the color, and then puts the marble back in the bag. The frequency table shows how many times he picked each color. Color Purple Pink Orange White

Frequency 12 10 15 13

Find the experimental probability for each color.

P P P P

(purple)  12  24% 50 10 (pink)  50  20% (orange)  15  30% 50 13 (white)  50  26%

Students at Prince Middle School were asked about their weekly allowance. Use the frequency table to calculate the experimental probability for each amount. Show your work. Allowance

# of students

$15.00

9

$20.00

11

$25.00

12

$30.00

8

Spectrum Critical Thinking for Math  Grade 7

Lesson 7.2 Frequency Tables

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Lesson 7.2 Frequency Tables Answer the questions. A spinner with 4 equal sections was spun 78 times. Use the frequency table to calculate the experimental probability of spinning each number. Show your work. Round your answer to the nearest tenth of a percent. Number on Spinner

Frequency

1

21

2

22

3

18

4

17

What is the probability of not spinning a 3?

A coin was flipped 60 times. The experimental probability of each outcome is shown in the table below. Coin Lands On

Frequency

Heads

27

Tails

33

27

P (heads)  60  45% P (tails)  33  55% 60

Is this the probability that you expected? Compare the results to your expectations.

Spectrum Critical Thinking for Math Grade 7

Lesson 7.2 Frequency Tables 87

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Lesson 7.3 Calculating Probability Theoretical probability is the probability of an event occurring based on all the possible outcomes. Theoretical probability can be calculated this way:

P (event) 

number of ways the event can occur total number of possible outcomes

A spinner has 3 equally sized sections labeled A, B, and C. What is the probability that your spinner landed on section A? There are 3 possible outcomes, with one of them being A. P (A) 

1 3

A bag of marbles contains 5 green marbles, 8 red marbles, and 9 yellow marbles. Ella chooses one marble at random from the bag. What is the probability that she picks a green marble? Round your answer to the nearest tenth of a percent.

What is the probability that she does not pick a red marble? Round your answer to the nearest tenth of a percent.

Spectrum Critical Thinking for Math  Grade 7

Lesson 7.3 Calculating Probability

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Lesson 7.4 Probability Models When all outcomes of an experiment are equally likely, the event has uniform probability. This probability can be used to predict outcomes. Vick rolls a number cube. What is the probability that he rolls a prime number? If he rolls the number cube 30 times, how many times is he expected to roll a prime number? A number cube has 3 prime numbers (2, 3, 5). There are 6 possible outcomes. P (prime number)  36  50% 0.5  30  15 There is a 50% chance of rolling a prime number. If Vick rolls the number cube 30 times, it is expected that he will roll a prime number 15 times

Answer the questions. Show your work. A spinner has 20 equal sections, numbered 1 through 20. a. What is the probability that the spinner will land on a multiple of 3?

b. I f the spinner is spun 42 times, how many times can it be expected to spin a multiple of 3?

c. What is the probability that it will not spin a multiple of 4?

Spectrum Critical Thinking for Math Grade 7

Lesson 7.4 Probability Models 89

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Lesson 7.5 Other Probability Models When a probability event has unequal odds, the outcomes are not equally likely to occur. A spinner has 4 equal sections. 2 of the sections are yellow, one of the sections is purple, and the other section is green. What is the probability that the spinner lands on yellow? P (yellow)  24  50% What is the probability of not spinning purple?

P (not purple)  1 14 

3 4

 75%

Answer the questions. Show your work. Round your answers to the nearest tenth of a percent. A grocery store randomly selects an item to be on sale each day



Item

# of Days on Sale

Ice Cream

4

Oranges

5

Chicken

3

Chips

5

Eggs 4 a. What is the probability that the item on sale will be ice cream or chips?

b. What is the probability that oranges or chicken will not be on sale?

Spectrum Critical Thinking for Math  Grade 7

Lesson 7.5 Other Probability Models

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Lesson 7.5 Theoretical vs. Experimental Probability Theoretical probability is what is expected to happen based on likely outcomes. Experimental probability is what actually happens. Suppose you toss a coin 25 times, and it lands tails up 11 times. Compare the experimental probability and the theoretical probability. 1

Theoretical probability: 2  50% 11 Experimental probability: 25  44% The experimental probability is less than the theoretical probability. It is impossible to meet the experimental probability because there are an odd number of coin tosses. Thomas spins a spinner 40 times. The results are shown in the table. Based on the results of the experiment, use your best guess to draw the spinner. Number

Frequency

1

9

2

11

3

12

4

8

Spectrum Critical Thinking for Math Grade 7

Lesson 7.6 Theoretical vs. Experimental Probability 91

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Lesson 7.7 Understanding Compound Events When two or more things are happening at one time in an experiment, it is a compound event. The probability of each event is multiplied. What is the probability of rolling a 2 and then a 6 when rolling a number cube twice? 1 P (2)  6 P (6)  16 1 P (2, then 6)  16  16  36 Answer the questions. Show your work. Round your answers to the nearest tenth of a percent. A standard spinner is arranged so that the numbers 1 to 15 share equal space. a. What is the probability of getting a 9 on two consecutive spins?

b. What is the probability of not getting a 9 on two consecutive spins?

What is the probability of rolling a 2 on a standard number cube and then getting heads on a coin toss?

What is the probability of not rolling a 6 on a number cube and then getting heads on a coin toss?

Spectrum Critical Thinking for Math  Grade 7

Lesson 7.7 Understanding Compound Events

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Lesson 7.7 Understanding Compound Events The Fundamental Counting Principle says that when there are m ways to do one thing, and n ways to do another, then the product of m and n is the possible number of outcomes for both events. A tree diagram can help you visualize this. An ice cream shop offers vanilla, strawberry, and chocolate ice cream. A customer can choose a regular cone, a sugar cone, or a cup. What is the probability of getting strawberry ice cream on a sugar cone? There are 3 flavors and 3 serving options. 3 3 3  9, so there are nine possible outcomes. There is one possible combination of strawberry ice cream and sugar cone.

P (strawberry sugar cone)  19  11%

V S C

Reg Sugar Cup Reg Sugar Cup Reg Sugar Cup

Answer the questions. Show your work. Round your answers to the nearest tenth of a percent. A salad bar has croutons, raisins, sunflower seeds, and cranberries available as toppings. Teresa wants 2 different toppings on her salad. How many possible 2-topping combinations can Teresa choose? What is the probability of having croutons and sunflower seeds on her salad?

Spectrum Critical Thinking for Math Grade 7

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Lesson 7.8 Probability in the Real World Answer the questions. Show your work. Round your answers to the nearest tenth of a percent. A retail store is having a contest. The randomly selected prize will be a can opener, a gift card, or a set of towels. The store cashier will spin a spinner with the numbers 5–8 to see whether every 5th, 6th, 7th, or 8th customer will win a prize. a. Create a tree diagram to show all the possible outcomes in this situation.

b. What is the probability that every 5th person will win a can opener or a gift card?

c. What is the probability that every 6th or 7th person will win a set of towels?

d. What is the probability that a customer will not win a can opener?

Spectrum Critical Thinking for Math  Grade 7

Lesson 7.8 Probability in the Real World

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Check What You Learned Probability Answer the questions. Show your work. 1. An auto company conducted a survey with a random sample of 500 people to find out which type of vehicle they preferred to drive. The results are shown below. Number of People

Compact

75

Sedan

45

SUV

95

Pickup

90

Station Wagon

95

Minivan

100

CHAPTER 7 1 POSTTEST



Favorite Vehicle

a. What is the probability that a randomly selected survey participant prefers to drive an SUV? Write it as a decimal.

b. I f 1,500 people were surveyed, how many would you expect to prefer to drive an SUV? Explain your answer.

2. Mr. Rose randomly selects names to see who will give the first book report. There are 10 boys and 14 girls in his class. What is the probability that he will select a girl’s name?

Spectrum Critical Thinking for Math Grade 7

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Check What You Learned Probability Answer the questions. Show your work. Round your answers to the nearest tenth of a percent.

CHAPTER 7 POSTTEST

3. Of the original 56 signers of the Declaration of Independence, 4 represented North Carolina. If you selected 1 signer randomly, how likely is it that he represented North Carolina?

4. Every seventh-grade student is eating in the cafeteria. Juwarne is a seventh-grade student. How likely is it that she is in the cafeteria?

5. Kobe makes 15 of 20 free throws at basketball practice. What is the experimental probability that he will miss his next free throw?

6. At the barbershop, there are 2 chairs for customers to wait in. There is a rack with 5 magazines for customers to read while they wait. How many possible choices of chairs and magazines do the barbershop customers have?

Spectrum Critical Thinking for Math Grade 7

Chapter 7 Check What You Learned

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Final Test

Chapters 1–7

Answer the questions. Show your work. 1. These temperature changes in a vat of liquid were noted by a scientist performing a chemical experiment. What was the net temperature change from the first Monday to the second Monday? Monday Tuesday Wednesday Thursday Friday Saturday Monday

4.6 °C 10.2 °C 20.3 °C 23.5 °C 4.2 °C 14.4 °C 26.9 °C

2. Serena took care of Jason’s large fish tank while he was on vacation. The tank lost water through evaporation, and Serena added more water as shown in the table. In total, how much water will be gained or lost by the time Jason returns from vacation? Water Lost (in quarts)

Water Added (in quarts)

Mon.

3 4

5 8

Tue.

1 2

7 8

Wed.

5 8

1 2

Spectrum Critical Thinking for Math Grade 7

Chapters 1–7 Final Test

CHAPTERS 1–7 FINAL TEST

Day

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Final Test

Chapters 1–7

Answer the questions. Show your work. 3. The chart shows the high and low temperature in Anchorage for a week. Temperature in Anchorage (°F) Sun

Mon

Tues

Wed

Thurs

Fri

Sat

High





6°

7°



15°



Low

8°

12°

21°

17°

15°

25°

18°

a. Find the average of the high temperatures. Round your answer to the nearest tenth of a degree.

b. Find the average of the low temperatures. Round your answer to the nearest tenth of a degree.

CHAPTERS 1–7 FINAL TEST

4. The terms 8x, 5z, 15y, z, 2x and another term are added to form an expression. When simplified, this expression equals 2 (3z  5x). Identify the missing term and write the expression.

Spectrum Critical Thinking for Math Grade 7

Chapters 1–7 Final Test

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Final Test

Chapters 1–7

Answer the questions. Show your work. 5. The circumference of a circular garden is 42 meters. A gardener is digging a straight line along the diameter of the garden at a rate of 10 meters per hour. How many hours will it take the gardener to dig across the garden? Use 3.14 for p. Round your answer to the nearest hundredth.

6. Thomas spins a spinner 25 times. The results are shown in the table. Based on the results of the experiment and your best guess, how does the size of the section containing #5 compare to the size of the section containing #6? Number

Frequency

1

2

2

4

3

1

4

8

5

2

6

8

7. A number j is positive and another number k is negative. Based on this information, can you determine whether j  k is positive or negative? Explain.

