STEMscopes Georgia Math Student Notebook Grade 7 by acceleratelearning

Grade 7 Student Notebook

GEORGIA

Student Notebook – Grade 7 ISBN: 978-1-64861-274-9 Published by Accelerate Learning Inc., 5177 Richmond Ave, Suite 800, Houston, TX 77056. Copyright © 2023 by Accelerate Learning Inc. All rights reserved. No part of this publication may be reproduced or distributed in any form or by any means, or stored in a database or retrieval system, without prior written consent of Accelerate Learning Inc., including, but not limited to, in any network or other electronic storage or transmission, or broadcast for distance learning.

To learn more, visit us at www.stemscopes.com.

GEORGIA

Student Notebook - Grade 7

Table of Contents Scope Name

Page Number

Addition and Subtraction with Rational Numbers

Multiplication and Division with Rational Numbers

Rational Number Operations

Proportional Relationships

Understand Slope

111

Ratios, Rates, and Percents

129

Percent Application

153

Expressions

177

Solve Equations and Inequalities

195

Scaling

217

Angles

229

Angle Relationships

237

Circles

247

Surface Area

269

iii

GEORGIA

Student Notebook - Grade 7

Table of Contents (Cont.) Scope Name

Page Number

Volume

281

Probability

297

Informal Inferences

313

Skills Quizzes

343

Glossary of Terms

433

Workspace

469

Addition and Subtraction with Rational Numbers

Explore 1

Name: _______________________ Date: ___________

Addition of Integers with Counters Find the number of positively charged protons and negatively charged electrons in each atom. Match yellow (proton) and red (electron) counters to determine if the atom has a zero charge. Check your answers by using the horizontal or vertical number line to represent the addition of positive and negative values, and identify atoms that have zero charge. Atom 1 Atomic Charge Number of electrons: _____

Counters: 10

9 8

Number of protons: _____

7 6 5

Does the atom have a zero charge? _____________

Write an equation to represent the charge of the atom.

3 2 1

________ + ________ = ________

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

Addition and Subtraction with Rational Numbers | 3

Addition and Subtraction with Rational Numbers

Explore 1

Atom 2 Atomic Charge Counters:

Number of electrons: ______ Number of protons: ______ Number line:

-10 -9

-8

-7

-6

-5

-4

-3

-2

-1

Does the atom have a zero charge? ___________ Write an equation to represent the charge of the atom. _______ + _______ = _______

Atom 3 Atomic Charge Counters:

Number of electrons: ______ Number of protons: ______ Number line:

-10 -9

-8

-7

-6

-5

-4

-3

-2

-1

Does the atom have a zero charge? ___________ Write an equation to represent the charge of the atom. _______ + _______ = _______ 4 | Addition and Subtraction with Rational Numbers

Addition and Subtraction with Rational Numbers

Explore 1

Atom 4 Atomic Charge Counters:

Number of electrons: ______ Number of protons: ______ Number line:

-10 -9

-8

-7

-6

-5

-4

-3

-2

-1

Does the atom have a zero charge? ___________ Write an equation to represent the charge of the atom. _______ + _______ = _______

Atom 5 Atomic Charge Counters:

Number of electrons: ______ Number of protons: ______ Number line:

-10 -9

-8

-7

-6

-5

-4

-3

-2

-1

Addition and Subtraction with Rational Numbers | 5

Addition and Subtraction with Rational Numbers

Explore 1

Atom 6 Atomic Charge Counters:

Number of electrons: ______ Number of protons: ______ Number line:

-10 -9

-8

-7

-6

-5

-4

-3

-2

-1

Does the atom have a zero charge? ___________ Write an equation to represent the charge of the atom. _______ + _______ = _______

Atom 7 Atomic Charge Counters:

Number of electrons: ______ Number of protons: ______ Number line:

-10 -9

-8

-7

-6

-5

-4

-3

-2

-1

Does the atom have a zero charge? ___________ Write an equation to represent the charge of the atom. _______ + _______ = _______ 6 | Addition and Subtraction with Rational Numbers

Addition and Subtraction with Rational Numbers

Explore 1

Atom 8 Atomic Charge Number of electrons: _____

Counters: 10

9 8

Number of protons: _____

7 6 5

Does the atom have a zero charge? _____________

Write an equation to represent the charge of the atom.

3 2 1

________ + ________ = ________

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

Addition and Subtraction with Rational Numbers | 7

Explore 1

Addition and Subtraction with Rational Numbers

Reflect 1. What is the charge of electrons? What is the charge of protons?

2. What did you notice about the number of electrons and protons when the charge of an atom was zero?

3. How did the counters help you see the equation and the answer?

4. How did the horizontal and vertical number lines help you see the equation and the answer?

5. How did you determine the equation?

6. In the real world, when do people add positive and negative numbers to see if the sum is zero? Give an example.

8 | Addition and Subtraction with Rational Numbers

Addition and Subtraction with Rational Numbers

Explore 2

Name: _______________________ Date: ___________

Rational Number Addition with Number Lines Part I Shufﬂe the cards, and put them in a pile facedown. Take turns drawing a card, reading it aloud, and deciding whether the real-world scenario results in a situation where opposites combine to make zero. Use the Horizontal Number Line or the Vertical Number Line and a dry-erase marker to help you solve the problems. Find the row for each card, write the addition equation, and circle “Yes” or “No.” Card

Addition Equation

______ + ______ = ______

Yes

______ + ______ = ______

Yes

______ + ______ = ______

Yes

______ + ______ = ______

Yes

______ + ______ = ______

Yes

______ + ______ = ______

Yes

______ + ______ = ______

Yes

______ + ______ = ______

Yes

______ + ______ = ______

Yes

______ + ______ = ______

Yes

______ + ______ = ______

Yes

______ + ______ = ______

Yes

Make Zero?

Addition and Subtraction with Rational Numbers | 9

Explore 2

Addition and Subtraction with Rational Numbers

Reflect 1. What were some action pairs in real-life situations that led to opposites combining to make zero?

2. How did you know you would be adding?

3. If the ﬁrst addend is a negative number, does the second addend need to be negative or positive to make zero? Why?

4. How did using the Horizontal Number Line or the Vertical Number Line help you determine the solution to each of the Alaska Cards?

10 | Addition and Subtraction with Rational Numbers

Addition and Subtraction with Rational Numbers

Explore 2 Part II

Color the word problem the same color as the corresponding equation and number line. Check your work with the Horizontal Number Line and the Vertical Number Line. Word Problem

Equation

Nukilik was ﬁshing. It was 6°C outside. Then, it dropped 12°C. What was the temperature?

−2.5 + (−7) = −9.5

An orca swam 2.5 feet below the surface, and then it dove down 7 feet. What was its depth?

9.75 + (−4.50) = 5.25

Nukilik had $9.75 in his store account. He made a purchase for $4.50. How much money was in his store account?

−3 2 + (−2.75) = −6.25

Nukilik owed his brother Panuk a dime. He found a nickel in his snowmobile and gave it to Panuk. How much did Nukilik still owe Panuk?

6 + (−12) = −6

Nukilik was at the beach. He dug a hole through ice 1 that was 3 2 feet below sea level. Panuk dug it 2.75 feet deeper. What’s its depth?

−10 + 5 = −5

Number Line

–10 –9 –8 –7 –6

–5 –4 –3 –2 –1

–10 –9 –8 –7 –6

–5 –4 –3 –2 –1

–10 –9 –8 –7 –6

–5 –4 –3 –2 –1

–10 –9 –8 –7 –6

–5 –4 –3 –2 –1

–10 –9 –8 –7 –6

–5 –4 –3 –2 –1

Addition and Subtraction with Rational Numbers | 11

Explore 2

Addition and Subtraction with Rational Numbers

Reflect 1. Which way do you move on the number line in each of the following situations?

• Negative number + negative number _________________________ • Negative number + positive number _________________________ • Positive number + negative number _________________________ • Positive number + positive number

_________________________

2. Can you tell if the answer will be positive or negative before you actually ﬁnd the sum of the numbers? Explain.

3. Temperature, altitude, and bank account balances are all examples of real-world situations that can have negative values. What are some examples of real-world situations that cannot have negative values? Explain.

12 | Addition and Subtraction with Rational Numbers

Addition and Subtraction with Rational Numbers

Explore 3

Name: _______________________ Date: ___________

Integer Subtraction with Counters and Number Lines One partner is the optimist; the other is the pessimist. Each partner rolls the dice each round. Even numbers are positive; odd numbers are negative. Subtract the second roll from the ﬁrst roll. Draw counters, and write corresponding equations to solve. If a solution is positive, the optimist gets points. If it is negative, the pessimist gets points. Total the points, and record them using tally marks. Optimist Points (+)

Draw the Counters

Pessimist Points (−)

_____ – _____ = _____

Addition and Subtraction with Rational Numbers | 13

Addition and Subtraction with Rational Numbers

Explore 3

For this portion of the game, continue to roll the dice and draw the counters. Then, draw the corresponding problem on the number line. Finally, write the equation, and record the points as tally marks. Optimist Points (+)

Mark the Number Line

Pessimist Points (−)

Counters:

Number line:

-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

_______ – _______ = _______ Counters:

Number line:

-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

_______ – _______ = _______ 14 | Addition and Subtraction with Rational Numbers

Addition and Subtraction with Rational Numbers

Explore 3 Optimist Points (+)

Mark the Number Line

Pessimist Points (−)

Counters:

Number line:

-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

_______ – _______ = _______ Counters:

Number line:

-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

Addition and Subtraction with Rational Numbers | 15

Addition and Subtraction with Rational Numbers

Explore 3

Take turns rolling the dice. Write corresponding equations for each turn. Use the Subtraction Counters Mat and counters or the Horizontal Number Line or Vertical Number Line and a dry-erase marker to ﬁnd solutions. Record points in the columns using tally marks. Total your points to ﬁnd the winner. Optimist Points (+)

Pessimist Points (−)

Write the Equation ______ – ______ = ______ ______ – ______ = ______ ______ – ______ = ______ ______ – ______ = ______ ______ – ______ = ______ ______ – ______ = ______ ______ – ______ = ______ ______ – ______ = ______ ______ – ______ = ______ ______ – ______ = ______ Total Scores Equations

Team

Counters Score

Number Line Score

Equations Score

Total

Optimist Pessimist

16 | Addition and Subtraction with Rational Numbers

Explore 3

Addition and Subtraction with Rational Numbers

Reflect 1. How did the counters help you visualize the solution?

2. What did you do when you did not have enough red or yellow counters to take away the number required?

3. How did the number line help you understand that adding a negative number is the same as subtracting a positive number?

4. Did you notice any rules or patterns that helped you successfully subtract numbers with different signs and values? Give examples.

Addition and Subtraction with Rational Numbers | 17

Addition and Subtraction with Rational Numbers

Explore 4

Name: _______________________ Date: ___________

Rational Number Subtraction with Number Lines Part I: Matching Place all 24 Matching Addition and Subtraction Cards facedown in a 6 × 4 array. Flip two cards over. If the expressions are equal, record the expressions on the number line, addition in blue and subtraction in green. Write an equation using both expressions and the solution. Keep the pair, and take another turn. If cards do not match, ﬂip the cards back over, and it is your partner’s turn. Card 1 Expression: ___________________

−10 −9 −8 −7 −6 −5 −4 −3 −2 −1 0

Card 2 Expression: ___________________

9 10

Equation: _____________ = _____________ Solution: _______ Expression: ___________________

−10 −9 −8 −7 −6 −5 −4 −3 −2 −1 0

Expression: ___________________

9 10

Addition and Subtraction with Rational Numbers | 19

Addition and Subtraction with Rational Numbers

Explore 4 Card 1 Expression: ___________________

−10 −9 −8 −7 −6 −5 −4 −3 −2 −1 0

Card 2 Expression: ___________________

9 10

Equation: _____________ = _____________ Solution: _______ Expression: ___________________

−10 −9 −8 −7 −6 −5 −4 −3 −2 −1 0

Expression: ___________________

9 10

Equation: _____________ = _____________ Solution: _______ Expression: ___________________

−10 −9 −8 −7 −6 −5 −4 −3 −2 −1 0

Expression: ___________________

9 10

Equation: _____________ = _____________ Solution: _______ 20 | Addition and Subtraction with Rational Numbers

Addition and Subtraction with Rational Numbers

Explore 4 Card 1 Expression: ___________________

−10 −9 −8 −7 −6 −5 −4 −3 −2 −1 0

Card 2 Expression: ___________________

9 10

Equation: _____________ = _____________ Solution: _______ Expression: ___________________

−10 −9 −8 −7 −6 −5 −4 −3 −2 −1 0

Expression: ___________________

9 10

Equation: _____________ = _____________ Solution: _______ Expression: ___________________

−10 −9 −8 −7 −6 −5 −4 −3 −2 −1 0

Expression: ___________________

9 10

Addition and Subtraction with Rational Numbers | 21

Addition and Subtraction with Rational Numbers

Explore 4 Card 1 Expression: ___________________

−10 −9 −8 −7 −6 −5 −4 −3 −2 −1 0

Card 2 Expression: ___________________

9 10

Equation: _____________ = _____________ Solution: _______ Expression: ___________________

−10 −9 −8 −7 −6 −5 −4 −3 −2 −1 0

Expression: ___________________

9 10

Equation: _____________ = _____________ Solution: _______ Expression: ___________________

−10 −9 −8 −7 −6 −5 −4 −3 −2 −1 0

Expression: ___________________

9 10

Equation: _____________ = _____________ Solution: _______ 22 | Addition and Subtraction with Rational Numbers

Addition and Subtraction with Rational Numbers

Explore 4 Card 1 Expression: ___________________

−10 −9 −8 −7 −6 −5 −4 −3 −2 −1 0

Card 2 Expression: ___________________

9 10

Equation: _____________ = _____________ Solution: _______

Reflect 1. What did you notice about the matches when you put them on the number line?

2. What rule did you learn about subtraction expressions and expressions using additive inverses?

3. What would be the equivalent equation to 7 − 4 = 3 using the additive inverse?

4. What did you observe about subtracting negative numbers?

5. Based on your experience with the matching game, explain which way to move on the number line in the following situations: •

Adding a positive number:

•

Adding a negative number:

•

Subtracting a positive number:

•

Subtracting a negative number:

Addition and Subtraction with Rational Numbers | 23

Addition and Subtraction with Rational Numbers

Explore 4 Part II: Subtraction Word Problems

Draw a Subtraction Word Problems Addition and Subtraction card. Match the card to the corresponding number line model. Circle the subtraction expression represented by the number line. Complete the addition equation by ﬁlling in the missing addend. Card Number

Subtraction Expression

Missing Addend

Solution ______ Explain:

-12 -11 -10 -9 -8 -7 -6 -5 -4

Card ____

−8 – 2

-3 -2 -1

−8 – (−2)

−6 – 2

10 11 12

−8 + ____ ______ Explain:

-12 -11 -10 -9 -8 -7 -6 -5 -4

Card ____

−12 – 8

-3 -2 -1

−8 – 4

−12 – (−8)

10 11 12

−12 + ____ ______ Explain:

−10

−9

−8

Card ____

−7

−6

−5

−4

−3 −2

−1

10.75 – 8.25 8.25 – 10.75 8.25 – (−10.75)

24 | Addition and Subtraction with Rational Numbers

Addition and Subtraction with Rational Numbers

Explore 4 Card Number

Subtraction Expression

Missing Addend

Solution ______ Explain:

−10 −9

−8

Card ____

−7

−6

−5

−4

−3

−2

−1

7.3 – 11.25 11.25 – 7.3 7.3 – (−11.25)

7.3 + _____

______ Explain: -12 -11 -10 -9 -8 -7 -6 -5 -4

Card ____

-3 -2 -1

−8 – (−5)

5–8

−8 – 5

10 11 12

−8 + _____

______ Explain: −12 −11 −10 −9 −8 −7 −6 −5 −4 −3 −2 −1

Card ____

1.5 – 10 5

10 11 12

10 5 – (−1.5) 3

−1.5 – 10 5

−1.5 + _____

Addition and Subtraction with Rational Numbers | 25

Explore 4

Addition and Subtraction with Rational Numbers

Reflect 1. What did you notice about the subtraction expressions you circled and the addition expressions in which you ﬁlled in the missing addend?

2. What do you think it means to say, “The difference of two numbers can be positive or negative, but the distance between two numbers is always positive”?

3. How did you use the number line to ﬁgure out the subtraction and addition expressions?

26 | Addition and Subtraction with Rational Numbers

Addition and Subtraction with Rational Numbers

Explore 5

Name: _______________________ Date: ___________

Using the Properties to Solve Part I: Who Owns That Property? Take turns drawing Who Owns That Property? Cards. Examine the equation on each card, and sort it into the correct property category. Record each equation under its category in the table. Write the solution. Commutative

Associative

Additive Inverse

Distributive

Solution: _____

Addition and Subtraction with Rational Numbers | 27

Explore 5

Addition and Subtraction with Rational Numbers

Reflect 1. Explain which of the properties work with subtraction. • Commutative:

• Associative:

• Additive inverse:

• Distributive:

2. Does the sign of a number affect whether a property works?

3. What is a rational number?

4. Do you think these properties just work for integers? Explain.

28 | Addition and Subtraction with Rational Numbers

Addition and Subtraction with Rational Numbers

Explore 5 Part II: Property Management

Read the word problems. Circle the equation or equations that correspond to the scenario. Circle the property being showcased. Use part-whole reasoning to solve where necessary. A client wants 2.5 liters of punch and 3.25 liters of lemonade for her daughter’s party. The birthday girl wants 3.25 liters of lemonade and 2.5 liters of punch. How many liters of drinks did they want? 2.5 – 3.25 = 3.25 – 2.5 Commutative

2.5 + 3.25 = 3.25 + 2.5

Associative

Additive inverse

Distributive

Part-whole reasoning (choose one): 2.5 + 3.25 = 2.75 + 3

2.5 + 3.25 = 2 + 3.75

2.5 + 3.25 = 2 + 3 + .75

They wanted ________ liters.

An engaged couple put $500.00 in their account for wedding decorations. They spent $621.03. What was their account balance?

500.00 – 621.03 = 500.00 + (−621.03) Commutative

Associative

500.00 + 621.03 = 500.00 – 621.03 Additive inverse

Distributive

Part-whole reasoning (choose one): 500 – 621.03 = 500 – 500 – 121.03

500 – 621.03 = 500 – 600 – 20 – 1 – .03

Addition and Subtraction with Rational Numbers | 29

Addition and Subtraction with Rational Numbers

Explore 5

Ms. Shelley saw a set of a dozen cookies. 5 were pumpkin and 7 were apple. She bought 10 sets for a client’s annual Thanksgiving Turkey Trot event. How many of each cookie did she buy? 10(5 + 7) = 50 + 70 Commutative

Associative

10(5 · 7) = 10 · 35 Additive inverse

Distributive

She bought _______ pumpkin cookies and _______ apple cookies.

A children’s hospital planned a picnic for its patients and their families. They ordered 1 200 cookies and 150 cupcakes. They wanted 5 of the desserts to be chocolate. How many items were chocolate? 1 1 1 5 (200 + 150) = ( 5 · 200) + ( 5 · 150)

Commutative

Associative

1 1 5 (200 – 150) = 3 (50)

Additive inverse

Distributive

_______ dessert items were chocolate.

30 | Addition and Subtraction with Rational Numbers

Addition and Subtraction with Rational Numbers

Explore 5

A local bank was having worker appreciation day and wanted Ms. Shelley to run it. The president wanted a donut box with 1 of a dozen sprinkled and 3 of a dozen 2

jelly-ﬁlled donuts and another box with 1 4 of a dozen glazed donuts. Her assistant 1

wanted to order a box with 2 of a dozen sprinkled donuts and a box with 4 of a 1 dozen jelly-ﬁlled and 1 4 of a dozen glazed donuts. Who wanted more donuts? 6 + 9 + 15 = 15 + 9 + 6 Commutative

(6 + 9) + 15 = 6 + (9 + 15)

Associative

Additive inverse

Distributive

Part-whole reasoning (show your work):

The president wanted _____ donuts. The assistant wanted _____ donuts. So ________ wanted more donuts because __________________.

Ms. Shelley planned a graduation party for the Garcia triplets. Mrs. Garcia told her to 1

order 10 2 cheese pizzas, 7 4 pepperoni pizzas, and 3 4 Hawaiian pizzas. Mr. Garcia 3

requested 7 4 pepperoni pizzas, 3 4 Hawaiian pizzas, and 10 2 cheese pizzas. How much pizza did they want? 1

10 2 + 7 4 + 3 4 = 7 4 + 3 4 + 10 2 Commutative

Associative

(10 2 + 7 4 ) + 3 4 = 10 2 + (7 4 + 3 4 ) Additive inverse

Distributive

Part-whole reasoning (show your work):

The Garcia parents wanted _________ pizzas.

Addition and Subtraction with Rational Numbers | 31

Addition and Subtraction with Rational Numbers

Explore 5

The Patel family was throwing a 100th birthday party for Grandma Patel. The family invited 200 people, but they only expected 100 guests to come. They saved money and opened a party account with $2,500.00 deposited. Each guest cost $20.50. Instead of 100 guests, 144 guests said they were coming. When the Patels paid the bill, what was their new account balance? 2,500 – 20.50 = 2,500 – 20.50(144) Commutative

Associative

2,500 – 2,952 = 2,500 + (−2,952) Additive inverse

Distributive

Their new account balance was ______________.

Reflect 1. What is the difference between an integer and a rational number?

2. Did the commutative, associative, additive inverse, and distributive properties work for all rational numbers?

3. Besides being a party planner, can you think of any other careers that use properties such as commutative, associative, additive inverse, distributive, or others as part of the job?

32 | Addition and Subtraction with Rational Numbers

Multiplication and Division with Rational Numbers

Explore 1

Multiplication and Division with Rational Numbers

Name: _______________________ Date: ___________

Integer Multiplication with Counters

Part I Use the Racer Cards to represent and solve the problems in the tables. Racer 1 Draw a model of the cups you created with the two-color counters.

Problem:

Solution:

Racer 2 Draw a model of the cups you created with the two-color counters.

Problem:

Solution:

Multiplication and Division with Rational Numbers | 35

Explore 1

Multiplication and Division with Rational Numbers

Racer 3 Draw a model of the cups you created with the two-color counters.

Problem:

Solution:

Racer 4 Draw a model of the cups that you created with the two-color counters.

Problem:

36 | Multiplication and Division with Rational Numbers

Solution:

Explore 1

Multiplication and Division with Rational Numbers

Reflect 1. How did you know how many groups you needed for each situation?

2. When multiplying integers, what type of integers will result in a negative product?

3. When multiplying integers, what type of integers will result in a positive product?

4. What rule can be used to determine if the product is going to be positive or negative?

Multiplication and Division with Rational Numbers | 37

Explore 1

Multiplication and Division with Rational Numbers

Part II Use the Team Cards to represent and solve the problems in the tables. Team 1 Draw a model of the cups you created with the two-color counters.

Problem:

Solution:

Team 2 Draw a model of the cups you created with the two-color counters.

Problem:

38 | Multiplication and Division with Rational Numbers

Solution:

Explore 1

Multiplication and Division with Rational Numbers

Team 3 Draw a model of the cups you created with the two-color counters.

Problem:

Solution:

Team 4 Draw a model of the cups you created with the two-color counters.

Problem:

Solution:

Multiplication and Division with Rational Numbers | 39

Explore 1

Multiplication and Division with Rational Numbers

Reflect 1. Why do you cancel out or cross off a pair of red and white counters?

2. Why do team 1 and team 2 have the same solution?

3. What happens to a multiplication or division expression when you add a negative sign? Give an example.

4. How does the distributive property of multiplication work?

40 | Multiplication and Division with Rational Numbers

Multiplication and Division with Rational Numbers

Explore 2

Name: _______________________ Date: ___________

Rational Number Multiplication with Number Lines Use the Team Cards to represent and solve each problem using the number line. Team 1 Represent a model of the scenario using a number line.

-16 -15 -14 -13 -12 -11 -10 -9

-8

-7

-6

-5

-4

-3

-2

-1

Problem:

10 11 12 13 14 15 16

Solution:

Team 2 Represent a model of the scenario using a number line.

-16 -15 -14 -13 -12 -11 -10 -9

-8

-7

-6

-5

Problem:

-4

-3

-2

-1

Solution:

Multiplication and Division with Rational Numbers | 41

Multiplication and Division with Rational Numbers

Explore 2

Team 3 Represent a model of the scenario using a number line.

-16 -15 -14 -13 -12 -11 -10 -9

-8

-7

-6

-5

-4

-3

-2

-1

Problem:

10 11 12 13 14 15 16

Solution:

Team 4 Represent a model of the scenario using a number line.

-16 -15 -14 -13 -12 -11 -10 -9

-8

-7

-6

-5

-4

-3

Problem:

42 | Multiplication and Division with Rational Numbers

-2

-1

Solution:

Explore 2

Multiplication and Division with Rational Numbers

Reflect 1. What pattern of the products did you notice after using number lines?

