STEMscopes Georgia Math Student Notebook Grade 6 by acceleratelearning

Grade 6 Student Notebook

GEORGIA

Student Notebook – Grade 6 ISBN: 978-1-64861-273-2 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 6

Table of Contents Scope Name

Page Number

Add and Subtract Fractions

Multiplication and Division Problem Solving Using Fractions

Add and Subtract Decimals

Multiply and Divide Decimals

Integers

Rational Numbers

Equivalent Numerical Expressions

107

Algebraic Expressions

139

Equations and Inequalities

159

Ratios, Rates, and Unit Rates

185

Percents

219

Measurement Conversions

237

Coordinate Planes

255

Coordinate Plane Problem Solving

265

iii

GEORGIA

Student Notebook - Grade 6

Table of Contents (Cont.) Scope Name

Page Number

Area and Volume

275

Surface Area

313

Represent and Interpret Data

327

Summarize Numerical Data

347

Skills Quizzes

369

Glossary of Terms

459

Workspace

495

Add and Subtract Fractions

Explore 1

Name: _______________________ Date: ___________

Add and Subtract Fractions Read each Scenario Card. In the workspace area, use a strategy of your choice to solve part of the scenario. Record your answer in the solution area.

Scenario 1: The East Field Workspace:

Solution Area available for beans:

Total area used for corn and wheat:

Add and Subtract Fractions | 3

Add and Subtract Fractions

Explore 1 Scenario 2: The Garden Workspace:

Solution Total area used for lettuce and beets:

Area available for tomatoes:

Scenario 3: The West Field Workspace:

Solution Total area used for squash and peas:

4 | Add and Subtract Fractions

Area available for okra:

Add and Subtract Fractions

Explore 1 Scenario 4: The Barn Workspace:

Solution Total area used for cotton and trees:

Area available for barn:

Scenario 5: The Pasture Workspace:

Solution Difference between aunt and uncle:

Area with trees and bushes:

Add and Subtract Fractions | 5

Add and Subtract Fractions

Explore 1 Scenario 6: The Pond Workspace:

Solution Total fraction of catﬁsh and smallmouth bass:

Fraction of carp:

Reflect 1. How is multiplication used in adding and subtracting fractions with different denominators?

2. What strategy do you use the most when adding and subtracting fractions? Explain why you choose this strategy.

6 | Add and Subtract Fractions

Add and Subtract Fractions

Explore 2

Name: _______________________ Date: ___________

Add and Subtract Mixed Numbers and Improper Fractions Sai and Cadence are driving to the Grand Canyon. On the ﬁrst day, Sai drove 2 1 hours and Cadence drove 1 1 hours. How much longer did Sai 3

drive than Cadence?

Number sentence:

Workspace:

Equation:

Solution statement:

Add and Subtract Fractions | 7

Add and Subtract Fractions

Explore 2 Scenario 1 Number sentence:

Workspace:

Equation:

Solution statement:

Scenario 2 Number sentence:

Workspace:

Equation:

Solution statement:

8 | Add and Subtract Fractions

Add and Subtract Fractions

Explore 2 Scenario 3 Number sentence:

Workspace:

Equation:

Solution statement:

Scenario 4 Number sentence:

Workspace:

Equation:

Solution statement:

Add and Subtract Fractions | 9

Add and Subtract Fractions

Explore 2 Scenario 5 Number sentence:

Workspace:

Equation:

Solution statement:

Scenario 6 Number sentence:

Workspace:

Equation:

Solution statement:

10 | Add and Subtract Fractions

Add and Subtract Fractions

Explore 2 Scenario 7 Number sentence:

Workspace:

Equation:

Solution statement:

Reflect 1. When looking at the fractions you added or subtracted, what did you notice about the relationship between the original denominators and the common denominators?

2. Why is it necessary to ﬁnd common denominators when adding or subtracting fractions?

3. Why do you sometimes need to regroup when you are subtracting mixed numbers?

Add and Subtract Fractions | 11

Multiplication and Division Problem Solving Using Fractions

Explore 1

Name: _______________________ Date: ___________

Multiply Fractions by Whole Numbers Part I Lemon Juice

Meaning:

________ groups of ________

Model:

The class needs __________ gallons of lemon juice for 5 batches. Multiplication expression: Strawberries

Meaning:

________ groups of ________

Model:

Multiplication and Division Problem Solving Using Fractions | 15

Explore 1

Multiplication and Division Problem Solving Using Fractions

Part II Grapeade Pouches

Meaning:

________ of the ________

Model:

The class has __________ boxes of grapeade pouches. Multiplication expression:

Kooky Kiwi Punch

Meaning:

________ of the ________

Model:

The class has __________ cartons of kiwi juice for the Kooky Kiwi Punch. Multiplication expression:

16 | Multiplication and Division Problem Solving Using Fractions

Multiplication and Division Problem Solving Using Fractions

Explore 1

Strawberry-Banana Smoothies

Meaning:

________ of the ________

Model:

The class used _____________ fresh bananas for the smoothies. Multiplication expression:

Wacky Watermelon Smoothies

Meaning:

________ groups of ________

Model:

The Wacky Watermelon recipe uses ______________ of fresh watermelon. Multiplication expression:

Multiplication and Division Problem Solving Using Fractions | 17

Explore 1

Multiplication and Division Problem Solving Using Fractions

Reflect 1. Why is a product not always bigger than the factors?

2. Why is the product larger than the fraction factor?

3. Explain why the denominator did not change.

18 | Multiplication and Division Problem Solving Using Fractions

Explore 2

Multiplication and Division Problem Solving Using Fractions

Name: _______________________ Date: ___________

Multiply Fractions and Mixed Numbers Pretzel Path The friends ran 1 of the path. 3

Model:

Multiplication equation:

Distance the friends ran:

Delicate Dip Trail The friends ran 5 of the path. 6

Model:

Multiplication equation:

Distance the friends ran:

Multiplication and Division Problem Solving Using Fractions | 19

Explore 2

Multiplication and Division Problem Solving Using Fractions

Mount Cisco The friends ran 5 of the path. 8

Model:

Multiplication equation:

Distance the friends ran:

Twisted Trail 3

The friends ran 4 of the path. Model:

Multiplication equation:

Distance the friends ran:

20 | Multiplication and Division Problem Solving Using Fractions

Explore 2

Multiplication and Division Problem Solving Using Fractions

Devil’s Landing The friends ran 4 of the path. 5

Predict whether the product of the two fractions will be less than or greater than the original factor of 1 3 . 8

Model:

Multiplication equation:

Distance the friends ran:

Multiplication and Division Problem Solving Using Fractions | 21

Explore 2

Multiplication and Division Problem Solving Using Fractions

Reflect 1. What connections did you make during this activity?

2. What observations did you make about the product when multiplying a fraction by a fraction? Why do you think that is?

3. Explain why the denominator changed in the product.

22 | Multiplication and Division Problem Solving Using Fractions

Explore 3

Multiplication and Division Problem Solving Using Fractions

Name: _______________________ Date: ___________

Modeling Fraction Division Use Cuisenaire RodsTM to determine how many individual packages can be made from each whole dessert type. Chocolate Cake Model:

Solution equation:

Solution statement:

Vanilla Cake Model:

Solution equation:

Solution statement:

Multiplication and Division Problem Solving Using Fractions | 23

Explore 3

Multiplication and Division Problem Solving Using Fractions

German Chocolate Cake Model:

Solution equation:

Solution statement:

Banana Bread Model:

Solution equation:

Solution statement:

24 | Multiplication and Division Problem Solving Using Fractions

Explore 3

Multiplication and Division Problem Solving Using Fractions

Strawberry Cake Model:

Solution equation:

Solution statement:

Red Velvet Cake Model:

Solution equation:

Solution statement:

Multiplication and Division Problem Solving Using Fractions | 25

Explore 3

Multiplication and Division Problem Solving Using Fractions

Reflect 1. What do you notice about the quotient when you divide a fraction by a fraction?

2. Why would dividing by a fraction give you a larger number?

3. What does a fraction in the quotient mean?

4. What patterns do you notice when you divide a whole number by 3 ?

5. How does this pattern change when it is a fraction divided by 3 ?

6. What changes in the pattern when you divide a fraction by 3 ?

26 | Multiplication and Division Problem Solving Using Fractions

Multiplication and Division Problem Solving Using Fractions

Explore 4

Name: _______________________ Date: ___________

Division of Fractions Use a number line to ﬁnd out how many tablecloths can be made in each color of fabric. Use what you know about reciprocals (or the multiplicative inverse) to solve. Write a solution statement to explain your answer. Blue Fabric Divide ____ ____

Equation:

____ yards into groups of

____ .

Model:

How many parts are in each group of 3 ? 4

How many groups of

There are ____

3 4 can you make?

____ of another tablecloth left over.

Use what you know about reciprocals to solve.

Solution statement:

Solution equation:

Multiplication and Division Problem Solving Using Fractions | 27

Explore 4

Multiplication and Division Problem Solving Using Fractions

Orange Fabric Divide ____ ____

____ of a yard into groups of

Equation:

____ .

Model:

Solution statement:

Solution equation:

Silver Fabric Divide ____ ____

____ yards into groups of

Equation:

____ .

Model:

Solution statement:

28 | Multiplication and Division Problem Solving Using Fractions

Solution equation:

Explore 4

Multiplication and Division Problem Solving Using Fractions

Pink Fabric Divide ____ ____

____ yards into groups of

Equation:

____ .

Model:

Solution statement:

Solution equation:

Yellow Fabric Divide ____ ____

____ yards into groups of

Equation:

____ .

Model:

Solution statement:

Solution equation:

Multiplication and Division Problem Solving Using Fractions | 29

Explore 4

Multiplication and Division Problem Solving Using Fractions

Green Fabric Divide ____ ____

____ yards into groups of

Equation:

____ .

Model:

Solution statement:

Solution equation:

Reflect 1. What is the reciprocal (multiplicative inverse)?

2. Why is it helpful to use the reciprocal (multiplicative inverse) when dividing?

30 | Multiplication and Division Problem Solving Using Fractions

Add and Subtract Decimals

Explore 1

Name: _______________________ Date: ___________

Add and Subtract Multi-Digit Decimal Numbers Part I As you participate in the class discussion, show your work below. Revise your work as needed before writing your class consensus. Then, answer the questions that follow.

Sum

Class Consensus

2,412 + 176

241.2 + 176

241.2 + 17.6

Reflect 1. What do you notice about the digits of the addends?

Add and Subtract Decimals | 33

Add and Subtract Decimals

Explore 1

Use the Grocery Ad and Bake Sale Cards to complete each table by listing the ingredients or items needed for each bake sale item and their cost. Then, answer the questions that follow. Show vertical calculations. Team A: Lemonade Stand Ingredient/ Item Price

You need 9 lemons for the recipe. How much will 9 lemons cost?

What is the cost for lemons, sugar, water, and ice?

What is the cost for cups, napkins, and a lemonade pitcher?

What is the total cost for lemonade?

34 | Add and Subtract Decimals

Add and Subtract Decimals

Explore 1 Team B: Chocolate Chip Cookies Ingredient/Item

Price in Ad

What is the cost to buy the ingredients to make one batch of chocolate chip cookies?

What is the total cost if you also need to purchase bags for packaging?

Team C: Sugar Cookies

What is the cost to buy the ingredients to make one batch of sugar cookies?

What is the total cost if you also need to purchase bags for packaging?

Add and Subtract Decimals | 35

Add and Subtract Decimals

Explore 1 Team D: Brownie Bites Ingredient/Item

What is the cost to buy the ingredients to make one batch of brownies?

Price in Ad

What is the total cost if you also need to purchase bags for packaging?

Reflect 1. What do you have to do to add decimal numbers together?

2. What is the total cost of all of the supplies needed for the bake sale? Include buying bags for each team that will need them.

36 | Add and Subtract Decimals

Add and Subtract Decimals

Explore 1 Part II

As you participate in the class discussion, show your work below. Revise your work as needed before writing your class consensus. Then, answer the questions that follow.

Difference

Class Consensus

2,412 – 176

241.2 – 176

241.2 – 17.6

Reflect 1. What do you notice about the digits when subtracting?

Add and Subtract Decimals | 37

Add and Subtract Decimals

Explore 1

Determine how much money each team has remaining after purchasing enough supplies for one batch of each recipe. Include the cost of the bags in the total for each batch. Lemonade

Chocolate Chip Cookies

Sugar Cookies

Brownies

Workspace:

Money remaining:

Reflect 1. What do you do to subtract from 0?