Chapters 1–7 Final Test

CHAPTERS 1–7 FINAL TEST

Spectrum Critical Thinking for Math Grade 7

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Final Test

Chapters 1–7

Answer the questions. Show your work. 8. A college football stadium holds 25,000 fans. In a random sample of 30 fans, 26 were wearing the colors of the home team. Predict the number of fans who are wearing the colors of the home team.

9. If it takes Joe 15 hours to make 3 cornhole boards, how long will it take him to make 11 cornhole boards?

CHAPTERS 1–7 FINAL TEST

10. J ack bought 4 turkey sandwiches and 2 bags of apple slices for $22.60. If the apple slices cost $0.75 per bag, how much did each sandwich cost?

Spectrum Critical Thinking for Math Grade 7

Chapters 1–7 Final Test

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Final Test

Chapters 1–7

Answer the questions. Show your work. 11. Identify the mistake that was made in simplifying the expression. Then, correctly simplify the expression. 5 (a  3)  (6a  12)  7a  5a 1 2  6a  12  7a  (5a  6a  7a)  (2  12)  4a  10

12. On a road map, the distance between two cities is 12.6 centimeters. What is the actual distance if the scale on the map is 2 cm:50 mi. How long would it take a driver traveling 70 miles per hour to go from one city to the next city?

13. Several puppies from 2 different breeds were weighed. The puppies’ weights in pounds are shown in the table. What can you infer from the data?



1

2

3

4

5

6

Spectrum Critical Thinking for Math Grade 7

7

8

9

10 11

12

13 14

15 16

17 18

Chapters 1–7 Final Test

CHAPTERS 1–7 FINAL TEST

Breed A Breed B

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Answer Key Page 5 CHAPTER 1 PRETEST

Adding and Subtracting Rational Numbers Compare the values using , , or . 1


|7 23 | because |–8 13 | is further from zero on the number line than |7 23 |.

No . Juan earned 28 + 15 for a total of + $43 . He spent 20 + 22, for a total of –$42 . He did not spend the additive inverse because 43 + (–42) ≠ 0 .

F

Point E is 2 units from 0 and point F is 3 units from 0. Spectrum Critical Thinking for Math Grade 7

Additive Inverse

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?

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.

E

________________________________________________________________________________

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

Answer the questions.

|E| 800 40 40 l > 20

Spectrum Critical Thinking for Math Grade 7

Page 35 NAME

5x + 3 ≤ 38 – 3 – 3 5x ≤ 35 5x 35 x ≤ 7 ≤ 5 5

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

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?

She was putting into the account

________________________________________________________________________________

Check What You Know

P 5 2l 1 2w 5 2(5x 1 2) 1 2(3x 2 4) 10x 1 4 1 6x 2 8 10x 1 (4 1 6x) 2 8 10x 1 (6x 1 4) 2 8 (10x 1 6x) 1 (4 2 8) x (10 1 6) 1 (4 2 8)

Distributive Property Associative Property Commutative Property Associative Property Distributive Property

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

Commutative Property

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

Associative Property: Commutative Property: Associative Property: Distributive Property: Commutative Property: 1 4 (x

212) 

1 2

(x  8)

Distributive Property: Associative Property: Commutative Property: Associative Property: Distributive Property: Commutative Property: Spectrum Critical Thinking for Math Grade 7

17x + (6 + 13x) – 3 17 x + (13x + 6) – 3 (17x + 13x) + (6 – 3) x(17 + 13) + (6 – 3) 30x + 3 1 x–3+ 1x+4 4 2 1 x + (–3 + 1 x) + 4 4 2 1 x + ( 1 x – 3) + 4 4 2 1 1 ( x+ x) + (–3 + 4) 4 2 3 x+1 Lesson 3 .1 4 Properties and Equivalent Expressions 35

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.

n – 0.25n n(1– 0.25) Distributive Property Commutative Property 0.75n

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.

Commutative Property Associative Property Distributive Property Commutative Property Spectrum Critical Thinking for Math Grade 7

(10 + 5c) + (7 + 3c) (10 + 5c) + (3c + 7) 10 + (5c + 3c) + 7 10 + (5 + 3) c + 7 10 + 7 + 8c 17 + 8c Lesson 3.2

Creating Expressions to Solve Problems

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Lesson 3.3

Page 38 Lesson 3.4

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

Darryl spent $165 .75 .

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

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?

cost of supplies 3 .20(3 + 2) + 0 .75(4 + 3) + 4 .50(2 + 1) 3 .20(5) + 0 .75(7) + 4 .50(3) 16 + 5 .25 + 13 .5 34 .75 The total cost was $34 .75 .

Total: 6 cups Before the double: 6 = 3 2 Before adding 2 cups: 3 – 2 = 1 cup

Lesson 3 .3 Using Variables and Expressions 37

Spectrum Critical Thinking for Math Grade 7

Page 40 NAME

________________________________________________________________________________

Equations in the Real World

Lesson 3.6

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 x  3x  100 5 1400 4x  100 5 1400 100 5  100 51,500 4x

5 1500 4 x 5 375

3(x  5)  1  77 3x  15  1  77 3x  14  77 14  14 3x  63 3x 63  3  3 x  21

Jesse scored 375. Jerri scored 3(375)  100 5 1,025.

–67r –16.75 = – 67 –67 r = 0.25

The dress was marked down 0.25  100 = 25%.

m = Sayeed's magazine sales 0.75m = Robyn's magazine sales m = 52; Sayeed raised $52 and Robyn raised 0.75  52 = $39 in magazine subscriptions. Lesson 3.5 Equations in the Real World 39

10 is not a part of the solution. The inequality sign had to change because we had to divide by a negative number.

Solve each inequality. Show your work. Is 20 a part of the solution? 3(p  3)  5p  23

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 Grade 7

Using Variables to Express Inequalities

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

4x 4

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

1.75m + 36.75 = 127.75 – 36.75 – 36.75 1.75m =91 1.75m = 91 1.75 1.75

________________________________________________________________________________

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.

Write and solve an equation for each problem.

67(1 – r) = 50.25 67 – 67r = 50.25 – 67 – 67 – 67r = – 16.75

Lesson 3.4 Numeric and Algebraic Solutions

38

Page 39 NAME

2(x + 2) = 6 2x + 4 = 6 –4 –4 =2 2x 2x = 2 x=1 2 2

3n – 5 = 25 5 less than what number is 25: +5 +5 25 + 5 = 30 =30 3n 3 times what number is 30: 30 = 10 3n = 30 3 3 3 n = 10

1 c = 16(1 – 0 .25) + 5 .25(1– 3 ) + 13 .50 2 c = 16(0 .75) + 5 .25( 3 ) + 13 .50 c = 12 + 3 .50 + 13 .50 c = $29 savings = 34 .75 – 29 .00 They will save $5 .75 . Spectrum Critical Thinking for Math Grade 7

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?

How would the amount that Bruno and Mark spent change if the pencils were 25% off and the paper was 13 off?

Lesson 3.5

Solve with equation: 615  1 13 (615)  x  2035 615  820  x  2035 1435  x  2035 1435 1435 x  600

Solve working backward: Day 1: 2035  615  1420 miles remaining 1 Day 2: 1 3 (615)  820 1420  820  600 miles Chelsea drove 600 miles the 3rd day.

s  amount spent s  2(31 .25)  3(17 .50)  50 .75 s  62 .50  52 .50  50 .75  165 .75

= = = = =

________________________________________________________________________________

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 13 times as far. How far did she drive the 3rd day?

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?

c c c c c

NAME

________________________________________________________________________________

Using Variables and Expressions

Yes, –20 is a part of the solution. Is 100 a part of the solution? 1.25(x  16)  140.75

No, 100 is not a part of the solution. Spectrum Critical Thinking for Math Grade 7

3p – 9 – 5p > 23 – 2p – 9 > 23 +9 +9 –2p > 32 –2p > 32 –2 –2 p < –16 1.25x + 20 ≤ 140.75 – 20 – 20 1.25x ≤ 120.75 1.25x ≤ 120.75 1.25 1.25 x ≤ 96.6 Lesson 3.6 Using Variables to Express Inequalities

40

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Lesson 3.7

Page 42 NAME

________________________________________________________________________________

Inequalities in the Real World

Expressions, Equations, and Inequalities

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?

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

CHAPTER 3 POSTTEST

Niki will have to work no more than 11 hours to earn enough money .

132  12h  264 2132 2132 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?

8x 8x 8x 8x 8x 8x

+ (2y – 4) + 3y – 3 + (–4 + 2y) + 3y – 3 – 4 + (2y + 3y) –3 – 4 + (2 + 3) y – 3 – 4 – 3 + 5y + 5y – 7

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

3 . Tia has 4 more than 12 the number of pairs of earrings that Ebony has . Together, they have 25 pairs of earrings . How many pairs of earrings does each girl have?

e = Ebony's earrings; e + 1 e + 4 = 25 2 3 e + 4 = 25 2 –4 –4 3e = 21 2 2  3 e = 21  2 3 2 3

You would need to play a minimum of 19 games in order for the unlimited plan to be the best deal . Lesson 3 .7 Inequalities in the Real World

Ebony has 14 pairs of earrings and Tia has 1 (14) + 4 = 11 pairs of 2 earrings . Chapter 3 Check What You Learned

Page 44 NAME

________________________________________________________________________________

CHAPTER 4 PRETEST

Check What You Learned Expressions, Equations, and Inequalities 4 . Solve for x: 26x  1 .5  8 .7

CHAPTER 3 POSTTEST

–6x + 1 .5 = 8 .7 – 1 .5 – 1 .5 –6x = 7 .2 –6x = 7 .2 –6 –6 x = –1 .2

1 5 . Is 0 a part of the solution? 2 3 (x 2 27) 2 17  1

x ≥ –27 0 is a part of the solution .

Ratios and Proportional Relationships 1 . Cheyenne can type 12 of a page of her essay in can she type in 1 hour?

Cheyenne can type 1 page in an hour .

of an hour . How many pages

1 ÷ 1 = 1  2 = 2 =1 2 2 2 1 2 1  12 = 2 1 1 =6 2

2 3

4 6

8 12

The values represent a proportional relationship .