2. Give an example of a real-world situation of multiplying two negative integers that results in a positive integer.

3. What is the sign of the product of two or more integers with an even number of negative signs? Why?

4. What is the sign of the product of two or more integers with an odd number of negative signs? Why?

Multiplication and Division with Rational Numbers | 43

Multiplication and Division with Rational Numbers

Explore 3

Name: _______________________ Date: ___________

Integer Division with Counters Part I Use the Score Cards to answer the questions. Jonathan Draw a model of the counters used to solve the problem.

Expression:

Solution:

Cassandra Draw a model of the counters used to solve the problem.

Expression:

Solution:

Multiplication and Division with Rational Numbers | 45

Explore 3

Multiplication and Division with Rational Numbers

Frederick Draw a model of the counters used to solve the problem.

Expression:

Solution:

Harrietta Draw a model of the water balloons with the two-color counters.

Expression:

46 | Multiplication and Division with Rational Numbers

Solution:

Explore 3

Multiplication and Division with Rational Numbers

Reflect 1. How did the signs of the dividend and divisor affect the sign of the quotient?

2. Explain why the expressions −(12 ÷ 4), (−12) ÷ 4, and 12 ÷ (−4) all have the same quotient.

3. How are the rules in multiplying and dividing integers similar?

Multiplication and Division with Rational Numbers | 47

Explore 3

Multiplication and Division with Rational Numbers

Part II Look at the scenarios, and solve the problems to settle the tabs. Holly started with $36. She decided to split that money between all the food trucks. She couldn’t remember how many trucks there were. If each truck got $9, how many food trucks were there? Draw a model of the equation using the two-color counters.

Expression:

Solution:

36 ÷ ? = 9 or 36 ÷ 9

Melanie owed $27 at the end of the day. She did not remember how many trucks she visited. If she owed $9 to each truck, how many trucks did she visit? Draw a model of the equation using the two-color counters.

Expression:

Solution:

−27 ÷ ? = −9 or −27 ÷ (−9) 48 | Multiplication and Division with Rational Numbers

Explore 3

Multiplication and Division with Rational Numbers

Lauren has $18. She owes money to 2 different food trucks. How much will she pay to each truck? Draw a model of the equation using the two-color counters.

Expression:

Solution:

18 ÷ −2 = ?

Jackson and Jillian owed $9 to Tasty Tacos. If they bought 1 platter to share, how much did they pay for the platter? Draw a model of the equation using the two-color counters.

Expression:

Solution:

−9 ÷ 1 = ?

Multiplication and Division with Rational Numbers | 49

Explore 3

Multiplication and Division with Rational Numbers

Reflect 1. How does the quotient help in ﬁnding the sign of the divisor?

2. How do you model dividing by negative numbers when using number counters?

50 | Multiplication and Division with Rational Numbers

Explore 4

Multiplication and Division with Rational Numbers

Name: _______________________ Date: ___________

Rational Number Division with Number Lines Part I Read each player’s card, and write an expression. Locate the matching Solution Card, and glue it in the box. Use the number line to ﬁnd each quotient. Quotient: Player 1 averaged −2 yards for a total of −10 yards. How many carries did they have? Expression:

Number line solution:

Multiplication and Division with Rational Numbers | 51

Explore 4

Multiplication and Division with Rational Numbers

Quotient: Player 2 ran a total of 4 yards over 4 carries. How many yards did they average? Expression:

Number line solution:

Player 3 took a short break before making any more passes to players. When he returned, he had 10 balls that he owed to 2 players. How many balls are due to each player?

Quotient:

Expression:

Number line solution:

52 | Multiplication and Division with Rational Numbers

Explore 4

Multiplication and Division with Rational Numbers

Quotient: Player 4 had 8 balls to be thrown to 4 players. How many balls are due to each player? Expression:

Number line solution:

Quotient: Player 5 had 9 balls. He owed throws to 3 players. How many balls are due to each player? Expression:

Number line solution:

Multiplication and Division with Rational Numbers | 53

Explore 4

Multiplication and Division with Rational Numbers

Quotient: Player 6 ran a total of −10 yards and averaged −5 yards per carry. How many carries did he have? Expression:

Number line solution:

Quotient: Player 7

−8 ÷ 2

Number line solution:

54 | Multiplication and Division with Rational Numbers

Explore 4

Multiplication and Division with Rational Numbers

Quotient: Player 8 had a total of −4 yards. If he ran 4 times, how many yards did he average per carry? Expression:

Number line solution:

Multiplication and Division with Rational Numbers | 55

Explore 4

Multiplication and Division with Rational Numbers

Reflect 1. How did the signs of the dividend and divisor affect the sign of the quotient?

2. What is the sign of the quotient of two or more integers with an even number of negative signs? Why?

3. What is the sign of the quotient of two or more integers with an odd number of negative signs? Why?

56 | Multiplication and Division with Rational Numbers

Multiplication and Division with Rational Numbers

Explore 4 Part II

Use the number line to ﬁnd each player’s kick return yards per second. Scorecard 1 expression:

Score:

−10 ÷ 2.5

Number line solution:

-10 -9

-8

-7

-6

-5

-4

-3

-2

-1

Scorecard 2 expression:

Score:

7÷1 3 4

Number line solution:

-10 -9

-8

-7

-6

-5

-4

-3

-2

-1

Multiplication and Division with Rational Numbers | 57

Multiplication and Division with Rational Numbers

Explore 4 Scorecard 3 expression:

Score: 8

−9 ÷ 1 10

Number line solution:

-10 -9

-8

-7

-6

-5

-4

-3

-2

-1

Scorecard 4 expression:

Score:

9 ÷ 2.25

Number line solution:

-10 -9

-8

-7

-6

-5

-4

-3

-2

58 | Multiplication and Division with Rational Numbers

-1

Explore 4

Multiplication and Division with Rational Numbers

Reflect 1. What is the sign of the quotient when both the dividend and divisor are negative?

2. Which of the two methods of dividing integers helped you understand the rules better?

Multiplication and Division with Rational Numbers | 59

Multiplication and Division with Rational Numbers

Explore 5

Name: _______________________ Date: ___________

Using Properties to Solve Part I Use the Matching Scorecards to calculate the scores and match scorecard 1 with the corresponding scorecard 2. Group each set of cards based on their properties. Commutative Property Matching Scorecards Card 1

Card 2

Card 1

Card 2

Score:

SCORECARD 2

7(6)

Score:

Multiplication and Division with Rational Numbers | 61

Multiplication and Division with Rational Numbers

Explore 5

Associative Property Matching Scorecards Card 1

Card 2

Card 1

Card 2

Score:

SCORECARD 2

−2(4 • 5)

Score:

62 | Multiplication and Division with Rational Numbers

Explore 5

Multiplication and Division with Rational Numbers

Distributive Property Matching Scorecards Card 1

Card 2

Card 1

Card 2

Score:

Multiplication and Division with Rational Numbers | 63

Multiplication and Division with Rational Numbers

Explore 5

Cards with No Matching Properties Matching Scorecards Card 1

Card 2

Card 1

Card 2

Score:

64 | Multiplication and Division with Rational Numbers

Explore 5

Multiplication and Division with Rational Numbers

Reflect 1. How many points did the player with the most points get?

2. Does division have a commutative property? Why or why not?

3. Does division have an associative property? Why or why not?

4. When multiplying terms using the commutative or associative property, does the sign of the ﬁnal product change based on the grouping or the order?

Multiplication and Division with Rational Numbers | 65

Explore 5

Multiplication and Division with Rational Numbers

Part II Use the Team Scorecards to determine whether scorer 1 has correctly calculated the scores for each team. Team 1

Expression:

Show your calculations here.

Did scorer 1 calculate the score correctly? Why or why not?

Team 2

Expression:

Show your calculations here.

Did scorer 1 calculate the score correctly? Why or why not?

66 | Multiplication and Division with Rational Numbers

Explore 5 Team 3

Multiplication and Division with Rational Numbers

Expression:

Show your calculations here.

Did scorer 1 calculate the score correctly? Why or why not?

Team 4

Expression:

Show your calculations here.

Did scorer 1 calculate the score correctly? Why or why not?

Multiplication and Division with Rational Numbers | 67

Explore 5

Multiplication and Division with Rational Numbers

Reflect 1. For which team(s) did scorer 1 get the calculations correct?

2. What rules do rational numbers have when multiplying or dividing negative numbers?

3. When simplifying multistep expressions, what is a common mistake students make?

4. What is an important reminder when solving multistep problems involving properties of rational numbers?

68 | Multiplication and Division with Rational Numbers

Rational Number Operations

Explore 1

Name: _______________________ Date: ___________

Convert between Forms Use the Fraction, Percent, and Sign Cards to convert fractions to decimals. Use long division to divide numerators by denominators. Use a calculator to check your answers. Card: ___ Sign: ___

Percent: _____

Card: ___ Sign: ___

Fraction: _____

Repeat/Terminate

Decimal: _____

Repeat/Terminate

Fraction: _____ Decimal: _____

Checked with calculator: ____ Card: ___ Sign: ___

Checked with calculator: ____

Percent: _____

Card: ___ Sign: ___

Fraction: _____

Repeat/Terminate

Decimal: _____

Repeat/Terminate

Fraction: _____ Decimal: _____

Checked with calculator: ____ Rational Number Operations | 71

Rational Number Operations

Explore 1 Card: ___ Sign: ___

Fraction: _____

Card: ___ Sign: ___

Percent: _____

Fraction: _____ Decimal: _____

Repeat/Terminate

Checked with calculator: ____ Card: ___ Sign: ___

Fraction: _____

Decimal: _____

Repeat/Terminate

Checked with calculator: ____ Card: ___ Sign: ___

Percent: _____

Fraction: _____ Decimal: _____

Repeat/Terminate

Checked with calculator: ____ 72 | Rational Number Operations

Decimal: _____

Repeat/Terminate

Explore 1

Rational Number Operations

Reflect 1. When setting up long division to convert a fraction to a decimal, why is the numerator the dividend and the denominator the divisor?

2. How can you tell if a fraction or percent converts to a terminating decimal?

3. How can you tell if a fraction or percent converts to a repeating decimal?

4. Is a decimal with more digits of a greater value than a decimal with fewer digits?

5. How should negative or positive signs in fractions be handled when converting fractions to decimals or percents to fractions and decimals?

6. What are some common errors that happen with long division when converting fractions and percents to decimals?

Rational Number Operations | 73

Rational Number Operations

Explore 2

Name: _______________________ Date: ___________

Solving with Complex Fractions Use the Word Problem Cards to write the complex fractions, division expressions, corresponding multiplication expressions, and solutions in the table. Complex Fraction

Division Expression

Multiplication Expression

Solution

Rational Number Operations | 75

Rational Number Operations

Explore 2 Complex Fraction

Division Expression

Multiplication Expression

Solution

76 | Rational Number Operations

Explore 2

Rational Number Operations

Reflect 1. How are complex fractions different from regular fractions?

2. Give an example of a pair of inverse fractions.

3. How could you check to make sure two fractions are inverse?

4. What do you notice about the relationship between the value of the divisor and the value of the quotient? Support your idea with an example.

5. What are some common errors that occur when solving complex fraction problems, such as fractions divided by fractions?

Rational Number Operations | 77

Rational Number Operations

Explore 3

Name: _______________________ Date: ___________

Multistep Rational Number Operations Look at the table comparing expenses and earnings for Priscilla’s Pies during two events, the county fair and the farmers’ market. Then, answer the questions, remembering the order of operations when ﬁnding solutions.

Priscilla’s Pies County Fair

Farmers’ Market

Supply cost: $850 per week

Supply cost: $750 per week

Employee wages • Week 1: $480 • Week 2: $640

Employee wages • $600 weekly

Employee bonuses • $200 per week

Employee bonuses • Week 1: $100 • Week 2: $0

Miscellaneous expenses • Week 1: $250 • Week 2: $125

Miscellaneous expenses • Week 1: $180 • Week 2: $100

Pie sales • Week 1: $1,800 • Week 2: $2,350

Pie sales • Week 1: $700 • Week 2: $940

Rational Number Operations | 79

Rational Number Operations

Explore 3

Use the information from the Priscilla’s Pies chart to solve the problems on the Scenario Cards. Use the spaces below to record your work. Scenario Card 1

Scenario Card 2

Expression:

Solution:

Scenario Card 3

Scenario Card 4

Expression:

Solution:

80 | Rational Number Operations

Rational Number Operations

Explore 3 Scenario Card 5

Scenario Card 6

Expression:

Solution:

Scenario Card 7

Scenario Card 8

Expression:

Solution:

Rational Number Operations | 81

Rational Number Operations

Explore 3 Reflect

1. What was most difﬁcult about ﬁnding solutions to the questions: setting up the expressions or solving them?

2. What types of interactions are negative (or subtraction)?

3. What types of interactions are positive (or addition) mathematically?

4. Were there any expressions that you could write a different way?

82 | Rational Number Operations

Proportional Relationships

Explore 1

Name: _______________________ Date: ___________

Proportionality vs. Non-proportionality Part I Use the graphs to complete the values in the tables. Using the tables, identify the ratio between the salary and the number of hours. Determine which offers show a proportional relationship. Offer 1: Proportional or Non-proportional

Hours

Salary ($)

$ hr

Offer 2: Proportional or Non-proportional

Hours

Salary ($)

$ hr

Proportional Relationships | 85

Proportional Relationships

Explore 1 Offer 3: Proportional or Non-proportional

Hours

Salary ($)

$ hr

Offer 4: Proportional or Non-proportional

Hours

Salary ($)

$ hr

86 | Proportional Relationships

Proportional Relationships

Explore 1 Part II

Use any two rates from the Job Listing Cards to plot points on the graph. Draw a line crossing the two points to check whether the job listing offers a signing bonus. Use the graph to determine whether the relationship is proportional. Job Listing 1

250

Proportional or Non-proportional

Use the table below to determine whether you have equivalent ratios in the table.

200 $ hr

150 100

What is the signing bonus?

x 1

9 10

Job Listing 2

250

Proportional or Non-proportional

Use the table below to determine whether you have equivalent ratios in the table.

200 $ hr

150 100

What is the signing bonus?

x 1

9 10 Proportional Relationships | 87

Proportional Relationships

Explore 1 Job Listing 3

250

Proportional or Non-proportional Use the table below to determine whether you have equivalent ratios in the table.

$ hr

200 150 100 50

What is the signing bonus?

x 1

9 10

Job Listing 4

250

Proportional or Non-proportional Use the table below to determine whether you have equivalent ratios in the table.

$ hr

200 150 100

What is the signing bonus?

x 1

88 | Proportional Relationships

9 10

Explore 1

Proportional Relationships

Reflect 1. What did you notice about the graphs of the offers with proportional relationships?

2. How did you determine which tables have proportional relationships?

3. What does the point on the y-axis represent?

4. What does it mean when the starting rate is at (0, 0)?

Proportional Relationships | 89

Proportional Relationships

Explore 2

Name: _______________________ Date: ___________

Unit Rates Use the runners’ information to determine the unit rate for each runner. Show your strategy for ﬁnding the unit rate. Show your work here.

Runner 1

Minutes

Miles

1.05

1.4

1.75

2.1

Miles per hour:

Proportional Relationships | 91

Proportional Relationships

Explore 2

Show your work here.

Runner 2

Miles

12 9 6 3 0

x 10 20 30 40 50 Minutes Miles per hour: Show your work here.

Runner 3

Roger runs 10 miles in 2.5 hours.

Miles per hour: 92 | Proportional Relationships

Proportional Relationships

Explore 2

Show your work here.

Runner 4

Each part represents a one-mile run.

Miles per hour: Show your work here.

Runner 5

Minutes

Miles

Proportional Relationships | 93

Proportional Relationships

Explore 2

Show your work here.

Runner 6

Miles

2.5 2 1.5 1 0.5 0

x 5

10 15 20 25 30 Minutes Miles per hour: Show your work here.

Runner 7

Stacey runs 7.5 miles in 1.5 hours.

Miles per hour: 94 | Proportional Relationships

Proportional Relationships

Explore 2

Show your work here.

Runner 8

y = 4.5x,

where x is the number of hours and y is the number of miles

Miles per hour: Show your work here.

Runner 9

y = 1.2x,

where x is the number of hours and y is the number of miles

Proportional Relationships | 95

Proportional Relationships

Explore 2

Show your work here.

Runner 10

Each part represents a one-mile run. Miles per hour:

96 | Proportional Relationships

Explore 2

Proportional Relationships

Reflect 1. How is a ratio different from a unit rate?

2. How do you determine the unit rate given a ratio?

3. How do you determine the unit rate in an equation?

4. For which representation was ﬁnding the unit rate the easiest?

Proportional Relationships | 97

Total Registration Fees ($)

Students Registered 2

High School Summer Camp Registration

Total Registration Fees ($)

Students Registered

Middle School Summer Camp Registration

Show your work here.

Registration fee: _______

Show your work here.

Registration fee: _______

Use the Camp Flyers to complete the tables and ﬁnd the constant of proportionality.

Part I

Proportional Relationships | 99

Name: _______________________ Date: ___________

Proportional Relationships with Equations

Explore 3

Proportional Relationships

100 | Proportional Relationships

4. What value remained the same throughout the high school summer camp registration?

3. What value remained the same throughout the middle school summer camp registration?

2. If you were only given the table and not the registration fee, how would you ﬁnd the registration fee per student?

1. What does it mean for the variable to be dependent?

Reflect

Explore 3

Proportional Relationships

Constant of proportionality:

Show your work here.

Equation:

Constant of proportionality:

Show your work here.

Equation:

Dependent variable: _______

Equation:

Proportional Relationships | 101

Show your work here.

Independent variable: _______

Jumps

Independent variable: _______

Sprints

Independent variable: _______

Hurdles

Use the Camp Fee Cards to ﬁnd the equation for each event.

Part II

Explore 3

Proportional Relationships

Dependent variable: _______ Constant of proportionality:

Show your work here.

Equation:

Dependent variable: _______

Constant of proportionality:

Show your work here.

Equation:

102 | Proportional Relationships

Independent variable: _______

Middle Distance

Independent variable: _______

Throws

Explore 3

Equation:

Show your work here.

Constant of proportionality:

Dependent variable: _______

Independent variable: _______

Pole Vault

Proportional Relationships

Proportional Relationships | 103

2. What is the product of the independent variable and the constant of proportionality equal to?

1. When looking at a graph or table, how do you ﬁnd the constant of proportionality?

Reflect

Explore 3

Proportional Relationships

Describe Points on a Graph

Name: _______________________ Date: ___________

If there were 60 student athletes who participated, how many schools brought student athletes?

How many students can each school bring to participate in the track meet?

If zero schools attend the event, how many students would participate in the track meet?

Question and Answer

Proportional Relationships | 105

Explanation Using Points from the Graph

Student Athletes

Use the points on the graphs on the Track Meet Scenario Cards to answer the following questions. Explain which point on the graph was used to determine each answer.

Explore 4

Proportional Relationships

106 | Proportional Relationships

How many guests can 5 students invite?

Explanation Using Points from the Graph

Event Guests

How many guests can each student invite to the track meet?

If 0 athletes participated in the event, how many guests would have been present at the event?

Question and Answer

Explore 4

Proportional Relationships

If there were 10 events in total for the track meet, how long would the whole event last?

How long does each event last?

Proportional Relationships | 107

Explanation Using Points from the Graph

Event Schedules

If 0 events were competed in at the track meet, how many minutes would it take to complete the events?

Question and Answer

Explore 4

Proportional Relationships

108 | Proportional Relationships

How many awards will be given out to 8 schools?

How many student winners will be awarded from each school?

Explanation Using Points from the Graph

Event Winners

If 0 athletes participated in the events, how many students would get an award?

Question and Answer

Explore 4

Proportional Relationships

Proportional Relationships | 109

3. How is knowing the values of the points on the graph of proportional relationships beneﬁcial in the real world?

2. What point on the graph shows the unit rate?

1. What does the origin mean in the graphs from the Track Meet Scenario Cards?

Reflect

Explore 4

Proportional Relationships

Understand Slope

111

Understand Slope

Explore 1

Name: _______________________ Date: ___________

Similar Triangles

Use the Similar Triangles Cards to determine the slope of the line, and then use similar triangles to justify that the slope is the same between any two points. Identify the slope as positive or negative. Card 1 Find the ratio of the length of the vertical side to the length of the horizontal side for △ABC and △WXY.

Does this slope represent a positive or negative slope?

Card 2 Find the ratio of the length of the vertical side to the length of the horizontal side for △JKL and △MNP.

Does this slope represent a positive or negative slope?

Understand Slope | 113

Understand Slope

Explore 1 Card 3

Find the ratio of the length of the vertical side to the length of the horizontal side for △DEF and △PQR.

Does this slope represent a positive or negative slope?

Card 4 Find the ratio of the length of the vertical side to the length of the horizontal side for △CDE and △ABC.

Does this slope represent a positive or negative slope?

114 | Understand Slope

Understand Slope

Explore 1

Use the coordinate plane to draw a triangle using a different pair of points on the same line. 10

y A

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

B 2 3

9 10

-2 -3 -4 -5 -6 -7 -8 -9 -10

Find the ratio of the length of the vertical side to the length of the horizontal side for both triangles.

Are the triangles similar? Explain.

Understand Slope | 115

Understand Slope

Explore 1 Reflect

1. What can you conclude about the slope between any two points on a line?

2. How can you determine whether a slope is positive or negative?

116 | Understand Slope

Understand Slope

Explore 2

Name: _______________________ Date: ___________ FOOD BANK

Determine the Rate of Change Part I: Determining Rate of Change from a Table and a Graph Use the Food Bank Volunteer Cards to determine the rate of change. Donations at the Food Bank Workspace: x₁: _____

x₂: ______

y₁: ______

y₂: _____

What is the rate of change, and what does the rate of change represent in this situation?

Canned Goods at the Food Bank Workspace: x₁: _____

x₂: ______

y₁: ______

y₂: _____

What is the rate of change, and what does the rate of change represent in this situation?

Understand Slope | 117

Understand Slope

Explore 2 People Greeted at the Door Workspace: x₁: _____

x₂: ______

y₁: ______

y₂: _____

What is the rate of change, and what does the rate of change represent in this situation?

Items Stocked in the Pantry Workspace: x₁: _____

x₂: ______

y₁: ______

y₂: _____

What is the rate of change, and what does the rate of change represent in this situation?

118 | Understand Slope

Explore 2

Understand Slope

1. What are the independent and dependent variables on Card 1?

2. What are the independent and dependent variables on Card 2?

Reflect 1. What happens when the rate of change is positive?

2. What happens when the rate of change is negative?

3. How do you choose the points to calculate the rate of change from graphs and tables?

Understand Slope | 119

Understand Slope

Explore 2

Part II: Determining Rate of Change Using Verbal Descriptions and Linear Functions Use the Food Bank Volunteer Cards to determine the rate of change. Write the equation of the linear function that models the relationship on the cards. Card 1 x

x₁: ______

Workspace:

x₂: ______

1 y₁: ______

y₂: ______

Equation:

What is the rate of change, and what does the rate of change represent in this situation?

Card 2 x

x₁: ______

Workspace:

x₂: ______

1 2 3

y₁: ______ y₂: ______

Equation:

What is the rate of change, and what does the rate of change represent in this situation?

120 | Understand Slope

Understand Slope

Explore 2 Card 3 x

x₁: ______

Workspace:

x₂: ______

2 y₁: ______

y₂: ______

Equation:

What is the rate of change, and what does the rate of change represent in this situation?

Card 4 x

x₁: ______

Workspace:

x₂: ______

3 6 9

y₁: ______ y₂: ______

Equation:

What is the rate of change, and what does the rate of change represent in this situation?

Understand Slope | 121

Understand Slope

Explore 2 Reflect

1. Does a proportional relationship have a constant rate of change? Explain.

2. What is an example of rate of change used in a real-world situation?

122 | Understand Slope

Understand Slope

Explore 3

Name: _______________________ Date: ___________

Slope Part I Use the Slope Scenario Cards to represent m and b, and write the equation that is represented by the graph. Card 1 What is the value of m?

What does b represent? What is the value of b?

Write the equation that is represented by the graph.

Understand Slope | 123

Understand Slope

Explore 3 Card 2 What is the value of m?

What does b represent? What is the value of b?

Write the equation that is represented by the graph.

Card 3 What is the value of m?

What does b represent? What is the value of b?

Write the equation that is represented by the graph.

124 | Understand Slope

Understand Slope

Explore 3 Card 4 What is the value of m?

What does b represent? What is the value of b?

Write the equation that is represented by the graph.

Reflect 1. Why is y = mx + b called the slope-intercept form of the equation of a line?