2. How could you check your answer to see if you made any mistakes?

38 | Add and Subtract Decimals

Multiply and Divide Decimals

Explore 1

Name: _______________________ Date: ___________

Multiply Decimals – Place Value Complete the table below. Use base ten blocks to represent each order, and ﬁnd the total amount of cake.

Order

Represent

Solve

One group of one chocolate cake Expression:

One-tenth of one strawberry cake Expression:

One-tenth of one-tenth of a vanilla cake Expression:

How did you ﬁnd one-tenth of one-tenth?

Multiply and Divide Decimals | 41

Multiply and Divide Decimals

Explore 1

Find the total amount of cake for the ﬁrst two orders without the base ten blocks. Use base ten blocks to model the rest of the orders, and then solve. Order

Represent

Solve

Six groups of 50 cakes Expression:

Six groups of ﬁve cakes Expression:

Six groups of 0.5 of each cake ﬂavor Expression:

Six groups of 0.05 of each cake ﬂavor Expression:

As you moved through the orders, what changed about the digit 5?

How did changing the place value of the digit 5 affect the place value of the product?

42 | Multiply and Divide Decimals

Multiply and Divide Decimals

Explore 1 Use base ten blocks to model the orders, and then solve. Order

Represent

Solve

Four groups of three cakes Expression:

Four groups of 0.3 of each cake ﬂavor Expression:

0.4 of 0.3 of each cake ﬂavor Expression:

Multiply and Divide Decimals | 43

Multiply and Divide Decimals

Explore 1 Reflect 1. How did you ﬁnd the total amount of cake for the ﬁnal order?

2. Complete the following statements by ﬁlling in what size pieces the product will have. Whole number × whole number = ________________________________________ Whole number × tenths = ______________________________________________ Whole number × hundredths = __________________________________________ Tenths × tenths = _____________________________________________________

3. What do you need to do if you have more than 10 tenths or hundredths in your product?

4. What did you notice about your product when multiplying by decimals less than 1? Explain.

44 | Multiply and Divide Decimals

Multiply and Divide Decimals

Explore 2

Name: _______________________ Date: ___________

Multiply Decimals – Arrays and Area Models Part I: Paper Measurement Tools Use the space below to trace each base ten block. These blocks will be used as paper measuring tools. Label the ﬂat, the rod, and the unit with the length, width, and area if the ﬂat is 1 square yard.

Make-a-Statement Sign Company

Reflect 1. How did you know the width of the rod?

2. How did you know the length and width of the unit cube?

Multiply and Divide Decimals | 45

Multiply and Divide Decimals

Explore 2 Part II: Sign Station

Use the base ten blocks as paper measurement tools to build a model of each sign. Create an area model based on the blocks. Use the models to ﬁgure out the total amount of paper needed for each sign. Sign Size

Array

Area Model

Paper Needed

Length: 3.2 yd. Width: 2 yd.

Length: 2.5 yd. Width: 1.3 yd.

Length: 0.5 yd. Width: 1.3 yd.

46 | Multiply and Divide Decimals

Multiply and Divide Decimals

Explore 2 Sign Size

Array

Area Model

Paper Needed

Length: 2.4 yd. Width: 3.1 yd.

Length: 0.4 yd. Width: 0.8 yd.

Reflect 1. How did you know what size pieces to use in each section of your model?

2. How did you use your model to ﬁnd the ﬁnal product?

3. How is this process similar to or different from multiplying whole numbers?

Multiply and Divide Decimals | 47

Multiply and Divide Decimals

Explore 2 Part III: Big Bucks Billboards

As a group, draw an area model that represents each billboard. Record the area models below, and ﬁnd the area of each section. Use the models to ﬁgure out the total amount of paper needed for each billboard. Sign Size

Area Model

Paper Needed

Length: 8.33 yd. Width: 6 yd.

Length: 4.27 yd. Width: 10.53 yd.

48 | Multiply and Divide Decimals

Multiply and Divide Decimals

Explore 2 Sign Size

Area Model

Paper Needed

Length: 0.72 yd. Width: 13.4 yd.

Length: 9.03 yd. Width: 5 yd.

Multiply and Divide Decimals | 49

Multiply and Divide Decimals

Explore 2 Sign Size

Area Model

Paper Needed

Length: 7.5 yd. Width: 14.3 yd.

Reflect 1. How are area models similar to arrays?

2. How is an area model helpful?

3. How did you know the place value of the digits in your product?

50 | Multiply and Divide Decimals

Multiply and Divide Decimals

Explore 3

Name: _______________________ Date: ___________

Multiply Multi-Digit Decimal Numbers Part I • • •

Look at the length and width of each sign. Use the Area Model Template to create an area model to solve the problem. Record your work below. Write each partial product in the box below it. Lemonade Sign Length = 2.5 yd. Width = 0.75 yd.

Chocolate Chip Cookies Sign Length = 3.25 yd. Width = 1.2 yd.

Area model:

Partial products:

___ ___ ___

× ___ ___ ___

× ___ ___ _ ___

___ ___ ___ ___ ___ ( ___ × ___ ) ___ ___ ___ ___ ___ ( ___ × ___ ) ___ ___ ___ ___ ___ ( ___ × ___ ) + ___ ___ ___ ___ ___ ( ___ × ___ ) ___ ___ ___ ___ ___

___ ___ ___ ___ ___ ( ___ × ___ ) ___ ___ ___ ___ ___ ( ___ × ___ ) ___ ___ ___ ___ ___ ( ___ × ___ ) ___ ___ ___ ___ ___ ( ___ × ___ ) ___ ___ ___ ___ ___ ( ___ × ___ ) + ___ ___ ___ ___ ___ ( ___ × ___ ) ___ ___ ___ ___ ___

Multiply and Divide Decimals | 51

Multiply and Divide Decimals

Explore 3 Sugar Cookies Sign Length = 2.75 yd. Width = 0.62 yd.

Brownie Bites Sign Length = 2.83 yd. Width = 1.13 yd.

Area model:

Partial products:

___ ___ ___

× ___ ___ ___

___ _ ___ ___ ___ ___ ( ___ × ___ )

___ ___ ___ ___ ___ ( ___ × ___ ) _

___ ___ ___ ___ ___ ( ___ × ___ )

+ ___ ___ ___ ___ ___ ( ___ × ___ )

___ ___ ___ ___ ___ ( ___ × ___ )

___ ___ ___ ___ ___

___ ___ ___ ___ ___ ( ___ × ___ ) ___ ___ ___ ___ ___ ( ___ × ___ ) + ___ ___ ___ ___ ___ ( ___ × ___ ) ___ ___ ___ ___ ___

52 | Multiply and Divide Decimals

Explore 3

Multiply and Divide Decimals

Part II • • •

Read each problem, and set up the equation. Use the Standard Algorithm Work Mat to solve each problem. Record your work in the space provided. Lemonade Price before tax = $20.81 Sales tax = $0.083

Chocolate Chip Cookies Price before tax = $30.98 Sales tax = $0.083

____ ____ ____ ____

× ____ ____ ____ ____

____ ____ ____ ____ ____ ____

+ ____ ____ ____ ____ ____ ____

____ ____ ____ ____ ____ ____

How many total decimal places are in the factors?

Move the decimal the same number of places in the product. What is your product?

How many decimal places do we use in money?

Determine the amount of sales tax that is added to the total.

Multiply and Divide Decimals | 53

Explore 3 Sugar Cookies Price before tax = $24.00 Sales tax = $0.083

Multiply and Divide Decimals

Brownie Bites Price before tax = $30.88 Sales tax = $0.083

____ ____ ____ ____

× ____ ____ ____ ____

____ ____ ____ ____ ____ ____

+ ____ ____ ____ ____ ____ ____

____ ____ ____ ____ ____ ____

Move the decimal the same number of places in the product. What is your product?

Determine the amount of sales tax that is added to the total.

Reflect 1. How do you determine where the decimal point should go when multiplying two decimal numbers together?

2. How many decimal places are used for money? How do you determine what the digit for the hundredths place for money will be?

54 | Multiply and Divide Decimals

Multiply and Divide Decimals

Explore 4

Name: _______________________ Date: ___________

Divide Decimals – Place Value Part I: Pet Food Patterns Complete the table below. Use base ten blocks to represent the amount of dog food in each bag if 1 pound is divided up evenly.

Bag Size

Represent

Amount of Food in Each Bag

1 lb. bags using 1 pound of food Expression:

One-tenth lb. bags using 1 pound of food Expression:

One-hundredth lb. bags using 1 pound of food Expression:

Multiply and Divide Decimals | 55

Multiply and Divide Decimals

Explore 4 Part II: Pet Food Orders

Model the amount of dog food needed using the base ten blocks, and then ﬁnd the size of bag needed for each order. Each ﬂat equals 1 pound of food. Order

Represent

Bag Size

15 pounds of food in 5 bags Expression:

1.5 pounds of food in 5 bags Expression:

0.15 pounds of food in 5 bags Expression:

As you moved through the different sizes of bags, what changed about the digits 1 and 5 in the dividend?

How did changing the place value of the dividend affect the place value of the quotient?

56 | Multiply and Divide Decimals

Multiply and Divide Decimals

Explore 4 Order

Represent

Bag Size

4 pounds of food in 2 bags Expression:

0.4 pounds of food in 2 bags Expression:

0.04 pounds of food in 2 bags Expression:

Reflect 1. How did you ﬁnd the size of the bags needed for these orders?

2. What happened to the size of the bags when you had smaller amounts of food?

3. What do you think the size of the bags would be if you had 0.004 pounds of food separated into two bags?

Multiply and Divide Decimals | 57

Multiply and Divide Decimals

Explore 5

Name: _______________________ Date: ___________

Divide Decimals — Arrays and Area Models Part I: Booth Measurement Tools Use the space below to trace each base ten block. These blocks will be used as paper measuring tools. Label the ﬂat and rod with the length, width, and area if the ﬂat is equal to 1 square yard.

Reflect 1. How did you know the width of the rod?

Multiply and Divide Decimals | 59

Multiply and Divide Decimals

Explore 5 Part II: Small Booths

Use the base ten blocks to build a model of each carnival booth with the total area, using the length measurement. Create an area model based on the blocks. Use the models to ﬁgure out the width of the booths. Booth Area and Length

Array

Area Model

Solution

Total area: 8.4 square yards

Length: 4 yards

Total area: 1.52 square yards

Length: 0.4 yards

Total area: 3.36 square yards

Length: 0.8 yards

60 | Multiply and Divide Decimals

Multiply and Divide Decimals

Explore 5 Booth Area and Length

Array

Area Model

Solution

Total area: 14.8 square yards

Length: 4 yards

Total area: 7.2 square yards

Length: 2 yards

Reflect 1. How did you know what size pieces to use in each section of your model?

2. How did you use your model to ﬁnd the quotient?

3. How is this process similar to or different from dividing whole numbers?

Multiply and Divide Decimals | 61

Multiply and Divide Decimals

Explore 5 Part III: Large Booths

As a group, draw an area model that represents each large carnival booth. Record the area models below, and ﬁnd the quotient of each section. Use the models to ﬁgure out the width of the booths. Booth Area and Length

Area Model

Solution

Area: 328.75 square yards

Length: 25 yards

Area: 429.4 square yards

Length: 3.8 yards

Area: 204.8 square yards

Length: 1.6 yards

62 | Multiply and Divide Decimals

Multiply and Divide Decimals

Explore 5 Booth Area and Length

Area Model

Solution

Area: 840.4 square yards

Length: 2.2 yards

Area: 30.68 square yards

Length: 1.3 yards

Reflect 1. How are area models similar to arrays?

2. How is an area model helpful?

3. How did you know the place value of the digits in your quotient?

Multiply and Divide Decimals | 63

Multiply and Divide Decimals

Explore 6

Name: _______________________ Date: ___________

Divide Multi-Digit Decimal Numbers Complete the tables to solve. Brownie Bites Total: $54.75 Price for each: $0.75 Division equation:

Solve without the decimal point in the dividend or divisor below.

Now, let’s try it with the decimals!

To make the divisor (0.75) a whole number, we can–

5,475 ÷ 75 Can we leave the dividend alone if we change the divisor?

Show the steps to change the dividend and divisor.

Solve:

How many brownie bites were sold?

Multiply and Divide Decimals | 65

Multiply and Divide Decimals

Explore 6 Chocolate Chip Cookies Total: Price for each: Division equation:

Solve without the decimal point in the dividend or divisor below.