12 11 10 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 y

43

1 2

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

325 + 20g + 35g ≤ 435 325 + 55g ≤ 435 – 325 – 325 Rhonda can buy at most 55g ≤ 110 g≤2 2 of each type of game . Chapter 3 Check What You Learned

________________________________________________________________________________

Check What You Know

x y

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 Grade 7

e = 14

42

Page 43

– 1 (x – 27) – 17 ≤ 1 3 – 1 x + 9 – 17 ≤1 3 –1x–8≤1 3 +8+8 – 1 x≤9 3 –3  – 1 x ≤ –3  9 3

1 e + 4 = Tia's earrings 2

Spectrum Critical Thinking for Math Grade 7

41

NAME

Associative Property Commutative Property Associative Property Distributive Property Commutative Property Commutative Property

2(3x – 1) + 2x + 2 6x – 2 + 2x + 2 8x

g = number of games 6g + 35 ≥ 149 – 35 – 35 6g ≥ 114 6g ≥ 114 6 6 g ≥ 19

Spectrum Critical Thinking for Math Grade 7

________________________________________________________________________________

Check What You Learned

40 30

80 60

120 90

Spectrum Critical Thinking for Math Grade 7

k = 30 = 3 40 4

Chapter 4 Check What You Know

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Page 46 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 .

k= 5 7 w= 5j 7

5 . Given the graph, what is the constant of proportionality? 10 9 8 7 6 5 4 3 2 1

k= 1 2

0

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 . 1

Carson can read 5 2 pages of his history textbook in pages can he read in 1 hour? 1 1 52 4 6 6 11 2  1 66  33 2

4,2 2,1

8,4

Penny is comparing two recipes . One recipe calls for 14 stick of butter for 34 cups of 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?

1 ÷ 3 1 stick of 1 ÷ 5 3 stick of 4 4 3 2 3 10 butter per 1 butter per 1 1  4 = 1 1  3 = 3 4 3 3 cup of milk . 2 5 10 cup of milk . 1  3 the first recipe has more butter per cup of milk . 3 10 1 2 1 3

1 2 3 4 5 6 7 8 9 10

Fran ran 4 2 miles in fastest?

9 ÷ 2 2 5 9  5 2 2 45 = 11 1 4 4 Fran ran the

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

y = 1 (50) 2 y = 25 Spectrum Critical Thinking for Math Grade 7

Chapter 4 Check What You Know 45

5

of an hour . Fred ran 6 2 miles in

NAME

Does this represent a proportional relationship? Pages read

6 12

The graph is forms a straight line that goes through the origin. It is proportional.

Is

costs

0

1 2 3 4 5 6 7 8 9 10

pounds

The graph forms a straight line that goes through the origin. It is proportional. Spectrum Critical Thinking for Math Grade 7

0

Testing Proportional Relationships

4 .5 2

,

6 .75 3

proportional?

4 .5  3  2  6 .75? 13 .5  13 .5?

Yes, the relationship is proportional .

1 2 3 4 5 6 7 8 9 10

Time (hours) Time

Georgia uses 4 pencils in 2 weeks, 7 pencils in 3 weeks, and 8 pencils in 4 weeks. 10 9 8 7 6 5 4 3 2 1

________________________________________________________________________________

Cross-multiply to determine if each relationship is proportional . Time (minutes)

6,7.50

3,3.75 2,2.50

NAME

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 .

13 12 11 10 9 8 7 6 5 4 3 2 1 0

pencils

10 9 8 7 6 5 4 3 2 1

Lesson 4 .1 Comparing Unit Rates

46

Lesson 4.2

Graph these relationships to determine if they are proportional. 2 3 6 2.50 3.75 7.50

Fred ran 10 5 miles 6 per hour

fastest .

Spectrum Critical Thinking for Math Grade 7

________________________________________________________________________________

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.

Number of Pounds Cost

of an hour . Who ran the

Page 48

Testing Proportional Relationships

4 8

5

13 ÷ 3 2 5 13  5 2 3 65 =10 5 6 6

Fran ran 11 1 miles 4 per hour .

Page 47

2 4

of an hour . How many

Carson can read 33 pages in 1 hour .

y= 1x 2

Time (minutes) Pages read

1 6

Solve the problems . Show your work .

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

Lesson 4.2

________________________________________________________________________________

Comparing Unit Rates

Lesson 4.1

2 116

3 .5 238

5 340

2 = 3 .5 116 238 2(238) = 3 .5(116) 476 = 406

4,8 3,7 2,4

Not proportional Mike is trying to choose a data plan for his phone . 3GB costs $27, 4GB costs $36, and 7GB costs $63 .

1 2 3 4 5 6 7 8 9 10

weeks

The graph does not form a straight line that goes through the origin. It is not proportional. Lesson 4.2 Testing Proportional Relationships 47

3 = 4 27 36 3(36) = 4(27)

4 = 7 36 63 4(63) = 7(36)

108 = 108

252 = 252

Proportional Spectrum Critical Thinking for Math Grade 7

Lesson 4 .2 Testing Proportional Relationships

48

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Answer Key Page 49 NAME

Lesson 4.3

Page 50

k

output input ;

3 96 .75

k  96 .75  32 .25; 3

161 .25 5

5 161 .25

 32 .25;

The constant of proportionality can be used to write an equation to represent the relationship. A recipe calls for 12 cup of sugar for every 1 13 cups of flour. Write an equation to calculate how much flour to use for each cup of sugar. How many cups of flour 1 will be used if 2 2 cups of sugar are used?

6 193 .50 193 .25 5

 32 .25

k  cups of

The constant of proportionality is 32 .25 . The cost of renting a car is $32 .25 per day .

3 6 .54

8 17 .44

0 .25

2 .5

21

3

30

There are 880 feet in

Lesson 4 .3 Constants of Proportionality 49

Lesson 4.5

1 6

Proportions on the Coordinate Plane

Value ($)

 0 .10

1 .25 0 .75 0 .25 0

1 2 3 4 5 6 7

150 value = 3 game tokens = 0 .50 Each game token is worth $0 .50 . 7 tokens would be worth 0 .50  7 = $3 .50 .

k=

Spectrum Critical Thinking for Math Grade 7

6

2 3

cups of flour

acre of grass. How much gas does it take to

of a mile. How many feet are in 1 13 miles?

Spectrum Critical Thinking for Math Grade 7

Lesson 4.4 Using Equations to Represent Proportions

50

NAME

Lesson 4.6

________________________________________________________________________________

Proportions in the Real World

Answer the questions. Show your work. Juice is sold at the grocery store in several different sizes. The prices are shown in the table. Size (fl. oz.) 16 32 64 128

0 .90 0 .70 0 .50 0 .30

0

Price ($) 1.89 3.49 7.59 9.99

Unit Price ($)

0.12 0.11 0.12 0.08

a. Complete the table with the unit prices of each size. Round to the nearest hundredth. 1.89 7.59

1 2 3 4 5 6 7 8

# of dimes

= 0.12 16 3.49 = 0.11 32

11 10 9 8 7 6 5 4 3 2 1

= 0.12 64 9.99 = 0.08 128

b. If you wanted 64 fluid ounces of juice, which would be the best way to purchase it?

It would be best to buy 2 32-oz. bottles for 3.49  2 = $6.98 instead of 1 64-oz bottle for $7.59.

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

How much would 7 tokens be worth?

2

 23

880 feet = = 5280 1 mile 6 f = 5280m 1 f = 5280 (1 ) = 7,040 feet 3

Calculate the constant of proportionality for each graph . What does the constant of proportionality represent? 12 1 .75

8 3

0 .10

The constant of proportionality is 0 .10 . It represents the value of each dime: $0 .10 .

2 .25

1 2

________________________________________________________________________________

Given the graph, calculate the constant of proportionality . What does the constant of proportionality represent on the graph? 0 .50 5



40 6

Page 52

A constant of proportionality can be found using the graph of a proportional relationship . Identify an ordered pair (x,y) on the line . The constant of proportionality is k  y/x .

k



1 1 gas gallons k= = 2 =3 1 acres of grass 2 g = 3a 1 1 g = 3 (2 ) = 7 gallons 2 2

Page 51 NAME

1 2 5 2 )

k=

k = 21 = 12; 3 = 12; 30 = 12 2 .5 1 .75 0 .25 The constant of proportionality is 12 . The person was biking 12 miles per hour . Spectrum Critical Thinking for Math Grade 7

8 3

13

Write an equation using the constant of proportionality to represent each relationship described. It takes 1 12 gallon of gas to mow mow 2 12 acres of grass?

15 32 .70

1 .75

1



f  (2 23 )s f  (2 ) (2 )  ( ) (

k = 6 .54 = 2 .18; 17 .44 = 2 .18; 32 .70 = 2 .18 15 3 8 The constant of proportionality is 2 .18 . Milk costs $2 .18 per gallon . Time (hours) Distance Biked (miles)

flour sugar

1 2

2 3

Use the information to calculate the constant of proportionality for each table . What does the constant of proportionality mean in the context of the data given? Gallons of Gas Price

________________________________________________________________________________

Using Equations to Represent Proportions

Lesson 4.4

A unit rate can also be called a constant of proportionality . The constant of proportionality, k, is the ratio of the output variable to the input variable . Days of Car Rental Cost of Car Rental

NAME

________________________________________________________________________________

Constants of Proportionality

How many blue marbles are there if there are 3 red marbles?

12 4 = k = blue = 9 3 red There are 4 blue marbles for every 3 red marbles . There would be 43 (3) = 4 blue marbles if there are 3 red marbles . Lesson 4 .5 Proportions on the Coordinate Plane 51

Amount of Bill

19.00

35.00

72.00

Tip

3.42

6.30

12.96

Calculate the constant of proportionality shown in the table above. What does this constant mean within the context of the data given? How much would the tip be if the bill was $95.00?

k = 3.42 = 0.18 19.00 The constant of proportionality is 0.18. This means that the customers paid an 18% tip.

tip = 0.18  95.00 = $17.10 Spectrum Critical Thinking for Math Grade 7

Lesson 4.6 Proportions in the Real World

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Answer Key Page 53 NAME

Lesson 4.6

Page 54 NAME

________________________________________________________________________________

Proportions in the Real World

Ratios and Proportional Relationships

Answer the questions. Show your work.

1 . Pool A is being filled with 23 gallon per 14 minute . Pool B is being filled with gallon per 35 minute . Which pool is being filled faster?

One half of a can of paint covers 150 square feet of a wall.

2 ÷ 1 3 4 2  4 3 1 8 =22 3 3

a. Create a table to represent this relationship. 1 2

1 300

150

2 600 CHAPTER 4 POSTTEST

Cans of paint Area covered

b. Create a graph to represent this relationship

area covered (square feet)

100 900 800 700 600 500 400 300 200 100 0

150 = 300 1 2

Pool A is filling 2 2 3 gallons per minute .

1 2 3 4 5 6 7 8 9 10

Each can of paint can cover 300 square feet of the wall.

Spectrum Critical Thinking for Math Grade 7

Time

1 2

Amount Painted (square feet)

28

120 110 100 90 80 70 60 50 40 30 20 10

w = 300p w = 300(2 12 ) = 750 square feet 750 square feet of wall can be covered.