2. How do you know whether a graph shows a proportional relationship?

Understand Slope | 125

Understand Slope

Explore 3 Part II

Look at the data for each excursion. Find the unit rate, and decide which company has the best price. Circle the company with the best price. Island Boat Tour

160

Jet Ski Journey

140

Hours

Price

$90

$135

$180

$225

120

Price

100 80 60 40 20 0

x 2

Hours

Coordinates:

Slope:

Unit rate:

Equation:

126 | Understand Slope

Understand Slope

Explore 3 Off-Broadway Show

160

Broadway Show

140

Tickets

Price

$157

$235.50

$314

$392.50

120

Price

100 80 60 40 20 0

x 2

Hours

Coordinates:

Slope:

Unit rate:

Equation:

Reflect 1. How does the slope of the line compare to the unit rate on the graph?

2. How do you know whether a table is proportional?

Understand Slope | 127

Ratios, Rates, and Percents

129

Ratios, Rates, and Percents

Explore 1

Name: _______________________ Date: ___________

Unit Rates with Ratios of Fractions Use the Recipe Cards to determine how much flour each baker will need to bake cupcakes. Kiana’s Recipe Complete the table to find how many cups of flour Kiana will need for each batch of cupcakes.

Cups of Flour

1 8

Batches of Cupcakes

1 4

What is the unit rate for cups of ﬂour per batch?

Ratios, Rates, and Percents | 131

Ratios, Rates, and Percents

Explore 1 Tai’s Recipe

Complete the table to ﬁnd how many cups of ﬂour Tai will need for each batch of cupcakes.

Cups of Flour Batches of Cupcakes

What is the unit rate for cups of ﬂour per batch?

Valerie’s Recipe Complete the table to ﬁnd how many cups of ﬂour Valerie will need for each batch of cupcakes.

Cups of Flour Batches of Cupcakes

1 2

2 2

4 2

8 2

What is the unit rate for cups of ﬂour per batch?

132 | Ratios, Rates, and Percents

Ratios, Rates, and Percents

Explore 1 Neri’s Recipe

Complete the table to ﬁnd how many cups of ﬂour Neri will need for each batch of cupcakes.

Cups of Flour Batches of Cupcakes

3 2

1 2

What is the unit rate for cups of ﬂour per batch?

Neri’s Bonus Round Recipe Interpret the graph to ﬁnd out how many cups of ﬂour Neri will need for each batch of cupcakes in his bonus recipe. y 10

Batches of cupcakes

9 8 7 6 5 4 3 2 1

x 1

Cups of flour

What is the unit rate for cups of ﬂour per batch?

Ratios, Rates, and Percents | 133

Ratios, Rates, and Percents

Explore 1 Tai’s Bonus Round Recipe

Complete the double number line to ﬁnd out how many cups of ﬂour Tai will need for each batch of cupcakes. 1 6

2 6

5 6

Cups of Flour Batches of Cupcakes 1 4

3 4

6 4

What is the unit rate for cups of ﬂour per batch?

134 | Ratios, Rates, and Percents

Explore 1

Ratios, Rates, and Percents

Reflect 1. List three ways to ﬁnd a unit rate.

2. How can tables and graphs help to solve problems of unit rate with ratios of different units?

3. Why does it make sense to simplify a fraction when computing unit rate? Relate your answer to the context of Kiana’s problem.

Ratios, Rates, and Percents | 135

Ratios, Rates, and Percents

Explore 2

Name: _______________________ Date: ___________

Ratios of Length and Area

Use the Baking Notes to determine the missing values in ratios for each baker. Kiana’s Baking Notes Kiana is using her width-to-length ratio to determine the width of the cake she will make for the contest. Ratio

What are two other ways this ratio can be written?

Proportion

What is the width of the cake that will be made for the contest?

Ratios, Rates, and Percents | 137

Ratios, Rates, and Percents

Explore 2 Tai’s Baking Notes

Tai is using her width-to-length ratio to determine the length of the cake she will make for the contest. Ratio

Proportion

What are two other ways this ratio can be written?

What is the length of the cake that will be made for the contest?

Valerie’s Baking Notes Valerie is using her width-to-length ratio to determine the width of the cake she will make for the contest. Ratio

What are two other ways this ratio can be written?

138 | Ratios, Rates, and Percents

Proportion

What is the width of the cake that will be made for the contest?

Ratios, Rates, and Percents

Explore 2 Neri’s Baking Notes

Neri is using her width-to-length ratio to determine the length of the cake she will make for the contest. Ratio

What are two other ways this ratio can be written?

Proportion

What is the length of the cake that will be made for the contest?

1. How can you ﬁnd the area of each contestant’s cake?

width 2. If the width of Neri’s cake were 40 inches, using the same ratio, what is the length length of the cake?

Ratios, Rates, and Percents | 139

Ratios, Rates, and Percents

Explore 2

width ratio for each contestant’s cake. Use the width and length to ﬁnd the length area of each cake. Write the

Contestant

Width

Area of Cake (in.2)

Length

Kiana Tai Valerie Neri

Reflect 1. How did you solve for the missing value in the proportion?

2. Compare and contrast ratios and proportions.

140 | Ratios, Rates, and Percents

Ratios, Rates, and Percents

Explore 3

Name: _______________________ Date: ___________

Multistep Ratio Problems Use the Price Guide to compare prices to determine the best price for each item. Sugar Price Comparison Travis is looking to purchase 5 pounds of sugar and is comparing 2 options. Which option offers the best price? Option 1

Option 2

What is the cost for 5 pounds?

What is the price per pound?

How much money is saved per pound by purchasing the option with the best price?

Ratios, Rates, and Percents | 141

Ratios, Rates, and Percents

Explore 3 Butter Price Comparison

Travis is looking to purchase 8 pounds of butter and is comparing 2 options. Which option offers the best price? Option 1

Option 2

What is the cost for 8 pounds?

What is the price per pound?

How much money is saved per pound by purchasing the option with the best price?

142 | Ratios, Rates, and Percents

Ratios, Rates, and Percents

Explore 3 Cream Price Comparison

Travis is looking to purchase 100 ounces of cream and is comparing 2 options. Which option offers the best price? Option 1

Option 2

What is the cost for 100 ounces?

What is the price per ounce?

How much money is saved per ounce by purchasing the option with the best price?

Ratios, Rates, and Percents | 143

Ratios, Rates, and Percents

Explore 3 Egg Price Comparison

Travis is looking to purchase 4 dozen eggs and is comparing 2 options. Which option offers the best price? Option 1

Option 2

What is the cost for 4 dozen eggs?

What is the price per dozen eggs?

How much money is saved per dozen eggs by purchasing the option with the best price?

144 | Ratios, Rates, and Percents

Explore 3

Ratios, Rates, and Percents

Reflect 1. Why is unit rate helpful?

2. If you were given a $0.30 off coupon for 1 pound of option 2 sugar, which sugar would be the best price?

3. How much would the option 1 eggs be if there was a 10% off coupon for 1 dozen eggs included? Which would be the better option?

Ratios, Rates, and Percents | 145

Ratios, Rates, and Percents

Explore 4

Name: _______________________ Date: ___________

Solve Problems – Percents Use the Appliance Cards to calculate the percentage of the budget that will be spent on each appliance. Find the percentage of the money that will be used toward the purchase of the appliance and the part of the budget that will be spent on each appliance. Blenders If Liam’s total budget is $2,000, how much of the budget will be spent on blenders? Use a strip diagram and a proportion to solve the problem.

Budget for blenders: If Liam is given $180 to put toward the purchase of the blenders, what percentage of the cost of the blenders does it represent? Use a proportion to represent your answer.

Ratios, Rates, and Percents | 147

Ratios, Rates, and Percents

Explore 4 Juicers

If Liam’s total budget is $2,000, how much of the budget will be spent on juicers? Use a strip diagram and a proportion to solve the problem.

Budget for juicers: If Liam’s manager gives him $33 to put toward the purchase of the juicers, what percentage of the cost of the juicers does it represent?

148 | Ratios, Rates, and Percents

Ratios, Rates, and Percents

Explore 4 Ice Makers

If Liam’s total budget is $2,000, how much of the budget will be spent on ice makers? Use a strip diagram and a proportion to solve the problem.

Budget for ice makers: If Liam’s manager gives him $70 to put toward the purchase of the ice makers, what percentage of the cost of the ice makers does it represent?

Ratios, Rates, and Percents | 149

Ratios, Rates, and Percents

Explore 4 Food Processors

If Liam’s total budget is $2,000, how much of the budget will be spent on food processors? Use a strip diagram and a proportion to solve the problem.

Budget for food processors: If Liam’s manager gives him $75 to put toward the purchase of the food processors, what percentage of the cost of the food processors does it represent?

150 | Ratios, Rates, and Percents

Explore 4

Ratios, Rates, and Percents

Reflect 1. How do proportions relate to percents?

2. If Liam is giving a new budget and $2,000 represents 40% of the budget, what amount represents the new budget?

3. What are some examples of percents in the real world?

Ratios, Rates, and Percents | 151

Percent Application

153

Percent Application

Explore 1

Name: _______________________ Date: ___________

Tax Part I: Calculating Sales Tax Complete each table by calculating the sales tax for each scenario. If necessary, round to the nearest hundredth. Pots and Pans for the Kitchen Anthony is buying supplies for the kitchen at his new restaurant, the Yellow Rose Diner. He begins stocking his kitchen by buying pots and pans from a local restaurant supply company that cost $450 with a sales tax of 9%.

Sales tax formula: __________________ Cost of Pots and Pans

Sales Tax (%)

Formula

Sales Tax ($)

Cake Pans for the Kitchen Anthony plans to also go to the Home Warehouse to purchase several cake pans. The total cost of the cake pans will be $142 with a sales tax of 12%.

Sales tax formula: __________________ Cost of Cake Pans

Sales Tax (%)

Formula

Sales Tax ($)

Percent Application | 155

Percent Application

Explore 1 Silverware for the Kitchen

The manager at the Yellow Rose Diner needs to purchase silverware. The silverware costs $230, and the manager placed an order for silverware in a state with a sales tax of 6.5%.

Sales tax formula: __________________ Cost of Silverware

Sales Tax (%)

Formula

Sales Tax ($)

Measuring Utensils for the Kitchen The chef at the Yellow Rose Diner has placed an order for measuring utensils in a store in another state. The measuring utensils cost $118 with a sales tax of 8.25%.

Sales tax formula: __________________ Cost of Measuring Utensils

156 | Percent Application

Sales Tax (%)

Formula

Sales Tax ($)

Explore 1

Percent Application

Reflect 1. What is sales tax?

2. How do you calculate the amount of sales tax on a purchase?

3. Which purchase has the greater amount of sales tax: a $25 mixing bowl with a sales tax rate of 7% or a $22 baking sheet with a sales tax rate of 9%?

4. Why do you think we pay sales tax on the purchase of goods and services?

Percent Application | 157

Percent Application

Explore 1 Part II: Calculating Total Cost

Calculate the total cost including sales tax for each problem. Use a tape diagram and an equation to solve the problem. Pots and Pans for the Kitchen Pots and pans cost $450 with a sales tax of 9%. Calculate the total cost. Cost of Pots and Pans

Sales Tax (%)

Tape diagram:

Formula

Sales Tax ($)

Equation:

Total cost: Cake Pans for the Kitchen Cake pans cost $142 with a sales tax of 12%. Calculate the total cost. Cost of Cake Pans

Tape diagram:

Sales Tax (%)

Formula

Sales Tax ($)

Equation:

Total cost: 158 | Percent Application

Percent Application

Explore 1 Silverware for the Kitchen

Silverware costs $230 with a sales tax of 6.5%. Calculate the total cost. Cost of Silverware

Sales Tax (%)

Tape diagram:

Formula

Sales Tax ($)

Equation:

Total cost: Measuring Utensils for the Kitchen Measuring utensils cost $118 with a sales tax of 8.25%. Calculate the total cost. Cost of Measuring Utensils

Sales Tax (%)

Tape diagram:

Formula

Sales Tax ($)

Equation:

Total cost:

Percent Application | 159

Explore 1

Percent Application

Reflect 1. Including sales tax, how can you determine the total cost of a purchase?

2. Which purchase has the greater total cost: a $25 mixing bowl with a sales tax rate of 7% or a $22 baking sheet with a sales tax rate of 9%?

3. How is the tape diagram helpful in understanding the total cost of a purchase, including sales tax?

4. How would you calculate the total cost, including sales tax, when buying multiple items?

160 | Percent Application

Percent Application

Explore 2

Name: _______________________ Date: ___________

Percent Change Part I: Amount of Change and Percent Change Calculate the amount of change and the percent change in hours worked by the staff at the diner. Represent the problem using a model. Identify the change as a percent increase or percent decrease by circling the type of change each scenario represents. Hours Worked by Manager at the Diner Diner manager Lacey notices she’s spending more time getting the diner ready for opening day. Last week, she spent 20 hours preparing for the grand opening. This week, she spent 40% more hours getting the diner ready. How many hours were spent working at the diner this week? What is the amount of change? Model:

Workspace:

Hours working at diner: Percent increase or percent decrease

Amount of change:

Percent Application | 161

Percent Application

Explore 2 Hours Worked by Cashier at the Diner

Janice spent 40 hours in training during her ﬁrst week on the job. Next week, she will spend 25% fewer hours at the diner. How many hours will she spend next week at the diner? What is the amount of change? Model:

Workspace:

Hours working at diner: Percent increase or percent decrease

Amount of change:

Hours Worked by Chef at the Diner Chef Antonio is trying out new recipes at the diner. He worked 28 hours in week 1 and will need to work 30% more hours next week to ﬁnalize the diner menu. How many hours will he spend next week at the diner? What is the amount of change? Model:

Workspace:

Hours working at diner: Percent increase or percent decrease 162 | Percent Application

Amount of change:

Percent Application

Explore 2 Hours Worked by Assistant Manager at the Diner

Lacey’s assistant manager is conducting interviews for the custodial staff positions that are open. She spent 42 hours in week 1 and will be working 15% fewer hours next week because of a doctor’s appointment. How many hours will the assistant manager spend next week at the diner? What is the amount of change? Model:

Workspace:

Hours working at diner: Percent increase or percent decrease

Amount of change:

Percent Application | 163

Percent Application

Explore 2 Reflect 1. What is percent change?

2. How does a tape diagram help model and calculate percent change?

3. There were 20 employees who completed training at the diner during week 1. During week 2, there were 6 employees who completed training. Was there a percent increase or a percent decrease from week 1 to week 2?

4. There were 3 employees that volunteered to work the ﬁrst shift at the diner during week 1. During week 2, there were 6 employees that volunteered to work the ﬁrst shift at the diner. Was there a percent increase or a percent decrease from week 1 to week 2?

164 | Percent Application

Percent Application

Explore 2 Part II: Markups and Markdowns

1. A markup is an increase in the cost of an item to make a proﬁt. Given this information, use the word bank provided to ﬁll in the blanks in the tape diagram below. Not all words or phrases from the word bank will be used.

_____________________________ _____________________

________

Markup Word Bank Discount Loss

Selling price Sale price

Cost of seller to produce/buy Proﬁt

2. Lacey begins planning the Yellow Rose Diner’s menu. To earn a proﬁt from each item ordered, she sets the menu price of each item at 75% above the cost to prepare and cook each food. Use this information to complete the table below. Item

Cost to Prepare and Cook

Deluxe hamburger

Veggie lasagna

$3.20

Beef fajita taco

Chicken stir-fry

$2.80

Selling Price

Proﬁt

Percent Application | 165

Percent Application

Explore 2

3. A markdown is a decrease in the cost of an item. A markdown is also known as a discount. Given this information, use the word bank provided to ﬁll in the blanks in the tape diagram below. Not all words from the word bank will be used.

_____________________________ _____________________

________

Markdown Word Bank Discount Gain

Selling price Sale price

Original price Proﬁt

4. Lacey mails coupons to local residents to encourage them to come to the grand opening of the diner for a discounted price. Use the Discount Spinner to determine the discount that residents will receive, and ﬁll in the second column of the table below. After you complete one row, spin the spinner again to determine the discount for the next meal. Cost of Meal

Discount (%)

Discount ($)

Calculating the Cost of Meal

Total Cost of Meal

$13.00

$27.80

$35.00

$41.40 166 | Percent Application

Explore 2

Percent Application

Use the table in Part II to answer the following questions. 1. What do the markup and markdown percentages represent?

2. What operations are used to calculate the markup and markdown of an item?

3. How can you determine the sale price of a good or service?

4. Which menu item costs less to the customer: a $7 chicken sandwich with a 40% coupon or a $5 veggie wrap with a 15% discount?

Reflect 1. Why is knowing percents important for companies and merchants when marking down the price of an item?

2. When might a high markdown be an advantage? When might a low markup be a disadvantage?

Percent Application | 167

Percent Application

Explore 3

Name: _______________________ Date: ___________

Tips and Commissions Part I: Tips Use the Tip Task Cards to complete the table below. Round your answer to the nearest hundredth, if necessary. Tips at the Yellow Rose Diner Will is a waiter at the Yellow Rose Diner. He gets paid an hourly wage plus tips. Will’s customers generally determine his tip by calculating a percentage of the cost of their meal.

Amir

Monique

Marcel

Jaslene

Percent Application | 169

Percent Application

Explore 3 Reflect 1. Describe how to determine the total cost of a meal including the tip.

2. Describe how the tape diagram represents the total cost of a meal at the Yellow Rose Diner.

3. Which customer leaves Will a larger tip: Mario, ordering a $17.50 meal and leaving an 18% tip, or Marla, ordering a $16 meal and leaving a 20% tip?

4. If leaving a 15%–20% tip is standard in the restaurant industry, what would encourage you to leave a larger tip?

5. Besides the restaurant industry, what other professions can you think of that encourage leaving a tip?

170 | Percent Application

Percent Application

Explore 3 Part II: Commissions

Use the Bob’s Appliance Store Sales Ad to get the price of each appliance. Use the Commission Spinner to determine the commission % for each appliance. Next, calculate the commission earned for each appliance. Brooke’s Commissions Diner co-owner Troy is looking to add some appliances to the diner’s kitchen. Troy visits his friend, Brooke, who sells kitchen appliances at Bob’s Appliance Store. Brooke earns a commission on every appliance she sells. How much commission did Brooke earn on the stove, microwave, refrigerator, and dishwasher that she sold?

Stove

Microwave

Refrigerator

Dishwasher

Percent Application | 171

Percent Application

Explore 3 Reflect 1. Describe how to determine the amount of commission earned.

2. How much commission does Brooke earn on a $378.50 appliance at a 12% commission rate?

3. Why do you think a company pays commissions to its salespeople?

4. When would a lower commission rate be beneﬁcial? When would a higher commission rate be beneﬁcial?

172 | Percent Application

Percent Application

Explore 4

Name: _______________________ Date: ___________

Simple Interest When a customer borrows money from the bank, the bank charges interest on the money borrowed until the money is paid back by the customer. The simple interest formula, I = P · r · t, determines the amount of simple interest paid to the bank by the customer on the loan. I = interest P = principal (amount of loan) r = interest rate (expressed as a decimal) t = time/length of loan (expressed in years) Diner manager Sienna has some great news to report! Local residents love eating at the diner, and business is booming. Due to the success of the Yellow Rose Diner, Sienna decides it’s time to expand the business and open a second restaurant in a nearby town. Sienna decides to visit the local bank and apply for a $10,000 loan. Roll a number cube twice to ﬁll in the interest rate and time for each loan option.

Loan Option A: $10,000 Interest Rate

Time

Simple Interest Formula

Interest Paid

Total Amount Paid

Percent Application | 173

Percent Application

Explore 4 Loan Option B: $10,000 Interest Rate

Time

Simple Interest Formula

Interest Paid

Total Amount Paid

Loan Option C: $10,000 Interest Rate

Time

Simple Interest Formula

Interest Paid

Total Amount Paid

Loan Option D: $10,000 Interest Rate

Time

174 | Percent Application

Simple Interest Formula

Interest Paid

Total Amount Paid

Percent Application

Explore 4 Reflect 1. Why do you think banks charge interest on money customers borrow?

2. According to your calculations from the loan option A table, which loan option would be the best for Sienna? Explain your reasoning.

3. A savings account yields a simple interest rate of 4%. You invest $1,500 for 2 years. How much simple interest do you earn on this investment?

4. Using the amount of interest in question 3, without additional deposits or withdrawals, what is your total balance in the savings account at the end of 2 years?

5. When is having a low interest rate an advantage? When is having a low interest rate a disadvantage?

6. Explain how you would apply the simple interest formula when borrowing money from the bank for 6 months.

Percent Application | 175

Expressions

177

Model:

Given Expression

Equivalent Expression

Expressions | 179

Simply App charges $4 for each app you upload plus $3 to join the company. You are excited to see that you will receive $5 for each app purchased, until you ﬁnd out that you must pay the company $2 each time someone purchases your app. x represents each app.

Use the Expression Cards to match each expression to a company’s expression. Then, use your algebra tiles to model the expression and determine an equivalent expression.

Part I

Name: _______________________ Date: ___________

Combining Like Terms with Rational Coefficients

Explore 1

Expressions

180 | Expressions

Model:

Given Expression

Equivalent Expression

Extraordinary Apps charges $1 per app, x, to set up your account. You gain $3 for the ﬁrst app you upload. You are paid $7 each time someone purchases your app. It costs $2 to promote your app.

Explore 1

Expressions

Model:

Given Expression

Equivalent Expression

Expressions | 181

App City pays $4 for each app, x, you upload into their system. They will pay you $2 more for each app you upload on the ﬁrst day. Your setup fee for App City is $6. You are charged $7 each time your app is sold, but you do get a joining bonus of $12.

Explore 1

Expressions

182 | Expressions

Model:

Given Expression

Equivalent Expression

Apps R Us! charges $3 per app that you upload to sell, but they will give you $5 upfront to sell with them. You will receive $6 per app purchase for all apps that you have uploaded.

Explore 1

Expressions

Model:

Given Expression

Equivalent Expression

Expressions | 183

B Appy! pays you $2 per app you upload but charges you $8 to set up your account. You must set up advertisements if you go with B Appy!, which will cost you $5 per app. They will pay you $3 at the end of the ﬁrst month.

Explore 1

Expressions

184 | Expressions

1 x + 7.75 − 0.25x − 3 2

−6.15x x − 2.3 + 1.7x + 5.9

− 7 m+ 1 −2 3 − 9 m

2 b− 1 −b+ 3 b− 3 5 6 10 4

−3.75x x + 9 − 6x + 1.25x − 12

Given Expression

Look at the given expression, and generate an equivalent expression for it.

Part II

Explore 1

Equivalent Expression

Expressions

Expressions | 185

4. Why is understanding if expressions are equivalent important? When might you use this understanding in the real world?

3. How does the knowledge of how to combine like terms help you understand equivalent expressions?

2. What are some common mistakes that could be made when combining like terms?

1. What are some strategies you can use to generate equivalent expressions for a given expression?

Reflect

Explore 1

Expressions

Explore 2

Name: _______________________ Date: ___________

Distributive Property Part I: Using Area Models to Determine Equivalent Expressions Complete each of the area models to determine equivalent expressions for each brother’s proﬁt expression. Joshia’s expression: 4(2x x – 5) 2x

−5

Equivalent expression using the area model: Joshia’s expression: −4(−2x x + 5)

Equivalent expression using the area model:

Reflect 1. What do you notice about the brothers’ expressions?

Expressions | 187

Expressions

Explore 2

Part II: Using the Distributive Property to Determine Equivalent Expressions Determine equivalent expressions for each level of app Joshia has created. Show your steps in the workspace provided, and explain the process of each step. App Level A: Proﬁt Expression Company’s Expression

5 – 2(−3x x + 2)

Workspace with Explanation

Joshia’s Equivalent Expression App Level A: Cost Expression Company’s Expression

−2 + 1 (10 (10x x – 6) 2

Workspace with Explanation

Joshia’s Equivalent Expression 188 | Expressions

Expressions

Explore 2 App Level B: Proﬁt Expression − 1 (−15 (−15x x + 9)

Company’s Expression

Workspace with Explanation

Joshia’s Equivalent Expression App Level B: Cost Expression Company’s Expression

0.75(x x – 24)

Workspace with Explanation

Joshia’s Equivalent Expression Joshia determined the proﬁt from his latest app to be 7 – 2 m. The company says his 10 5 expression is equivalent to the expression they use, which is p(7 – 4m). Use common factors to help Joshia determine the value of p.

Expressions | 189

Explore 2

Expressions

Reflect 1. Can you determine whether two expressions are equivalent without knowing the numerical value of a variable? Explain.

2. What strategies could you use to check your work with equivalent expressions?

3. What can be challenging about working with negative numbers in expressions?

190 | Expressions

Expressions

Explore 3

Name: _______________________ Date: ___________

Finding Equivalent Expressions Using Properties Part I Read the scenarios. Determine whether the two expressions in the table are equivalent. Record your work and answers, and explain your reasoning. Joshia wants to get a frame with tiles around it for four of his apps. The frames are priced by the tile. What expressions could determine how many tiles there are in 4 frames like the one shown on the right?

Equivalent?