Now, let’s try it with the decimals!

Show the steps to change the dividend and divisor.

15,450 ÷ 150

Solve:

How many chocolate chip cookies were sold?

66 | Multiply and Divide Decimals

Multiply and Divide Decimals

Explore 6 Sugar Cookies

Solve:

Total: Price for each: Division equation:

Show the steps to change the dividend and divisor.

How many sugar cookies were sold?

Lemonade

Solve:

Total: Price for each: Division equation:

Show the steps to change the dividend and divisor.

How many cups of lemonade were sold?

Multiply and Divide Decimals | 67

Multiply and Divide Decimals

Explore 6 Reflect

1. How are the division problem with the decimals and the division problem without the decimals the same?

2. Why do you think we should move the decimal to make a whole-number divisor?

3. Would you get the same answer if you only moved one of the decimals? Explain.

4. What if the dividend does not have a decimal in its number?

68 | Multiply and Divide Decimals

Integers

Explore 1

Name: _______________________ Date: ___________

A Number and Its Opposite Read each Rock Wall Scenario Card. Represent and label each rock-wall stone on the given number lines. Then, write each value as a positive or negative integer. Rock Wall 1 Integer Locations Negative integer:

Positive integer:

What is the distance of each yellow rock-wall stone from zero?

Rock Wall 2 Integer Locations

Negative integer:

Positive integer:

What is the distance of each red rock-wall stone from zero?

Integers | 71

Integers

Explore 1 Rock Wall 3 Integer Locations

Negative integer:

Positive integer:

What is the distance of each blue rock-wall stone from zero?

Rock Wall 4 Integer Locations Negative integer:

Positive integer:

What is the distance of each orange rock-wall stone from zero?

72 | Integers

Integers

Explore 1 Rock Wall 5 Integer Locations

Negative integer:

Positive integer:

What is the distance of each black rock-wall stone from zero?

Rock Wall 6 Integer Locations Negative integer:

Positive integer:

What is the distance of each green rock-wall stone from zero?

Integers | 73

Integers

Explore 1 Reflect 1. How did you use a number line to ﬁnd integers in each scenario?

2. How can you determine if the positive and negative integers are opposites?

3. Where are positive integers located on the number line?

4. Where are negative integers located on the number line?

5. What is the opposite of zero?

74 | Integers

Integers

Explore 2

Name: _______________________ Date: ___________

Compare and Order Integers Read each Elevation Card. Then, use the Elevation Card to work with your group to represent each card on the number line and answer the questions related to integer inequalities. Card 1 Represent the cities on the number line.

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

Which city is highest?

Which city is lowest?

The opposite integer of the highest city:

The opposite integer of the lowest city:

Compare the integers using an inequality.

Integers | 75

Integers

Explore 2 Card 2 Represent the cities on the number line.

−70

−60

−50

−40

−30

−20

−10

Which city is highest?

The opposite integer of the highest city:

Which city is lowest?

The opposite integer of the lowest city:

Compare the integers using an inequality.

Card 3 Represent the cities on the number line.

Which city is highest?

The opposite integer of the highest city:

Which city is lowest?

The opposite integer of the lowest city:

Compare the integers using an inequality.

76 | Integers

Integers

Explore 2 Card 4 Represent the cities on the number line.

Which city is highest?

Which city is lowest?

The opposite integer of the highest city:

The opposite integer of the lowest city:

20 15 10 5 0 −5

Compare the integers using an inequality.

−10 −15 −20

Card 5 Represent the cities on the number line.

−5

Which city is highest?

The opposite integer of the highest city:

Which city is lowest?

The opposite integer of the lowest city:

Compare the integers using an inequality.

Integers | 77

Integers

Explore 2 Card 6 Represent the cities on the number line.

Which city is highest?

Which city is lowest?

The opposite integer of the highest city:

The opposite integer of the lowest city:

70 60 50 40 30 20 10 0 −10 −20

Compare the integers using an inequality.

−30 −40 −50 −60 −70

78 | Integers

Explore 2

Integers

Reflect 1. How did you use a number line to compare the cities’ elevation levels?

2. Explain what would be the opposite of the opposite of a below-sea-level elevation.

3. What elevation would be the opposite of 68?

4. When comparing three integers, can you use two different inequality symbols when writing the inequality? Why or why not?

Integers | 79

__________.

The absolute value is __________.

How far is –1 away from 0? __________

How far is 8 away from 0? ___________

How far is −8 away from 0? __________

How far is 5 away from 0? ___________

How far is −5 away from 0? __________

How far is 10 away from 0? ___________

How far is −10 away from 0? __________

The absolute value is

How far is 1 away from 0? ___________

Use the integer football ﬁeld you created to answer the related questions below.

|−10 | = ____

| 10 | = ____

Integers | 81

Representing absolute value

|−5 | = ____

| 5 | = ____

Representing absolute value

|−8 | = ____

| 8 | = ____

Representing absolute value

|−1 | = ____

| 1 | = ____

Representing absolute value

Name: _______________________ Date: ___________

Absolute Value

Part I: Creating an Integer Football Field Number Line

Explore 3

Integers

82 | Integers

|5|=

__________.

–5

How far is −5 away from the line of scrimmage? _________

–10

The absolute value is

–15

How far is 5 away from the line of scrimmage? _________

–20

Represent the penalties on the number line.

Card 1

Read each Integer Football Scenario Card, and work with your group to represent each card on the Integer Football Number Line. Use the number line representation to draw a model below, and answer the questions related to absolute value.

Part II: Integer Football Penalties

Explore 3

Integers

__________.

–5

How far is −8 away from the line of scrimmage? _________

–10

The absolute value is

–15

How far is 8 away from the line of scrimmage? _________

–20

Represent the penalties on the number line.

Card 3

__________.

–5

How far is −10 away from the line of scrimmage? ________

–10

The absolute value is

–15

Represent the penalties on the number line.

Card 2

How far is 10 away from the line of scrimmage? _________

–20

Explore 3

Integers | 83

|8|=

= 10

| 10 | =

Integers

= 15

| 15 | =

84 | Integers

4. What is the absolute value of 0?

3. Can the absolute value of a number ever be negative? Explain.

2. How can a positive integer and a negative integer have the same absolute value?

1. Whether an integer is positive or negative, what do you notice about the absolute value of a number and its opposite?

Reflect

__________.

–5

How far is −15 away from the line of scrimmage? ________

–10

The absolute value is

–15

Represent the penalties on the number line.

Card 4

How far is 15 away from the line of scrimmage? _________

–20

Explore 3

Integers

Rational Numbers

Explore 1

Name: _______________________ Date: ___________

Absolute Value of Rational Numbers Read the Park Scenario Cards, and determine the location of each value on a number line. Then, use your number line to answer the questions. Bird Watching Model pair A on a number line.

Model pair B on a number line.

1. What is the absolute value for the blue jay?

2. What is the absolute value for the cardinal?

3. What is the absolute value for the robin?

4. What is the absolute value for the wren?

Which bird had the greatest absolute value? Explain how you know.

Rational Numbers | 87

Rational Numbers

Explore 1 The Climb Plot where each animal is on a vertical number line.

1. What is the absolute value for the squirrel?

2. What is the absolute value for the newt?

3. What is the absolute value for the caterpillar?

4. Who was the highest out of all three animals? Explain how you know.

5. Who was the lowest in the tree of all three animals? Explain how you know.

88 | Rational Numbers

Rational Numbers

Explore 1 Schooling Model pair A on a number line.

Model pair B on a number line.

1. What is the absolute value of the koi?

2. What is the absolute value of the minnows?

3. What is the absolute value of the sunﬁsh?

4. What is the absolute value of the goldﬁsh?

5. Which school of ﬁsh has the greatest absolute value? Explain how you know.

Rational Numbers | 89

Rational Numbers

Explore 1 Wet and Dry Seasons Plot each day’s water level on a vertical number line.

1. What is the absolute value of the water level for day 1 of the wet season?

2. What is the absolute value of the water level for day 1 of the dry season?

3. What is the absolute value of the water level for day 2 of the wet season?

4. What is the absolute value of the water level for day 2 of the dry season?

5. Which day had the smallest absolute value?

90 | Rational Numbers

Explore 1

Rational Numbers

Reflect 1. In the Park Scenario Card titled “The Climb,” which animal is closer to the ground, the caterpillar or the newt?

2. What do you notice about the absolute value of a fraction and its opposite?

3. In the Park Scenario Card titled “Wet and Dry Seasons,” which day has a greater magnitude, day 1 of wet season or day 1 of dry season?

Rational Numbers | 91

Rational Numbers

Explore 2

Name: _______________________ Date: ___________

Compare Rational Numbers Read each Animal Show Information Card, and locate the rational numbers on your ﬂoor number line. Then, use your number line to answer the questions. Jumping Contest Plot each jumper’s height on the number line.

1. Write an inequality statement to compare team A’s two jumpers.

2. Write an inequality statement to compare team B’s two jumpers.

3. Which team had the highest jumper? Explain how you know.

Rational Numbers | 93

Rational Numbers

Explore 2 Timed Swimming Contest Plot each swimmer’s distance on the number line.

1. Write an inequality statement to compare team C’s two swimmers.

2. Write an inequality statement to compare team D’s two swimmers.

3. Who swam the most out of both teams? Explain how you know.

4. Who swam the least out of both teams? Explain how you know.

94 | Rational Numbers

Rational Numbers

Explore 2 Proﬁts and Losses Plot each proﬁt and loss on the number line.

–40 –30 –20 –10

10 20

30 40

70 80

90 100 110 120 130

1. Write an inequality statement to compare day 1’s and day 2’s proﬁts and losses. Explain your answer in terms of the scenario.

2. Write an inequality statement to compare day 1’s and day 3’s proﬁts and losses. Explain your answer in terms of the scenario.

3. Which has a greater magnitude, the value of day 4 or the value of day 2?

4. Which day(s) did the animal show have a debt greater than $10?

Rational Numbers | 95

Rational Numbers

Explore 2 Temperatures Plot each temperature on the vertical number line.

1. Write an inequality statement to compare the warmest temperature and the coldest temperature.

13 12 11 10 9 8

2. Which day had the smallest absolute value?

7 6 5 4 3 2 1 0 –1 –2 –3

96 | Rational Numbers

Rational Numbers

Explore 2 Above and Below Plot each animal’s location on a vertical number line.

1. Write each animal’s location as a rational number.

12 10 8 6 4 2 0 –2

2. Write an inequality statement to compare the magnitude of the location for Grant the green sea turtle and Alec the Atlantic pufﬁn.

–4 –6 –8 –10 –12 –14 –16 –18 –20

3. Who is closer to sea level, Grant the green sea turtle or Alec the Atlantic pufﬁn?

–22 –24 –26 –28 –30 –32 –34 –36 –38 –40

Rational Numbers | 97

Rational Numbers

Explore 2 Tracking Allie the Atlantic Pufﬁn Plot Allie’s location for each day on the number line.

1. On which day(s) was Allie’s location below sea level?

12 10 8 6 4

2. On which day was Allie’s depth greater than ten feet below sea level?

2 0 –2 –4

3. On which day was Allie closest to sea level?

–6 –8 –10 –12 –14

98 | Rational Numbers

Rational Numbers

Explore 2 Reflect

1. How does using a number line help determine which value is smaller when both values are negative numbers?

2. In the Animal Show Information Card titled “Above and Below,” which animal had the greatest magnitude?

The table below has a snapshot of the animal show’s proﬁts and losses.

Line 1

−$12.80

Line 2

−$15.40

Line 3

−$8.90

Line 4

$19.30

Line 5

−$22.20

Line 6

$15.80

3. Which lines show debts of more than $10.00?

Rational Numbers | 99

Rational Numbers

Explore 3

Name: _______________________ Date: ___________

Model and Order Rational Numbers Part I: Hiking Trip Read each situation, and work with your group to determine a rational number that represents that situation. Record the approximate location of each rational number on the number line. Earning and Spending Express each situation as a rational number. • Jack spent $4.13 on hiking clothes.

_______________

• He earned $4.25 from selling some old toys.

_______________

• He spent $1.37 on ice cream.

_______________

• He earned $2.75 from selling desserts.

_______________

Locate and label each rational number on the number line.

-$4.50 -$3.50 -$2.50 -$1.50 -$0.50 $0.50 $1.50 $2.50 $3.50 $4.50 -$4.00 -$3.00 -$2.00 -$1.00 0 $1.00 $2.00 $3.00 $4.00 Compare the amount Jack spent on hiking clothes and the amount he spent on ice cream using < or >.