Lesson 4.6 Proportions in the Real World

25

NAME

________________________________________________________________________________

Mid-Test

1

3 4

3

175

350 CHAPTER 4 POSTTEST

The pay is $10 per hour .

Spectrum Critical Thinking for Math Grade 7

You would expect to burn 320 calories .

2

6

10

14

2 . Astrid played a board game in math class . She ended up with these cards: 3 .9 24 .2 23 .9 4 .1 6 .8 28 .5 210 .8 a . The score is the sum of the card values . What was her score? Show your work .

3 .9 + (–3 .9) + 4 .1 + (–4 .2) + 6 .8 + (–8 .5) + (–10 .8) 0 + (–0 .1) + 6 .8 + (–8 .5) + (–10 .8) 6 .7 + (–8 .5) + (–10 .8) –1 .8 + (–10 .8) –12 .6

CHAPTERS 1–4 MID-TEST

pay ($)

c = 100( 16 ) 5 c = 320

6 60

Chapters 1–4

57 – (–14) = 57 + 14 = 71 The total amount of the debits was $71 .

1 2

4 . Write an equation using the constant of proportionality in #3 on the previous 1 page . How many calories would you expect to burn if you walk 3 5 miles?

160 140 120 100 80 60 40 20 10 0

________________________________________________________________________________

1 . Yuri had a bank balance of $57 before he went shopping . After he used his debit card twice, his account was overdrawn by $14 . What was the total amount of the debits?

k = 25 ÷ 1 = 100 . The constant of 4 proportionality is 100 .

5 . Use the graph to create a table of values . Find the constant of proportionality and write an equation . What does it mean within the context of the graph? How much would the pay be after 25 hours of work?

Chapter 4 Check What You Learned

Page 56

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

Calories Burned

112

54

Ratios and Proportional Relationships

1 4

2

84

Spectrum Critical Thinking for Math Grade 7

Check What You Learned

Miles Walked

1 2

Yes, it is proportional because it forms a straight line that passes through the origin .

Page 55 NAME

1

1 2 3 4 5 6 7 8 9 10

53

Hours worked Pay (dollars) 60 k= = $10 6

Pool B is filling 1 gallon per minute .

3 ÷ 3 5 5 3  5 5 3 15 = 1 15

Pool A is being filled faster .

d. Write an equation to represent the relationship between cans of paint used and how much of the wall is covered. How much of the wall can be covered by 2 12 cans of paint?

c = 100m c = 100(3 1 ) 5

3 5

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

cans of paint c. What is the constant of proportionality? What does it represent?

k=

________________________________________________________________________________

Check What You Learned

18

b . In order to win the game, the absolute value of a score must be less than 12 . Is it possible for Astrid to win? Explain why or why not .

|–12 .6| = 12 .6 12 .6 > 12 Astrid cannot win . The absolute value of her score is greater than 12 .

hours worked (h)

10 100 p =10h p = 10(25) p = $250

14 140

3 . A number j is positive and another number k is negative . Based on this information, can you determine whether j 2 k is positive or negative? Explain .

j – k will always be positive . Since k is a negative number, its additive inverse will be added to j, resulting in a positive number . Chapter 4 Check What You Learned 55

Spectrum Critical Thinking for Math Grade 7

Chapters 1–4 Mid-Test

56

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Answer Key Page 57 NAME

Mid-Test

Page 58 NAME

________________________________________________________________________________

Mid-Test

Chapters 1–4

4. A mini-shelf in the food pantry can hold 3 34 pounds. If a can weighs 38 of a pound, how many cans can it hold? If you add more support so the shelf can hold 5 14 pounds, how many cans can the shelf hold now? Show your work.

3 3 ÷ = 3 4 8 15 8  = 4 3 120 = 10 12 The shelf can hold 10 cans.

7. Determine whether the expression 2(x  3) is equal to (4x  Identify the properties you used in your solution steps.

1 1 4x + – 8x + 5 2 2 1 1 4x – 8x + +5 21 21 (4x – 8x) + ( +5 ) 2 2 –4x + 6 = –2(2x – 3) –2(2x

1 3 ÷ = 5 4 8 21 8  = 4 3 168 = 14 12 The shelf can now hold 14 cans.

CHAPTERS 1–4 MID-TEST

CHAPTERS 1–4 MID-TEST

6. A number is multiplied by  4 , divided by 2 , and then divided by  8 . The resulting number is 96. Work backward to get the original number by performing opposite operations.

7 ) = –84 8 1 –84  = –42 2 3 4 –42 ÷ (– ) = –42  (– ) = 56 4 3 The original number is 56.

Candace can buy no more than 3 pillows.

Chapters 1–4 Mid-Test

Page 60 NAME

________________________________________________________________________________

CHAPTER 5 PRETEST

Chapters 1–4 15,000 14,000 13,000 12,000 11,000 10,000 9,000 8,000 7,000 6,000 5,000 4,000 3,000 2,000 1,000

________________________________________________________________________________

Check What You Know Geometry 1. Find the length of the missing side for the pair of similar triangles. 12 ft.

14 ft.

18 ft.

18 ft. 27 ft.

12 = 14 18 x 12x = 252 12x = 252 12 12 x = 21ft.

2. Can these lengths form a triangle? Side 1: 9 cm Side 2: 5 cm Side 3: 11 cm

1 2 3 4 5 6 7 8 9 1011121314

a . Create a table that represent the proportional relationship .

6 2000

The job took 3 hours.

58

10 . A museum is keeping track of the number of visitors per day .

# of days 3 # of visitors 1000

195 = 65h 195 = 65h 65 65 h=3

9. Candace has $65.25. She spent $31.50 on a new throw rug for her bedroom. She wants to buy some matching throw pillows. How many pillows can she buy if the pillows cost $11.25 each?

Page 59

0

).

Distributive Property – 3) = –2(x – 3)

Spectrum Critical Thinking for Math Grade 7

57

Mid-Test

1 2

Commutative Property Associative Property

31.50 + 11.25p ≤ 65.25 –31.50 –31.50 11.25p ≤ 33.75 11.25p ≤ 33.75 11.25 11.25 p≤3

Chapters 1–4 Mid-Test

NAME

)  (8x  5

Distributive Property

c = cost; h = hours c = 110 + 65(h –1) 240 = 110 + 65 (h – 1) 240 =110 + 65h - 65 240 = 65h + 45 –45 –45

96  (–

Spectrum Critical Thinking for Math Grade 7

1 2

8. A plumber charges $110 for a service call and $65 for each hour of work after the first hour. Let h represent the hours the electrician works on a service call. Write an expression to represent the cost. How many hours did it take the plumber to complete a job if the total cost is $240?

5. Mary Ellen says that the expression 5  7  8(4  2) simplifies to a negative number because if you multiply three negative numbers, the final answer will be negative. Is she correct? Show why or why not.

–5 + 7 + 8(–4  –2) = The answer is positive. Although the expression has 3 negative –5 + 7 + 8(8) = numbers, it is not purely a –5 + 7 + 64 = multiplication problem. This 2 + 64 = problem also has addition. You must use the order or operations 66 to simplify this expression. 7 3 1

________________________________________________________________________________

Chapters 1–4

9 3000

k = 100 = 333 .333 3 The museum has about 333 visitors per day . c . Write an equation to represent the relationship .

CHAPTERS 1–4 MID-TEST

b . What is the constant of proportionality to the nearest hundredth? What does it represent on the graph?

9 + 5 > 11 9 + 11 > 5 5 + 11 > 9 These sides can form a triangle.

3. Name the shape that is created by each cross section.

v = 333 .33d rectangle

d . Predict how many people will have visited the museum after 33 days .

triangle

v = 333 .33(33) v = 10999 .89 They can expect about 11,000 visitors after 33 days . Spectrum Critical Thinking for Math Grade 7

Chapters 1–4 Mid-Test 59

Spectrum Critical Thinking for Math Grade 7

Chapter 5 Check What You Know

60

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Answer Key Page 61 NAME

Page 62 CHAPTER 5 PRETEST

Geometry 4. Find the circumference and area of the circle. Use 3.14 for . Round answers to the nearest thousandth, if necessary. A  _______ square feet C  _______ feet

4.3 m

H

120 degrees HGE  _______

B G

C E F

The measure of HGE is also 120 degrees because they are vertical angles.

D

Scale drawings are used to represent an object. Scale drawings can be smaller, larger, or the same size as the original object. The scale factor shows the proportional relationship between the original object and the scale drawing.

4 ft. 

0.5 in. 1 f t.

 2 inches

8 ft. 

0.5 in. 1 f t.

 4 inches

7 in.

6 in.

The bed should be 2 inches by 4 inches on the diagram.

6. What is the volume of a rectangular prism with a length of 10mm, a width of 8mm, and a height of 5mm?

Solve the problem. Show your work. The scale in the drawing is 2 cm:5 m (2 cm  5 m). Find the length and width of the actual room. What is the area? What is the perimeter?

V = 10  8  5 = 400 mm3

20 cm

5 cm

7. What is the combined area of a rectangle with a length of 13.2 in. and a width of 4.1 in., and a triangle with a base of 13.2 in. and a height of 6.5 in.?

A = 13.2(4.1) = 54.12 in.2 A = 1 (13.2)(6.5) = 42.9 in.2 2 Total area = 54.12 + 42.9 = 97.02 in.2 Spectrum Critical Thinking for Math Grade 7

Chapter 5 Check What You Know 61

2 cm 20 cm 2 20 5m = ? m ; 5 = l 2l = 100; l = 50 m

NAME

2 cm = 5 cm; 2 = 5 5m ?m 5 w 2w = 25; w = 12.5 m

A = lw; A = 50(12.5) = 625 m2 P = 2l + 2w; P = 2(50) + 2(12.5) P = 125 m

Spectrum Critical Thinking for Math Grade 7

Lesson 5.1 Scale Drawings

62

Page 63 Lesson 5.2

________________________________________________________________________________

Scale Drawings

6 in. 0.5 in. scale factor  inches feet  12 ft.  1 f t. The scale factor can also be written as 0.5 in.:1 ft.

5. If AGB is 120 degrees, what is the measure of HGE? A

Lesson 5.1

Benita makes a scale drawing so she can rearrange her room. Her actual room is 12 feet by 14 feet. Her drawing is 6 inches by 7 inches. What is the scale factor? She wants to draw her bed in a new spot. If her bed is 4 feet by 8 feet, what size should she draw it on her diagram?

A = r2 A = 3.14  ( 4.3 )2 14.515 m2 2

C = d C = 3.14  4.3 C = 13.502 m

NAME

________________________________________________________________________________

Check What You Know

Page 64 NAME

________________________________________________________________________________

Forming Triangles

Lesson 5.3

The sum of the lengths of two sides of a triangle must be greater than the length of the third side.