Expression 1

Expression 2

4(4n + 4)

4(n + 1) · 4

Explanation:

The app company is running a sale today! Buy 4 apps, and get a 20% discount. What expressions could determine the total cost of buying 4 apps? 0.80(u · 4)

Equivalent?

(u · 4) – 0.20(u · 4)

Explanation:

Expressions | 191

Expressions

Explore 3

Joshia gets a deal on a new app management program. If he signs up today, he will get 10% off the total cost. The company charges $4 per app you have them manage.

Equivalent?

Expression 1

Expression 2

0.90(4 · t)

(4 · t) – 0.1(4 · t)

Explanation:

Joshia wants to add a rectangular background to his latest app. The length of the background is m inches, and the width will be 0.8 inches. Half of this area will be shaded in blue. He needs to ﬁnd the area of the blue part of the new background. 0.8(m) 2

Equivalent?

192 | Expressions

[ 2 (0.8)]( (0.8)](m)

Explanation:

Expressions

Explore 3 Part II

Solve each expression given. Then, generate an equivalent expression for the given expression using properties of operations. Expression 1

Expression 2

− 2 (12 (12x x − 2) 5

−6 3 − 2 ( 2 + y)

−3( 1 x + 2 1 ) 3

− a +1 6

Expressions | 193

Explore 3

Expressions

Reflect 1. How can you determine whether two expressions are equivalent for a word problem?

2. If two expressions are equivalent, does that mean the expressions correctly solve the word problem?

3. When you generated an equivalent expression for a word problem, did you have the same expression as all of your group members every time? Why?

194 | Expressions

Solve Equations and Inequalities

195

Solve Equations and Inequalities

Explore 1

Name: _______________________ Date: ___________

Construct Equations Use the work from your group’s Game Booth Cards to identify the variable, ﬁnd the model that matches your drawing, and select the correct equation.

Ring Toss Identify your variable.

p represents

Which diagram models this situation?

p 7

20 •

1 2

•

1 2

Which equation represents this situation? p = 16 + 7

(p + 7) ÷ 2 = 20

Justify your choice.

Solve Equations and Inequalities | 197

Solve Equations and Inequalities

Explore 1 Beanbag Toss Identify your variable.

l represents

Which diagram models this situation?

28 5

28 I

Which equation represents this situation? 5 + 5 + l + l = 28

5 · 5 · l · l = 28

Justify your choice.

Write a different equation that can be used to represent this situation.

198 | Solve Equations and Inequalities

Solve Equations and Inequalities

Explore 1 Basketball Shoot Identify your variable.

m represents

Which diagram models this situation?

22 m + m + m + m + 14

Write an equation to represent this situation.

Justify your equation.

Solve Equations and Inequalities | 199

Solve Equations and Inequalities

Explore 1 Putting Green Identify your variable.

p represents

Which diagram models this situation?

24 2(p + 3)

(p + 3)

Write an equation that can represent this situation.

Justify your equation.

200 | Solve Equations and Inequalities

Explore 1

Solve Equations and Inequalities

Reflect 1. How can you decide on which side of the equal sign you should put the variable?

2. How can you evaluate your equation to make sure it makes sense?

Solve Equations and Inequalities | 201

Solve Equations and Inequalities

Explore 2

Name: _______________________ Date: ___________

Solve and Compare Equations Carefully read and analyze each question. Identify the variable, write an equation, and model the problem using the algebra tiles and Algebra Equations Mat. Record your work and solution in the workspace provided.

Origami • Jaden made origami cranes for the craft booth. He sold them for $2 each. • Jaden spent $8 on materials in order to make the origami cranes. • After deducting how much Jaden spent on materials, he calculated a proﬁt of $24. • How many origami cranes (c) did Jaden sell? c represents Identify your variable.

Write an equation for the problem. Model the problem using algebra tiles and an Algebra Equations Mat. Record your model here.

Show your work and solution.

Solve Equations and Inequalities | 203

Solve Equations and Inequalities

Explore 2 Birdhouses

• Jamal made $8 from each birdhouse he sold at the craft booth. • Jamal will charge $3 for each hour he spent on his birdhouses and $2 for decoration. • How many hours did Jamal spend on his birdhouses (x)? x represents Identify your variable.

Write an equation for the problem. Model the problem using algebra tiles and an Algebra Equations Mat. Record your model here.

Show your work and solution.

204 | Solve Equations and Inequalities

Solve Equations and Inequalities

Explore 2 Balloon Animals

• Sangeeth made 2 types of balloon animals for the craft booth. • He sold 5 dogs and some giraffes. • He charged $4 for each balloon he sold. • Sangeeth made a total of $32 from selling balloon animals. • How many giraffes (g) did he make? g represents Identify your variable.

Write an equation for the problem. Model the problem using algebra tiles and an Algebra Equations Mat. Record your model here.

Show your work and solution.

Solve Equations and Inequalities | 205

Solve Equations and Inequalities

Explore 2 Art Boxes

• Zoe made art boxes with rectangular bases for the craft booth. • The length of one side of the base was 4 inches. • The perimeter of the base was 18 inches. • What was the width (w) of the base? w represents Identify your variable.

Write an equation for the problem. Model the problem using algebra tiles and an Algebra Equations Mat. Record your model here.

Show your work and solution.

206 | Solve Equations and Inequalities

Solve Equations and Inequalities

Explore 2 Jewelry Boxes • Heather made jewelry boxes for the craft booth.

• Heather had to pay a vendor fee of $6 in order to sell at the craft booth. • She sold each of her jewelry boxes for $3. • After deducting her vendor fee, Heather made a total of $27. • How many jewelry boxes (b) did Heather sell? b represents Identify your variable.

Write an equation for the problem. Model the problem using algebra tiles and an Algebra Equations Mat. Record your model here.

Show your work and solution.

Solve Equations and Inequalities | 207

Solve Equations and Inequalities

Explore 2 Reflect 1. Which side of the scale should you put the variable on?

2. Why is it important to identify the variable?

3. What are two ways to solve for the missing variable?

4. What changes to the birdhouse problem occur if Jamal found he is losing $3 for each hour he spent on birdhouses? Explain how it would affect solving the problem and the solution.

208 | Solve Equations and Inequalities

Solve Equations and Inequalities

Explore 3

Name: _______________________ Date: ___________

Construct Inequalities Carefully read and analyze each question. Identify the variable, write an inequality, and model the problem using the algebra tiles and Algebra Inequality Mat. Record your work in the workspace provided.

Baking Cupcakes Identify your variable.

d represents

Write an inequality for the problem. Model the problem using algebra tiles and an Algebra Inequality Mat. Record your model below.

Solve Equations and Inequalities | 209

Solve Equations and Inequalities

Explore 3 Personalized Cupcakes Identify your variable.

p represents

Write an inequality for the problem. Model the problem using algebra tiles and an Algebra Inequality Mat. Record your model below.

Eli’s Purchases Identify your variable.

c represents

Write an inequality for the problem. Model the problem using algebra tiles and an Algebra Inequality Mat. Record your model below.

210 | Solve Equations and Inequalities

Solve Equations and Inequalities

Explore 3 Fundraising Goals Identify your variable.

c represents

Write an inequality for the problem. Model the problem using algebra tiles and an Algebra Inequality Mat. Record your model below.

Reflect 1. How do you decide which Algebra Inequality Mat to use?

2. What words help you know if you are using > or ≥?

3. Why are the scales on the Algebra Inequality Mat at different levels?

Solve Equations and Inequalities | 211

Solve Equations and Inequalities

Explore 4

Name: _______________________ Date: ___________

Solve and Graph Inequalities Carefully read and analyze each question. Identify the variable, write an inequality, and model the problem. Record your work and graph your solution set in the workspace provided.

Purchasing Ducks • Players at the lucky duck booth will select a random duck for the chance to win a prize. • Players have to pay an entrance fee of $5 and $2.00 for each duck they choose. • Jasmine plans to spend less than $25 at the duck pond. • How many ducks (d) is Jasmine most likely going to buy?

Identify your variable.

d represents

Write an inequality for the problem.

Model the problem.

Solve algebraically.

Graph your solution set.

9 10 11 12 13 14 15 16 17 18 19 20 Solve Equations and Inequalities | 213

Solve Equations and Inequalities

Explore 4 Losing Tickets • Darian won some tickets at the lucky duck booth. • Javier lost twice as many tickets as Darian won.

• Javier found 7 more tickets that didn’t ﬂy away as he rushed to pick them up. • Now the boys have more than 34 tickets in all for the booth. • How many tickets (t) did Javier lose during the lucky duck booth?

Identify your variable.

t represents

Write an inequality for the problem.

Model the problem.

Solve algebraically.

Graph your solution set. -28 -26 -24 -22 -20 -18 -16 -14 -12 -10

214 | Solve Equations and Inequalities

-8

-6

-4

-2

14 16

22 24

Solve Equations and Inequalities

Explore 4 Calculating Results

• Rasul and David spent a lot of time at the lucky duck booth. • They noticed that prizes were awarded in a predictable way. • Rasul says that if you add 3 to every dollar spent (d) and multiply the total by 2, you will get a number greater than or equal to 14. • David says that if you multiply the number of dollars (d) by two and add six, you will get a number greater than or equal to 14. • The boys bring their math to you and ask you to ﬁnd out how many dollars must be spent.

Identify your variable.

d represents

Write an inequality to represent Rasul’s results. Write an inequality to represent David’s results.

Model both problems. Rasul’s Inequality

David’s Inequality

Solve Equations and Inequalities | 215

Solve Equations and Inequalities

Explore 4 Solve and graph both inequalities. Rasul’s Inequality

David’s Inequality

Summarize your ﬁndings.

Reflect 1. Why is it necessary to show your solution set with a ray?

2. What did you learn when you analyzed Rasul and David’s inequalities?

216 | Solve Equations and Inequalities

Scaling

217

Scaling

Explore 1

Name: _______________________ Date: ___________

Scale Drawings Match three pairs of National Park Sign Cards as scale drawings. Draw each pair of signs under its type. Use markers to identify corresponding sides. Determine ratios to prove the signs are scale drawings of each other. Create a third scaled sign that belongs in each sign category, and include the “between” and “within” ratios. Signs for “Hiking Only Trail”

“Between” Ratios

Simplify

“Within” Ratios

Simplify

3rd “Hiking Only Trail” Sign (reduction or enlargement)

“Between” Ratios

Simplify

“Within” Ratios

Simplify

Scaling | 219

Scaling

Explore 1 Signs for Trail Directions and Distances

“Between” Ratios

Simplify

“Within” Ratios

Simplify

3rd “Trail Directions and Distances” Sign (reduction or enlargement)

“Between” Ratios

Simplify

“Within” Ratios

Simplify

220 | Scaling

Scaling

Explore 1 Signs for “Take Only Pictures, Leave Only Footprints”

“Between” Ratios

Simplify

“Within” Ratios

Simplify

3rd “Take Only Pictures, Leave Only Footprints” Sign (reduction or enlargement)

“Between” Ratios

Simplify

“Within” Ratios

Simplify

Scaling | 221

Explore 1

Scaling

Reflect 1. What is the ratio of any two corresponding sides?

2. How did you ﬁnd the “between” ratios?

3. How did you ﬁnd the “within” ratios?

4. What can you say about rectangles that are scaled drawings of each other and the ratios of their corresponding sides?

5. How did you draw scaled rectangles that were either enlargements or reductions of given rectangles?

222 | Scaling

Perimeter and Area

New Dimensions

Original Dimensions Ratio 1

Simplify

Ratio 2

Scaling | 223

Simplify

Name: _______________________ Date: ___________

Use the National Park Task Cards to ﬁnd the scale factor of the maps and answer questions.

Part I: Finding the Scale Factor

Explore 2

Scaling

224 | Scaling

5. What numbers do you divide to ﬁnd the scale factor? Does it matter in what order you put the numbers?

4. What do you think happens when the scale factor is 1?

3. How does scale factor affect the size of two rectangular maps that are scale drawings of each other?

2. Describe the scale factor numbers when the maps get larger.

1. Describe the scale factor numbers when the maps get smaller.

Reflect

Explore 2

Scaling

Answer: ________

Finding the Perimeter of the Pool in Real Life Width

Answer: ________

Scaling | 225

_______ of fencing is needed to enclose the pool.

Use the perimeter formula for a rectangle to ﬁnd the perimeter of the pool. P = 2l + 2w

Length

Width

The area of the playground is _______________________.

Use the area formula for a rectangle to ﬁnd the area of the playground. A = l · w

Length

Finding the Area of the Actual Playground

Use Part II of the National Park Task Cards. Use a proportion to help ﬁnd the area and/or perimeter of places in the national park.

Part II: Finding Area and Perimeter of Rectangles Using Proportions

Explore 2

Scaling

Answer: ________

______ picnic tables are needed in the picnic area.

Answer: ________

Width

Answer: ________

________ yards

Perimeter:

226 | Scaling

_______ square yards

Area:

Use the perimeter formula for a rectangle to determine how much fencing is needed. P = 2l + 2w

Use the area formula for a rectangle to ﬁnd the area needed for the netting. A = l · w

Length

Finding the Real-World Area and the Perimeter of the Animal Rehabilitation Area

Determine how many tables will ﬁt in the picnic area.

Width

The area of the picnic area is _______________________.

Use the area formula for a rectangle to ﬁnd the area of the picnic area. A = l · w

Length

Finding the Area of the Picnic Area and Determining the Number of Tables Needed

Explore 2

Scaling

Scaling | 227

4. What information do you need to work backward and ﬁnd the dimensions such as length or width of something on a map?

3. How do you ﬁnd the area or perimeter of a rectangular feature in the real world from seeing it on a map or diagram?

2. Can you apply a scale factor to only one dimension and not the other?

1. When given a scale and measurements on a map, how do you create a proportion to ﬁnd measurements that are the actual length or width of something?

Reflect

Explore 2

Scaling

Angles

229

End Table

Sketch:

Coffee Table

Angles | 231

Name: _______________________ Date: ___________

Measuring in Nonstandard Units

Sketch each shape, and label the number of units for each angle.

Explore 1

Angles

232 | Angles

Sketch:

Kitchen Table

Explore 1

Sketch:

Couch

Angles

Rug

Sketch:

Explore 1

Sketch:

Bookshelf

Angles | 233

Angles

234 | Angles

3. In the real world, in what other situations would you need to measure angles?

2. What does this technique tell us about what it means to measure angles?

1. How do you use the circle to determine the number of units in the angle?

Reflect

Explore 1

Angles

Explore 2

Name: _______________________ Date: ___________

Measuring Angles Measure the angles of each puzzle piece. Find the puzzle piece with the same measurement, and record it on the table below. Number

Angle Measure

Letter

Angle Measure

Angles | 235

Explore 2

Angles

Reflect 1. Why are there two sets of numbers on a protractor?

2. How does using a protractor compare to using the circle units from the last Explore?

3. What do you need to do to get an accurate measurement when using a protractor?

236 | Angles

Angle Relationships

237

Angle Relationships

Explore 1

Name: _______________________ Date: ___________

Supplementary and Complementary Angles The map below shows the existing park. Use a protractor and a straightedge to sketch Jaime and Aliyah’s plans onto the map of Rustic Oak Park. Then, compare the plan to the city requirements to determine whether the plan meets the guidelines. Complete the analysis, and answer the reﬂection questions. Fountain

N W

E S

Sunset Trail

A B ck

Main Trail

Children’s Playground

Shady Trail

D Map

Duck Pond

i Tra

p Ha

Parking

ail

Jaime and Aliyah’s Bike Trail Proposal Construction – Use straight lines to add the trails as described. • Connect the intersection of Sunset Trail and Happy Trail with Main Trail with a line that is perpendicular to Main. Name it “Jaime Trail.” • Connect the north end of Shady Trail to the east end of Main Trail. Name it “Aliyah Trail.” • Connect the intersection of Main Trail and Shady Trail with Aliyah Trail. Name it “Sky Trail.” Beautiﬁcation – Add the features described. • Main Trail is crowded. Add a hiking lane on the edge of Main Trail. • Plant six shade trees along Happy Trail. • Add a water fountain on the north side of the playground. © Accelerate Learning Inc. – All Rights Reserved

Angle Relationships | 239

Angle Relationships

Explore 1 Use the protractor to measure each angle. ∠A _____°

∠B _____°

∠F _____°

∠G _____°

∠C _____°

∠H _____°

∠D _____° ∠J _____°

∠E _____°

∠K _____°

Fill in the table, and determine whether the bike trail proposal meets the guidelines. Provide evidence by listing the angles or features that prove the requirement is being met. City Park Improvement Guidelines for Rustic Oak Park Requirement

Evidence

Met?

The plan must include at least 4 sets of trails that form complementary angles that share a side. (Complementary angles are pairs of angles that have a sum of 90°.) The plan must include at least 2 pairs of trails that form supplementary angles. (Supplementary angles are pairs of angles that have a sum of 180°.) The plan must include at least two beautiﬁcation suggestions.

Reflect 1. Design a change to the bike trail proposal to meet the guidelines.

2. Compare and contrast supplementary and complementary angles.

240 | Angle Relationships

Angle Relationships

Explore 2

Name: _______________________ Date: ___________

Vertical and Adjacent Angles The map below shows the current design of Tall Pines Park. Use a protractor and a straightedge to sketch Min and Xavier’s plan onto the map. Then, compare the plan to the city’s requirements to determine whether the plan meets the guidelines. Complete the analysis, and answer the reﬂection questions. N W

E S

Parking A

B C

Main Trail

Pine Lake

ake

Big Lake Trail

rai tT

s We

Big Lake

il Tra

Swings P

Q R

W S

Back Trail

Min and Xavier’s Hiking Trail Proposal Construction – Use straight lines to add the trails as described. • Extend Pine Lake Trail to the parking lot. • Connect the south ends of West Trail and Big Lake Trail. • Make Big Lake Trail wider to better accommodate visitors walking at different speeds. Beautiﬁcation – Add the features described. • Add 3 reserved parking spots for mobility-aid users near Pine Lake Trail. • Add benches at locations A, C, W, and Z. • Add rest areas at locations K, N, P, and R. © Accelerate Learning Inc. – All Rights Reserved

Angle Relationships | 241

Angle Relationships

Explore 2 Use the protractor to measure each angle. ∠A ____°

∠D ____°

∠B ____°

∠C ____°

∠K ____°

∠N ____°

∠L ____°

∠M ____°

∠P ____°

∠S ____°

∠Q ____° ∠R ____°

∠W ____° ∠X ____°

Fill in the table, and determine whether the hiking trail proposal meets the guidelines. Provide evidence by listing the angles or features that prove the requirement is being met. City Park Improvement Guidelines for Tall Pines Park Requirement

Evidence

Met?

Benches must be added in pairs and placed on adjacent angles. (Adjacent angles are angles that have the same vertex and a common side.) Rest areas must be added in pairs and placed on vertical angles. (Vertical angles are angles that are opposite of each other when two lines cross. They share a vertex.) The plan must include at least two trail extensions that connect current trails.

Reflect 1. Design a change to the hiking trail proposal to meet the guidelines.

2. Compare and contrast adjacent and vertical angles.

242 | Angle Relationships

Angle Relationships

Explore 3

Name: _______________________ Date: ___________

Multistep Angle Problems The image below shows the current design of Redwood Park. Use a straightedge to sketch Gabriella and Jasmine’s plan onto the map. Then, compare the plan to the city’s requirements to determine whether the plan meets the guidelines. Complete the analysis, and answer the reﬂection questions. N

Scenic Overlook

E S

B C

North Trail D

L M

ckw

Tra i

Hydration Station

Ranger Station

rry

S T

i Tra

South Trail

Bla

Redwood Trail

Rest Area

X Z

Parking

Gabriella and Jasmine’s Nature Park Proposal Construction – Use straight lines to add the trails as described. • Extend Buckwheat Trail to the intersection of Redwood Trail and South Trail. • Extend Blackberry Trail to the scenic overlook. • Add safety railings to the scenic overlook. Beautification – Add the features described. • Add a wildflower garden between Redwood Trail and Blackberry Trail to attract butterflies and bees. • Add a water feature to provide fresh water for birds and insects. • Add an informational sign at the scenic overlook to identify common trees and wildlife in the park. © Accelerate Learning Inc. – All Rights Reserved

Angle Relationships | 243

Angle Relationships

Explore 3

Label the angles on the map using the chart below. Use the measurements provided and your knowledge of supplementary, complementary, vertical, and adjacent angles to determine the missing measurements. Angle

Measurement

Justiﬁcation and Equation

∠A

30°

Provided

x°

Provided

∠T

90°

Provided

x°

Provided

∠Q

2x°

Provided

∠B ∠C

∠D ∠K ∠L

∠M ∠N ∠P

∠P ∠Q

244 | Angle Relationships

Angle Relationships

Explore 3

Fill in the table, and determine whether the nature trail proposal meets the guidelines. Provide evidence by listing the angles or features that prove the requirement is being met. City Park Improvement Guidelines for Redwood Park Requirement

Evidence

Met?

Trails must create at least six pairs of vertical angles.

Trails must create at least two complementary angles.

The plan must include at least two natural enhancements.

Reflect 1. Design a change to the nature trail proposal to meet the guidelines.

2. Why is it important to understand adjacent, vertical, supplementary, and complementary angles?

Angle Relationships | 245

Circles

247

Circles

Explore 1

Name: _______________________ Date: ___________

Discovering Circumference

Part I: Parts of a Circle Every pizza made and sold at the pizzeria will be in the shape of a circle. Use the Deﬁnition Cards to identify and label the center, radius, diameter, and circumference of the circle below.

Circles | 249

Circles

Explore 1 Diego’s Pizzeria The pizza makers have learned how to make their ﬁrst pizza and have recorded measurements for the pizza. The measurements are 14 inches, 44 inches, and 7 inches. What parts of the circle could be represented by each measurement? Explain.

Reflect 1. What is the relationship between the circle’s radius and its diameter?

2. Do you think there is a relationship between the circle’s diameter and its circumference?

250 | Circles

Circles

Explore 1 Part II: Discovering Pi as a Constant

The pizzeria will make and sell 4 different-sized pizzas: a small pizza, a medium pizza, a large pizza, and a mega pizza. Use a ruler to measure the diameter, and use string to measure the circumference of each pizza on the Pizza Cards to the nearest inch. Use the measurements to complete the table below. Round your answer to the nearest hundredth. Pizza Size

Diameter

Circumference

Circumference Diameter

Decimal Form

Small

Medium

Large

Mega

Reflect 1. What do you notice about the information in the table?

2. Do you think there is a relationship between a circle’s diameter and its circumference? If so, what is it?

Circles | 251

Circles

Explore 2

Name: _______________________ Date: ___________

Circumference

Part I: Connecting to the Formulas Diego is planning to order pizza pans, and the pizza makers need to measure the circumference of the pizza. Use string and the Pizza Cutout to help Diego measure the diameter and circumference of the pizza. Diego’s Pizza Pans Diego is placing his order for pizza pans and needs help understanding what the measurements represent. 1. How many string diameters were cut from the string circumference?

2. What fraction of the string was left over?

Circles | 253

Circles

Explore 2 Reflect

1. What do you think the value of the number of diameters needed to go around the circumference represents?

2. If you are given the measure of a circle’s diameter, how could you calculate the circumference of the circle?

3. If you are given the measure of a circle’s radius, how could you calculate the circumference of the circle?

254 | Circles

Circles

Explore 2 Part II: Calculating Circumference

Determine which expressions can be used to calculate the circumference of each pizza pan needed. Use the Pizza Circumference Cards to match each pizza with the equations that can be used to determine its circumference. Write the matching equation that could be used to calculate the circumference of each pizza. Calculate the circumference of each pizza using 3.14 for 𝜋. Record the information in the table. Small Pizza

Radius = __________

Diameter = ________

Circumference equation = __________

Circumference = ________

Medium Pizza Radius = __________

Diameter = ________

Circumference equation = __________

Circumference = ________

Large Pizza Radius = __________

Diameter = ________

Circumference equation = __________

Circumference = ________

Mega Pizza Radius = __________

Diameter = ________

Circumference equation = __________

Circumference = ________

Circles | 255

Circles

Explore 2

Calculate the circumference of the pepperoni and pizza for the pizza makers at Diego’s Pizzeria. Use 3.14 for 𝜋. Diego’s Pizzeria

1. The pizzeria will offer several specialty pizzas. Most of the specialty pizzas will be made with pepperoni along with other toppings. Each pepperoni has a diameter of 5.3 centimeters before it is cooked on the pizza. What is the circumference of each pepperoni before it is cooked?

2. Pepperoni shrinks while cooking. After being cooked, the pepperoni has a radius of 2.25 centimeters. What is the circumference of a pepperoni after it has been cooked?

3. Before putting toppings on a pizza, the pizza makers must use sauce to make a circle on the pizza dough, leaving a 1 in. ring of dough around the pizza. What is the circumference of the circle that is made of sauce on the mega pizza? Explain your reasoning.

14 in.

256 | Circles

Explore 2

Circles

Reflect 1. What formula should be used to calculate the circumference of a circle if given the circle’s diameter?