Use the number line to write the values in order from least to greatest.

Rational Numbers | 101

Rational Numbers

Explore 3 Hiking

4 3

• Jack climbed 1 1 miles up the mountain on day 1. 2

2 1

• He climbed 3 miles down the mountain on day 2.

0 -1 -2 -3 -4

• He climbed 2 3 miles up the mountain on day 3. 4

• He climbed 1 of a mile down the mountain on day 4. 4

Compare Jack’s climbing heights on day 1 and day 3 using < or >.

Use the number line to write the values in order from least to greatest.

102 | Rational Numbers

Rational Numbers

Explore 3 Part II: Fishing Trip Plot each rational number on the vertical number line. Fishing

• Jack decided to go ﬁshing. The boat sits on the water at sea level. Mark this spot on the vertical number line.

10 8

• The anchor stopped at 7.8 meters below sea level at the ﬁrst stop.

6 4 2 0 -2 -4

• He decided to put up the sunshade. The sunshade 1 rose 2 5 meters above the boat.

• The ﬁshing line drops to 4 3 meters below the 5 boat.

• The top of the ﬁshing pole is 1.2 meters above the water.

-6 -8

Use the number line to write the values in order from least to greatest.

Rational Numbers | 103

Rational Numbers

Explore 3 Temperatures

5 4 3 2 1

• Jack recorded the temperatures on different days on his trip.

• Friday, the temperature was 4 10 °C.

• Saturday, the temperature dropped to −2.7°C.

0 • Sunday, the temperature was −0.9°C.

-1 -2 -3

Use the number line to write the values in order from warmest to coldest.

-4 -5 104 | Rational Numbers

Explore 3

Rational Numbers

Reflect 1. How is plotting rational numbers on a number line similar to plotting whole numbers on a number line?

2. How do you create your own number lines for rational numbers?

3. How can you represent rational numbers with fractions and rational numbers with decimals on the number line together?

4. How can a number line help to order rational numbers?

Rational Numbers | 105

Equivalent Numerical Expressions

107

Name: _______________________ Date: ___________

Greatest Common Factors

Factors of 18:

Factors of 12:

Greatest common factor of 12 and 18:

Common factors of 12 and 18:

Arrangement groups:

Equivalent Numerical Expressions | 109

Blue Balloon Arrangements

Arrangement groups:

Red Balloon Arrangements

Read the red and blue Balloon Arrangements Card with the class. Use your linking cubes to model the different arrangements that can be made. Write out the different arrangements that can be made of red balloons and the different arrangements that can be made of blue balloons. Complete the remaining questions as you discuss with the class.

Part I: Greatest Common Factors

Explore 1

Equivalent Numerical Expressions

Common Factors

_____ balloon arrangements can be made.

Factors

Greatest Common Factor

Common Factors

_____ balloon arrangements can be made.

Factors

Greatest Common Factor

110 | Equivalent Numerical Expressions

There will be _____ purple balloons and _____ green balloons in each arrangement.

Green Balloons: _____

Purple Balloons: _____

Card 2

There will be _____ black balloons and _____ gold balloons in each arrangement.

Gold Balloons: _____

Black Balloons: _____

Card 1

Read each Balloon Arrangements Card. Find the factors of each balloon set to determine the greatest number of balloon arrangements that can be made. Write how many balloons of each color will be in each arrangement.

Part II: Balloon Arrangements

Explore 1

Equivalent Numerical Expressions

Common Factors

_____ balloon arrangements can be made.

Factors

Greatest Common Factor

Common Factors

_____ balloon arrangements can be made.

Factors

Greatest Common Factor

Equivalent Numerical Expressions | 111

There will be _____ yellow balloon and _____ green balloons in each arrangement.

Green Balloons: _____

Yellow Balloons: _____

Card 4

There will be _____ orange balloons and _____ blue balloons in each arrangement.

Blue Balloons: _____

Orange Balloons: _____

Card 3

Explore 1

Equivalent Numerical Expressions

112 | Equivalent Numerical Expressions

3. Explain how ﬁnding all of the factors of each number helps you determine the greatest common factor.

2. How can you determine what the factors are for each number given?

1. What is a factor?

Reflect

Explore 1

Equivalent Numerical Expressions

Explore 2

Name: _______________________ Date: ___________

Prime Factorization Part I: Factor Trees Follow your class discussion to complete the table below, and then answer the questions that follow. 120 Factor tree:

Factor tree:

Expression:

1. What do you notice about all of the circled numbers of the factor tree?

2. Are the prime factorization expressions different if you use different factors for the ﬁrst branch?

Equivalent Numerical Expressions | 113

Equivalent Numerical Expressions

Explore 2 Part II: Cookies

Use the Cookie Scenario Cards to ﬁnd the prime factors of each set. Write the prime factorization of each cookie type. Find the greatest common factor to determine the greatest number of cookie snack bags that can be made. Party A 30

Factor tree:

Prime factorization:

Multiply all of the common prime factors together.

GCF:

What is the greatest number of cookie snack bags that can be made for each type of cookie? _____

114 | Equivalent Numerical Expressions

Equivalent Numerical Expressions

Explore 2 Party B 64

Factor tree:

Prime factorization:

Multiply all of the common prime factors together.

GCF:

What is the greatest number of cookie snack bags that can be made for each type of cookie? _____

Equivalent Numerical Expressions | 115

Equivalent Numerical Expressions

Explore 2 Party C 40

Factor tree:

Prime factorization:

Multiply all of the common prime factors together.

GCF:

What is the greatest number of cookie snack bags that can be made for each type of cookie? _____

Reflect 1. What is a prime number?

2. Explain how ﬁnding the prime factors helps to determine the greatest common factor.

116 | Equivalent Numerical Expressions

Equivalent Numerical Expressions

Explore 2 Part III: Tables and Chairs

Find the greatest common factor for the number of adults and the number of children. Then, use the distributive property to help determine how many tables are needed if the same number of adults and children will sit at each table. Party A Adults

Children

How Many? GCF

Area Model

Equivalent Expressions

There will be ____ tables for adults. There will be ____ tables for children. There will be ____ adults at each table. There will be ____ children at each table. © Accelerate Learning Inc. – All Rights Reserved

Equivalent Numerical Expressions | 117

Equivalent Numerical Expressions

Explore 2 Party B Adults

Children

How Many? GCF

Area Model

Equivalent Expressions

There will be ____ tables for adults. There will be ____ tables for children. There will be ____ adults at each table. There will be ____ children at each table.

118 | Equivalent Numerical Expressions

Equivalent Numerical Expressions

Explore 2 Party C Adults

Children

How Many? GCF

Area Model

Equivalent Expressions

There will be ____ tables for adults. There will be ____ tables for children. There will be ____ adults at each table. There will be ____ children at each table.

Equivalent Numerical Expressions | 119

Equivalent Numerical Expressions

Explore 3

Name: _______________________ Date: ___________

Least Common Multiples Part I Read each Food Supply Card. Use your Cuisenaire RodsTM to build a model showing an equal number of each item. Draw a model, and list the multiples for each number of items. Use this information to determine how many packages of each item are needed with none left over. Hot Dogs and Hot Dog Buns Model:

List of multiples:

What is the smallest number of packages Chao will need to buy to have one hot dog per hot dog bun?

How many total hot dogs will there be?

Hamburger Patties and Hamburger Buns Model:

List of multiples:

What is the smallest number of packages Chao will need to buy to have one hamburger patty per bun?

Equivalent Numerical Expressions | 121

Equivalent Numerical Expressions

Explore 3 Spoons and Forks Model:

List of multiples:

What is the smallest number of packages Chao will need to buy to have one spoon for every fork?

How many total spoons will there be?

Plates and Cups Model:

List of multiples:

What is the smallest number of packages Chao will need to buy to have one cup per plate?

How many total cups will Chao buy? 122 | Equivalent Numerical Expressions

Equivalent Numerical Expressions

Explore 3

Hamburger Patties and Cheese Slices Model:

List of multiples:

What is the smallest number of packages Chao will need to buy to have one slice of cheese per hamburger patty?

How many total patties with cheese will there be?

Cupcakes and Toppers Model:

List of multiples:

What is the smallest number of packages Chao will need to buy to have one topper per cupcake?

Equivalent Numerical Expressions | 123

Explore 3

Equivalent Numerical Expressions

Reflect 1. How did you make 12 with your rods?

2. How did you know when to stop adding rods to your models?

Part II Camila says you can ﬁnd the least common multiple by using the prime factors of both numbers. Examine her work below to determine the relationship between prime factors and the least common multiple. Hot Dogs in One Package

Hot Dog Buns in One Package

Prime factors: 2 ∙ 5

Prime factors: 2 ∙ 2 ∙ 2

Common prime factors: 2

Remaining factors: 5, 2, 2

Equation to ﬁnd the least common multiple: 2 ∙ 2 · 2 · 5 = 40

1. How are prime factors related to the least common multiple?

124 | Equivalent Numerical Expressions

Equivalent Numerical Expressions

Explore 3

2. Use Camila’s strategy to determine the least common multiple of cupcakes and cupcake toppers.

Cupcakes in One Package

Cupcake Toppers in One Package

______

Prime factors:

Common prime factors:

Remaining factors:

Equation to ﬁnd the least common multiple:

Reflect 1. How can ﬁnding the prime factors of each number help us ﬁnd the least common multiple?

2. Why would we need to ﬁnd the least common multiple when buying food for a party?

Equivalent Numerical Expressions | 125

Exponents

Name: _______________________ Date: ___________

64 chocolate cupcakes

32 strawberry cupcakes

16 vanilla cupcakes

8 confetti cupcakes

4 fudge cupcakes

2 red velvet cupcakes

Total Cupcakes Needed Expanded Form

How many 2s?

Equivalent Numerical Expressions | 127

Exponential Expression

Use prime factorization to write each cupcake order in expanded form. Use the information to rewrite each order as an exponential expression.

Part I: Cupcake Orders

Explore 4

Equivalent Numerical Expressions

Expanded Form

128 | Equivalent Numerical Expressions

Order Letter

Total Pounds of Cookies Needed

Exponential Expression

Cookie Orders by the Pound

Read each Order Card. Write the expression and exponential expression for each order. Then, answer the reﬂection questions.

Part II: Orders

Explore 4

Equivalent Numerical Expressions

Expanded Form

Order Letter

Explore 4

Exponential Expression

Chair Rental Orders

Equivalent Numerical Expressions | 129

Total Chairs Needed

Equivalent Numerical Expressions

Expanded Form

130 | Equivalent Numerical Expressions

Order Letter

Explore 4

Total Flowers Needed

Exponential Expression

Flower Orders

Equivalent Numerical Expressions

Equivalent Numerical Expressions | 131

5. Jazmine has a disagreement with her coworker. Her coworker says that the pounds of cookies are incorrect because the total pounds of cookies are all less than the orders. How can Jazmine explain that she is correct?

4. Shawn says that any time you have an exponent of 1, 1 item is ordered. Is this true?

3. How do you write an expression of repeated multiplication when given an exponential expression?

2. What is the exponent in an exponential expression?

1. What is the base in an exponential expression?

Reflect

Explore 4

Equivalent Numerical Expressions

Order of Operations

Name: _______________________ Date: ___________

Expression

Order Card Letter Workspace

Equivalent Numerical Expressions | 133

Solution

Record the expression from each Which Order Is Correct? card. Solve to determine which order card shows an order of 225 plates.

Part I

Explore 5

Equivalent Numerical Expressions

134 | Equivalent Numerical Expressions

3. What is the correct order of operations now that we have added exponents to our operations?

2. How are the solutions for the expressions on card A and card B different?

1. Which order card shows the correct expression for 225 plates? Explain your strategy for solving.

Reflect

Explore 5

Equivalent Numerical Expressions

Show your work.

Catering Order Card Expression

Catering Order Card Number

Show your work.

Catering Order Card Expression

Equivalent Numerical Expressions | 135

Catering Order Card Number

In the space provided, solve each of the Catering Order Card expressions. Then, match the expression with its correct Catering Order Card number.

Part II

Explore 5

Equivalent Numerical Expressions

136 | Equivalent Numerical Expressions

Show your work.

Catering Order Card Expression

Explore 5 Catering Order Card Number

Show your work.

Catering Order Card Number

Catering Order Card Expression

Equivalent Numerical Expressions

Show your work.

Catering Order Card Expression

Explore 5 Catering Order Card Number

Show your work.