________________________________________________________________________________

Cross Sections of 3-Dimensional Figures

A cross section is the intersection of a 3-dimensional figure and a plane . Here are some examples:

Jorge is planning to build a plant box in the shape of a triangle. He has 3 planks of wood that are 4 feet, 6 feet, and 3 feet long. Will he be able to build a rectangular plant box? 463 634 346 Jorge will be able to build a triangular box using these three planks because each sum of two sides is greater than the length of the remaining side. Don has 3 straws with lengths of 3 cm, 4 cm, and 9 cm. He is trying to make a triangle with the straws. Will he be successful? If not, which straw should he replace? What is the minimum length of the replacement?

3+44 4 + 9 >3

Don will not be successful. He could replace the 3 cm straw with a straw longer than 5 cm (x + 4 > 9) so that he could make the triangle.

There are 3 line segments with lengths x  2, x 2 3,and x  1. What is the minimum value of x that allows these line segments to form a triangle? Assume that x is an integer.

x+2+x+1>–3 2x + 3 > x – 3 x>–6

x+2+x–3>x+1 2x – 1 > x + 1 x>2

Intersect each 3-D figure with the given 2-D shape . A rectangle

A square

A quadrilateral

A triangle

x+1+x–3>x+2 2x – 2 > x + 2 x>4

The minimum value of x is 5. Spectrum Critical Thinking for Math Grade 7

Lesson 5.2 Forming Triangles 63

Spectrum Critical Thinking for Math Grade 7

Lesson 5 .3 Cross Sections of 3-Dimensional Figures

64

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Answer Key Page 65 NAME

Page 66 NAME

________________________________________________________________________________

Circles: Circumference

Lesson 5.4

Lesson 5.5

The area of a circle is A  pr2, where p  3.14.

The perimeter of a circle is called the circumference .

C  2r (r  radius) or C  d (d  diameter), where   3 .14

A circular pool has a radius of 10 feet. What is the area of the pool? A  3.14(10)2 A  314 square feet

A plate has a diameter of 10 inches . What is the circumference of the plate?

C  3 .14(10)  31 .4 inches

Answer the questions. Show your work. Use 3.14 for p.

Answer the questions . Show your work . Use 3 .14 for  . In college basketball rules, the ball can have a maximum circumference of 30 inches . What is the maximum diameter of a basketball to the nearest hundredth?

30 ≤ 3 .14d 30 ≤ 3 .14d 3 .14 3 .14 d ≤ 9 .55 inches

A playground area is circular with a diameter of 32 feet. What is the area of the playground? Round your answer to the nearest tenth.

2 A = 3.14( 32 ) 2 A = 3.14(256) A = 803.8 square feet

A frying pan has a diameter of 11 inches. What is the area to the nearest square inch of the smallest cover that will fit on top of the frying pan?

The diameter has to be 9 .55 inches or less . A round stained-glass window has a circumference of 195 inches . What is the radius of the window to the nearest inch?

2(3 .14)r = 195 6 .28r = 195 6 .28r = 195 6 .28 6 .28 r = 31 inches

2 A = 3.14( 11 ) 2 A = 3.14(30.25) A = 95 square inches

Justin just got his driver’s license. His parents are giving him permission to drive within a 25-mile radius of his home. What is the area Justin is restricted to when driving? Round your answer to the nearest tenth.

2

A = 3.14(25) A = 3.14(625) A = 1,962.5 square miles

The radius of the window is 31 inches . Spectrum Critical Thinking for Math Grade 7

Lesson 5 .4 Circles: Circumference 65

Spectrum Critical Thinking for Math Grade 7

Lesson 5.5 Circles: Area

66

Page 67 NAME

Page 68 Lesson 5.7

Find the area of the composite shape:

p

3m

What is the value of p? The angles are supplementary, so the sum is 180.

11.4 m

Area = area of rectangle + area of semicircle area of rectangle  lw  (11.4)(3)  34.2 m2

p  82  180 82 82 p  98 degrees

Area of semicircle  12 pr2  34.2  3.53  37.73 m2

If 4 is a right angle and 5 measures 40 degrees, find the measures of the remaining angles. G L

5 K

4I

1

2

J

5 ft.

1 + 2 = 180 (supplementary) 1 + 40 = 180 1 = 140°

3 ft.

(3.14) ( 32 )2  3.53 m2

Lesson 5.6 Angle Relationships 67

1 bh = ( 1 )(3)(5) = 7.5 ft.2 2 2 Area of semicircle = 1 pr2 = 1 (3.14) ( 5 )2 = 9.81 ft.2 2 2 2 Sum of each area = 7.5 + 9.81 = 17.31 ft.2

3.1 cm

2 + 3 = 90 (complementary) 40 + 3 = 90 3 = 50° Spectrum Critical Thinking for Math Grade 7

1 2

Find the area of the composite shapes. Show your work. Use 3.14 for p. Round answers to the nearest hundredth.

5 = 2 (vertical angles) 2 = 40°

H

3

________________________________________________________________________________

Area of Composite Shapes

Shapes that are composed of other shapes are called composite shapes The area of a composite shape is equal to the sum of each shape it is made of.

When two lines intersect, they form angles that have special relationships. • Vertical angles have the same measure. • Supplementary angle are two angles with the sum of 180 degrees • Complimentary angles are two angles with the sum of 90 degrees. 828

NAME

________________________________________________________________________________

Angle Relationships

Lesson 5.6

________________________________________________________________________________

Circles: Area

7.4 cm

4.2 cm

bh = ( 1 )(4.2)(7.4) = 15.54 cm2 2 lw = (7.4)(3.1) = 22.94 cm2 1 pr2 = 1 (3.14) ( 7.4 )2 = 21.49 cm2 2 2 2 1/2 pr2 = 1 (3.14) ( 3.1 )2 = 3.77 cm2 2 2 15.54 + 22.94 + 21.49 + 3.77 = 63.74 cm2

Spectrum Critical Thinking for Math Grade 7

Lesson 5.7 Area of Composite Shapes

68

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Answer Key Page 69 NAME

Lesson 5.8

Page 70 NAME

________________________________________________________________________________

Volume of Rectangular Prisms

A triangular prism is a prism whose base is a triangle. The volume of a triangular prism is the product of the area of the base and the height of the prism.

Volume is the amount of space an object occupies. The volume of a rectangular prism can be calculated using the formula V  bh, where b  area of the base and h  height. The area of the base is b  lw, where l  length and w  width.

Volume  bh, where b  12 bh The triangular base has a height of 3 cm and a base of 8 cm. The height of the prism is 12 cm.

The volume of a rectangular prism is 210 cm3. If it has a length of 5 cm and a width of 6 cm, what is the height?

b  12 (8)(3)  12 (24)  12 cm2 V  (12)(12)  144 cm3

V  210 cm3 210  (5)(6)h 210  30h h  70 cm

Find the volume of each figure. Show your work.

Penny is using colored sand to fill a jar that is shaped like a rectangular prism. The bag of sand contains 150 cubic inches. The base of the prism is 6.5 inches by 7.4 inches. The height of the box is 2.2 inches. Will all the sand fit in the jar?

3

V = 6.5  7.4  2.2 = 105.82 in. 150 – 105.82 = 44.18 cubic inches of the sand will not fit into the jar.

6 cm

12 4 cm

4 cm

Volume of rectangular prism = lwh

4 cm 12.5 cm 3 cm

Spectrum Critical Thinking for Math Grade 7

Spectrum Critical Thinking for Math Grade 7

Lesson 5.8 Volume of Rectangular Prisms

Page 71 NAME

Lesson 5.9 Volume of Triangular Prisms

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Check What You Learned

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Check What You Learned

Geometry

Geometry

No, this cannot be done because the cross-products are not equal.

2. If a triangle XYZ has two sides with lengths of 5 cm and 8 cm, what is the maximum and minimum length of the third side? Explain your answer. Assume that the length of the third side is an integer.

5+x>8 –5 –5 x>3

x+8>5 –8 –8 x > –3

The maximum length is 12 cm, and the minimum length is 4 cm.

CHAPTER 5 POSTTEST

4 = 20 6 25 100 ≠ 120

4. A mini pancake has a circumference of 3 centimeters. A regular pancake has a circumference of 6 centimeters. Is the area of the regular pancake twice the area of the mini pancake? Use 3.14 for .

C = 2r 3 = 2r 3 = 2r 2 2 r = 3 cm 2 2 A = r 2 A = ( 3 ) 2 A = 9  = 7.065 cm2 4

CHAPTER 5 POSTTEST

1. At the photo lab, a customer brings in a photograph that is 4 inches wide by 6 inches high. The customer wants the photograph enlarged to 20 inches wide by 25 inches high. Can this be done? Explain your reasoning.

C

5

A

1

= 150 cm3

B

4 2 3

D

E

F

prism = bh

A cross section that is parallel to the base (the base is the triangle) will form a triangle. A cross section that is perpendicular to the base will form a rectangle. Spectrum Critical Thinking for Math Grade 7

Chapter 5 Check What You Learned 71

C = 2r 6 = 2r 6 = 2r 2 2 r = 3 cm A = r2

No, the area of the larger pancake is 4 times the area of the mini pancake.

A = (3)2 A = 9 = 28.26 cm2

5. If 4 is a right angle and 5  40°, find the measure of the remaining angles.

3. What are two possible shapes that can be formed by a cross section of this shape? Describe the angle of the cross section.

5 = 225 cm3

cm

(4)(3)(12.5) = 150 cm3

966 = (4)(14)w 966 = 56w 966 = 56w 56 56 17.25 ft. = w

6)(12.5) = 75 cm3

V = bh b = 1 bh 2 b = 1 (6)(4) = 12 cm2 2 V = (12)(12) = 144 cm3

Volume of triangular prism = bh b = 1 (3)(4) = 6 = (6)(12.5) = 75 cm3 2 Total area = 150 + 75 = 225 cm3

A rectangular prism has a volume of 966 ft3. The prism’s height is 4 feet, and its length is 14 feet. What is its width?

ar prism

12 cm

3 cm 8 cm

Answer the questions. Show your work.

5+8>x 13 > x

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Volume of Triangular Prisms

Lesson 5.9

Spectrum Critical Thinking for Math Grade 7

3 + 4 + 5 = 180 3 + 90 + 40 = 180 3 = 50° 5 + 1 = 180 40 + 1 = 180 1 = 140° 1 + 2 = 180 140 + 2 = 180 2 = 40° Chapter 5 Check What You Learned

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CHAPTER 6 PRETEST

Check What You Learned Geometry 6. Find the volume of the figure.

V = sum of volume of each section (2 mm)(10 mm)(2 mm) = 40 mm3 (9 mm)(2 mm)(5 mm)= 90 mm3

2 mm

10 mm

5 mm 2 mm 2 mm

Statistics 1. Are the following samples biased or random? Explain your answer. a. Wendy wants to find out the favorite sports of the students at her school. She asks 25 students who were at the basketball team tryouts.