2. What formula should be used to calculate the circumference of a circle if given the circle’s radius?

3. Why are there two formulas for calculating the circumference of a circle?

Circles | 257

Circles

Explore 3

Name: _______________________ Date: ___________

Area of a Circle Part I: Connecting to the Area Formula Use the Decomposing a Circle handout, and cut out the parts of the pizza. Rearrange the parts of both pizzas to form parallelograms. Label the parts below to show how the area of a circle formula is generated.

Diego’s Pizza Dough After decomposing the pizzas, Diego and the pizza makers would like to understand how the area of the pizza dough connects to the pizza when it is rearranged to form a parallelogram. What are the dimensions of the parallelogram that is formed?

Circles | 259

Circles

Explore 3 Reflect

1. How is the radius of the circle connected to the height of the parallelogram created from the decomposed circle?

2. Why does the base of the parallelogram represent only half of the circumference of the circle?

3. How are the area of the parallelogram and the area of the circle related?

4. Based on the parallelogram, what formula can be used to calculate the area of a circle?

260 | Circles

Circles

Explore 3 Part II: Calculating the Area of a Circle

Determine which expressions can be used to calculate the area of each pizza needed. Use the Pizza Area Cards to match each pizza with the equation that can be used to determine its area, and write the matching equation in the spaces below. Calculate the area of each pizza using 3.14 for 𝜋. Record the information in the table. Small Pizza

Radius = __________

Radius2 = ________

Area equation = __________

Area = ________ Medium Pizza

Radius = __________

Radius2 = ________

Area equation = __________

Area = ________ Large Pizza

Radius = __________

Radius2 = ________

Area equation = __________

Area = ________ Mega Pizza

Radius = __________

Radius2 = ________

Area equation = __________

Area = ________

Circles | 261

Circles

Explore 3

Calculate the area of the pepperoni and pizza for the pizza makers at Diego’s Pizzeria. Use 3.14 for 𝜋. Specialty Pizzas

The pizzeria will offer several specialty pizzas. Most of the specialty pizzas will be made with pepperoni along with other toppings. Each pepperoni has a diameter of about 5 centimeters before it is cooked on the pizza. What is the approximate area of each pepperoni before it is cooked?

Pizza Sauce on the Mega Pizza Before putting toppings on a pizza, the pizza maker must use sauce to make a circle on the pizza dough, leaving a 1 in. ring of dough around the pizza as shown in the diagram.

14 in.

What is the area of the circle that is made of sauce on the mega pizza? Explain your reasoning.

262 | Circles

Explore 3

Circles

Reflect 1. What part of a circle is needed to calculate the area of the circle?

2. What expression can be used to ﬁnd the area of a personal-sized pizza that has a diameter of 7 inches?

Circles | 263

Circles

Explore 4

Name: _______________________ Date: ___________

Area and Circumference Problem Solving Determine whether the scenario represents area or circumference. Write the matching equation that could be used to calculate the circumference or area in each scenario. Calculate the circumference or area of each pizza using 3.14 for 𝜋. Record the information in the table. Pizza with Pepperoni The pizza makers are trying to determine the amount of pizza covered up by a pepperoni. How much pizza is covered up by a pepperoni?

Radius: 3 cm Solve: Area or Circumference

Answer: Red Peppers on Pizza Crust Diego is measuring the length of red peppers around the crust of a personal pan pizza. What is the length of red peppers around the crust of the pizza?

Diameter: 30 cm Solve: Area or Circumference

Circles | 265

Circles

Explore 4 Flavored Butter on Pizza Crust

Diego is trying to determine the amount of ﬂavored butter that is brushed around the ring of pizza crust. What amount of ﬂavored butter will be brushed around the ring of pizza crust?

Diameter: 7 in. Solve:

Area or Circumference

Answer:

Pizza Seasoning on Pizza The pizza makers are trying to keep track of the amount of pizza seasoning needed to cover a pizza. How much pizza seasoning is needed to cover a pizza?

Diameter: 8 in. Solve:

Area or Circumference

Answer: 266 | Circles

Circles

Explore 4 Pizza Sauce on Pizza Dough

Diego is calculating the amount of pizza sauce that is used to cover the dough. How much pizza sauce will be used to cover the dough?

Radius: 5 in. Solve:

Area or Circumference

Answer:

Cheese in Pizza Crust The pizza makers are measuring the length of cheese string inside the crust of a pizza. What is the length of cheese string that will be placed inside the crust of a pizza?

Radius: 4 in. Solve:

Area or Circumference

Circles | 267

Circles

Explore 4 Reflect

1. If a circle has a radius of 3 cm and you are ﬁnding the circumference and area of the circle, how would the circumference and area of the circle be affected if the radius doubled?

2. Compare using the area and circumference formulas when given the radius or diameter.

3. What are examples of how the area and circumference of a circle can be used in the real world?

268 | Circles

Surface Area

269

Slicing 3-D Figures

Name: _______________________ Date: ___________

Vertically

Right rectangular prism Right pyramid Cube Cylinder Cone Sphere

Horizontally

How was it sliced?

Right rectangular prism Right pyramid Cube Cylinder Cone Sphere

3-D Figure

Card

Surface Area | 271

____________________

2-D Figure (Draw and label.)

Choose a Slicing 3-D Figures Scenario Card. Using modeling clay, sculpt the 3-D figure. Then, follow the directions to slice the figure correctly. Complete the table below by circling the 3-D figure and determining how it was sliced. Sketch the 2-D figure, and label the figure.

Explore 1

Surface Area

Vertically

Right rectangular prism Right pyramid Cube Cylinder Cone Sphere

Horizontally

How was it sliced?

Vertically

3-D Figure

Right rectangular prism Right pyramid Cube Cylinder Cone Sphere

272 | Surface Area

Card

Explore 1

____________________

2-D Figure (Draw and label.)

Surface Area

Vertically

Right rectangular prism Right pyramid Cube Cylinder Cone Sphere

Horizontally

How was it sliced?

Right rectangular prism Right pyramid Cube Cylinder Cone Sphere

3-D Figure

Card

Explore 1

Surface Area | 273

____________________

2-D Figure (Draw and label.)

Surface Area

Vertically

Right rectangular prism Right pyramid Cube Cylinder Cone Sphere

Horizontally

How was it sliced?

Vertically

3-D Figure

Right rectangular prism Right pyramid Cube Cylinder Cone Sphere

274 | Surface Area

Card

Explore 1

____________________

2-D Figure (Draw and label.)

Surface Area

5. Can you think of any other pairs of 3-D ﬁgures with 2-D plane sections?

4. Could a rectangular prism that is not a square ever have a 2-D plane section that is a square?

Surface Area | 275

3. Did you notice anything special about the triangle that was revealed when the right pyramid was sliced vertically? Explain.

2. What kind of two-dimensional ﬁgures were found when plane sections were sliced from right rectangular prisms (including cubes) and right pyramids?

1. What is the relationship between a right rectangular prism and a cube? What is the relationship between a rectangle and a square?

Reflect

Explore 1

Surface Area

Name: _______________________ Date: ___________

Surface area = _______ square inches

Faces:

Surface area = _______ square inches

Faces:

Workspace:

Faces:

Box 2

Workspace:

Bases:

Box 1

Surface Area | 277

Mrs. Lopez noticed that some of her shipping boxes ripped and fell apart if they got wet during shipping. She created a spray-on product to waterproof her boxes. One spray bottle covers 10,000 square inches of surface area. She asked Maria to measure the dimensions of the six different sizes of shipping boxes she sells and ﬁgure out the surface area of each.

Explore 2

Surface Area

278 | Surface Area

Surface area = _______ square inches

Faces:

Surface area = _______ square inches

Bases:

Box 4

Workspace:

Faces:

Box 3

Workspace:

Bases:

Explore 2

Surface Area

Surface area = _______ square inches

Workspace:

Faces:

Workspace:

Bases:

Box 6

Faces:

Box 5

Faces:

Bases:

Explore 2

Surface Area | 279

Surface Area

280 | Surface Area

5. In the real world, when might people need to determine the surface area of complex three-dimensional ﬁgures?

4. Mrs. Lopez wants to create a display of boxes covered in the protective coating by spraying one of each of the 6 boxes she sells. Does she need more than one bottle to coat the boxes? (Remember that one bottle coats 10,000 square inches.)

3. What is important to remember about the base of any triangular prism?

2. What are the similarities and differences of the surface area formulas for a rectangular prism and a cube?

1. What were the surface area formulas you used for rectangular prisms and cubes?

Reflect

Explore 2

Surface Area

Volume

281

Volume of Rectangular Prisms

Name: _______________________ Date: ___________

Volume = ___________ cubic inches

Workspace:

Side:

Box 2

Side:

Volume = ___________ cubic inches

Workspace:

Side:

Height:

Length:

Width:

Sketch:

Box 1

Find the volume of each box by sketching it and using the correct volume formula.

Volume | 283

To keep the contents of packages from being damaged, Mrs. Lopez ﬁlls her shipping boxes with environmentally friendly, biodegradable packing peanuts. To have enough packing peanuts on hand, she needs to know how many packing peanuts she uses in a month. She can determine that by ﬁguring out the volume of each of her six shipping boxes and applying that information to how many boxes she sells and ships in a month.

Part I: Box Volume

Explore 1

Volume

284 | Volume

Volume = ___________ cubic inches

Workspace:

Width:

Box 4

Height:

Volume = ___________ cubic inches

Workspace:

Length:

Side:

Sketch:

Box 3

Sketch:

Explore 1

Volume

Volume = ___________ cubic inches

Workspace:

Width:

Box 6

Height:

Volume = ___________ cubic inches

Workspace:

Length:

Side:

Sketch: (Part 2)

Length:

Height:

Length:

Width:

Sketch: (Part 1)

Box 5

Sketch: (Part 1)

Explore 1

Volume | 285

Volume

Formula:

286 | Volume

Difference in Box Volumes

Formula:

Shipping box

Volume of Large Box

Box containing ceramic box

Solution: Mrs. Lopez needs to use __________ cubic inches of packing peanuts.

Formula:

Volume of Small Box

Look at the diagrams of the boxes being used below, and determine the volume of packing peanuts Mrs. Lopez must use to protect the package.

A customer brings in a fragile ceramic box packed in a small cubic box with all edges 15 inches long. She wants it packed in another, larger cubic box with lots of packing peanuts to cushion it.

Part II: Customer Service

Explore 1

Volume

Formula:

Volume of Large Box

Volume | 287

Difference in Box Volumes Formula:

Solution: Mrs. Lopez needs to use __________ cubic inches of packing peanuts.

Formula:

Volume of Small Box

Draw diagrams of the boxes you would choose from Mrs. Lopez’s collection, and determine the volume of packing peanuts Mrs. Lopez must use to protect the package.

A customer brings in a fragile family heirloom to send across the country. It measures 8 inches long, 3 inches wide, and 6 inches tall. She wants it carefully wrapped and packed in a small box. Then, she wants it put in a larger box full of packing peanuts to cushion it.

Explore 1

Volume

288 | Volume

4. In the real world, why might people need to ﬁnd the volume of rectangular prisms or cubes?

3. How would you ﬁnd the volume of packing peanuts needed for a box with the dimensions 20 inches long, 10 inches wide, and 10 inches tall containing another box shaped like a cube with all edges 5 inches?

2. Maria tells her mom that ﬁguring out the volume of a box does not really explain the volume of peanuts a box will use when something is being shipped inside the box. Is Maria correct? Explain what she means.

1. What were the volume formulas you used for rectangular prisms and cubes?

Reflect

Explore 1

Volume

Volume of Triangular Prisms

Name: _______________________ Date: ___________

Base height:

Box 2

Volume | 289

Prism height:

Volume = ___________ cubic inches

Workspace:

Volume = ___________ cubic inches

Formula:

Base length:

Prism height:

Shape of the base:

Shape of the base: Base height:

Sketch:

Box 1

Find the volume of each box by sketching it and using the correct volume formula. Then, solve the problems.

Explore 2

Volume

290 | Volume

Base height:

Box 4

Prism height:

Volume = ___________ cubic inches

Workspace:

Volume = ___________ cubic inches

Formula:

Base length:

Prism height:

Shape of the base:

Shape of the base: Base height:

Sketch:

Box 3

Sketch:

Explore 2

Volume

Base height:

Box 6

Volume | 291

Prism height:

Volume = ___________ cubic inches

Workspace:

Volume = ___________ cubic inches

Formula:

Base length:

Prism height:

Shape of the base:

Shape of the base: Base height:

Sketch:

Box 5

Sketch:

Explore 2

Volume

292 | Volume

4. In the real world, why might people need to ﬁnd the volume of triangular prisms?

3. What are some differences between triangular prisms and rectangular prisms?

2. What are some similarities between triangular prisms and rectangular prisms?

1. What were the volume formulas you used for triangular prisms?

Reflect

Explore 2

Volume

Volume of Cylinders

Name: _______________________ Date: ___________

Column height:

Column B

Volume = ___________ cubic inches

Workspace:

Volume = ___________ cubic inches

Formula:

Column radius:

Column height:

Sketch:

Column A

Volume | 293

Find the volume of each column by sketching it and using the correct volume formula. Then, solve the problems.

Explore 3

Volume

294 | Volume

Column height:

Column D

Volume = ___________ cubic inches

Workspace:

Volume = ___________ cubic inches

Formula:

Column radius:

Column height:

Sketch:

Column C

Sketch:

Explore 3

Volume

Column height:

Column F

Volume = ___________ cubic inches

Workspace:

Volume = ___________ cubic inches

Formula:

Column radius:

Column height:

Sketch:

Column E

Sketch:

Explore 3

Volume | 295

Volume

296 | Volume

3. In the real world, why might people need to ﬁnd the volume of cylinders?

2. How is that formula similar to the formula used for prisms?

1. What were the volume formulas you used for cylinders?

Reflect

Explore 3

Volume

Probability

297

Probability

Explore 1

Name: _______________________ Date: ___________

Probability Part I: Discovering Likelihood Use the mystery bag shown below to determine how likely it is to select a certain marble color from the bag. Complete the table, and include the fraction, the percent, and the likelihood.

1. List all possible outcomes of picking a marble from the mystery bag.

Color

Fraction

Percent

Likelihood

Green Yellow Red Purple Blue Red or blue marble 2. If the mystery bag included 20 red marbles, what is the likelihood a red marble would be selected?

Probability | 299

Probability

Explore 1 Reflect 1. What is the range of values for the likelihood of a predicted event?

2. What are the numerical values of probability associated with each category of likeliness? Explain. • Impossible:

• Unlikely:

• Equally likely:

• Likely:

• Certain:

300 | Probability

Probability

Explore 1 Part II: Assessing Probability and Likelihood

Use the Probability Task Cards to determine the probability and likelihood of each scenario. Card Number

Probability

Likelihood

Probability | 301

Probability

Explore 1 Reflect 1. What steps did you take to ﬁnd the probability of an event?

2. Once you knew the probability, how did you choose the likelihood of an event?

3. Create an example of events with probabilities of 0, 2 , and 1.

302 | Probability

Probability

Explore 2

Name: _______________________ Date: ___________

Predicting Probability Part I: Flip a Coin Predict how many heads and tails will be tossed. Toss a coin 20 times, and record each toss with a tally mark.

I predict that I will toss _______ heads and _______ tails.

Heads

Tails

Total: ______

1. What was the result of your ﬁrst coin toss?

2. Did you get exactly 10 heads and 10 tails?

3. What do you predict the results would be if you did 1,200 coin tosses?

__________ tails Probability | 303

Probability

Explore 2 Reflect

1. What is the difference between theoretical probability and experimental probability?

2. How do you make predictions using theoretical probability?

3. If the theoretical probability was the same, why did you not get exactly the same results as another pair of students who ﬂipped their coins?

304 | Probability

Probability

Explore 2 Part II: Flip a Coin (Long-Run Relative Frequency)

Predict how many heads and how many tails will be tossed. Flip a coin 10 times, and record each toss.

I predict that I will toss ____ heads and ____ tails.

Flip a coin 10 times, and record your data from each coin toss on the next page in the Outcome column. After 10 coin ﬂips, come back to this page and complete the chart below.

Heads/Tails

Frequency

Frequency Written as a Fraction

Heads

Tails

Probability | 305

Probability

Explore 2

Using your data from the 10 coin ﬂips, calculate the relative frequency for heads and tails. Represent the relative frequency as a fraction and a decimal.

Toss

Outcome

Total Number of Heads So Far

Relative Frequency of Heads So Far (to the nearest hundredth)

Total Number of Tails So Far

Relative Frequency of Tails So Far (to the nearest hundredth)

1 2 3 4 5 6 7 8 9 10

306 | Probability

Probability

Explore 2 Reflect

1. What do you notice in the table about the changes in the relative frequency of the number of tails as the number of tosses increases?

2. What is long-run relative frequency, and why does it usually come closer to theoretical probability than a one-time short experiment?

3. What is an example of ﬁnding the relative frequency of a coin toss?

Probability | 307

Probability

Explore 3

Name: _______________________ Date: ___________

Probability Models Use the Probability Model Cards to determine the experimental probability and theoretical probability of each event. Record your results from the experiment in the designated table for each card. Design a probability model, and compare the probabilities for your model.

Card 1 What is the probability of pulling a yellow counter from the brown bag?

Probability: ________

Card 2 What is your group’s experimental probability of pulling a yellow counter?

Outcomes

Frequency

Frequency (fraction)

Probability | 309

Probability

Explore 3 Card 3 What is the probability that the ﬁfth student will pull a yellow color counter?

Probability: ________

Card 4 What is the probability of spinning the spinner and landing on the color blue?

Probability: ________

Card 5 What is your group’s experimental probability of spinning the spinner and landing on the color blue?

Outcomes

Frequency

Frequency (fraction)

Probability: ________

310 | Probability

Probability

Explore 3 Card 6 Explain the model your group developed using the brown paper bag and color counters or the spinner. Create a table to show the recorded results of the experiment.

Probability: ________

Comparing Probabilities Using Card 6 Red

Yellow

Blue

Green

Theoretical Probability Experimental Probability

Probability | 311

Probability

Explore 3 Reflect

1. Is the experimental probability always the same as the theoretical probability?

2. Based on the theoretical probability of the spinner landing on the color blue, how many spins out of 40 spins should land on the color blue?

3. How do probability models help with comparing probabilities?

312 | Probability

Informal Inferences

313

Informal Inferences

Explore 1

Name: _______________________ Date: ___________

Variability in Data Part I: Understanding a Survey Follow the steps below to conduct a survey of your classmates on a question of your choice. Use the data collected to answer your question, and then answer the reﬂection questions related to the survey. Step 1: Write a question that can be answered with data. Question:

Step 2: Ask classmates the above question. Organize the data in the table below. Classmate Responses and Data

Informal Inferences | 315

Informal Inferences

Explore 1 Step 3: Create a dot plot, and record the data.

Step 4: Use the data you collected to answer your question from step 1. Answer:

Reflect 1. Was there only one response to the question, or did the responses vary?

2. What is the difference between numerical data and categorical data?

316 | Informal Inferences

Informal Inferences

Explore 1 Part II: Understanding Statistical Questions

Determine whether the questions asked for each scenario are statistical questions or nonstatistical questions. Record and explain your thinking for each scenario below. Survey 1 Letter(s) ____________ is/are statistical because Statistical Questions

Letter(s) ____________ is/are nonstatistical because Nonstatistical Questions

Survey 2 Letter(s) ____________ is/are statistical because Statistical Questions

Letter(s) ____________ is/are nonstatistical because Nonstatistical Questions

Informal Inferences | 317

Informal Inferences

Explore 1 Survey 3 Letter(s) ____________ is/are statistical because Statistical Questions

Letter(s) ____________ is/are nonstatistical because Nonstatistical Questions

Survey 4 Letter(s) ____________ is/are statistical because Statistical Questions

Letter(s) ____________ is/are nonstatistical because Nonstatistical Questions

318 | Informal Inferences

Informal Inferences

Explore 1 Survey 5 Letter(s) ____________ is/are statistical because Statistical Questions

Letter(s) ____________ is/are nonstatistical because Nonstatistical Questions

Survey 6 Letter(s) ____________ is/are statistical because Statistical Questions

Letter(s) ____________ is/are nonstatistical because Nonstatistical Questions

Informal Inferences | 319

Explore 1

Informal Inferences

Reflect 1. How would you describe the difference between a statistical question and a nonstatistical question?

2. What are the two types of responses to a statistical or nonstatistical question?

3. How can collecting data and answering questions be helpful in a real-world situation?

320 | Informal Inferences

Informal Inferences

Explore 2

Name: _______________________ Date: ___________

Valid Generalizations Part I After your group has made a decision about each Sample Population Card, record your decisions in the chart below. The assistant principal asks ____________ which type of pizza they prefer.

Valid

Invalid

One student 7th graders with odd birthdays 25 students in Mr. Valdez’s 1st-period English class Every ﬁfth student who walks in the front door The teachers on campus The girls’ basketball team Every ﬁfth person on the 7th-grade attendance roster All of the students who earned a 100 on the last math test 1. Write an example of a different valid sample population the assistant principal could use.

2. What makes a sample population a valid or invalid sample?

3. Consider the valid inferences. Could potential limitations exist because of how the sample was selected or how the question was asked? Explain.

Informal Inferences | 321

Informal Inferences

Explore 2 Part II Read through the information on each Sample Data Card. Make 3 inferences about all 200 7th graders. 1.

Draw 3 conclusions about all of the 7th graders. 1.

322 | Informal Inferences

Explore 2

Informal Inferences

Make 3 generalizations about all of the 7th graders. 1.

Make 3 predictions about the entire season. 1.

How can valid sample population data be useful when making decisions about an entire population?

Informal Inferences | 323

Explore 2

Informal Inferences

Reflect 1. How can the sample data be useful when making decisions for the whole group?

2. When might someone in the real world use data from a sample population?

3. Explain the process you would use to make a prediction about how many of the 200 7th graders would be expected to make a selection based on a given value and total sample size.

324 | Informal Inferences

Informal Inferences

Explore 3

Name: _______________________ Date: ___________

Use Data to Make Inferences Read each Theater Survey Result Data Card, and review its graph. Then, answer each of the questions for that card. Days of the Week Last week, 3,750 people came to the movie theater. How many of those people would you expect to visit the theater on Saturday and Sunday?

On Thursday of this week, 180 people visited the movie theater. Based on this information, how many visitors did they have during the entire week?

Ms. Harshan concludes that the theater should be closed every Tuesday since no guests purchased tickets for that day. Do you think this is a reasonable conclusion? Explain your reasoning.

Informal Inferences | 325

Informal Inferences

Explore 3 Popcorn Preference

Ms. Harshan plans to place an order for 560 popcorn containers. How many large popcorn containers should she order?

If 960 people bought popcorn at the movie theater this week, how many of those visitors could you expect to buy a medium popcorn?

Neysa is working at the concession stand. Out of the next 30 customers who purchase popcorn, how many will most likely purchase a small container of popcorn?

Movie Madness Cinema Star 16 sold 15,600 tickets in the past 6 months. How many of those tickets were sold to people who only watched one movie?

Jerry the ticket collector records the age of each person coming into the theater one week. If there are 700 visitors to the theater that week, how many visitors should be between 26 and 43 years old?

326 | Informal Inferences

Informal Inferences

Explore 3 Snack Time!

Based on the results of the three data samples, what percentage of the people who purchased items at the concession stand this month can be expected to prefer popcorn?

A. Exactly 38% B. Between 35% and 41% C. More than 35% D. Less than 41%

Next month, the theater expects to serve 2,400 items at the concession stand. Jes, the assistant manager, wants to place an order for exactly 216 pretzels. Ivan, the night manager, estimates that it will be somewhere between 192 and 240 pretzels and wants to order 230 pretzels for next month. Who do you think has the better plan? Why do you think it’s better?

Informal Inferences | 327

Informal Inferences

Explore 3 Reflect

1. Why do we use surveys instead of collecting data from the entire population?

2. When using a proportion to make predictions about a population, how do you determine where the values go?

3. Does it make sense to base inferences/predictions on biased data?

4. Why might the predicted result based on a survey differ from the actual population?

328 | Informal Inferences

Compare Data

Name: _______________________ Date: ___________

Boys’ heights (in inches):

Girls’ heights (in inches):

Informal Inferences | 329

Use the data collected to list the heights of the girls and the heights of the boys from least to greatest.

Hypothesis statement:

Will there be a visible difference between the heights of the girls and the heights of the boys when displayed in a dot plot? If so, which gender do you think will be taller?

Part I

Explore 4

Informal Inferences

330 | Informal Inferences

Boys’ heights (in inches):

Girls’ heights (in inches):

Use the ordered list to determine the ﬁve-number summary in order to create a box plot for each data set.

Explore 4

Informal Inferences

2. After looking at the data presented in the dot plots, what observations have you made?

Informal Inferences | 331

July High Temperatures Use the information on the July High Temperatures Dot Plot Data Card to answer the following questions. 1. Write down your predictions about the July high temperatures in Los Angeles, CA, and Little Rock, AR.