Catering Order Card Expression

Equivalent Numerical Expressions | 137

Catering Order Card Number

Equivalent Numerical Expressions

138 | Equivalent Numerical Expressions

2. In the expression 23 · 15 – (8 + 14), what would be the ﬁrst step when solving? What would be the last step when solving?

1. Why does order matter when solving expressions?

Reflect

Explore 5

Equivalent Numerical Expressions

Algebraic Expressions

139

Algebraic Expressions

Explore 1

Name: _______________________ Date: ___________

Write Expressions

Part I The announcer at Melbourne’s Kangaroo Jump Fundraiser announces the distance each kangaroo jumps. The distances are announced in relation to the average jump length for a kangaroo, x, which is represented by the bar below. Use your math operation terms to write an algebraic expression for each kangaroo. Use the model provided to guide your thinking.

Average Jump Length, x

1. “Karly’s jump was the average length, x, plus 4.”

2. “Kaden’s jump was the average length, x, decreased by 1.”

x–1

x 3. “Kuta’s jump was 3 times the average length, x.”

4. “Kin’s jump was the average length, x, divided by 3.”

Algebraic Expressions | 141

Algebraic Expressions

Explore 1 Part II

Read each statement from the Statement Cards. Write an algebraic expression to represent the length of each kangaroo’s jump. Record your expressions in the table provided. Sketch the model for each of the jump lengths.

Kangaroo

Model

Jump Length

Katia

Krishna

Kam

Kawa

Kellah

Kris

142 | Algebraic Expressions

Algebraic Expressions

Explore 1 Kangaroo

Model

Jump Length

Kang

Koﬁ

Kimba

Kai

1. Joni wrote Kellah’s distance as 9 + x. Jasmyn wrote Kellah’s distance as x + 9. Which student is correct? Explain.

2. Sarah wrote Kawa’s distance as (x x + 2) + (x + 2). Is Sarah’s expression equivalent to yours? Why or why not?

Algebraic Expressions | 143

Algebraic Expressions

Explore 1 Reflect

1. Write at least three different verbal statements that represent the expression x – 4.

2. Sai and Annika wrote expressions to represent the following phrase: the quotient of the average length, x, and 5. Sai: 5 x

Annika: x

Which student was correct? Explain.

144 | Algebraic Expressions

Algebraic Expressions

Explore 2

Name: _______________________ Date: ___________

Simplify Expressions Part I For each habitat, use coefﬁcients and variables to create an expression to represent the zookeeper’s count. Use a different color of linking cubes for each animal, and model the terms of the zookeeper’s count. Combine like terms, and write a simpliﬁed expression. Finally, write the number of each animal that was visible to guests at the end of the 30 minutes for the zookeeper’s report. Expression

Simpliﬁed Expression

Animal Counts

D. The zookeeper tried his hand at simplifying an expression for habitat E, which contained cockatoos, c, kookaburras, k, and magpies, m. He read his notes for the animals that were visible from the viewing deck and wrote the expression below. He decided to put matching symbols around like terms to make them easier to combine. Continue the zookeeper’s strategy by underlining terms for cockatoos and circling terms for magpies. Then, combine like terms to simplify the expression, and write the animal count.

Algebraic Expressions | 145

Algebraic Expressions

Explore 2 Part III

The side lengths are labeled on the enclosures pictured below. Using the diagrams for each animal enclosure, write an expression representing the perimeter of each enclosure. Then, give the simplest form of each expression by combining like terms.

g + 2h g+h

• Shape – trapezoid

g+h

Kookaburra Enclosure

9g + h

a. Expression representing perimeter:

b. Perimeter in simplest form:

5x + y

• Shape – parallelogram

3y + z

Platypus Enclosure

a. Expression representing perimeter:

b. Perimeter in simplest form:

146 | Algebraic Expressions

Algebraic Expressions

Explore 2 3b + 3 c

• Shape – triangle

5a + b

Crocodile Enclosure

4 7a +

a. Expression representing perimeter:

b. Perimeter in simplest form:

Reflect 1. Explain how to simplify expressions. Use the words coefﬁcient, variable, and term in your explanation.

2. Does it matter which terms you combine ﬁrst when simplifying? Why or why not?

Algebraic Expressions | 147

Algebraic Expressions

Explore 3

Name: _______________________ Date: ___________

Simplify Using Properties Part I Create an expression representing the area of each rectangular feeding tray using the length and width dimensions. Label the area model to create an equivalent expression for each tray. Adolescent Parrot Feeding Tray

Length = 5 Width = (x x + 4)

Expression representing area:

Equivalent expression representing area:

Adult Parrot Feeding Tray

Length = 12 Width = (x x + 7)

Expression representing area:

Equivalent expression representing area:

Algebraic Expressions | 149

Algebraic Expressions

Explore 3 Part II

Help the parrot sanctuary team sort through the algebraic expressions on the Expression Cards to identify which expressions are equivalent. Match a lettered expression with a numbered expression. Draw an area model to simplify, using the distributive property when applicable to prove your thinking. Area Model Workspace

Recorded Distance A.

Equivalent Expression 1.

x 3(x x + 3)

3 9

3x x+9

150 | Algebraic Expressions

Algebraic Expressions

Explore 3 Recorded Distance

Area Model Workspace

Equivalent Expression

Algebraic Expressions | 151

Algebraic Expressions

Explore 3 Part III

Canopies are needed to contain the parrots in their appropriate zones. Using the given formulas and values, evaluate the areas of the canopies needed. 1. Square canopy for zone 1: A = s2 What is the area of the square canopy if s = 13 meters?

2. Rectangular canopy for zone 2: A = lw What is the area of the rectangular canopy if l = 13 meters and w = 8 meters?

3. Rectangular canopy for zone 3: A = lw What is the area of the rectangular canopy if l = 12.5 meters and w = 9 meters?

152 | Algebraic Expressions

Explore 3

Algebraic Expressions

Reflect 1. Does order matter when simplifying expressions?

2. What does the term distribute mean in relation to math?

3. What do mathematicians mean when they say to simplify?

4. Another ﬂight distance was recorded as 3x x + 3(x + 2). Margot decided the equivalent expression would be 6x x + 6. Dani decided the equivalent expression would be 6(x x + 1). Which student was correct? Justify your answer.

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Algebraic Expressions

Explore 4

Name: _______________________ Date: ___________

Evaluate Expressions Part I Use the Outback Snack Shack Menu to evaluate each expression to determine the total each group will spend. If a group’s order is over the $40 limit, suggest an item or items to remove from the order to get the total down to $40. Group 1 Expression representing the order: 3s + g + 2b + r + 3

Group 1

Total cost of order: Item(s) to be removed from the order (if needed): Group 2 Expression representing the order: 3(h + ff) + 3b + 3

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Algebraic Expressions

Explore 4 Group 3 Expression representing the order: 2(g + a) + 2h + 2d d+3

Total cost of order: Item(s) to be removed from the order (if needed): Group 4 Expression representing the order: h + 3(s + b) + 2(g + r) + 3

Total cost of order: Item(s) to be removed from the order (if needed): Group 5 Expression representing the order: 2(rr + s + g + f + a + m) + 3

Total cost of order: Item(s) to be removed from the order (if needed): 156 | Algebraic Expressions

Explore 4

Algebraic Expressions

Part II Groups 7 and 8 lost their lunch orders, but they know how much they spent in relation to group 6. Use the verbal descriptions to write algebraic expressions to represent the total spent by each group. Then, ﬁnd the total of all three groups.

Group 6 spent d dollars. Group 7 spent $3 more than group 6. Group 8 spent twice as much as group 6. Group 6: d Group 7: Group 8: Total for groups 6, 7, and 8 in simplest form:

Reflect 1. In Part I, why was there no variable on the last term, 3, in each expression?

2. The order for group 1, 3s + g + 2b + r + 3, represented 3 salads, 1 grilled cheese, 2 bottles of water, 1 root beer, and a $3 tip. Create an order that would match the expression 4(h + f + d).

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Equations and Inequalities

159

Equations and Inequalities

Explore 1

Name: _______________________ Date: ___________

Add and Subtract Equations Read the information on each Amusement Park Card. Write an equation and draw two models to represent the information on each card.

Identify your variable. Total amount of money family brought for snacks: Monday Equation:

Algebra tiles model:

Tape diagram model:

Total amount of money:

Equations and Inequalities | 161

Equations and Inequalities

Explore 1 Identify your variable. Price per popcorn bucket: Tuesday Equation:

Algebra tiles model:

Tape diagram model:

Cost per family bucket of popcorn:

162 | Equations and Inequalities

Equations and Inequalities

Explore 1 Identify your variable. Price per popcorn bucket: Wednesday Equation: Model:

Cost per family bucket of popcorn: Identify your variable. Price per popcorn bucket: Saturday Equation: Model:

Equations and Inequalities | 163

Equations and Inequalities

Explore 1 Identify your variable. Total amount of money family brought for snacks: Friday Equation: Model:

Total amount of money: Identify your variable. Price per popcorn bucket: Sunday Equation: Model:

Cost per family bucket of popcorn: 164 | Equations and Inequalities

Explore 1

Equations and Inequalities

Reflect 1. How can you ﬁnd the value of the variable when using a model?

2. What operations are you doing to ﬁnd the value of the variable in the equation for snacks?

3. When solving equations, what is the relationship between addition and subtraction problems?

Equations and Inequalities | 165

Equations and Inequalities

Explore 2

Name: _______________________ Date: ___________

Multiply and Divide Equations Read the ticket information on each Amusement Park Card. Write an equation and draw two models to represent the information on each card. Solve to determine the price of one ticket on each day.

Identify your variable. Price per ticket: _______________ Monday Equation: Algebra tiles model:

Tape diagram model:

Price per ticket: _______________

Equations and Inequalities | 167

Equations and Inequalities

Explore 2 Identify your variable. Total cost: _______________ Tuesday Equation: Algebra tiles model:

Tape diagram model:

Total cost: _______________

168 | Equations and Inequalities

Equations and Inequalities

Explore 2 Identify your variable. Price per ticket: _______________ Wednesday Equation: Model:

Price per ticket: _______________ Identify your variable. Total cost: _______________ Thursday Equation: Model:

Equations and Inequalities | 169

Equations and Inequalities

Explore 2 Identify your variable. Price per ticket: _______________ Saturday Equation: Model:

Price per ticket: _______________ Identify your variable. Price per ticket: _______________ Sunday Equation: Model:

Price per ticket: _______________ 170 | Equations and Inequalities

Explore 2

Equations and Inequalities

Reflect 1. How can you ﬁnd the value of the variable when using models?

2. What operation are you using to ﬁnd the value of the variable in the equation for tickets?

3. What operation are you using to ﬁnd the value of the variable in the equation for total cost?

4. What relationship do you notice with multiplication and division equations?

5. What is a coefﬁcient?

6. What is a variable?

Equations and Inequalities | 171

Equations and Inequalities

Explore 3

Name: _______________________ Date: ___________

Write and Solve Equations For each scenario, write an equation and solve for the missing information. Identify your variables. Parking price per hour: _____________ Price per pizza: _____________ Tuesday’s Deals Equation:

Show your work.

Parking

Parking price per hour: _____________ Equation:

Show your work.

Pizza

Equations and Inequalities | 173

Equations and Inequalities

Explore 3 Wednesday’s Deals Equation:

Show your work.

Parking

Parking price per hour: _____________ Equation:

Show your work.

Pizza

Price per pizza: _____________ 174 | Equations and Inequalities

Equations and Inequalities

Explore 3 Thursday’s Deals Equation:

Show your work.

Parking

Parking price per hour: _____________ Equation:

Show your work.

Pizza

Equations and Inequalities | 175

Equations and Inequalities

Explore 3 Friday’s Deals Equation:

Show your work.

Parking

Parking price per hour: _____________ Equation:

Show your work.

Pizza

Price per pizza: _____________ 176 | Equations and Inequalities

Explore 3

Equations and Inequalities

Reflect 1. Which day had the best deal for parking?

2. Which day had the best deal for pizza?

3. How do you divide a fraction by another fraction?

4. If the equation is using addition, why do you need to use subtraction in order to solve?

Equations and Inequalities | 177

Equations and Inequalities

Explore 4

Name: _______________________ Date: ___________

Write and Model Inequalities Write a symbol for each variable below. •

Circular rides: _____

•

Roller-coaster rides: _____

•

Water rides: _____

•

Food tickets: _____

•

Total tickets: _____

Read each Ticket Scenario Card. Complete the sentence frame by using the phrases less than, greater than, less than or equal to, or greater than or equal to. Write an inequality to describe each scenario, and graph the solution on a number line. Use your model to answer the question. Card 1 The number of tickets is _________________________ the number ______. Inequality:

Model:

Is it possible that Emily has used 20 tickets for circular rides? Explain your answer.