This is a biased sample. The answers are more likely to be “basketball” since she is asking people who are at basketball tryouts.

CHAPTER 5 POSTTEST

40 mm3 + 90 mm3 = 130 mm3

b. Jake wants to know how many students are interested in buying a yearbook this year. He used a random number generator to randomly select 25 students from each grade level.

This is a random sample. Technology is used to randomly select students.

9 mm

7. Bill wants to fill this triangular prism need?

2 3

2. The graph represents a sample of football players’ heights. If there were 100 players, how many players could be expected to be 70 inches tall?

full of water. How much water does he

V = bh b = 1 bh 2 b = 1 (12)(16) = 96 m2 2 V = 96 m2 (20 m) = 1920 m3

16 m

Football Players’ Heights (in.) Number of Players

20 m

12 m

If you only want to fill this up 2 of the 3 way, you will need: 2 (1920 m3) = 1280 m3 3 Spectrum Critical Thinking for Math Grade 7

2

1

62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78

Heights (in.)

Spectrum Critical Thinking for Math Grade 7

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Page 76 CHAPTER 6 PRETEST

Check What You Know

3 . A sample of people were asked how far they drive to work . What percentage of people drive 6 miles to work? Round your answer to the nearest tenth of a percent .

2

3

4

5

6

7

8

9

NAME

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Statistics

1

Chapter 6 Check What You Know

74

Page 75 NAME

3 = x 10 100 3  10 = 30 10  10 100 x = 30 players

3

0

Chapter 5 Check What You Learned

Distance to Work from Home

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Check What You Know

10

5 = x 13 100 13x = 500 13x = 500 13 13 x = 38 .5%

Lesson 6.1

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Sampling and Drawing Inferences

When a population has a large number of data points, a sample can be taken to help summarize information and make inferences about the entire population. A random sample has individuals who are chosen by chance, and each member of the population has an equal chance of being included. In a biased sample, some members of the population are less likely to be chosen. Samples that are random are better predictors of trends for the bigger population. Rosewood Middle School has 714 students. Susan surveys a random sample of 34 students and finds that 9 of them play a sport outside of school. How many students at the school are likely to play a sport outside of school? 9  s ; 34s  (9) (714)  6426 34 714 34s  6426; s  189 34 34

4 . What can you infer from this data collected about the number of apps on a sample of smart phones?

189 students are likely to play sports outside of the school. 1

2 3 4

5 6 7

8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25

The median number of apps is 10 . The minimum is 1 and the largest number is 25 . The middle 50% have between 5 and 18 apps .

5 . A factory produces 92,000 tubes of toothpaste each day . The quality manager claims that fewer than 750 defective tubes are produced each day . In a random sample of 420 tubes of toothpaste, 3 are defective . Is the quality manager’s claim correct? Explain your answer .

x = 3 92000 420 420x = 276,000 x = 657 .14

Spectrum Critical Thinking for Math Grade 7

The manager’s claim is correct . There are likely to be about 657 defective tubes of toothpaste a day . Chapter 6 Check What You Know 75

Answer the questions. Show your work. A high-tech company makes 3,500 widgets a day. The quality department chooses a random sample of 50 widgets and finds that 3 are defective. How many high tech widgets per day are likely to be defective?

10500 3 = x 50x = 3(3500) 50x = 50 3500 50 50 x = 210 50x = 10500

Grace hears that the average gas price has risen to $2.89 during the gas shortage. She checks gas prices at stations near her school, and finds that the average is $3.20. Why are the averages different?

The averages are different because Grace’s sample was biased. She did not randomly select the gas stations in the city. She only took data from the gas stations that were in her area. Spectrum Critical Thinking for Math Grade 7

Lesson 6.1 Sampling and Drawing Inferences

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Comparing Similar Data Sets

Lesson 6.2

Lesson 6.2

What can you infer from the two histograms?

Class 1: Number of People in Household Class 2: Number of People in Household

In class 1, no students were shorter than 34 inches or taller than 46 inches . In class 2, the range of heights is 25 inches, but the range in class 1 is just 12 inches . The median for both classes is 42 inches . 50% of the students in class 1 are between 42 and 46 inches . 12-

10

84

40-

6-

6

62-

8-

0 30 34

0 38

42

46

4

4

0-

50

30

35

Heights of Class #1

40

45

50

55

Heights of Class #2

54

4-

4 3

8-

3

210-

0

0

4-

0-

150 200 250 300 350 400 450

Weights of Chickens: Soybean Diet (dkg)

Class 1: Teacher Donations to Charity Fund Class 2: Student Donations to Charity Fund

2

20

1

1

1

150 200 250 300 350 400 450

Weights of Chickens: Sunflower Diet (dkg)

Sunflowers: There are 12 values . The median value will be between 300 and 350 .

What percentage of the chickens are between 300 dkg and 350 dkg for each type of feed? Round your answers to the nearest percent .

The teachers donated more to charity. They had a higher mean and median with a mode of $20.

3 7 soybean: = .21 = 21% sunflower: = 0 .58 = 58% 14 12 Spectrum Critical Thinking for Math Grade 7

Lesson 6 .2 Comparing Similar Data Sets 77

Spectrum Critical Thinking for Math Grade 7

NAME

Page 80 NAME

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Comparing Similar Data Sets

Lesson 6.3

Box-and-whisker plots can help you interpret the distribution of data. Each section of a box and whisker plot contains 25% of the data points. Active time data is collected from a group of high school students and a group of elementary students.

54

56

58

60

62

64

66

68

70

72

74

76

78

The population is the 180 students who were invited. This is not a random sample. She only chose people who lived in her neighborhood. This could be biased because these students may be more likely to come to the party because they live closer to her brother. These results are likely too high.

The double box and whisker plot shows that the high school students are overall less active with a median of 59 minutes a day. The middle 50% of students sampled are active between 54 and 63 minutes a day. The elementary students are more active. The median is 64 minutes a day. The middle 50% exercise between 56 and 72 minutes. This double box-and-whisker plot displays the test scores of students who studied alone and the scores of students who studied with a study group. Use it to compare the data sets.

b. Tameka decides to look at the graduation program and call every 10th person on the list of graduates to see if they plan to come. She calls 18 people and 8 of them say that they will be able to come. How many people can she expect to come to the party?

w/o study group with study group 10

15

20

25

30

35

40

45

50

55

60

65

70

75

80

85

90

95

8 = x 180 18 18x = 1440  x = 80

100

The students who studied alone had a median of 75 for their test grade. The grades ranged from 55 to 95. The middle 50% made between a 62 and an 85. The students who studied in a group had a median of 90 with grades that ranged from 65 to 95. The middle 50% made between an 82 and 92. I can infer that the students that studied in a group did better. Spectrum Critical Thinking for Math Grade 7

Data in the Real World

a. Tameka decides to ask the 20 students who live in her neighborhood. 12 of them say that they will be able to come to the party. What is the population in this event? Is this a random sample? Could this sample be biased? Are these results too low or too high? Explain.

Elementary Students 52

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Tameka is planning a party for her brother. She invites 180 of the people in her brother’s class. When she sent out invitations, she listed the wrong phone number for RSVP, so she will not be getting any responses. She is trying to figure out how many people are planning to come to the party.

High School Students

50

Lesson 6.2 Comparing Similar Data Sets

78

Page 79 Lesson 6.2

2 3 7 8 10 11 12 14 15 20 17 20 14 12 11 12 20 20 20 20 1 2 5 7 8 9 1 11 14 15 17 19 19 17 11 8 2 2 11

mean = 2 + 3 + 7 + 8 + 10 + 2(11) + 3(12) + 2(14) + 15 + 17 + 6(20) = 13.4 20 median = 13 mode = 20 mean = 2(1) + 3(2) + 5 + 7 + 2(8) + 9 + 3(11) + 14 + 15 + 2(17) + 2(19) = 9.55 19 median = 9 mode = 2, 11

In which range will the median occur for each diet?

Soybeans: There are 14 values . The median value will be between 200 and 250 .

Class 2 Mean: 3.8; Median: 4 Mode: 4; Range: 3 This data is more compact and closer to the center. There is less variety in sizes. Four is the most common size.

Find the mean, median, and mode of each set of data. How do the data sets compare?

7

6Frequency

Frequency

3-

3, 5, 4, 4, 5, 4, 3, 2, 4, 4

Class 1 Mean: 4; Median: 4 Mode: 2, 4, 5; Range: 7 This data is spread out fairly evenly between 1 and 8. There are a variety of household sizes in this class.

1

2-

8, 2, 5, 5, 3, 1, 6, 2, 4, 4

Find the mean, median, and mode of each set of data. How do the data sets compare?

6

5

4-

Frequency

Frequency

10-

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Comparing Similar Data Sets

Lesson 6.2 Comparing Similar Data Sets 79

She can expect 80 people to come to the party.

c. Is this a random sample? Could this sample be biased? Compare these results to the results from the first sample.

This is a random sample. This is more likely not to be biased. This is a lower number of expected guests than the first sample. This is more realistic since it was a random sample. Spectrum Critical Thinking for Math Grade 7

Lesson 6.3 Data in the Real World

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Check What You Learned Statistics

Statistics

1 . Will wants to survey a sample of students at his school to find out how many play musical instruments . He surveys students coming out of band class . Is Will’s sample biased or random? Why?

4. The graph shows a sample of heights of plants that were grown with no fertilizer and plants that were grown with fertilizer. What can you infer from the box and whisker plots? (no fertilizer) (w/fertilizer)

This is a biased sample . Everyone coming out of band class plays an instrument . 6

5

5

4

4

3

3

2

2

1

CHAPTER 6 POSTTEST

6

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

CHAPTER 6 POSTTEST

2 . The graph shows a sample of heights of sixth graders and eighth graders . Compare the data . What can you infer?

0

The plants that were grown with no fertilizer were shorter and more consistent in height. The median height was about 4.4 inches, with the middle 50% being between 4 and about 5.4 inches tall. The plants grown with fertilizer are taller. The median was 9 inches, with the middle 50% being between about 8.1 and about 10.4 inches.

1 0

50-59 60-69 Heights of 6th Graders

5. The graph shows a sample group of girls and a sample group of boys, and the number of books they read during the school year. If there are 200 boys and 200 girls at the school, how many girls and boys read 10 books?

50-59 60-69 70-79 Heights of 8th Graders

The median height for 6th graders is between 50 and 59 inches . There are no students taller than 69 inches . The median height for 8th grade students is between 60 and 69 inches . About 43% of the 8th graders are taller than 69 inches .