Part II

1. Use the median and interquartile range to determine whether there is a meaningful difference between the heights of the girls and the heights of the boys in the data sets. Discuss median and interquartile range to explain.

Reflect

Explore 4

Informal Inferences

Mean absolute deviation:

332 | Informal Inferences

Mean:

Little Rock, AR

Mean:

Los Angeles, CA

Use the space below to determine the mean and mean absolute deviation for each dot plot.

Explore 4

Informal Inferences

2. How does the data presented in the dot plot compare to your initial thoughts?

Informal Inferences | 333

1. How do you think the amount of sodium in regular sodas will compare to the sodium in diet sodas?

Sodium Content in Soda Use the information on the Sodium Content in Soda Dot Plot Data Card to answer the following questions.

4. Based on these data sets, do you think latitude is a good way to determine if two areas will have similar weather?

3. Make further comparisons between the data using the mean and the mean absolute deviation (MAD) for each of the two cities.

Explore 4

Informal Inferences

Mean absolute deviation:

334 | Informal Inferences

Mean:

Diet Soda

Mean:

Regular Soda

Use the space below to determine the mean and mean absolute deviation for each dot plot.

Explore 4

Informal Inferences

2. How can mean and MAD help you compare data?

Informal Inferences | 335

1. Does knowing the variability (MAD) of a set of data give you enough information to make conclusions about the data?

Reflect

4. What do the means and MADs tell you about the sodium content in regular soda versus diet soda?

3. Make further comparisons between the data using the means and the MADs for regular soda and diet soda.

Explore 4

Informal Inferences

Mean:

Median:

Tyson Intermediate boys:

Hawkeye Junior High boys:

Use the information on the Basketball Cards to ﬁnd the measures of center.

Median:

Informal Inferences | 337

Name: _______________________ Date: ___________

Centers and Measures of Variability

Part I: Measures of Center

Explore 5

Informal Inferences

Median:

338 | Informal Inferences

3. The school district ﬁrst completed an estimate of each girls’ team by rounding to the nearest ten using every other card when the cards were placed in numerical order. What sample mean did they calculate for each girls’ team? Gauge how far off the estimate is from the actual mean.

2. Which measure of center would be the best to compare the data sets that represent the boys’ teams? Explain.

1. Which measure of center would be the best to compare the data sets that represent the girls’ teams? Explain.

Mean:

Median:

Columbus City girls:

Hope Middle School girls:

Explore 5

Informal Inferences

Interquartile range:

Range:

Hope Middle School girls:

Range:

Columbus City girls:

MAD:

Use the information on the Basketball Cards to ﬁnd the measures of variability.

Part II: Measures of Variability

Explore 5

Informal Inferences | 339

Informal Inferences

Interquartile range:

340 | Informal Inferences

Range:

Interquartile range:

Hawkeye Junior High boys:

Range:

Tyson Intermediate boys:

Explore 5

MAD:

Informal Inferences

3. Why was it important also to look at the measures of variability when choosing the best team?

Informal Inferences | 341

2. How did you know which measure of center to use when choosing between the median and the mean?

1. Looking at all 4 schools, which school had the greatest variability?

Reflect

2. Based on the data, which boys’ team should be chosen to represent Columbus City in the basketball shooting competition? Explain.

1. Based on the data, which girls’ team should be chosen to represent Columbus City in the basketball shooting competition? Explain.

Explore 5

Informal Inferences

Skills Quizzes

343

Addition and Subtraction with Rational Numbers

Skills Quiz

Name: _______________________ Date: ___________

Addition and Subtraction with Rational Numbers Directions: Solve each problem. Show or explain your mathematical thinking.

1. Find the difference of 524.3 – 94.16.

2. What is the sum of 23 2 + (−5 3 )? 5

17 6

29 6

18 5

−17 5

1 3 6

Addition and Subtraction with Rational Numbers | 345

Addition and Subtraction with Rational Numbers

Skills Quiz 2

3. Find the difference of − 9 − (−3 3 ).

4. Find the sum of −16.72 + 5.89.

5. Find the difference of −2 6 − 7 3 . 5

9 6

5 6

−9 6

−5 6

3 5 3

346 | Addition and Subtraction with Rational Numbers

Skills Quiz

Addition and Subtraction with Rational Numbers

6. 0.9 – 3.2 = A.

4.1

−2.3

2.3

−4.1

7. 7.6 – (−9.7) =

8. −5 5 + (2 2 + 3 10 ) =

Addition and Subtraction with Rational Numbers | 347

Skills Quiz

Addition and Subtraction with Rational Numbers

9. −3.25 – (−7.62) = A.

10.87

−4.37

−10.87

4.37

10. −2 3 + (−1 5 ) = 4

A. B. C. D.

−4 3

8 −4 2 3 −3 1 8 −3 2 3

348 | Addition and Subtraction with Rational Numbers

Multiplication and Division with Rational Numbers

Skills Quiz

Name: _______________________ Date: ___________

Multiplication and Division with Rational Numbers Directions: Solve each problem. Show or explain your mathematical thinking. 1

1. Solve the expression −8 2 · (−2 5 ).

2. Solve the expression − 4 ÷ 5. 5

−4

4 25

− 4

25 1 −6 4

3. Solve the expression 4.9 · −5.

Multiplication and Division with Rational Numbers | 349

Multiplication and Division with Rational Numbers

Skills Quiz 4. Solve the expression −2.5 · −8.2.

5. Solve the expression 54 · 1 . 8

6. Solve the expression −85 ÷ 12.5. A.

−0.68

6.8

−6.8

−680

7. Solve the expression 8 ÷ 1 5 .

350 | Multiplication and Division with Rational Numbers

Multiplication and Division with Rational Numbers

Skills Quiz 8. Solve the expression −2.3 · 8.

9. Solve the expression 67 2 (− 3 ). 1

−22 2

22 2

−202 1

202 1

1 2

10. Solve the expression 125.4 . −0.6

Multiplication and Division with Rational Numbers | 351

Rational Number Operations

Skills Quiz

Name: _______________________ Date: ___________

Rational Number Operations Directions: Solve each problem. Show or explain your mathematical thinking.

1. Convert 37.5% to a fraction and a decimal.

2. Convert −4 1 to a decimal rounded to the nearest hundredth. 6

3. Convert − 5 to a decimal.

Rational Number Operations | 353

Rational Number Operations

Skills Quiz 2

4. Convert 3 to a decimal rounded to the nearest hundredth.

5. 9 3 ÷ 8 =

6. (94.72 – 85) ÷ 2.16 =

7. 75 + 3(40) =

354 | Rational Number Operations

Skills Quiz

Rational Number Operations

8. 8(−2) + 35(12.75) =

9. 12 5 – [(2 8 ) (3)] =

10. 33 4 ÷ 2 4 5

Rational Number Operations | 355

Proportional Relationships

Skills Quiz

Name: _______________________ Date: ___________

Proportional Relationships Directions: Solve each problem. Show or explain your mathematical thinking.

1. Does the table below represent a proportional relationship? Explain your reasoning.

2. Solve for x.

Notepad(s)

Cost ($)

Proportional Relationships | 357

Proportional Relationships

Skills Quiz

3. Determine whether the graph below represents a proportional relationship. Explain your reasoning. y

10 9 8 7 6 5 4 3 2 1 0

1 2 3 4 5 6 7 8 9 10

4. Select the best equation for a proportional relationship, using the following information. Total cost: t Number of bracelets: n $5.32 for each bracelet A.

n = 5.32t

t = 5.32 + n

t = 5.32n + 5.32

t = 5.32n

358 | Proportional Relationships

Proportional Relationships

Skills Quiz

5. Determine whether the table below represents a proportional relationship. Explain your reasoning. x

6. Represent the equation y = 3.5x on a graph. Show at least 2 points.

8 7 6 5 4 3 2 1 –1

–1

x 1

Proportional Relationships | 359

Proportional Relationships

Skills Quiz 7. Write an equation to represent the relationship in the table.

y = x + 1.50

y = x – 1.50

x = 1.50y

y = 1.50x

360 | Proportional Relationships

Number of Brownies

Price

$1.50

$3.00

$4.50

$6.00

Proportional Relationships

Skills Quiz 8. Write an equation for the graph. y 5 4 3 2 1 –5

–4

–3

–2

–1

x 1

–2 –3 –4 –5

y=x–2

y = 2x

y = −2x

x = −2y

Proportional Relationships | 361

Proportional Relationships

Skills Quiz For questions 9 and 10, use the table to answer. Hours (x)

0.50

1.50

Miles (y)

5.25

10.50

15.75

21.00

9. Write an equation to represent the relationship in the table.

10. Describe the signiﬁcance of the point (1, 10.50) when graphed.

362 | Proportional Relationships

Understand Slope

Skills Quiz

Name: _______________________ Date: ___________

Understand Slope Directions: Solve each problem. Show or explain your mathematical thinking.

Determine the rate of change and the initial value of each situation presented. Make sure to label the units based on each situation. Use the graph below to answer questions 1 and 2. y 90 80

Cost ($)

70 60 50 40 30 20 10

x 1

Time (h)

1. What is the rate of change?

2. What is the initial value?

Understand Slope | 363

Understand Slope

Skills Quiz

3. The graph shows two similar triangles with the hypotenuse of each lying on the same line.

14 12 10 8 6 4 2 0

x 2

Which proportion proves that the slope of the line is the same for any two points on the line? A.

2−0 3−0

6−2 9−3

2−0 3−0

2−6 3−9

6−2 9−3

2−0 3−0

2+0 3+0

6+2 9+3

364 | Understand Slope

Understand Slope

Skills Quiz Use the table to answer questions 4 and 5.

Cups

Weight (lb.)

4. What is the rate of change?

5. Does the table have a higher rate of change than the equation y = 0.4x? Explain.

Understand Slope | 365

Understand Slope

Skills Quiz Use the given situation for questions 6 and 7. A golden eagle can ﬂy 200 miles in 2.5 hours.

6. Create a graph to represent this rate.

7. How does the slope of the graph compare to the unit rate of how far a golden eagle can travel in one hour?

366 | Understand Slope

Understand Slope

Skills Quiz 8. Two similar triangles are shown in the graph. y

Which statement best compares the slopes of AC and XZ? A.

The slope of line XZ is less than the slope of line AC.

The slope of line AC is less than the slope of line XZ.

The slope of line AC is the same as the slope of line XZ.

The slopes of the lines cannot be compared because they are on the same line.

Understand Slope | 367

Understand Slope

Skills Quiz

9. A bakery sold 325 cookies in 5 hours. Which graph has a slope that represents the number of cookies the bakery sold each hour?

400

300 Cookies sold

Cookies sold

300

200

100

x 2

400

6 8 Hours

400

6 8 Hours

Cookies sold

300

200

100

300 Cookies sold

200

100

200

100

x 2

368 | Understand Slope

6 8 Hours

x 2

6 8 Hours

Understand Slope

Skills Quiz

10. The graph and table show the pay rate that Jacob could earn at each of his new jobs.

Job 1

Job 2

Hours

Money

Money earned

150

100

x 0

Hours worked

9 10 11 12

Which statement best describes which job pays Jacob more? A.

Jacob makes more money at Job 1 at $12.50 per hour.

Jacob makes more money at Job 2 at $12.50 per hour.

Jacob makes more money at Job 2 at $13.00 per hour.

Jacob makes more money at Job 1 at $13.00 per hour.

Understand Slope | 369

Ratios, Rates, and Percents

Skills Quiz

Name: _______________________ Date: ___________

Ratios, Rates, and Percents Directions: Solve each problem. Show or explain your mathematical thinking.

1. Identify the unit rate represented in the table below.

Cost ($)

Gallons

2. Determine the unit rate, given the scenario below. 1

8 yards per 3 hour A.

8 yards per hour

16 yards per hour

24 yards per hour

8 yards per hour 3

Ratios, Rates, and Percents | 371

Ratios, Rates, and Percents

Skills Quiz 3. Identify the unit rate below. 3 miles : 15 hours 5 miles : 25 hours 9 miles : 45 hours

4. A shirt is on sale for 30% off. The sale price of the shirt is $14. How much money do you save from the original price?

5. Find the total price of a meal that is $67.25 plus 8% tax and 20% tip.

372 | Ratios, Rates, and Percents

Ratios, Rates, and Percents

Skills Quiz

6. What is the percent increase in the price of tickets between 2017 and 2018?

Year

2016

2017

2018

Price

$22

$28

$35

7. Use the graph to identify the unit rate per cup of ﬂour.

9 8

Cups of flour

7 6 5 4 3 2 1 0

x 1

Ratios, Rates, and Percents | 373

Ratios, Rates, and Percents

Skills Quiz

8. The current cost of gasoline is $2.89 a gallon. If that cost is expected to increase by 48% by November, what will a gallon of gas cost in November?

9. On Saturday, a shirt was on sale for $18. Today, the sale is over, and the same shirt is $25. What was the percent decrease while the shirt was on sale?

374 | Ratios, Rates, and Percents

Ratios, Rates, and Percents

Skills Quiz 10. Identify the unit rate of speed in the table below.

3 miles per hour

2.5 hours per mile

2 of a mile per hour 5

2.5 miles per hour

Hours

Miles

12.5

Ratios, Rates, and Percents | 375

Percent Application

Skills Quiz

Name: _______________________ Date: ___________

Percent Application Directions: Solve each problem. Show or explain your mathematical thinking.

1. Find the sales price of a $28 item with a 15% discount.

2. Determine the percent change, given the information below. Original amount: 20 New amount: 4 A.

80%

20%

−80%

−400%

Percent Application | 377

Skills Quiz

Percent Application

3. Calculate the amount of simple interest earned on a $3,400 investment for 3 years with an interest rate of 4%.

4. Determine the percent increase if the original amount is 80 and the new amount is 120.

5. What is the total cost of a $64 purchase with a sales tax rate of 5.5%?

378 | Percent Application

Skills Quiz

Percent Application

6. Find the total cost of a $32.50 meal with an 8% tip. A.

$2.60

$26.00

$29.90

$35.10

7. Determine the total cost of your $96 shopping trip with a tax rate of 8.5%.

8. Calculate the amount of simple interest paid on an $8,000 loan for 2 1 years at an 2

interest rate of 5%. A.

$800

$1,000

$9,100

$11,000

Percent Application | 379

Percent Application

Skills Quiz

9. Calculate the commission earned from a $12,000 sale with a 3% commission rate.

10. What is the cost of a $55 purchase with a 25% off coupon? A.

$13.75

$41.25

$68.75

$137.50

380 | Percent Application

Expressions

Skills Quiz

Name: _______________________ Date: ___________

Expressions Directions: Solve each problem. Show or explain your mathematical thinking.

1. Factor the expression −8y y – 2.

2. Simplify the expression −2.5rr – 0.25r. A.

−2.25r

−2.75r

2.25r

2.75r

3. Expand the expression 1 ((a + 8). 4

Expressions | 381

Skills Quiz

Expressions

4. Factor the expression 6a – 21. A.

3(2a – 7)

3(2a – 21)

−3(2a + 7)

6(a – 21)

5. Factor the expression −3p + 18.

6. Simplify the expression d + 0.75d.

382 | Expressions

Expressions

Skills Quiz 7. Expand the expression 2.5(b – 6). A.

−2.5b + 15

2.5b – 6

2.5b – 15

2.5b + 15

8. Factor the expression 0.25x x – 0.75.

9. Expand the expression − 1 ((tt – 15). 3

10. Factor the expression 10e + 25.

Expressions | 383

Solve Equations and Inequalities

Skills Quiz

Name: _______________________ Date: ___________

Solve Equations and Inequalities Directions: Solve each problem. Show or explain your mathematical thinking.

1. Solve for x, given 6x x + 7 = 22.

2. Solve for x.

Key =x

= −1 =1

Solve Equations and Inequalities | 385

Solve Equations and Inequalities

Skills Quiz 3. Solve the inequality −9x x + 12 > −78. A.

x < −10

x < 10

x > 10

x > −10

4. Circle the number line that represents the solution for the inequality 2x x + 7 < −3.

5 5 5 B.

386 | Solve Equations and Inequalities

-5 -5 -5 5 5 5 -5 -5 -5

Skills Quiz

Solve Equations and Inequalities

5. Solve for x, given 4(x x + 7) = 56.

6. Solve for x, given 8x x + 74 ≤ 138.

7. Solve for m, given 15m + 30 = 195.

Solve Equations and Inequalities | 387

Solve Equations and Inequalities

Skills Quiz For questions 8–10, use the key below.

KEY =x = −x

= −1 =1

8. Draw a model to represent 3x x + 9 < 12.

9. Draw a model to represent 4x x – 6 = 18.

388 | Solve Equations and Inequalities

Skills Quiz

Solve Equations and Inequalities

10. Draw a model to represent −5x x + 16 ≥ −13.

Solve Equations and Inequalities | 389

Scaling

Skills Quiz

Name: _______________________ Date: ___________

Scaling Directions: Solve each problem. Show or explain your mathematical thinking.

1. Actual height: 90 feet Scaled height: 6 inches What scale is being used? A.

1 in. = 90 ft.

15 in. = 1 ft.

1 in. = 15 ft.

1 in. = 6 ft.

2. If 1 inch = 4 feet in this model, what is the actual area of the triangle?

5.5 in. 2 in.

Scaling | 391

Scaling

Skills Quiz 3. What is the length of the missing side, x?

3 cm 4.5 cm

x in.

Scale: 1 centimeter = 3 inches

4. Given the information below, what is the actual distance between the two cities? Map scale: 1 inch = 12 miles Distance between 2 cities on map: 4.5 inches apart

392 | Scaling

Scaling

Skills Quiz

5. Given the information below, what is the actual distance between the two landmarks? Map scale: 1 centimeter = 3 1 feet 4

Distance between two landmarks: 8 centimeters A.

26 feet

2.46 feet

11.25 feet

24 feet

6. What is the perimeter of the actual rectangle?

5 cm 2 cm Scale is 1 centimeter = 5.5 feet.

Scaling | 393

Scaling

Skills Quiz 7. How wide is the living room on the blueprint? Blueprint scale: 1 centimeter = 2 feet Living room’s width: 21 feet

8. If 1 inch = 1 of a foot on the scale drawing of the rectangle below, what is the area 2

of the actual rectangle?

10 in. 5 in.

394 | Scaling

Scaling

Skills Quiz

9. A scaled drawing is featured below. What is the actual distance between the two houses?

6 inches

Scale: 1 inch = 4.25 yards

10. Read the information below about two towns to determine the scale used on the map. Actual distance between two towns: 120 miles Distance on a map: 5 inches

Scaling | 395

Angles

Skills Quiz

Name: _______________________ Date: ___________

Angles Directions: Solve each problem. Show or explain your mathematical thinking.

1. Use the circle units below to measure the angle to the closest unit.

1 unit

2 units

3 units

4 units

2. Anglecia used a protractor to measure an angle that was 45°. Which angle below could be the one Anglecia measured? A.

Angles | 397

Angles

Skills Quiz 3. What is the angle measure of the angle below?

10 2 0 170 16 3 01 0 4 50 14 0 0 0 180

65°

125°

75°

115°

70 60 1 0 01 15 20 1 0 0 14 0 3 4

80 90 100 110 70 12 80 7 0 1 0 0 0 6 10 0 1 6 0 0 130 50 0 12 50 13

0 180

4. Case was trying to measure the corner of his room for new ﬂooring. He didn’t have a protractor, so he drew part of a circle from one wall to the intersecting wall. Is this an effective way to measure angles? Why or why not? A.

Yes, angles are measured in degrees, which are part of a circle.

Yes, circles are easier to draw than squares.

No, you must use a protractor to measure an angle.

No, the part of the circle on the wall is different from the circle on the ﬂoor.

398 | Angles

Angles

Skills Quiz 5. Use your protractor to ﬁnd the angle measure of ∠ABC. A

B A.

120°

100°

50°

60°

6. Martha and Thomas were measuring angles with the circle units shown below. How many units would a right angle be?

5 units

4 units

3 units

2 units

Angles | 399

Angles

Skills Quiz 7. Use your protractor to measure the angle below.

55°

65°

125°

135°

8. Oakley measured an angle that was 2 units of a circle. Look at the angles below. Which one could be the angle she measured? Explain.

Angle A because it’s less than 90°.

Angle B because it’s 90°.

Angle A because the 2 parts are equal.

Angle B because the 2 parts are not equal.

400 | Angles

Angles

Skills Quiz 9. Use your protractor to ﬁnd the measure of ∠DEF.

G A.

105°

75°

85°

115°

10. What is the angle measure of the angle below?

10 2 0 170 16 3 01 0 4 50 14 0 0

0 180

70°

60°

120°

130°

70 60 1 0 01 15 20 1 0 0 14 0 3 4

80 90 100 110 70 12 80 7 0 0 0 60 110 10 6 0 0 130 50 0 12 50 13

0 180

Angles | 401

Angle Relationships

Skills Quiz

Name: _______________________ Date: ___________

Angle Relationships Directions: Solve each problem. Show or explain your mathematical thinking.

1. Angles S and T are vertical angles. Determine the measure of angle T if angle S measures 52°.

2. Solve for x if angles A and B are supplementary.

72°

Angle Relationships | 403

Skills Quiz

Angle Relationships

3. Angles G and H are adjacent angles. Angle H measures 28°. What is the measure of angle G? A.

152°

28°

62°

Not enough information to answer

4. Angles X and Y are vertical angles. Determine the measure of angle X if angle Y measures 112°. A.

68°

22°

112°

Not enough information to answer

5. Angles J and K are supplementary angles. The sum of the measures of angles J and K is ___________.

404 | Angle Relationships

Angle Relationships

Skills Quiz

.5 °

6. Solve for r if angles E and F are complementary angles.

r E

7. Angles C and D are complementary angles. Find the measure of angle C if angle D measures 61.4°.

8. Angles P and Q are supplementary angles. Determine the measure of angle P if angle Q measures 95.7°.

Angle Relationships | 405

Angle Relationships

Skills Quiz 9. Solve for x.

62°

15.5°

31°

62°

28°

10. Solve for x.

406 | Angle Relationships

36°

Circles

Skills Quiz

Name: _______________________ Date: ___________

Circles Directions: Solve each problem. Show or explain your mathematical thinking.

1. Find the area of the circle below.

2. What is the circumference, in units, of this circle?

10.99 sq. units

21.98 sq. units

43.96 sq. units

153.86 sq. units

Circles | 407

Circles

Skills Quiz 3. Find the area of this circle in terms of 𝜋.

50 cm

A. B. C. D.

50𝜋 cm2

2,500𝜋 cm2

100𝜋 cm2

1,000𝜋 cm2

4 ft.

4. Find the area of the white part of this circle.

2 ft.

113.04 ft.2

12.56 ft.2

100.48 ft.2

100.96 ft.2

408 | Circles

Circles

Skills Quiz 5. Explain how the area and circumference of the same circle are related.

6. What is the circumference of this circle?

9 cm

28.26 cm

63.585 cm

14.13 cm

56.52 cm

7. Circle A has a diameter of 18 inches. Circle B has a diameter of 10 inches. How much smaller is the area of Circle B than the area of Circle A?

Circles | 409

Circles

Skills Quiz 8. What is the area, in square units, of this circle?

31.4 sq. units

62.8 sq. units

78.5 sq. units

314 sq. units

9. A large circle has a diameter of 17 inches. What is the length around the circle?

10. Explain the relationship between the circumference and diameter of the same circle.

410 | Circles

Surface Area

Skills Quiz

Name: _______________________ Date: ___________

Surface Area Directions: Solve each problem. Show or explain your mathematical thinking.

1. Find the surface area of this prism. 11

7 6

2. Find the surface area of this prism. 2 12.5 10

340 sq. units

170 sq. units

250 sq. units

290 sq. units

Surface Area | 411

Surface Area

Skills Quiz

3. This three-dimensional solid is sliced perpendicular to one of its faces. Which of the following best describes the shapes of its possible plane sections?

Triangles

Rectangles

Both triangles and rectangles

Neither triangles nor rectangles

4. Find the surface area of this prism.

5 3

9 4

412 | Surface Area

Surface Area

Skills Quiz 5. What is the height of the ﬁgure shown below? Surface area = 100.53 in.2

Radius = 2 in.

2 in.

2 inches

8 inches

12 inches

6 inches

6. If you cut through the ﬁgure from question 5 vertically, what is the resulting shape? A.

Circle

Rectangle

Triangle

Square

Surface Area | 413

Surface Area

Skills Quiz

7. For the ﬁgure below, what three-dimensional ﬁgure could have the given cross section parallel to its base?

Rectangular prism

Cylinder

Cube

Pyramid

8. What is the surface area of a cylinder with the following dimensions? Height = 16 inches A.

502.4 in.2

401.92 in.2

100.48 in.2

128 in.2

414 | Surface Area

Diameter = 8 inches

Surface Area

Skills Quiz

9. Which of the following statements about the cross sections of spheres and cones is true?

Sphere

Cone

The horizontal cross section of a sphere is a circle, and the horizontal cross section of a cone is a triangle.