Equations and Inequalities | 179

Equations and Inequalities

Explore 4 Card 2

The number of tickets is _________________________ the number ______. Inequality:

Model:

Is it possible that Emily has used 40 tickets for roller-coaster rides? Explain your answer.

Card 3 The number of tickets is _________________________ the number ______. Inequality:

Model:

Is it possible that Emily has used 35 tickets for water rides? Explain your answer.

180 | Equations and Inequalities

Equations and Inequalities

Explore 4 Card 4

The number of tickets is _________________________ the number ______. Inequality:

Model:

Can Emily spend 17 tickets on food items? Explain your answer.

Card 5 The number of tickets is _________________________ the number ______. Inequality:

Model:

Is it possible that Emily used 42 tickets today? Explain your answer.

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Equations and Inequalities

Explore 4 Card 6

The number of tickets is _________________________ the number ______. Inequality:

Model:

Is it possible that Sam has used 10 tickets for circular rides? Explain your answer.

Card 7 The number of tickets is _________________________ the number ______. Inequality:

Model:

Could Sam have used 52 tickets for roller-coaster rides? Explain your answer.

182 | Equations and Inequalities

Equations and Inequalities

Explore 4 Card 8

The number of tickets is _________________________ the number ______. Inequality:

Model:

Is it possible that Sam has used 40 tickets for water rides? Explain your answer.

Card 9 The number of tickets is _________________________ the number ______. Inequality:

Model:

Could Sam have used 25 tickets for food? Explain your answer.

Equations and Inequalities | 183

Equations and Inequalities

Explore 4 Card 10

The number of tickets is _________________________ the number ______. Inequality:

Model:

Is it possible that Sam purchased only 55 tickets today? Explain your answer.

Reflect 1. When do you use an open circle to model an inequality, and when do you use a closed circle to model an inequality?

2. How are inequalities different from equations?

3. What inequality did you write to represent Emily’s food tickets in scenario 4?

4. Why are both f ≥ 2 and 2 ≤ f acceptable for writing the inequality for Emily’s food tickets?

184 | Equations and Inequalities

Ratios, Rates, and Unit Rates

185

Ratios

Name: _______________________ Date: ___________

Draw a model of blackberries to raspberries.

Write the ratio two ways.

Draw a model of strawberries to blueberries.

Write the ratio two ways.

Ratios, Rates, and Unit Rates | 187

raspberries.

blueberries.

There are _____ blackberries for every _____

For every _____ strawberries, there are _____

_____-to-_____

_____ raspberries _____ blackberries

_____ strawberries _____ blueberries

_____ : _____

Week 2

Week 1

Look at the Fruit Stand Card for each week. Record how many of each type of fruit will be taken to the farmers’ market. Draw a tape diagram to represent the ratio between the types of fruit. Write the ratio two ways.

Explore 1

Ratios, Rates, and Unit Rates

blueberries.

There are _____ strawberries for every _____

_____-to-_____

Write the ratio two ways.

188 | Ratios, Rates, and Unit Rates

1. Are the ratios for weeks 1–3 comparing part-to-part relationships or part-to-whole relationships? Explain how you know.

Draw a model of strawberries to blueberries.

blackberries.

For every _____ blueberries, there are _____

_____ : _____

Write the ratio two ways.

_____ strawberries _____ blueberries _____ blackberries

Week 3

Draw a model of blueberries to blackberries.

Explore 1

Ratios, Rates, and Unit Rates

Week 4

Draw a model of total fruit to blackberries.

Draw a model of raspberries to strawberries.

blackberries.

Ratios, Rates, and Unit Rates | 189

For every _____ pieces of fruit, _____ will be

_____-to-_____

Write the ratio two ways.

strawberries.

For every _____ raspberries, there are _____

_____ : _____

Write the ratio two ways.

_____ strawberries _____ raspberries _____ blackberries _____ total fruit

Explore 1

Ratios, Rates, and Unit Rates

Week 5

190 | Ratios, Rates, and Unit Rates

Draw a model of raspberries to total fruit.

Draw a model of blueberries to total fruit.

Draw a model of blackberries to strawberries.

_____ out of _____ pieces of fruit are raspberries.

_____ : _____

Write the ratio two ways.

fruit.

There are _____ blueberries out of _____ pieces of

_____-to-_____

Write the ratio two ways.

strawberries.

There are _____ blackberries for every _____

_____ : _____

Write the ratio two ways.

_____ strawberries _____ raspberries _____ blackberries _____ blueberries _____ total fruit

Explore 1

Ratios, Rates, and Unit Rates

5. What is the difference between a part-to-part ratio and a part-to-whole ratio?

Ratios, Rates, and Unit Rates | 191

4. If you were given a total of two fruits and the exact number of one fruit, how could you determine the ratio between the two different fruits?

3. How did you determine what numbers to use in each ratio?

2. List two different ways to represent a ratio.

1. In your own words, deﬁne the word ratio.

Reflect

Explore 1

Ratios, Rates, and Unit Rates

Explore 2

Name: _______________________ Date: ___________

Ratio Tables and Graphs Part I Fruit bushes are sold in packs at the store. Use the Purchasing Fruit Bushes Cards – Part I to determine how many of each type of fruit bush come in one pack. Then, answer the questions, and record your solutions in the tables. Blueberry Bushes

Strawberry Bushes

1. Write the number of blueberry bushes and the number of strawberry bushes that come in one pack in row 1 of the table.

2. What is the ratio of blueberry bushes in one pack to strawberry bushes in one pack? _____ : _____ 3. Trixie decides that 4 blueberry bushes are not enough, so she purchases 8 blueberry bushes. How many total strawberry bushes will she buy if she keeps the same ratio? Record these numbers in row 2 of the table.

4. If Trixie wants to purchase 36 strawberry bushes, how many blueberry bushes would she purchase? Record these numbers in row 3 of the table.

5. After reviewing her sales of blueberries in the past year, Trixie decides she needs to buy 40 blueberry bushes. How many strawberry bushes would she purchase to keep the same ratio? Record these numbers in row 4 of the table. © Accelerate Learning Inc. – All Rights Reserved

Ratios, Rates, and Unit Rates | 193

Ratios, Rates, and Unit Rates

Explore 2 Raspberry Bushes

Blackberry Bushes

12 24 36 32 6. Write the number of raspberry bushes and the number of blackberry bushes that come in one pack in row 1 of the table.

7. What is the ratio of raspberry bushes in one pack to blackberry bushes in one pack? _____ : _____ 8. Trixie will buy 12 blackberry bushes, which is triple the number of blackberry bushes in one pack. To keep her ratio of raspberry bushes to blackberry bushes consistent, she must buy triple the raspberry bushes, too. How many raspberry bushes will Trixie buy? Record this number in row 2 of the table.

9. When purchasing 32 blackberry bushes, Trixie will need to purchase how many raspberry bushes? Record this number in row 5 of the table.

10. Use the information in your table to record the missing number of bushes in rows 3 and 4.

Reflect 1. Could Trixie buy 30 raspberry bushes and 18 blackberry bushes while keeping the same ratio? Justify your answer.

194 | Ratios, Rates, and Unit Rates

Ratios, Rates, and Unit Rates

Explore 2 Part II

Use the Purchasing Fruit Bushes Cards – Part II to determine the ratio for raspberry bushes to blueberry bushes and the ratio for strawberry bushes to blackberry bushes that Trixie has in her garden. Then, use these ratios to complete the ratio tables for each comparison, and answer the questions that follow. Raspberry Blueberry

27 20

Strawberry Blackberry

24 12

1. Trixie wants to have the same number of raspberry bushes as she has blackberry bushes. Using the ratio tables, determine how many of each type of bush Trixie will need so that there are equal numbers of raspberry bushes and blackberry bushes.

2. Trixie knows she needs more strawberry bushes and blackberry bushes. She wants to graph the equivalent ratios for strawberry bushes to blackberry bushes to determine how many new bushes she can plant in the open area of her garden. Write each equivalent ratio from the strawberry to blackberry table as an ordered pair.

Ratios, Rates, and Unit Rates | 195

Explore 2

Ratios, Rates, and Unit Rates

3. Graph the ratio between strawberry bushes and blackberry bushes.

4. Trixie wants to buy half of the number of strawberry bushes and blackberry bushes she already has planted in her garden. How many strawberry bushes and how many blackberry bushes will Trixie need to buy?

5. If the ratio must stay the same, is it possible for Trixie to buy 10 strawberry bushes? Use the graph to justify your answer.

6. Can Trixie buy 8 blackberry bushes? Use the graph to justify your answer.

196 | Ratios, Rates, and Unit Rates

Explore 2

Ratios, Rates, and Unit Rates

Reflect 1. If you increase the number of one type of fruit bush, what must you do to the other number in order to keep the ratio equivalent?

2. If you were given a ratio of 3 strawberries for every 5 blueberries, how can you determine the number of blueberries if the number of strawberries increased to 12?

3. How can you determine from a table whether two numbers you are looking for are an equivalent ratio to what you have?

4. How can you determine from a graph if a pair of numbers you are looking for is part of the equivalent ratio?

Ratios, Rates, and Unit Rates | 197

Ratios, Rates, and Unit Rates

Explore 3

Name: _______________________ Date: ___________

Rates and Unit Rates Part I Using the information in the table below, write the rate at which Trixie picks each basket of berries.

Type of Berry

Number of Baskets Picked

Time Spent Picking Baskets

Strawberry

Blueberry

Raspberry

Blackberry

Rate of Baskets Picked (minutes per basket)

Create a double number line to determine how many minutes it took Trixie to pick one basket of strawberries. Write your unit rate solution. Strawberry

_______ minutes per basket

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Ratios, Rates, and Unit Rates

Explore 3

Draw a ratio table to determine how many minutes it took Trixie to pick one basket of blueberries. Write your unit rate solution. Blueberry

_______ minutes per basket

Draw a tape diagram to determine how many minutes it took Trixie to pick one basket of raspberries. Write your unit rate solution. Raspberry

_______ minutes per basket

200 | Ratios, Rates, and Unit Rates

Ratios, Rates, and Unit Rates

Explore 3

Draw a double number line to determine how many minutes it took Trixie to pick one basket of blackberries. Write your unit rate solution. Blackberry

_______ minutes per basket

Reflect The following rates are unit rates. 8 minutes per basket

12 berries per bush

4 dollars per basket

1. Write your own deﬁnition of the term unit rate.

2. What strategies can you use to ﬁnd unit rate?

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Ratios, Rates, and Unit Rates

Explore 3 Part II

Read each scenario. Use a ratio table, tape diagram, or double number line to solve for the unit rate of the money Trixie earned for selling each basket of berries. Use the unit rate to solve for the additional rates. Trixie sold ten baskets of strawberries for $32.50. Unit rate model and solution:

_________ per basket At this rate, how much would Trixie earn if she sold 15 baskets of strawberries?

Trixie sold 18 baskets of blueberries for $67.50. Unit rate model and solution:

_________ per basket At this rate, how much would Trixie earn if she sold 26 baskets of blueberries?

202 | Ratios, Rates, and Unit Rates

Ratios, Rates, and Unit Rates

Explore 3

Trixie sold 12 baskets of raspberries for $30.00 and 22 baskets of blackberries for $75.68. Unit rate model and solution for raspberries:

_________ per basket Unit rate model and solution for blackberries:

_________ per basket

Next weekend, for raspberries and blackberries, she only wants to sell the fruit that gives her the highest selling price per basket. Which fruit should she sell next weekend? Explain.

Ratios, Rates, and Unit Rates | 203

Ratios, Rates, and Unit Rates

Explore 3 Reflect 1. What is the difference between a rate and a unit rate?

2. How can you use a rate to determine the unit rate?

3. Trixie wanted to ﬁnd the highest unit rate for determining which fruit would sell for the most money per basket. What fruit has the highest unit rate?

204 | Ratios, Rates, and Unit Rates

Ratios, Rates, and Unit Rates

Explore 4

Name: _______________________ Date: ___________

Comparing Ratios and Using Rates to Make Predictions Part I: Comparing Ratios Include the missing information in the tables to represent the ratios of blueberries to strawberries. Answer the questions in the table, and compare the ratios for the berry bags using the signs <, >, or =. Berry Bag 1 Blueberries

Berry Bag 2

Strawberries

Blueberries

Strawberries

15 Berry Bag 1

Berry Bag 2

For every ____ blueberries, there are

____ strawberries.

What is the ratio of blueberries to strawberries in berry bag 1?