0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20

# of books read during school year (boys)

3 . A factory produces 74,000 sets of headphones each day . The quality manager claims that fewer than 600 defective tubes are produced each day . In a random sample of 310 sets of headphones, 3 are defective . Is the quality manager’s claim correct? Explain your answer .

x = 3 74000 310 x = 716 .1

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Check What You Learned

0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20

# of books read during school year (girls)

310x = 222000

Boys: 2 = x 14x = 400 Girls: 1 = x 14x = 200

14

The manager’s claim is incorrect . There are likely to be about 716 defective sets of headphones a day . Spectrum Critical Thinking for Math Grade 7

200

82

Page 83

Page 84 NAME CHAPTER 7 PRETEST

Probability 1. Of the 50 U.S. states, 13 were the original colonies. If you select 1 state randomly, how likely is it to be one of the original colonies?

13 = 26% 50

CHAPTER 7 PRETEST

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Check What You Know

Check What You Know Probability 5 . The school picnic is a two-day weekend event . It has been scheduled for May . The area routinely gets 16 rainy days in May . What is the probability that the weekend will be dry? Round your answer to the nearest percent .

6 . In basketball, Alan makes 1 out of every 4 free throws he tries . What is the probability that Alan will make his next 3 free throws? Round your answer to the nearest tenth of a percent .

P(miss) = 1 – 6 = 14 = 70% 20 20

P(three free throws) = 1  1  1 = 1 = 1 .6% 4 4 4 64 1 .6% chance

3. Evan hits 6 out of 14 pitches during practice. What does an experimental probability of 47 describe?

P(misses pitch) It is the probability that he misses a pitch. 4. At Luvski Ski Resort, there are two chair lifts to the top of the mountain. There are five ski trails to the bottom of the mountain. What is the probability of riding on Chair 1 and skiing on Trail 3?

Chair 2

T1 T2 T3 T4 T5

7 . Gregg has 12 cards . Half are black, and half are red . He picks 2 cards out of the deck . What is the probability that both cards are red?

P(red, red) = 1  1 = 1 = 25% 2 2 4 25% chance

8 . Lucy places 5 cards face down on the table and mixes them up . The cards are numbered 1 through 6 . What is the likelihood that her friend Harry will draw an even-numbered card?

P(even) = 3 = 50% 6 50% chance

P(chair 1, trail 3) = 1 = 10% 10 Spectrum Critical Thinking for Math Grade 7

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15  15 = 225 = 23% 31 31 961

2. David takes 20 shots and scores 6 goals at soccer practice. What is the experimental probability that he will miss his next shot?

Chair 1

Chapter 6 Check What You Learned

Spectrum Critical Thinking for Math Grade 7

81

T1 T2 T3 T4 T5

200

x = 28.6; about 29 boys x = 14.3; about 14 girls

Chapter 6 Check What You Learned

NAME

14

Chapter 7 Check What You Know 83

Spectrum Critical Thinking for Math Grade 7

Chapter 7 Check What You Know

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Page 86

Understanding Probability

Lesson 7.1

Lesson 7.2

The probability of an event measures the likelihood that the event will occur.

ssible

ely

Unlik

Impo

e lly Lik

1 4

0 0%

1 2

0.25 25%

0.50 50%

in

Likely

Certa

3 4

1

0.75 75%

100%

Frequency Tables

Marvin has a bag of marbles. He removes a marble, records the color, and then puts the marble back in the bag. The frequency table shows how many times he picked each color. Color Purple Pink Orange White

The complement of an event is the set of all outcomes not included in the event. Answer the questions.

Find the experimental probability for each color.

Frequency 12 10 15 13

P P P P

What is the sum of the probabilities of an event and its complement?

The sum is 1. The event has a 100% chance of either occurring or not occurring.

It is as likely as not likely to pick a white piece of paper out of the hat.

Allowance

# of students

$15.00

9

$20.00

11

$25.00

12

$30.00

8

Where would the probability of picking a white piece of paper fall on the number line if there were 6 pieces of white paper and 2 pieces of black paper in the hat?

The probability of picking a white piece of paper would be likely. Lesson 7.1 Understanding Probability 85

Lesson 7.2

86

Lesson 7.3

Calculating Probability

P (event) 

P(1) = 21 = 26.9% 78 22 P(2) = = 28.2% 2 22 78 3 18 4 17 P(3) = 18 = 23.1% 78 P(4) = 17 = 21.8% 78 What is the probability of not spinning a 3? P(not 3) = 1 – 0.231 = 0.769 = 76.9% 21

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Theoretical probability is the probability of an event occurring based on all the possible outcomes. Theoretical probability can be calculated this way:

A spinner with 4 equal sections was spun 78 times. Use the frequency table to calculate the experimental probability of spinning each number. Show your work. Round your answer to the nearest tenth of a percent. Frequency

NAME

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Answer the questions.

1

Lesson 7.2 Frequency Tables

Page 88

Frequency Tables

Number on Spinner

P($15) = 9 = 22.5% 40 P($20) = 11 = 27.5% 40 P($25) = 12 = 30% 40 P($30) = 8 = 20% 40

Spectrum Critical Thinking for Math Grade 7

Page 87 NAME

(purple)  12  24% 50 (pink)  10  20% 50 (orange)  15  30% 50 (white)  13  26% 50

Students at Prince Middle School were asked about their weekly allowance. Use the frequency table to calculate the experimental probability for each amount. Show your work.

Students in Ms. Baldwin’s class are picking numbers out of a hat. The hat has 8 pieces of paper. Four pieces of the paper are black, and the other pieces are white. Where does the probability of picking a white piece of paper out of the hat fall on the number line above?

Spectrum Critical Thinking for Math Grade 7

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The experimental probability of an event is found by comparing the number of times the event occurs to the total number of trials. A frequency table is used to keep track of the trials.

ly

like ly/Un

Equa

NAME

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number of ways the event can occur total number of possible outcomes

A spinner has 3 equally sized sections labeled A, B, and C. What is the probability that your spinner landed on section A? There are 3 possible outcomes, with one of them being A. P (A) 

1 3

A bag of marbles contains 5 green marbles, 8 red marbles, and 9 yellow marbles. Ella chooses one marble at random from the bag. What is the probability that she picks a green marble? Round your answer to the nearest tenth of a percent.

P(green) = 5 = 22.7% 22

A coin was flipped 60 times. The experimental probability of each outcome is shown in the table below. Coin Lands On

Frequency

Heads

27

Tails

33

27

P (heads)  60  45% P (tails)  33  55% 60

What is the probability that she does not pick a red marble? Round your answer to the nearest tenth of a percent.

Is this the probability that you expected? Compare the results to your expectations.

Since there were 2 possible outcomes, a 50% was expected for each side of the coin. This did not happen because this is experimental probability. Spectrum Critical Thinking for Math Grade 7

Lesson 7.2 Frequency Tables 87

P(not red) = 1 – 8 = 1 – .364 = .636 = 63.6% 22

Spectrum Critical Thinking for Math Grade 7

Lesson 7.3 Calculating Probability

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Lesson 7.4

Page 90 NAME

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Probability Models

Lesson 7.5

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Other Probability Models

When all outcomes of an experiment are equally likely, the event has uniform probability . This probability can be used to predict outcomes .

When a probability event has unequal odds, the outcomes are not equally likely to occur.

Vick rolls a number cube . What is the probability that he rolls a prime number? If he rolls the number cube 30 times, how many times is he expected to roll a prime number?

A spinner has 4 equal sections. 2 of the sections are yellow, one of the sections is purple, and the other section is green. What is the probability that the spinner lands on yellow? P (yellow)  24  50%

A number cube has 3 prime numbers (2, 3, 5) . There are 6 possible outcomes . P (prime number)  36  50% 0 .5  30  15 There is a 50% chance of rolling a prime number . If Vick rolls the number cube 30 times, it is expected that he will roll a prime number 15 times

What is the probability of not spinning purple?

P (not purple)  1 14 

 75%

Answer the questions. Show your work. Round your answers to the nearest tenth of a percent.

Answer the questions . Show your work .

A grocery store randomly selects an item to be on sale each day

A spinner has 20 equal sections, numbered 1 through 20 . a . What is the probability that the spinner will land on a multiple of 3?

P(multiple of 3) = 6 = 30% 20 There are 6 multiples of 3 between 1 and 20 (3, 6, 9, 12, 15, 18)

b . If the spinner is spun 42 times, how many times can it be expected to spin a multiple of 3?

There are 5 multiples of 4 (4,8,12,16,20) . P(not multiples of 4) = 1 – 5 20 1 – 5 = 15 = 75% 20 20

Spectrum Critical Thinking for Math Grade 7

Item

# of Days on Sale

Ice Cream

4

Oranges

5

Chicken

3

Chips

5

Eggs

4

a. What is the probability that the item on sale will be ice cream or chips?

0 .3  42 = 12 .6; about 13 times

P(ice cream or chips) = 4 + 5 = 9 = 42.9% 21 21

c . What is the probability that it will not spin a multiple of 4?

b. What is the probability that oranges or chicken will not be on sale?

P(no oranges or chicken) = 1 – 5 + 3 = 1 – 8 = 61.9% 21 21

Lesson 7 .4 Probability Models 89

Spectrum Critical Thinking for Math Grade 7

NAME

Lesson 7.5 Other Probability Models

90

Page 91 Lesson 7.5

3 4

Page 92 NAME

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Theoretical vs . Experimental Probability

Lesson 7.7

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Understanding Compound Events

Theoretical probability is what is expected to happen based on likely outcomes . Experimental probability is what actually happens .

When two or more things are happening at one time in an experiment, it is a compound event . The probability of each event is multiplied .

Suppose you toss a coin 25 times, and it lands tails up 11 times . Compare the experimental probability and the theoretical probability .

What is the probability of rolling a 2 and then a 6 when rolling a number cube twice? 1 P (2)  6 P (6)  16 1 P (2, then 6)  16  16  36

1

Theoretical probability: 2  50% 11 Experimental probability: 25  44% The experimental probability is less than the theoretical probability . It is impossible to meet the experimental probability because there are an odd number of coin tosses .

A standard spinner is arranged so that the numbers 1 to 15 share equal space .

Thomas spins a spinner 40 times . The results are shown in the table . Based on the results of the experiment, use your best guess to draw the spinner . Number

a . What is the probability of getting a 9 on two consecutive spins?

P(two consecutive numbers) = 1  1 = 1 = 0 .4% 15 15 225

Frequency

1

9

2

11

3

12

4

8

b . What is the probability of not getting a 9 on two consecutive spins?

P(1) = 9 = .225 .225(360°) = 81° 40 P(2) = 11 = .275 .275(360°) = 99° 40 12 P(3) = = .30 .30(360°) = 108° 40 8 P(4) = = .20 .20(360°) = 72° 40 All of the experimental probabilities are close to 25% . This indicates that each section of the spinner is 1 of the area . 4 Spectrum Critical Thinking for Math Lesson 7 .6 Grade 7

Answer the questions . Show your work . Round your answers to the nearest tenth of a percent .