The horizontal cross sections of both spheres and cones are circles.

The vertical cross sections of both spheres and cones are circles.

The vertical cross sections of both spheres and cones are triangles.

10. Calculate the surface area, in square units, of this prism.

4 5

7.5

Surface Area | 415

Volume

Skills Quiz

Name: _______________________ Date: ___________

Volume Directions: Solve each problem. Show or explain your mathematical thinking.

1. Find the volume of this prism. 11

7 6

2. Find the volume of this prism.

2 12.5 10

340 cubic units

170 cubic units

250 cubic units

290 cubic units

Volume | 417

Volume

Skills Quiz 3. The volume of the triangular prism below is 48 cm3. If the height of the prism is 8 cm, what is the area of the base of the prism?

9 cm2

8 cm2

6 cm2

5 cm2

4. Find the volume of this prism.

5 3

9 4

418 | Volume

Volume

Skills Quiz 5. What is the height of the ﬁgure below? Volume = 96 in.3

Base area = 16 in.2

4 in. A.

4 inches

8 inches

12 inches

6 inches

6. Find the volume of the cylinder.

8 ft.

15 ft.

Volume | 419

Volume

Skills Quiz 7. Find the volume of the ﬁgure below. 22 yards

7 yards 7 yards

49 cubic yards

1,078 cubic yards

539 cubic yards

154 cubic yards

8. What is the volume of a cylinder with the following dimensions? Height = 16 inches

803.84 in.3

502.4 in.3

100.48 in.3

128 in.3

420 | Volume

Diameter = 8 inches

Volume

Skills Quiz 9. Find the volume of the ﬁgure below. 11

119 cubic units

269.5 cubic units

145.25 cubic units

310.86 cubic units

10. Calculate the volume, in cubic units, of this prism.

4 5

7.5

Volume | 421

Probability

Skills Quiz

Name: _______________________ Date: ___________

Probability Directions: Solve each problem. Show or explain your mathematical thinking.

1. What is the theoretical probability of the spinner landing on red? Write your answer as a fraction.

2. A six-sided number cube labeled 1–6 is rolled. The chance of landing on an even number is– A.

unlikely.

likely.

neither likely nor unlikely.

certain.

Probability | 423

Probability

Skills Quiz

3. The likelihood of an event occurring can best be described by which of the following number lines? A.

1 1 1 1

2 2 2 2

0 0 0 0

1 1 1 1

0 0 0 0

-1 -1 -1 -1

0 0 0 0

1 12 12 12 2

⁄ ⁄ ⁄⁄

4. What is the chance of picking a multicolored marble? Write your answer as a percentage.

424 | Probability

Skills Quiz

Probability

5. This spinner is spun 16 times and lands on green twice. What is the experimental probability of landing on green? Express your answer in decimal form.

6. Which number line represents an unlikely event? A.

Probability | 425

Probability

Skills Quiz

7. A number cube labeled 1–6 is rolled 36 times. Estimate the amount of times the number cube is expected to land on a number greater than 3. A.

8. A six-sided number cube labeled 1–6 is rolled. The chance of landing on a number less than two is– A.

likely.

unlikely.

neither likely nor unlikely.

impossible.

426 | Probability

Probability

Skills Quiz

9. The chance of picking a chocolate-chip cookie from a cookie jar is 3 . If the jar 5

contains 90 cookies in total, which number best predicts the number of chocolatechip cookies in the jar? A.

10. A bag contains the marbles shown below. The chance of selecting a dark gray marble has the same probability as–

rolling a 4 or greater on a number cube labeled 1–6.

ﬂipping a coin and landing on tails.

landing on yellow using a spinner containing four equal sections of blue, green, yellow, and red.

selecting one student at random from a group of 8.

Probability | 427

Informal Inferences

Skills Quiz

Name: _______________________ Date: ___________

Informal Inferences Directions: Solve each problem. Show or explain your mathematical thinking.

For questions 1–4, refer to the dot plots below.

Dot Plot A

10 11 12

Dot Plot B

1. Calculate the difference between the averages of dot plot A and dot plot B.

2. What is the mean absolute deviation in dot plot A?

3. What is the mean absolute deviation in dot plot B?

Informal Inferences | 429

Skills Quiz

Informal Inferences

4. About how many times greater is set B’s mean absolute deviation when compared to set A’s mean absolute deviation? A.

2 times

3 times

4 times

5 times

5. A poll from Mr. Henderson’s 7th-grade math class shows that 27% of his 3rd-period students dislike math as a school subject. Bobby, who is in Mr. Henderson’s math class, chooses to write an essay on how 7th graders dislike math as a school subject.

Explain why this is or is not a valid topic to write his essay on. If it is not a valid topic, suggest a better way to gather data. What possible limitations exist to this question?

430 | Informal Inferences

Informal Inferences

Skills Quiz

6. Kellen wants to know if students in 6th-grade classes like the current grading policies. Select which sample of students Kellen should choose to poll. A.

Students sitting randomly in the cafeteria

Students randomly selected from all 6th-grade teachers’ classes

Students selected at random in PE class

Students selected from the entire middle school

7. Based on the survey information below, how many of the total 150 members can be expected to eat a sundae?

Students Who Gave Survey

Students Who Ate Sundaes

Total Students Surveyed

Anika

Informal Inferences | 431

Skills Quiz

Informal Inferences

Use the visual representation below for questions 8–10.

8th graders 6th graders

51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68

8. Which grade level appears to have the larger average height? Explain your reasoning.

9. Which grade level appears to have the greatest variability? Explain your reasoning.

10. How many 8th graders from a sample of 100 students can be expected to be 65 inches or taller?

432 | Informal Inferences

GLOSSARY OF TERMS absolute value

angle-angle criterion

absolute value: the distance a number

additive inverse: what must be added to

is from zero on a number line; also

a number in order for the sum of the two

called the magnitude of a number; never

numbers to be zero

negative adjacent angles: two angles that have absolute value equation: an equation

the same vertex and a common ray but

in which x is c units from b in either

no interior common points

direction algebraic expression: numbers, absolute value function: a function that

variables, and symbols grouped together

contains an algebraic expression within

without an equal sign to show a

absolute value symbols

relationship

absolute value inequality: an inequality

algorithm: a step-by-step method for a

in which the distance from x to b is less

solution

than/greater than c altitude: the height of a polygon acute angle: an angle that measures less than 90°

amplitude: the height from the center line to the peak (or to the trough)

acute triangle: a triangle where every angle measures less than 90°

angle: a geometric figure formed by two rays with the same endpoint (vertex)

addends: the numbers added together to form a sum; any numbers being added

angle-angle criterion: the criterion which states that if two triangles have

addition property of equality: the

two pairs of congruent angles, then the

mathematical property which states that

triangles are similar

433

GLOSSARY OF TERMS angle measure

bar graph

angle measure: the measure of the

association: the form (linear/nonlinear),

angle formed by the two rays from a

direction (positive/negative/none), and

common vertex

strength (weak/moderate/strong) seen between two variables in a scatterplot

angle sum theorem: the theorem which states that the sum of the three interior

associative property of addition: the

angles of a triangle is equal to 180°

mathematical property which states that when adding three or more numbers, the

approximate: to find a number that is

placement of the grouping symbols does

close to the given number on a number line

not affect the sum, e.g., (a + b) + c = a + (b + c)

arc: a part of the circumference of a circle or a section of a curve

associative property of multiplication: the mathematical property which states

area: the number of square units it takes

that when multiplying three or more

to cover the two-dimensional surface of

numbers, the placement of the grouping

an object

symbols does not affect the product, e.g.,

area model: a model where the length

(a × b) × c = a × (b × c)

and width represent the factors and

asymptote: a line that a graph

are configured through the operation of

approaches but never crosses as the

multiplication

value of a variable becomes extremely

arithmetic pattern: a number pattern

large or small

that changes at the same rate, either

axis of symmetry: the line that divides

increasing or decreasing

a figure into two identical parts that are

arithmetic sequence: a sequence where

mirror images of each other

the successive terms differ by the same

bar graph: a graph that uses horizontal

number d, called the common difference,

or vertical rectangular bars to show each

where d ≠ 0

category of qualitative data

434

GLOSSARY OF TERMS base

chance

base: (1) the lower number of an

budget: a financial plan that estimates

exponent that is multiplied by itself;

expenditure for a certain period of time

(2) the surface that a solid object stands on categorical data: a type of data that can base of a polygon: the polygon side that

be divided into groups

is perpendicular to the altitude categorical variable: nonnumerical data base of a triangle: the triangle side that

represented by a letter or symbol

is perpendicular to the altitude category: a collection of objects with benchmark fraction: a familiar fraction

shared attributes

used as a reference point in order to measure, compare, and assess the

causation: the action of one event

reasonableness of a fractional value

causing another event to occur

binomial: a polynomial expression

center: referring to measures of center in

containing two terms

data collection

bivariate categorical data: data for two

center of a circle: the point that is an

nonnumerical variables

equal distance from any point on the circle

bivariate data: data for two variables that are paired to each other

center of a data set: a value in the middle of a distribution that represents a

boundary line: a line that corresponds

typical value of the data set

to the function that divides the coordinate plane into two halves

central angle: an angle in a circle with its corner in the circle’s center

box plot: a diagram that shows the fivenumber summary of a distribution

chance: the possibility of something happening

435

GLOSSARY OF TERMS circle

commutative property of addition

circle: a closed round figure in which

commission: money earned for selling a

every point on the boundary is equidistant

product, usually earned as a percentage

from the center

of the sales

circumference: the distance around a

common coefficient: when a variable

circle

has the same coefficient in two or more equations regardless of the sign

classify: to arrange into groups according to shared characteristics

common denominator: a denominator that is the same in two or more fractions

clockwise rotation: rotating in the direction in which the hands of a clock

common difference: the nonzero

normally move

constant difference, d, of any term and the previous term in an arithmetic

cluster: a group of data occurring closely

sequence

together on a graph common factor: a factor that two or coefficient: the number placed directly

more numbers share

before a variable that tells you to multiply that number by the variable

common multiple: a multiple that two or more numbers share

coinciding lines: lines that lie one on top of the other; the same line with the

common ratio: the ratio of each term

equations expressed in different forms

of a geometric progression to the term preceding it

combine like terms: to add together terms that have the same variable(s),

commutative property of addition: the

including their exponent

mathematical property which states that when adding two or more numbers, the order of the addends does not affect the sum; a + b = b + a

436

GLOSSARY OF TERMS commutative property of multiplication

constant

commutative property of

compound event: a combination of two

multiplication: the mathematical

or more simple events (with two or more

property which states that when

outcomes)

multiplying two or more numbers, the order of the factors does not affect the

compound interest: interest calculated

product; a × b = b × a

multiple times in a given time period so that interest is calculated on the original

complementary angles: two acute

amount and previous interest

angles that, when added, make 90°; two angles whose sum is 90°

conditional relative frequency: the fraction used to express the ratio of the

complete the square: the process used

number of participants in a group that

to form a perfect square trinomial for

meet a certain qualification

the purpose of finding the solution(s) by taking the square root

cone: a solid (three-dimensional) shape that has a flat, circular base joined to a

complex fraction: a fraction where the

point (vertex) by a curved side

numerator and/or the denominator are fractions

congruent: having exactly the same shape and size; being identical; congruent

complex solutions of a quadratic

objects coincide when they overlap.

equation: in the form a + bi; solutions that occur when the value under the

congruent angles: angles that have the

radical of the quadratic formula is less

same measure

than zero congruent figures: figures with the composite figure: a figure that consists

same size and shape

of two or more geometric shapes constant: a fixed number that stands composite number: a number with more

alone in an equation or expression

437

GLOSSARY OF TERMS constant of proportionality

correlation coefficient

constant of proportionality: the

convert: to change the form of a

positive constant, usually denoted k, that

measurement using different units

relates two quantities in the form y = kx

without changing the size or amount of the quantity being measured

constant of variation: the constant (unchanged) ratio of two quantities; in

coordinates: a pair of numbers that

direct variation, it is usually denoted as k.

provides the location of a point along the coordinate plane using the values of the

constant rate of change: a rate of

x-axis and y-axis

change that does not vary coordinate pair: the location of a single constant speed: the rate of fixed speed

point on a coordinate plane where the

per time

first and second values represent the position relative to the x-axis and y-axis,

constraint: a condition that the solution

respectively (x, y); also known as ordered

must satisfy

pair

continuous: data that can contain any

coordinate plane: two perpendicular

real number value between data points;

number lines, called the x-axis and the

data points can be connected.

y-axis, that intersect at the point (0, 0)

converse of the Pythagorean theorem: the theorem which states that

and create four quadrants; also called a graph, coordinate grid, or Cartesian plane

if the square of the length of the longest

correlation: the relationship between two

side of a triangle is equal to the sum of

variables that vary together

the squares of the other two sides, then the triangle is a right triangle; if c² = a²

correlation coefficient: a number r

+ b², then it is a right triangle.

that describes how closely the points in a scatterplot are related, where −1 ≤ r ≤ 1

438

GLOSSARY OF TERMS corresponding angles

data point

corresponding angles: angles in the

cube root: a number that, when

same position in different plane figures

multiplied by itself three times, produces the given number

corresponding congruent angles: angles in identical positions formed by a

cube root function: a function of the

transversal line cutting through two lines

form f( f x) =

corresponding sides: two sides that are

cubic number: a number to the power

in the same position in different plane

of three, i.e., 2³ represents the cubic

figures; in scale drawings, these sides will

number 8 and can be read as two cubed

have a proportional relationship.

or two to the power of three.

corresponding similar sides: sides in

cylinder: a solid (three-dimensional)

matching positions of similar figures that

shape that has two flat, circular, parallel

have a proportional relationship

bases joined by a curved surface at a

fixed distance counterclockwise rotation: rotating in the opposite direction in which hands of a

data: a collection of organized facts,

clock normally move

usually in numerical form, words, measurements, or descriptions

credit: a positive money value data distribution: a function or a listing cross-section: a two-dimensional shape

that shows all the possible values (or

that is created when a three-dimensional

intervals) of the data

shape is sliced data point: a point on a scatterplot that cube: a solid figure with six congruent

represents the data

square faces

439

GLOSSARY OF TERMS data set

difference

data set: a collection of organized facts,

degree (°): the unit of measure for an

usually in numerical form, but can also

angle

be given in words, measurements, or descriptions

degree of a polynomial: the largest exponent or the largest sum of exponents

debt: money that is owed; describes a

of a term within a polynomial

person’s bank account balance when it is less than zero

denominator: the bottom number within a fraction; the number that represents

decimal: a number that uses a decimal

the whole and how many parts total are

point followed by digits that show a value

in the whole

smaller than one, in powers of ten that decrease; a number with one or more

dependent variable: a variable,

digits to the right of the decimal point

often y, that relies on the value of the independent variable

decimal expansion: the decimal form of a number

deposit: a sum of money that is put into a bank account

decimal notation/decimal form: a number that uses a decimal point followed

deviation: the amount by which a single

by digits showing values less than one

measurement differs from a fixed value

decompose: to separate into parts or

diameter: any straight line segment that

elements (e.g., geometric figures or

passes through the center of the circle

numbers)

and has endpoints that lie on the circle

decreasing: the measure of the

difference: a number that is the result of

steepness of a line that shows the slant

subtraction

downward from left to right

440

GLOSSARY OF TERMS difference of two squares

dividend

difference of two squares: the

distance: a measurement of the length

difference of two squares, such as a² – b²

between two points

being factored into (a + b)(a – b) distance formula: the formula used to digit: any one of the numbers 0–9 dilation: a type of transformation where

find the distance, d, between two points (x1, y1) and (x2, y2) on the coordinate plane;

a scale factor is used to enlarge or reduce the distances in the original image

distance-time graph: a graph that shows the distance traveled by an object

dimension: something measurable (such

against the time it takes; any given

as length, width, and height)

point represents the speed of the object

direct variation: a relationship between two variables including a constant (k) discount: the amount subtracted from the original cost of an item discrepancy: a lack of compatibility or similarity between two or more things discrete: data that cannot contain the real number values between data points; data points are not connected. discriminant: the expression under the square root of the quadratic formula that determines the types of solutions of a quadratic equation

(distance per time). distribution: a list of all the possible values of the data and how often they occur distributive property: the mathematical property which states that multiplying the sum or difference of a group of terms by a number or variable is the same as multiplying each term by a number or variable and then adding or subtracting the products dividend: the number you divide into; a quantity that is to be divided by another quantity; a number that shows the amount of equal parts of a whole; the numerator (top number) that tells the number or quantity; a quantity to be divided

441

GLOSSARY OF TERMS division property of equality

equivalent

division property of equality: the

elimination method: a method of

mathematical property that states that

solving systems by adding or subtracting

dividing both sides of an equation by

equations to eliminate a variable

the same number gives us an equivalent equation

end behavior: the trend the graph follows as x approaches infinity in the

divisor: the quantity by which another

negative and positive directions

quantity is to be divided endpoint: the point at the end of a line domain: the set of all possible input

segment or ray

(x x values) of a function enlarge: to create a similar image that is dot plot: a method of visually displaying

now larger than the original image

a distribution of data values where each data value is shown as a dot or mark

equal sign: the symbol used to show

above a number line

that two quantities or expressions are the same

double number line diagram: a pair of parallel number lines used to represent

equal to (=): having exactly the same

equivalent ratios

amount or value

downward: the direction a parabola

equation: a mathematical statement that

opens when the value of a < 0

shows that two expressions are equal to each other

edge: a line at which a space or shape terminates, where two faces of a 3-D solid

equilateral triangle: a triangle with

intersect

three congruent sides and three congruent angles

element: any distinct number or value that is part of a set

442

equivalent: equal in value or amount

GLOSSARY OF TERMS equivalent expressions

expression

equivalent expressions: expressions

exponent: a mathematical notation that

that name the same number no matter

indicates the number of times the base

what value is substituted for the variable

number is multiplied by itself; also called power

equivalent ratios: two or more ratios that are equal; two different ratios

exponential decay: the process of

representing the same value

reducing an amount by a consistent percentage rate over a period of time

estimate: an approximation of an overall amount or value

exponential expression: an expression involving a term with a variable as an

evaluate: to determine or calculate the

exponent; 2x for example

numerical value of something exponential function: a function in the even function: when x is replaced with

form of f( f x) = abx where a and b are real

−x x in a function and the function is

numbers and a ≠ 0, b ≠ 1, and b > 0

simplified, the resulting function will be identical to the original function.

exponential growth: the change that occurs when an original amount is

event: one (or more) outcome(s) of an

increased by a consistent rate over a

experiment

period of time

experimental probability: the ratio that

exponential notation: an expression

compares the number of occurrences to

that takes the form aⁿ, where a is

the number of trials

multiplied by itself n times

explicit formula: a formula to find the

expression: numbers, variables, and

nth term of a sequence

symbols grouped together without an equal sign to show a relationship

443

GLOSSARY OF TERMS exterior angle of triangles theorem

gap

exterior angle of triangles theorem: the

force of gravity: the universal force of

mathematical theorem which states that

attraction acting between all matter

an exterior angle is equal to the sum of the two opposite interior angles of a triangle

formula: a mathematical statement or rule written with symbols

factor: A number or algebraic expression that another number or algebraic

fraction: a number that shows a part of a

expression can be divided by without

whole or part of a set

having a remainder frequency: how often a number occurs in factors: expressions that are multiplied

a data set

together to get a polynomial; factors that appear in the form of ax + b and cannot

frequency table: a table that lists

be factored further

outcomes and the number of times that they occur

factor pair: a set of two factors that multiply to give a particular product;

function: a special relationship between

listing factor pairs is a strategy used to

values; each input value gives back

determine all the factors of a number.

exactly one output value.

factor tree: a mathematical tool to help

function notation: a way of representing

break down a number into its prime

y, the dependent value in a relationship,

factorization

as f( f x), read “ff of x” where f names the function

figure: a two-dimensional shape function rule: the dependent variable five-number summary: the five values

(range, output, y value) expressed

used to make a box plot, including the

in terms of the independent variable

lowest value, lower quartile, median,

(domain, input, x value)

upper quartile, and highest value gap: a missing range of values in a data set 444

GLOSSARY OF TERMS geometric sequence

horizontal reflection

geometric sequence: a sequence in

half-plane: a planar region consisting of

which the ratio of successive terms is

all points on one side of an infinite straight

a constant r, called the common ratio,

line, and no points on the other side

where r ≠ 0 and r ≠ 1 height: the perpendicular distance from a graph: a visual representation of data

vertex to the opposite side of a figure

graph of a quadratic function:

height (3-D figure): the vertical

the attributes of a quadratic function

distance from the top of an object or

including the vertex, the y-intercept, the

figure to its base

x-intercepts, and the axis of symmetry histogram: a special type of bar graph graphing method: a method of solving

with numerical intervals as its labels

systems by graphing horizontal: describes the direction of gratuity: money given above the amount

a line that travels from left to right,

charged for a service; tip

perpendicular to a corresponding vertical line; from left to right; parallel to the

greater than (>): more than another

horizon

(e.g., 49 > 12) horizontal dilation: the act of expanding greater than or equal to (≥): more

or contracting in the horizontal direction

than or the same as another horizontal number line: describes the greatest common factor: the largest

direction of a horizontal number line that

same factor of two or more numbers

travels from left to right, perpendicular to a corresponding vertical line; from left to

grouping symbols: symbols that help to

right: parallel to the horizon

organize mathematical expressions; braces { }, brackets [ ], and parentheses ( )

horizontal reflection: a reflection over a vertical line such as the y-axis

445

GLOSSARY OF TERMS horizontal shift

inequality phrase

horizontal shift: a change in a function

increasing slope: the measure of the

that moves the function left or right

steepness of a line that shows the slant upward from left to right

horizontal translation: a shift in the base of the graph to the left or right

increasing/decreasing: a function is increasing if f( f b) > f( f a) and decreasing

hundredths: the second digit to the right

if f( f b) < f( f a) for any two input values a

of the decimal point; a hundredth is one

and b.

out of 100 equal parts of a whole. increments: the evenly spaced and scaled hypotenuse: the longest side of the right

markings used to locate and plot points

triangle, the side opposite of the right angle independent variable: a variable, often identity property of addition: the

x, that does not rely on the value of

mathematical property which states that

another variable

adding zero to a number does not change the value

index: a number indicating how many of a kind you need to put together to be able

identity property of multiplication: the

to move that number or variable from

mathematical property which states that

inside the radical to outside the radical

multiplying 1 by any number does not change the value

inequality: a mathematical sentence that uses symbols such as <, ≤, >, or ≥ to

image: the new figure in a transformation

compare two quantities

improper fraction: a fraction that has a

inequality notation: notation in

numerator that is greater than or equal to

which the solution is represented by an

the denominator

inequality statement

increasing: when the y value increases

inequality phrase: phrase representing

as the x value increases

each of the inequalities

446

GLOSSARY OF TERMS inference

inverse function

inference: a conclusion based on the

integer exponent: a positive or negative

given data

whole number or zero that tells the number of times a base is multiplied by

infinite: having an unlimited number of

itself

values intercept: the point where the line on a infinite number: the concept of

graph crosses the x-axis or y-axis

something that is unlimited, endless, without bound

interest: money that is a percentage of an original amount typically owed as part

infinite solutions: in systems of

of a debt

equations, coinciding lines have infinite solutions.

interquartile range (IQR): the difference between the upper quartile

input: the set of values supplied to a

(Q3) and the lower quartile (Q1)

function intersecting lines: lines that cross at a input-output pair: an ordered pair

point

in which the input corresponds to the independent variable in the left column

intersection: the point at which two lines

of a function table and the output

cross

corresponds to the right column of a function table; an ordered pair is

interval: the set of continuous input

determined by evaluating the function

values on which a function’s outputs could

using the input.

be increasing, decreasing, or constant

integer: any one of the positive whole

inverse: the opposite number or

numbers, negative whole numbers, and

operation

zero; any member of the set of all whole numbers and their opposites

inverse function: a function that undoes the action of another function

447

GLOSSARY OF TERMS inverse operation

less than or equal to (≤)

inverse operation: the operation that

laws of exponents – multiplication

reverses the effect of another operation

of same bases: can be rewritten as the base raised to the sum of the powers

inverse property of addition: the mathematical property that states that

laws of exponents – negative

when you add a number to its opposite,

exponents: can be rewritten as the

you will always get zero as the sum

multiplicative inverse of the base raised to the positive opposite of the power

inverse property of multiplication: the mathematical property that states

laws of exponents – zero exponents:

that when you multiply a number by its

the mathematical law which states that

reciprocal, you will always get 1

any number raised to the power of zero equals one

irrational number: a decimal number that cannot be expressed as a fraction,

least common multiple: the smallest

is not imaginary, and does not repeat or

multiple that is the same in a set of two

terminate

or more numbers

isosceles triangle: a triangle with two

leg: either of the two sides in a right

or more congruent sides where angles

triangle that form the right angle and are

opposite of the congruent sides are

opposite of acute angles

congruent angles length: the measure of an object from joint frequency: the ratio of the

end to end; the distance from one end to

frequency in a particular category and the

the other end of an object

total number of data values less than (<): smaller than another laws of exponents – division of same