What is the ratio of blueberries to strawberries in berry bag 2?

Compare the ratios. Berry bag 1

Berry bag 2

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Ratios, Rates, and Unit Rates

Explore 4 Berry Bag 3 Raspberries

Berry Bag 4

Strawberries

Raspberries

Strawberries

Berry Bag 3

Berry Bag 4

For every____ raspberries, there are

____ strawberries.

What is the ratio of raspberries to strawberries in berry bag 3?

What is the ratio of raspberries to strawberries in berry bag 4?

Compare the ratios. Berry bag 3

206 | Ratios, Rates, and Unit Rates

Berry bag 4

Ratios, Rates, and Unit Rates

Explore 4 Trixie’s and Aanya’s Berry Bags

Trixie is preparing a berry bag that calls for 4 cups of blueberries and 6 cups of raspberries. Aanya is helping Trixie prepare berry bags and uses 6 cups of blueberries and 9 cups of raspberries. 1. Find the ratio of cups of blueberries and cups of raspberries in Trixie’s berry bag.

2. Find the ratio of cups of blueberries and cups of raspberries in Aanya’s berry bag.

3. Did Aanya use the correct ratio of blueberries to raspberries? Explain.

Ratios, Rates, and Unit Rates | 207

Explore 4

Ratios, Rates, and Unit Rates

Reflect 1. How can you check to see if the ratio 2 : 3 is equivalent to another ratio?

2. How did you use equivalent ratios to compare ratios?

3. If an extra cup of raspberries and an extra cup of strawberries were added to berry bag 4, what would the ratio of raspberries to strawberries be? Compare berry bag 3 to berry bag 4.

208 | Ratios, Rates, and Unit Rates

Ratios, Rates, and Unit Rates

Explore 4 Part II: Making Predictions

Use the scenarios on the Berry Farms Cards to represent the rate on a double number line. Use the double number line to make predictions, and answer the questions in the table. Berry Farms Card 1 Represent the rate on a double number line to make predictions.

1. How many miles does Trixie drive in 1 hour?

2. At this rate, can Trixie drive more than 450 miles in 12 hours? Explain.

3. How would Trixie’s total distance change if she drove for 12 hours at an increased rate of speed?

4. In this situation, which unit of measure represents the independent variable? Which unit of measure represents the dependent variable? Explain.

Ratios, Rates, and Unit Rates | 209

Ratios, Rates, and Unit Rates

Explore 4 Berry Farms Card 2

Represent the rate on a double number line to make predictions.

1. How long does it take Aanya to travel 1 mile?

2. At this rate, can Aanya travel 6 miles in one half of an hour? Explain.

3. How would Aanya’s rate change if she were riding at a decreasing rate instead of a steady rate?

4. In this situation, which unit of measure represents the independent variable? Which unit of measure represents the dependent variable? How do you know?

210 | Ratios, Rates, and Unit Rates

Explore 4

Ratios, Rates, and Unit Rates

Reflect 1. How did you use the double number line to make predictions about the rates?

2. Give an example of why someone would want to make predictions about rates.

3. In ten minutes, Trixie can pack 2 berry bags. How many berry bags can she pack in 70 minutes? Use a double number line to ﬁnd the answer.

Ratios, Rates, and Unit Rates | 211

Ratios, Rates, and Unit Rates

Explore 5

Name: _______________________ Date: ___________

Solving Proportions FRESH PRODUCE

Part I: Solving Proportions Using Ratios Use the scenarios on the Farmers’ Market Cards to represent and solve the proportions. Farmers’ Market Card 1

Workspace:

Solution:

Farmers’ Market Card 2 Workspace:

Ratios, Rates, and Unit Rates | 213

Ratios, Rates, and Unit Rates

Explore 5 Farmers’ Market Card 3 Workspace:

Solution:

Farmers’ Market Card 4 Workspace:

Solution:

214 | Ratios, Rates, and Unit Rates

Ratios, Rates, and Unit Rates

Explore 5 Part II: Solving Proportions Using Unit Rates

Use the scenarios on the Berry Festival Cards to determine the cost of each individual berry in a pack using proportions. Show your work in each table below, and write a solution statement to answer the question on each Berry Festival Card. Strawberries Workspace:

Solution statement:

Ratios, Rates, and Unit Rates | 215

Ratios, Rates, and Unit Rates

Explore 5 Raspberries Workspace:

Solution statement:

Blueberries Workspace:

Solution statement:

216 | Ratios, Rates, and Unit Rates

Ratios, Rates, and Unit Rates

Explore 5 Blackberries Workspace:

Solution statement:

Reflect 1. How are proportions helpful in ﬁnding an equivalent rate or ratio?

2. What strategies were used to solve the proportions?

Ratios, Rates, and Unit Rates | 217

Percents

219

Percents

Explore 1

Name: _______________________ Date: ___________

Represent Percents Using a Hundreds Grid

Part I: Understanding Percents Using a Hundreds Grid Read each Part I Mowing Card. Shade in each model with a colored pencil to represent the lawns that were mowed by each lawn-care company. Record each percent as a partto-whole ratio in fraction form and decimal form. Scott’s Mowing Service Model

Number of Units Shaded in Model

Fraction

Decimal

What percentage of lawns were mowed by this lawn-care company?

Percents | 221

Percents

Explore 1 Green Lawn Care Model

Number of Units Shaded in Model

Fraction

Decimal

What percentage of lawns were mowed by this lawn-care company?

Best Grass-Mowing Company Model

Number of Units Shaded in Model

Fraction

Decimal

What percentage of lawns were fertilized by this lawn-care company?

222 | Percents

Percents

Explore 1 Sunny Lawns and Gardens Company Model

Number of Units Shaded in Model

Fraction

Decimal

What percentage of lawns were mowed by this lawn-care company?

Percents | 223

Percents

Explore 1 Part II: Using a Hundreds Grid to Represent Percents

Read each Part II Mowing Card, and create a model using different colors or designs to shade the hundreds grid to represent the percentage of lawns mowed in each neighborhood for each week. Use that information to shade the model and complete the tables. Week 1 Neighborhood

Percent

Fraction

Number of Lawns Mowed

Fraction

Number of Lawns Mowed

Hawk’s Landing

Fire Ranch

Sundown

Week 2 Neighborhood

Percent

Mason Terrace

Dayspring

Sunrise

224 | Percents

Percents

Explore 1 Week 3 Neighborhood

Percent

Fraction

Number of Lawns Mowed

Fraction

Number of Lawns Mowed

Old Road

Oak Falls

Granger

Week 4 Neighborhood

Percent

Nest Lake

Waterstone

Hawthorne

Percents | 225

Percents

Explore 1 Reflect 1. How do you represent a fractional percent between 0% and 1%?

2. How do models help in understanding percents?

3. How are percents, fractions, and decimals related to one another?

4. How did the value of the whole change when using a 10-by-10 grid to represent percents, decimals, and fractions?

226 | Percents

Percents

Explore 2

Name: _______________________ Date: ___________

Solving Percent Problems Using Benchmark Fractions and Percents

Part I: Benchmark Fractions and Percents Using Models Use the shaded area of the tape diagram to determine the fraction and the percent that are represented in the tape diagram. Fairﬁeld Castle

Fraction:

Percent:

Chateau Providence

Fraction:

Percent:

Percents | 227

Percents

Explore 2 Bailey Manor

Fraction:

Percent:

Sterling Estate

Fraction:

228 | Percents

Percent:

Percents

Explore 2 Part II: Solving Percent Problems Using Models

Read each Landscaping Mansions Card. Draw and label a tape diagram or double number line that represents how the benchmark percent is used to calculate the amount. Spycastle Manor

Calculate the number of acres the Mowing Mansions Lawn-Care Company will mow.

Northwind Estate

Calculate the number of trees the Mowing Mansions Lawn-Care Company will remove.

Percents | 229

Percents

Explore 2 Greenbriar Mansion

Calculate the number of acres the Mowing Mansions Lawn-Care Company will mow.

Chateau Fieldstone

Calculate the number of acres the Mowing Mansions Lawn-Care Company will mow.

230 | Percents

Percents

Explore 2 Moore Mansion

Calculate the number of shrubs the Mowing Mansions Lawn-Care Company will replace.

Hardows Residence

Calculate the number of bushes the Mowing Mansions Lawn-Care Company will remove.

Percents | 231

Percents

Explore 2 Reflect 1. How do benchmark percents help with calculating percents?

2. Why is 10% a useful benchmark percent to use when solving percent problems?

3. How do you ﬁnd the amount for percents that are greater than 100%?

232 | Percents

Percents

Explore 3

Name: _______________________ Date: ___________

Finding the Price and Discount s a l e

sale

Use the Sale Cards to record the given information for each sale. Create a model to ﬁnd the original price, discount, or sale price of each item. After solving for the original price, discount, or sale price, complete the solution statement. Sale 1 Item for sale:

Model to solve for the original price:

Discount:

Sale price:

Solution statement:

Percents | 233

Percents

Explore 3 Sale 2 Item for sale:

Model to solve for the sale price:

Discount:

Original price:

Solution statement:

Sale 3 Item for sale:

Model to solve for the discount:

Sale price:

Original price:

Solution statement:

234 | Percents

Percents

Explore 3 Sale 4 Item for sale:

Model to solve for the original price:

Discount:

Sale price:

Solution statement:

Sale 5 Item for sale:

Model to solve for the sale price:

Discount:

Original price:

Solution statement:

Percents | 235

Percents

Explore 3 Sale 6 Item for sale:

Model to solve for the original price:

Discount:

Sale price:

Solution statement:

Reflect 1. How are models useful when solving percent problems?

2. Why is it important to know multiple strategies to solve various percent problems?

236 | Percents

Measurement Conversions

237

Measurement Conversions

Explore 1

Name: _______________________ Date: ___________

One-Step Measurement Conversions Read each Petting Zoo Card. Record the given information from each scenario in the conversion table. Solve by creating a model using ratio reasoning to convert each given measurement into an equivalent measurement. Petting Zoo Update 1: New Fence Given Measurement

Equivalent Measurement

Conversion 3 feet = 1 yard

Ratio table model: Feet

Yards

Petting Zoo Update 2: Sheep Pen Given Measurement

Conversion

Equivalent Measurement

1 meter = 100 centimeters Double number line model:

Meters Centimeters

Measurement Conversions | 239

Measurement Conversions

Explore 1 Petting Zoo Update 3: Goat Run Given Measurement

Equivalent Measurement

Conversion 1 km = 1,000 m

Tape diagram model:

Petting Zoo Update 4: Chicken Feed Given Measurement

Conversion

Equivalent Measurement

1 ton = 2,000 pounds Proportions model:

240 | Measurement Conversions

Measurement Conversions

Explore 1 Petting Zoo Update 5: Water Barrel Given Measurement

Equivalent Measurement

Conversion 1 L = 1,000 milliliters

Tape diagram model:

Petting Zoo Update 6: Goat Bottles Given Measurement

Equivalent Measurement

Conversion 8 ﬂuid ounces = 1 cup

Ratio table model:

Fluid Ounces

Cup

Measurement Conversions | 241

Measurement Conversions

Explore 1

Petting Zoo Update 7: Chicken Coop Platforms Given Measurement

Equivalent Measurement

Conversion 1 pound = 16 ounces

Double number line model:

Pounds Ounces

Petting Zoo Update 8: Duck Feed Barrels Given Measurement

Equivalent Measurement

Conversion 1 kilogram = 1,000 grams

Proportions model:

242 | Measurement Conversions

Explore 1

Measurement Conversions

Reflect 1. How are conversions between measurements the same as equivalent ratios?