Theoretical vs . Experimental Probability 91

P(not two consecutive numbers) = 1 – 1 = 224 = 99 .6% 225 225 What is the probability of rolling a 2 on a standard number cube and then getting heads on a coin toss?

P(2, heads) = 1  1 = 1 = 8 .3% 6 2 12 What is the probability of not rolling a 6 on a number cube and then getting heads on a coin toss?

P(not 6, then heads) = 5  1 = 5 = 41 .7% 6 2 12 Spectrum Critical Thinking for Math Grade 7

Lesson 7 .7 Understanding Compound Events

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Lesson 7.7

NAME

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Understanding Compound Events

Lesson 7.8

Answer the questions . Show your work . Round your answers to the nearest tenth of a percent .

The Fundamental Counting Principle says that when there are m ways to do one thing, and n ways to do another, then the product of m and n is the possible number of outcomes for both events . A tree diagram can help you visualize this .

A retail store is having a contest . The randomly selected prize will be a can opener, a gift card, or a set of towels . The store cashier will spin a spinner with the numbers 5–8 to see whether every 5th, 6th, 7th, or 8th customer will win a prize .

An ice cream shop offers vanilla, strawberry, and chocolate ice cream . A customer can choose a regular cone, a sugar cone, or a cup . What is the probability of getting strawberry ice cream on a sugar cone?

3 3 3  9, so there are nine possible outcomes .

S

There is one possible combination of strawberry ice cream and sugar cone .

C

P (strawberry sugar cone)  19  11%

a . Create a tree diagram to show all the possible outcomes in this situation .

Reg Sugar Cup Reg Sugar Cup Reg Sugar Cup

V

There are 3 flavors and 3 serving options .

5th 6th

Answer the questions . Show your work . Round your answers to the nearest tenth of a percent .

can opener gift card set of towels can opener gift card set of towels

can opener gift card set of towels can opener gift card set of towels

7th 8th

b . What is the probability that every 5th person will win a can opener or a gift card?

P(5th, can opener or giftcard) = 2 = 16 .7% 12

A salad bar has croutons, raisins, sunflower seeds, and cranberries available as toppings . Teresa wants 2 different toppings on her salad . How many possible 2-topping combinations can Teresa choose? What is the probability of having croutons and sunflower seeds on her salad?

c . What is the probability that every 6th or 7th person will win a set of towels?

1 = 8 .3% raisins sunflower seeds 12 cranberries croutons raisins sunflower seeds cranberries croutons sunflower seeds raisins cranberries croutons cranberries raisins Spectrum Critical Thinking for Math Lesson 7 .7 sunflower seeds Understanding Compound Events Grade 7

P(6th or 7th, towels) = 2 = 16 .7% 12

croutons

d . What is the probability that a customer will not win a can opener?

P(not can opener) = 8 = 66 .7% 12 Spectrum Critical Thinking for Math Grade 7

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Lesson 7 .8 Probability in the Real World

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Probability in the Real World

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Check What You Learned

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Check What You Learned

Probability

Probability

Answer the questions . Show your work .

Answer the questions . Show your work . Round your answers to the nearest tenth of a percent .

1 . An auto company conducted a survey with a random sample of 500 people to find out which type of vehicle they preferred to drive . The results are shown below .

75

Sedan

45

SUV

95

Pickup

90

Station Wagon

95

Minivan

100

4 . Every seventh-grade student is eating in the cafeteria . Juwarne is a seventh-grade student . How likely is it that she is in the cafeteria?

It is certain that she is in the cafeteria .

P(miss) = 1 – 15 = 5 = 25% 20 20

95 = 0 .19 500

b . If 1,500 people were surveyed, how many would you expect to prefer to drive an SUV? Explain your answer .

6 . At the barbershop, there are 2 chairs for customers to wait in . There is a rack with 5 magazines for customers to read while they wait . How many possible choices of chairs and magazines do the barbershop customers have?

19% of 1,500 people is 285 people .

2 . Mr . Rose randomly selects names to see who will give the first book report . There are 10 boys and 14 girls in his class . What is the probability that he will select a girl’s name?

P(girl) = 14 = 58 .3% 24 Spectrum Critical Thinking for Math Grade 7

4 = 7 .1% 56

5 . Kobe makes 15 of 20 free throws at basketball practice . What is the experimental probability that he will miss his next free throw?

a . What is the probability that a randomly selected survey participant prefers to drive an SUV? Write it as a decimal .

0 .19  1500 = 285

CHAPTER 7 POSTTEST

Number of People

Compact

CHAPTER 7 1 POSTTEST

Favorite Vehicle

3 . Of the original 56 signers of the Declaration of Independence, 4 represented North Carolina . If you selected 1 signer randomly, how likely is it that he represented North Carolina?

1st

M1 M2 M3 M4 M5

2nd

M1 M2 M3 M4 M5

There are 10 possibilities . Chapter 7 Check What You Learned 95

Spectrum Critical Thinking for Math Grade 7

Chapter 7 Check What You Learned

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Final Test

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Chapters 1–7

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Chapters 1–7

Final Test

Answer the questions . Show your work .

Answer the questions . Show your work .

1 . These temperature changes in a vat of liquid were noted by a scientist performing a chemical experiment . What was the net temperature change from the first Monday to the second Monday?

3 . The chart shows the high and low temperature in Anchorage for a week .

Monday Tuesday Wednesday Thursday Friday Saturday Monday

Temperature in Anchorage (°F)

4 .6 °C 210 .2 °C 20 .3 °C 223 .5 °C 4 .2 °C 214 .4 °C 226 .9 °C

Sun

Mon

Tues

Wed

Thurs

Fri

High





26°

27°



215°



Low

28°

212°

221°

217°

215°

225°

218°

a . Find the average of the high temperatures . Round your answer to the nearest tenth of a degree .

3 + 5 + (–6) + (–7) + 2 + (–15) + 1 = –2 .4°F 7

–26 .9 – 4 .6 = –31 .5°C 2 . Serena took care of Jason’s large fish tank while he was on vacation . The tank lost water through evaporation, and Serena added more water as shown in the table . In total, how much water will be gained or lost by the time Jason returns from vacation? Water Added (in quarts)

Mon .

3 4

5 8

Tue .

1 2

7 8

Wed .

5 8

1 2

4 . The terms 8x, 5z, 15y, z, 2x and another term are added to form an expression . When simplified, this expression equals 2 (3z  5x) . Identify the missing term and write the expression .

– 3 + 5 – 1 + 7 – 5 + 1 = 1 quarts 4 8 2 8 8 2 8 Spectrum Critical Thinking for Math Grade 7

(–8) + (–12) + (–21) + (–17) + (–15) + (–25) + (–18) = –16 .6°F 7

Chapters 1–7 Final Test

CHAPTERS 1–7 FINAL TEST

Water Lost (in quarts)

b . Find the average of the low temperatures . Round your answer to the nearest tenth of a degree .

CHAPTERS 1–7 FINAL TEST

Day

97

2(3z + 5x) = 6z + 10x 5z + z + 15y + 8x + 2x = 6z + 15y + 10x The missing term is –15y .

Spectrum Critical Thinking for Math Grade 7

NAME

Chapters 1–7 Final Test

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Final Test

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Chapters 1–7

Answer the questions . Show your work .

Answer the questions . Show your work .

5 . The circumference of a circular garden is 42 meters . A gardener is digging a straight line along the diameter of the garden at a rate of 10 meters per hour . How many hours will it take the gardener to dig across the garden? Use 3 .14 for  . Round your answer to the nearest hundredth .

8 . A college football stadium holds 25,000 fans . In a random sample of 30 fans, 26 were wearing the colors of the home team . Predict the number of fans who are wearing the colors of the home team .

C = d 42 = 3 .14d d =13 .38 m

26 = x 30x = 650000 30 25000 30 30 30x = (26)(25000) x = 21666 .67 30x = 650000 Approximately 21,667 fans are predicted to be wearing colors of the home team .

rt = d 10t = 13 .38 t = 1 .34 hours

6 . Thomas spins a spinner 25 times . The results are shown in the table . Based on the results of the experiment and your best guess, how does the size of the section containing #5 compare to the size of the section containing #6? Number

Frequency

1

2

2

4

3

1

4

8

5

2

6

8

P(5) = 2 = 8% 25 P(6) = 8 = 32% 25 Section #6 is 4 times as large as Section #5 .

9 . If it takes Joe 15 hours to make 3 cornhole boards, how long will it take him to make 11 cornhole boards?

15 = x 11 3 3x = 165

It will take 55 hours to make 11 cornhole boards . 10 . Jack bought 4 turkey sandwiches and 2 bags of apple slices for $22 .60 . If the apple slices cost $0 .75 per bag, how much did each sandwich cost?

Chapters 1–7 Final Test 99

CHAPTERS 1–7 FINAL TEST

No, it depends on whether the absolute value of j or the absolute value of k is larger . If the absolute value of j is larger, then the sum is positive . If the absolute value of k is larger, then the sum is negative .

CHAPTERS 1–7 FINAL TEST

7 . A number j is positive and another number k is negative . Based on this information, can you determine whether j  k is positive or negative? Explain .

Spectrum Critical Thinking for Math Grade 7

3x = 165 3 3 x = 55

4t + 2(0 .75) = 22 .6 4t + 1 .5 = 22 .6 – 1 .5 –1 .5 4t = 21 .1 4 4 Spectrum Critical Thinking for Math Grade 7

t = 5 .275 Each sandwich is about $5 .28 . Chapters 1–7 Final Test

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Answer Key Page 101 NAME

Final Test

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Chapters 1–7

Answer the questions. Show your work. 11. Identify the mistake that was made in simplifying the expression. Then, correctly simplify the expression.

5(a – 3) + (6a + 12) – 7a = 5a – 15 + 6a + 12 – 7a = (5a + 6a – 7a) + (–15 + 12) = 4a – 3 The 5 was added to the –3 instead of multiplied when the 5 was distributed. 5 (a  3)  (6a  12)  7a  5a  2  6a  12  7a  (5a  6a  7a)  (2  12)  4a  10

12. On a road map, the distance between two cities is 12.6 centimeters. What is the actual distance if the scale on the map is 2 cm:50 mi. How long would it take a driver traveling 70 miles per hour to go from one city to the next city?

2x = 630 2 2 x = 315 The cities are 315 miles apart.

2 cm = 12.6 x 50 mi. 2x = (50)(12.6) 2x = 630

13. Several puppies from 2 different breeds were weighed. The puppies’ weights in pounds are shown in the table. What can you infer from the data?

1

2

3

4

5

6

7

8

9

10 11

12

13 14

15 16

17 18

Breed B is larger than Breed A. Breed B also has a larger range of weights than Breed A. Spectrum Critical Thinking for Math Grade 7

Chapters 1–7 Final Test

CHAPTERS 1–7 FINAL TEST

Breed A Breed B

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Notes

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Notes

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Critical Thinking for Math

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