(e.g., 432 < 501)

bases: can be rewritten as the base raised to the difference of the powers

less than or equal to (≤): smaller than or the same as another

448

GLOSSARY OF TERMS like terms

magnitude

like terms: terms that have the same

linear expression: an expression in

variables, including their exponents

which all terms have an exponent of one

likelihood: the probability that an event

linear function: a relationship that when

will occur; also called chance

graphed is a straight line

line: a straight geometric element with

linear graph: a series of points

no thickness, extending endlessly in both

connected on the coordinate plane,

directions; the shortest distance between

forming a straight line that shows a

two points

relationship or rate of change

line of best fit (trend line): a line that

linear inequality: an inequality that

best represents the data on a scatterplot

involves a linear function

line plot: a graph that displays data as

linear parent function: the simplest

points above a number line, to show the

equation of the linear function, y = x or

frequency of each value

f x) = x f(

line segment: a section of a line with two

linear relationship: having a constant

distinct endpoints

rate of change between two quantities/ variables and making a straight line when

linear association: a proportional

graphed; a relationship that creates a

relationship that creates a straight line on

straight line

a graph long division: an algorithm used to find linear equation: an equation in which no

the quotient of two numbers

variable has a power greater than 1; the general form is y = mx + b, where m =

magnitude: the absolute value or

slope and b = y-intercept.

distance to zero

449

GLOSSARY OF TERMS mapping

measures of variability

mapping: a function represented by

measure: a number of units that shows

two sets of objects with arrows drawn

the amount or size of something

between them to show relationships between the objects or data

measure of center: a single value used to represent/summarize a collection

marginal frequency: the ratio of the

of data; three commonly used types

sum of the joint relative frequency in a

are mode, median, and mean; also

row or column and the total number of

called measures of central tendency or

data values

measures of average

markdown: a decrease in the cost of an

measurement: a number that shows the

item; a discount

size or amount of something

markup: an increase in the cost of an

measure of variation: a measure of

item to make a profit

how data is spread out, usually including range, interquartile range, variance, and

maximum: the greatest or highest

standard deviation

amount possible or attained measurement system: one of two main maximum value: the place where a

systems of measurement—the metric

function reaches its highest point, or

system and the standard or customary

vertex, on a graph

system, each of which uses different units to measure distance, mass, and volume

mean: the average of a set of numbers calculated by finding the sum of all data

measures of variability: measures of

and dividing by the number of data values

how data is spread out, usually including range, interquartile range, variance, and

mean absolute deviation: the average

standard deviation

difference between the mean and each data point

450

GLOSSARY OF TERMS median

multiplication property of equality

median: the middle number of a set of

mode: the number or value that appears

numbers when the numbers are arranged

the most frequently in a data set

from least to greatest, or the mean of the two middle numbers when the set has two

monomial: an expression containing only

middle numbers

one term

midpoint formula: the formula used to

multi-digit: a number that has more

calculate the point on a line segment that

than one digit

is equidistant from the endpoints (x1, y1) and (x2, y2) on the coordinate plane;

multiple: a product of two integers; one of the numbers that result from multiplying a whole number by the set of whole numbers

minimum: the least or smallest amount or

multiple representations: different

quantity possible, attainable, or required

mathematical ways to represent a relation or a function

minimum value: the place where a function reaches its lowest point, or

multiplicand: the number that is

vertex, on a graph

multiplied by another number; a quantity that is to be multiplied by another quantity

minuend: a number or quantity from which another number is to be

multiplication: a mathematical operation

subtracted; for example, in the equation

consisting of obtaining a product or

7 – 4 = 3, the number 7 is the minuend,

result by joining equal groups, repeated

the number you subtract from.

addition, or forming arrays

mixed number: a whole number and a

multiplication property of equality:

fraction combined; a number made up of

the mathematical property that states

a whole number and a fraction

that multiplying the same number by each side of an equation gives us an equivalent equation

451

GLOSSARY OF TERMS multiplicative comparison

non-proportional relationship

multiplicative comparison: shows the

negative exponent law: the

relationship between two amounts, where

mathematical law which states that any

one quantity is a certain number of times

nonzero number raised to a negative

as large as another quantity; a number is

exponent is equivalent to the reciprocal

multiplied by another number to result in

of the base raised to the opposite of the

a greater or lesser quantity.

negative exponent

multiplicative identity property: the

negative number: a number that is less

mathematical property which states that

than zero

the resulting product of any number and 1 is equal to the original number

negative reciprocal: the result of multiplying the reciprocal by −1

multiplicative inverse: one of two numbers whose product is 1; also called

negative slope: the measure of the

the reciprocal

steepness of a line that shows the slant downward from left to right

multiplier: the number you multiply by; the quantity that the multiplicand

net: a two-dimensional shape that when

is multiplied by; the number being

folded represents a three-dimensional

multiplied

figure

multistep problem: a mathematical

nonlinear association: a relationship

problem involving more than one

that does not create a straight line

operation nonlinear function: a relationship that negative association: a relationship

when graphed does not make a straight

between two variables that move in

line; a relationship that does not create a

opposite directions

straight line; nonlinear association non-proportional relationship: two quantities that do not have equal ratios

452

GLOSSARY OF TERMS nonvertical line

ordered pair

nonvertical line: a line that is horizontal

obtuse angle: an angle that measures

or diagonal

greater than 90°

no solution: in systems of equations,

obtuse triangle: a triangle that contains

parallel lines have no solution.

one obtuse angle and two acute angles

number line/number line diagram:

odd function: when x is replaced with

a line on which numbers are marked at

−x x in a function and the function is

intervals

simplified, the terms in the resulting function have the opposite signs of those

numerator: the top number within a

in the original function.

fraction, which represents the part of the whole

one solution: in systems of equations, intersecting lines have one solution (x, y).

numeric expression: a mathematical sentence that uses numbers and one or

opposites: numbers the same distance

more operation symbols

away from zero, located on different sides of zero

numerical data: data comprised of numbers, measurements, or quantities

order of operations: a set of rules that dictate which mathematical operation to

numerical radical expression: any

perform first, second, and so on when

numerical expression that contains a

evaluating a mathematical expression

radical ordered pair: the location of a single numerical reasoning: a process

point on a coordinate plane where

using numbers and quantities to draw

the first and second values represent

conclusions

the position relative to the x-axis and y-axis, respectively (x, y); also known as

observation: the value of what is being

coordinate pair

453

GLOSSARY OF TERMS organized data list

percent decrease

organized data list: elements listed in a

part-to-part ratio (comparison): a

particular sequence or order

relationship between one part of a whole and another part of a whole

origin: the center point of a coordinate plane, where the x-axis and y-axis

part-to-whole ratio (comparison): a

intersect, located at (0, 0)

relationship between one part of a whole and the total number of parts in the

outcome: the result of an event

whole

outlier: a number in a set of data that

partial product: the product of the

is much larger or smaller than other

multiplicand and one digit of the multiplier

numbers in the set pattern: a repeating arrangement of output: the result of the input placed in

numbers or shapes

the function pattern of association: a relationship parabola: the shape that a quadratic

between data sets

equation takes when graphed peak: the highest value(s) in a set of data parallel: existing in the same plane and equidistant and not intersecting

per (unit rate): a ratio for an amount for one unit of the other quantity

parallel lines: lines in the same plane that are equidistant and do not intersect

percent: a special ratio that compares a number to 100 using the percent symbol,

parallelogram: a quadrilateral with two

%; a rate per 100

sets of parallel sides percent decrease: the amount by which parameter: a quantity that influences

the cost decreased from the initial value,

the output or behavior of a mathematical

expressed as a percent

object but is viewed as being held constant 454

GLOSSARY OF TERMS percent error

plot

percent error: the measure of how far

periodicity: the tendency of a function

off an estimated value is from the true

to repeat itself in a regular pattern at

value, expressed as a percent

established intervals

percent increase: the amount by which

perpendicular: having the position of

the cost increased from the initial value,

two lines that intersect at a right angle;

expressed as a percent

intersecting at a 90° angle

percent rate of change: the percentage

perpendicular lines: two lines that

increase or decrease of an amount over a

intersect at a 90° angle

unit of time, denoted by r pi: a constant which is found by dividing percentage: a special ratio that

the circumference of a circle by its

compares a number to 100 using the

diameter; approximately 3.142

percent symbol, %; a rate per 100 piecewise function: a function that is perfect cube: an integer that is the result

defined by different formulas at different

of another integer times itself three times

inputs

perfect square: an integer that is the

place value: the numerical value that a

result of another integer times itself

digit has, based on its position within a number

perfect square trinomial: a trinomial whose factored form is the square of a

plane: a flat, two-dimensional surface

binomial; takes the form ax² + bx + c

that continues indefinitely

and satisfies the condition b² = 4ac plot: to indicate the position a number perimeter: the distance around the

is relative to zero on a number line or

outside of a figure or shape

relative to the origin on a coordinate plane

455

GLOSSARY OF TERMS point

power of a power law

point: a dot that represents a specific

positive association: a relationship in

spot on a number line or coordinate

which the values of one variable tend

plane; a geometric object with no

to increase as the values of the other

dimension used to indicate a location

variable increase

point of intersection: the point where

positive number: a number that is

two or more lines cross each other

greater than zero

point-slope form: an equation written

positive rational number: a number to

in the form of y – y1 = m(x x – x1), where

the right of (or greater than) zero that

m is the slope and (x1, y1) is any point

can be expressed as a fraction of two

contained in the line

integers

polygon: a closed figure that has three

positive slope: the measure of the

or more sides, no curved lines, and no

steepness of a line that shows the slant

intersections; a closed figure formed by

upward from left to right

line segments that meet at their endpoints power: a mathematical notation that polynomial: a mathematical expression

indicates the number of times the base

consisting of several terms

number is multiplied by itself; also called an exponent

population: a discrete group for the purposes of data collection and analysis

power law: the distribution of an exponent through multiplication to all

positive/negative interval: positive

parts of the base

intervals are those above the x-axis; negative intervals are those below the

power of a power law: the

x-axis.

mathematical law that states that when raising a base with an exponent to another exponent, the exponents are multiplied and the base stays the same

456

GLOSSARY OF TERMS power of one law

protractor

power of one law: the mathematical

probability model: a mathematical

law which states that any number to the

description of an experiment that lists

power of one is equal to that number

all of the possible outcomes and their probabilities

power of zero law: the mathematical law that states that any number to the

product: the solution when multiplying

power of 0 is equal to 1

two or more numbers; the answer to a multiplication problem

prediction: a reasonable guess as to what will happen

proof: evidence or argument establishing a fact or the truth of a statement

preimage: the original figure in a transformation

product of powers law: the mathematical law which states that when

prime number: a number with exactly

multiplying two exponents with the same

two factors—one and itself

base, the exponents are added together

prime factorization: a given set of prime

and the base stays the same

numbers that when multiplied together

proportion: two fractions or ratios that

equals the original number

are equal in value; a type of equation that

prism: a three-dimensional figure that

shows that two ratios are equal

has at least one set of congruent, parallel

proportional corresponding sides:

faces (bases) that are polygons with

sides in the same position in two similar

parallelograms as the remaining faces

polygons that are proportional

probability: the likelihood that something

proportional relationship: when two

will happen

quantities have the same ratio protractor: a mathematical tool for measuring and drawing angles

457

GLOSSARY OF TERMS pyramid

radius

pyramid: a three-dimensional figure in

quantitative data: numerical or

which the base is any polygon and the

measured data that is analyzed for

other faces are triangles that share a

statistical purposes

common vertex quantitative relationship: the Pythagorean theorem: a theorem that

relationship between magnitudes

states that the square of the hypotenuse is equal to the sum of the squares of the

quantity: a number or amount; an

other two sides of a right triangle; a² +

amount that tells how much

b² = c² quotient: the solution when dividing quadrant: one of four sections of

two numbers; the answer to a division

the coordinate plane, formed by the

problem; the result of the division of one

intersection of the x-axis and y-axis

quantity by another quantity

quadratic formula: the formula

quotient of powers law: the

, which gives the

mathematical law that states that when

solutions of equations in the form of ax²

dividing two exponents with the same

+ bx + c = 0, where a ≠ 0

base, one subtracts the exponents and keeps the base the same

quadratic function: a function that can be written in the form f( f x) = ax2 + bx +

radical: a symbol that indicates the root

c, where a, b, and c are real numbers and

of a quantity

a≠0 radicand: the value inside the radical quadratic parent function: the simplest

symbol

equation of the quadratic function, y = x² or f( f x) = x²

radius: the distance from the center of a circle or a sphere to any point that lies on

quadrilateral: a polygon with four sides

the circle or the sphere

and four angles 458

GLOSSARY OF TERMS random sampling/random sample

real-world problem

random sampling/random sample: a

ratio table: a list of pairs of equivalent

selection chosen by chance and which has

ratios used to determine the relationship

no predictability

between the ratios

range: (1) the difference between the

rational exponent: an exponent that can

maximum and minimum values within a

be expressed as

data set; (2) the set of all possible output,

a radical expression where m and n are

or y values, of a relation or function

integers and m represents the power

as a way to rewrite

of the base and n represents the root; rate: a type of ratio where the quantities have two different units rational number: a number that can rate of change: the rate that shows

be written as a fraction of integers a/b,

how one quantity changes in relation to

where b ≠ 0; a number that can be

another quantity

written as a ratio using two integers

ratio: a comparison of two quantities

ray: part of a line with a fixed starting

that shows their sizes in relation to one

point and no endpoint

another real number: any one of the set of all ratio language: language used to

rational and irrational numbers

mathematically describe the relationship between any two units that are being

real solution: a value that satisfies the

compared in a ratio using the phrase for

equation; called roots, x-intercepts, or

every… there are… or the word to

zeros

ratio relationship: equivalent ratios

real-world problem: a contextual-

form a ratio relationship between the two

based problem that can be interpreted,

quantities being compared

represented, and analyzed through the application of mathematics

459

GLOSSARY OF TERMS reciprocal

right angle

reciprocal: one of two numbers whose

relative frequency: how often a number

product is 1; also called the multiplicative

occurs in a data set divided by the total

inverse

number of outcomes

rectangle: a parallelogram with opposite

relative maximum: a point that is higher

equal sides and four right angles

than the points directly beside it on both sides

recursive formula: a formula that defines each term of a sequence using

relative minimum: a point that is lower

preceding term(s)

than the points directly beside it on both sides

recursive process: the calculation of the next number in a sequence by repeated

remainder: a leftover quantity resulting

application of a rule

from the quotient of 2 integers

reduction: the creation of a similar image

repeating decimal: a decimal number

that is now smaller than the original image

in which a digit or group of digits is repeated indefinitely, as in 0.333… or

reflect: to transform a point so that it is

1.851851851…

equidistant on opposite sides of the x- or y-axis

representative sample: a sample that matches or reflects a population

reflection: the mirror image of a figure; the flipping of a figure

residual: the difference between the observed y value (from the scatterplot)

regression: the process of drawing a line

and the predicted y value (from the

through data in a scatterplot

regression equation line)

relationship: the rule in a pattern

right angle: an angle that measures 90°

460

GLOSSARY OF TERMS right polygon

scientific notation

right polygon: a polygon with at least

scale: the representation of the

one right angle

relationship between a measurement on a model and the corresponding

right prism: a solid composed of a

measurement on the actual object

polygon as its base and vertical sides perpendicular to the base

scale drawing: a smaller or larger representation of an object that is

right rectangular prism: a prism with six

proportional to the original object

rectangular faces where the lateral edge is perpendicular to the plane of the base

scale factor: the ratio of corresponding side lengths in a scale drawing to those of

right triangle: a triangle with one 90º

the original figure

angle scaled interval: a measurement scale rotation: the turning of a figure around a

used on a graph with the distance

fixed point

between marks being equal and the marks counting by a constant value

rounding: the process of raising or lowering a number to a specific

scalene triangle: a triangle with no

place value position; representing an

congruent sides

approximate worth scatterplot: a series of plotted points ruler: a tool used to measure length and

that show the relationship between two

to draw straight lines

sets of data

sample: one part of the given population

scientific notation: a method of expression used to write very small and

sample space: all possible outcomes of

very large numbers by representing them

an experiment

with decimal numbers between 1 and 10, with each decimal being multiplied to a power of 10

461

GLOSSARY OF TERMS sequence

solution

sequence: an ordered arrangement of

simplest form: (1) the smallest possible

numbers or objects

way to write an equivalent fraction for the fraction given; (2) the smallest way to

set: (1) a collection of objects or things;

write an equivalent expression

(2) a group of unique numbers or objects called members or elements

simplify: to replace a numerical expression with the simplest name for its

shape: a description of the type of graph

value by using the substitution principle

seen, as symmetrical, peaks, skewed, or uniform

simulation: a model of random events

side: the line segment that connects two

skewed data: when data on a graph is

vertices in a figure

not symmetrical; when the graphed data shows a tail on one side or the other

signed number: a positive or negative number; a number that has the sign + for

slope: how steep a line is; represented as

positive or − for negative

m in the slope-intercept equation

similar figures: two or more figures that

slope formula: the formula used to find

are the same shape but different sizes

the slope between two points (x1, y1) and

similar triangles: two or more triangles

(x2, y2) ,

that have congruent angles and

slope-intercept form: a way to write

proportional sides

the equation of a line so that it is easy to view the slope and y-intercept of the line;

simple event: one event at a time with

y = mx + b

one single outcome solution: any number that makes an simple interest: a way to calculate

equation true

interest accrued using the formula I = Prt

462

GLOSSARY OF TERMS solution of a system of inequalities

stem-and-leaf plot

solution of a system of inequalities:

square unit: a unit of area, specifically

the overlapping region that makes both

square centimeters, inches, feet, and

inequalities true

meters

solution set: a set of numbers that

standard deviation: a measure of how

makes an inequality statement true

spread out numbers are; calculated by finding the square root of the variance

sphere: a three-dimensional round figure where every surface point is equidistant

standard form: a way to write numbers

from the center of the figure

by using the digits 0–9, with each digit having a place value

spread: a measure of how far the numbers in a data set are from the mean

standard form (linear): Ax + By = C,

or median; including the commonly used

where A, B, and C are constants and A

types range and quartiles; also known as

and B are not both 0

measures of variation or dispersion standard form (quadratic): y = ax² + square: any number or variable times itself

bx + c or ax² + bx + c = 0, where a, b, and c are constants and a ≠ 0

square number: a number to the power of 2, i.e., 3² represents the square

statistical question: a question that

number 9 and can be read as “three

anticipates differences in data

squared” or “three to the power of two.” statistics: the study of data and square root: a number that, when

collecting, organizing, representing, and

multiplied by itself, produces the given

interpreting data

number stem-and-leaf plot: a plot where each square root function: a function of the

data value is split into a “leaf” (usually

form f( f x) =

the last digit) and a “stem” (the other

, where x is greater than

digits) 463

GLOSSARY OF TERMS step function

system of equations

step function: a piecewise-defined

sum: the solution when adding two or

function where each piece’s formula is a

more numbers; the answer to an addition

constant

problem

straight angle: an angle that measures

supplementary angles: two adjacent

exactly 180°

angles that, when added, make 180°; two angles whose sum is 180°

strict inequality: an inequality that has no equality conditions; the strict inequality

surface area: the total area of each of the

is either greater than or less than.

faces and curved surfaces of a solid figure

subcategory: a category within a category;

survey: a data collection tool or list of

a collection of objects with even more

questions used to gather information

specific characteristics than a category

about individuals or groups of people

substitution: replacing letters in an

symbol: a mark or character used as a

algebraic expression with known values

representation of an object, function, or process

substitution method: a method of solving systems by substituting equations

symmetrical: the relationship between

within one another

objects that are the same size and shape after a flip, slide, or turn

subtraction property of equality: the mathematical property that states that

symmetrical distribution: data that is

subtracting the same number from each

in the shape of a bell; it can be equally

side of an equation gives us an equivalent

divided in half.

equation system of equations: two or more subtrahend: a quantity or number to

equations with two or more variables

be subtracted from another; the number being subtracted 464

GLOSSARY OF TERMS system of inequalities

triangle

system of inequalities: two or more

thousandths: the third digit to the right

inequalities with two or more variables

of the decimal point; a thousandth is one out of 1,000 equal parts of a whole.

table: a chart that uses rows and columns to organize information

three-dimensional figure: a solid having three measurable dimensions

tape diagram: a rectangular visual model that represents equal parts, used to

transformation: changing a shape

model word problems involving part-part-

through movement on a coordinate plane

whole relationships translation: moving a figure along a line tax: a fee added to a good or service,

for a specific distance

usually a percentage of the total transversal: a line that cuts through two tenths: the first digit to the right of the

or more lines in the same plane

decimal point; a tenth is one out of 10 equal parts of a whole.

trapezoid: a quadrilateral with one set of parallel sides

term: (1) a number, a variable, or a product of numbers and variables in

trend: the general direction that data

an expression separated by addition,

points seem to follow

subtraction, or sometimes division; (2) in an algebraic expression, a number

tree diagram: a diagram with connecting

or variable, or a product or quotient of

lines to calculate the number of possible

numbers and variables

outcomes of an event

terminating decimal: a decimal number

triangle: a polygon with exactly three

that has a finite number of digits

straight sides and three angles

465

GLOSSARY OF TERMS triangle angle sum property

triangle angle sum property: the

variation

union: a combination of two or more things

mathematical property of a triangle which states that the angles of a triangle always

unit: a type of measurement such as an

add up to 180°

inch, a pound, or a second

triangle inequality theorem: the

unit cube: a cube in which all sides have

theorem that states that the sum of any

a length of one unit

2 sides of a triangle must be greater than the measure of the third side

unit of measurement: a standard amount that is used to measure

trinomial: a polynomial expression containing three terms

unit price: the price of goods per one unit of measure

truncated decimal: a decimal number where some digits are left off and the

unit rate: a rate with a denominator of 1

number is approximated at a certain point

that shows how many units of the first type

without rounding

correspond to one unit of the second type

two-dimensional figure: a flat figure

upward: the direction a parabola opens

with two measurable dimensions

when the value of a > 0

two-way relative frequency table: a

variability: how spread out data is

two-way table that displays percentages or ratios, called relative frequencies

variable: a letter or symbol that takes

two-way table: a chart used to show

a letter that can stand for an unknown

the relationship between two categorical

number or a set of numbers

variables

the place of a number that can change;

variation: how spread out data is

undefined slope: the slope of a vertical line

466

GLOSSARY OF TERMS vertex/vertices

x-axis

vertex/vertices: the common point of

vertical number line: a number line that

two rays that form an angle; the common

travels up and down, perpendicular to a

point to any two sides of a polygon

corresponding horizontal line; from top to bottom; perpendicular to the horizon

vertex: the minimum or maximum point in a quadratic function; identified as (h, k)

vertical reflection: a reflection over a horizontal line such as the x-axis

vertex form: y = a(x – h)² + k, where a, h, and k are constants and a ≠ 0

vertical shift: a change in a function that moves the function up or down

vertical: describes the direction of a line that travels up and down, perpendicular

vertical translation: a shift in the base

to a corresponding horizontal line; from

of the graph up or down

top to bottom; perpendicular to the horizon

volume: the amount of space an object occupies; the measured amount of cubic

vertical angles: angles opposite from

units that fit inside a solid figure

one another when two lines cross; opposite congruent angles that are

whole number: a number zero or above

formed on either side of intersecting lines

that contains no fractional or decimal part; a positive number without a

vertical dilation: expansion or

fractional piece

contraction in the vertical direction width: how many units wide something is vertical line test: a visual way to tell whether a line is a function; if any vertical

withdrawal: a sum of money that is

line intersects the graph more than once,

taken out of a bank account

then the graph is not a function. x-axis: a horizontal number line on a coordinate plane

467

GLOSSARY OF TERMS x-coordinate

x-coordinate: the first term in an

zero slope

zero slope: the slope of a horizontal line

ordered pair; provides the location along the x-axis within the coordinate plane x-intercept: the x-coordinate or coordinates where a graph intersects the x-axis, identified as (x, 0) y-axis: a vertical number line on a coordinate plane y-coordinate: the second term in an ordered pair; provides the location along the y-axis within the coordinate plane y-intercept: the point on a graph of an equation where the line crosses the y-axis zero: (1) the only integer that is neither negative nor positive and is its own opposite; (2) the value of x where an expression is equal to zero; this is the x-coordinate of the x-intercept of the expression’s graph. zero product property: the mathematical property which states that when multiplying two numbers together results in zero, then either a, b, or both a and b are zero; if ab = 0, then either a = 0 or b = 0 or both 468

Workspace

469

I BELONG TO:

MY TEACHER IS:

A Part of STEMscopes Math Developed by Accelerate Learning Inc. 800-531-0864

ISBN: 978-1-64861-274-9

9 781648 612749