2. What strategies can be used to solve measurement conversion problems?

3. How can you tell if you need to multiply or divide?

Measurement Conversions | 243

Measurement Conversions

Explore 2

Name: _______________________ Date: ___________

Two-Step Measurement Conversions Read each Old Mac’s Petting Zoo Card. Record the given information from each scenario in the conversion table. Solve by creating a model using ratio reasoning to convert each given measurement into an equivalent measurement. Caring for the Animals 1: Ducks Given Measurement

Conversion

Equivalent Measurement

1,000 grams = 1 kilogram Ratio table model: Grams

Kilograms

Caring for the Animals 2: Chickens Given Measurement

Conversion

Equivalent Measurement

2 pints = 1 quart Double number line model:

Measurement Conversions | 245

Measurement Conversions

Explore 2 Caring for the Animals 3: Minipigs Given Measurement

Equivalent Measurement

Conversion 12 inches = 1 foot

Tape diagram model:

Caring for the Animals 4: Rabbits Given Measurement

Equivalent Measurement

Conversion 1 liter = 1,000 milliliters

Proportions model:

246 | Measurement Conversions

Measurement Conversions

Explore 2 Caring for the Animals 5: Horse Given Measurement

Equivalent Measurement

Conversion 1 ton = 2,000 pounds

Ratio table model:

Pounds

Tons

Caring for the Animals 6: Sheep Pen Given Measurement

Conversion

Equivalent Measurement

1 kilometer = 1,000 meters Tape diagram model:

Measurement Conversions | 247

Measurement Conversions

Explore 2 Caring for the Animals 7: Cows Given Measurement

Equivalent Measurement

Conversion 1 meter = 1,000 millimeters

Proportional model:

Caring for the Animals 8: Goats Given Measurement

Conversion

Equivalent Measurement

1 gallon = 4 quarts Double number line model:

248 | Measurement Conversions

Explore 2

Measurement Conversions

Reflect 1. How can you determine what equivalent ratio you should use to help ﬁnd the new equivalent measurement for Caring for the Animals 1: Ducks?

2. Explain how to change the supply amount from 5 inches to an equivalent amount in feet.

Measurement Conversions | 249

Measurement Conversions

Explore 3

Name: _______________________ Date: ___________

Multistep Conversions between Systems of Measurement Read each Petting Zoo Cleanup Day Card. Record the given information from each scenario in the conversion table. Solve by creating a model using ratio reasoning to convert each given measurement into an equivalent measurement. Cleaning Up the Petting Zoo 1: Fuel Given Measurement

Equivalent Measurement

Conversion 1 gallon = 3.785 liters

Model for one week:

Use the information in Cleaning Up the Petting Zoo 1 to complete the table. Week 1

Week 2

Gallons

Liters

Week 3

Week 4

22.71

Measurement Conversions | 251

Measurement Conversions

Explore 3 Cleaning Up the Petting Zoo 2: Painting Given Measurement

Conversion(s)

Equivalent Measurement

1 liter is about 2 pints. 1 pint costs $7.50. Model:

What will be the total cost of paint?

Cleaning Up the Petting Zoo 3: Ticket Booth Sign Given Measurement

Conversion(s)

Equivalent Measurement

1 centimeter is about 4 10 of an inch.

Model:

What is the length of the sign in centimeters?

252 | Measurement Conversions

Measurement Conversions

Explore 3

Cleaning Up the Petting Zoo 4: Grass Seed Given Measurement

Conversion(s)

Equivalent Measurement

1 pound is about 0.5 of a kilogram. A 1 lb. bag costs $7.50. Model:

How much will the bags of grass seed cost?

Cleaning Up the Petting Zoo 5: Pig Trough Given Measurement

Conversion(s)

Equivalent Measurement

1 yard is about 9 of a 10 meter. Model:

How many yards long will the pig trough be?

Measurement Conversions | 253

Measurement Conversions

Explore 3

Cleaning Up the Petting Zoo 6: Barn Doors Given Measurement

Equivalent Measurement

Conversion(s) 1 kilogram = 2.2 pounds

Model:

What is the total weight in kilograms of one door?

Reflect 1. How can you use multiplication and division to convert measurement units?

2. Describe how you transform the units of measure for the paint for the ticket booth.

254 | Measurement Conversions

Coordinate Planes

255

Explore 1

Coordinate Planes

Name: _______________________ Date: ___________

Number Lines and Coordinate Planes Part I: Number Lines to Quadrants The two main roads intersect at the courthouse. Plot the location of each important building using the Main Buildings Cards. Label each building.

Coordinate Planes | 257

Coordinate Planes

Explore 1 Circle the word that makes each statement true.

Traveling east from the courthouse will give you (positive / negative) numbers. Traveling west from the courthouse will give you (positive / negative) numbers. Traveling north from the courthouse will give you (positive / negative) numbers. Traveling south from the courthouse will give you (positive / negative) numbers. The school is located _____ blocks (east / west) of the courthouse and _____ blocks (north / south) of the courthouse. The ordered pair for the school will be ________.

Reflect 1. How are number lines and coordinate planes similar?

2. What is the ordered pair for the courthouse?

3. Do you think one number in the ordered pair must always be zero?

258 | Coordinate Planes

Coordinate Planes

Explore 1 Part II: Making the Map

Draw a model of the city map after being put together. Label the map with each additional building. y 12 11 10 9 8 7 6 5 4 3 2 1 –12 –11 –10 –9 –8 –7 –6 –5 –4 –3 –2 –1 0 –1

x 1

9 10 11 12

–2 –3 –4 –5 –6 –7 –8 –9 –10 –11 –12

The courthouse is at the center of the city. Locate and label the courthouse on the map.

This location on a coordinate plane is called the origin. What is the ordered pair for the courthouse?

Coordinate Planes | 259

Coordinate Planes

Explore 1 Reflect

1. When finding ordered pairs on the map, you must first move ______________ or ______________ , and then you will move ______________ or ______________. 2. The coordinate plane is divided into four quadrants. Refer to the Quadrant Cards as you reflect. a. What do you notice about all of the ordered pairs that were located in Quadrant I?

b. What do you notice about all of the ordered pairs that were located in Quadrant II?

c. What do you notice about all of the ordered pairs that were located in Quadrant III?

d. What do you notice about all of the ordered pairs that were located in Quadrant IV?

3. What is the name of the location (0, 0) on a coordinate plane?

4. A library is being built at (−6, 0). What quadrant would the library be located in?

260 | Coordinate Planes

Coordinate Planes

Explore 2

Name: _______________________ Date: ___________

Reflections on a Coordinate Plane Part I: Graphing in All Four Quadrants Plot and label each building on the city map. Mark the courthouse at the origin with a square.

Coordinate Planes | 261

Coordinate Planes

Explore 2 Reflect 1. Which quadrant has positive x values and positive y values?

2. Which quadrant has negative x values and negative y values?

3. How can the signs of ordered pairs help you determine if you correctly plotted the point on the graph?

Part II: Reflections Add the new building developments to the city map. y

262 | Coordinate Planes

Coordinate Planes

Explore 2 Reflect 1. How do signs in ordered pairs differ in reﬂections across the x-axis?

2. How do signs in ordered pairs differ in reﬂections across the y-axis?

3. What is the ordered pair for the reﬂection of (−3, 7) across the x-axis?

4. What is the ordered pair for the reﬂection of (8, 12) across the y-axis?

Coordinate Planes | 263

Coordinate Plane Problem Solving

265

Coordinate Plane Problem Solving

Explore 1

Name: _______________________ Date: ___________

Distances between Points Part I: Partner Competition The location of each partner is recorded in the table below. Plot the location of each partner on the coordinate plane, and calculate the distance between them. Team Name

Partner 1’s Coordinates

Partner 2’s Coordinates

Orange

(3, 3)

(3, 8)

Navy

(−3, 3)

(−9, 3)

Purple

(−4, −2)

(3, −2)

Green

(6, 1)

(6, −3)

Distance

1. How were you able to identify the team that won the competition?

2. How did you determine the distance between partners?

Coordinate Plane Problem Solving | 267

Explore 1

Coordinate Plane Problem Solving

Reflect 1. What do you notice about the coordinates for each pair on a team? What does this tell you about the location of the pair on the coordinate plane?

2. How are the coordinates related to the distance?

268 | Coordinate Plane Problem Solving

Coordinate Plane Problem Solving

Explore 1 Part II: Group Showcase

The locations of three members in each group are recorded in the table below. Plot the locations of the ﬁrst three members of each group on the coordinate plane below and determine the coordinates for the fourth member that would complete the rectangle. Record the coordinates in the table. Group Name

Member 1 Coordinates

Member 2 Coordinates

Member 3 Coordinates

Pink

(−3, 4)

(−3, 7)

(1, 7)

Blue

(6, −4)

(2, −4)

(2, 2)

Yellow

(−7, −6)

(−7, 5)

(−6, 5)

Member 4 Coordinates

1. What do you notice about the coordinates of the members within each group?

2. If three members of the red group were standing at (1, −8), (1, −2), and (3, −2), what would be the location of the fourth member to complete the rectangle?

Coordinate Plane Problem Solving | 269

Explore 1

Coordinate Plane Problem Solving

Reflect 1. If two ordered pairs share an x-coordinate, what does that tell you about the location of the points on the coordinate plane?

2. If two ordered pairs share a y-coordinate, what does that tell you about the location of the points on the coordinate plane?

3. How is absolute value related to ﬁnding the distance between two ordered pairs that share either an x-coordinate or a y-coordinate?

4. If you were asked to ﬁnd the fourth coordinate of a square, would the process change? Explain.

270 | Coordinate Plane Problem Solving

Explore 2

Coordinate Plane Problem Solving

Name: _______________________ Date: ___________

Polygons on a Coordinate Plane Part I: Pecan Park East Read each Pecan Park Card for Part I. Plot the coordinates for the vertices of each piece of equipment or facility area on the coordinate plane. Connect the vertices to form polygons. Then, use your completed park map to determine whether any of the equipment or facility areas overlap.

1. Do any of the pieces of equipment or facility areas overlap? If so, which items need to be adjusted?

Coordinate Plane Problem Solving | 271

Coordinate Plane Problem Solving

Explore 2 Part II: Pecan Park West

Read each Pecan Park Card for Part II. Plot the coordinates of the vertices for each facility area on the coordinate plane. Then, use your completed park map to determine the additional calculations required to complete the upgrades. Label each calculation in units.

1. A special nonslip coating must be applied to the entire ﬂoor of the basketball court to complete the upgrade. What is the area of the basketball court that will be coated?

Length: ___________

Width: ___________

Area: ___________

2. An ant-resistant border will be placed around the perimeter of the picnic area to complete the upgrade. What is the perimeter of the picnic area?

Length: ___________

272 | Coordinate Plane Problem Solving

Width: ___________

Area: ___________

Explore 2

Coordinate Plane Problem Solving

3. The sandpit requires a new wooden border around the outside and a weatherproof liner covering the entire bottom to complete the upgrade. Determine the amount of border and liner needed to complete the upgrade. Write an equation to show your solutions.

Length: _____________

Width: _____________

Border: _________________________________ Liner: __________________________________

Reflect 1. How is ﬁnding the coordinates (−1, 2) different from ﬁnding the coordinates (1, 2)?

2. How do you determine the length of a rectangle graphed on a coordinate plane?

3. How would you determine the side length of a rectangle given the vertices (−2, 4) and (4, 4)?

Coordinate Plane Problem Solving | 273

Area and Volume

275

Discovering Area Formulas

Name: _______________________ Date: ___________

Area:

Area and Volume | 277

Formula:

h = ____ units

Formula:

b = ____ units

How can you ﬁnd the area of this garden?

h = ____ units

New Garden 1

How can you ﬁnd the area of this garden?

b = ____ units

Garden 1

Describe how each figure can be decomposed and/or rearranged to find the area. Represent the decomposed and rearranged image in the grid for each new garden. Label the base and height. Write the formula, and then find the area in square units of each garden. Each grid square is 1 square unit.

Part II: Decomposing and Rearranging Figures to Discover Area Formula

Explore 1

Area and Volume

h = ____ units

278 | Area and Volume

Area:

Formula:

b = ____ units

Formula:

h = ____ units

New Garden 2

How can you ﬁnd the area of this garden?

Garden 2

How can you ﬁnd the area of this garden?

b = ____ units

Explore 1

Area and Volume

h = ____ units

Area:

Area and Volume | 279

Formula:

b = ____ units

Formula:

h = ____ units

New Garden 3

How can you ﬁnd the area of this garden?

Garden 3

How can you ﬁnd the area of this garden?

b = ____ units

Explore 1

Area and Volume

280 | Area and Volume

Formula:

Area:

How can you ﬁnd the area of this garden?

Area:

h = ____ units

b2 = ____ units

New Garden 4

h = ____ units

Garden 4

b1 = ____ units

Explore 1

Area and Volume

Area:

Area and Volume | 281

Formula:

h = ____ units

Formula:

h = ____ units

b = ____ units

How can you ﬁnd the area of this garden?

b2 = ____ units

b1 = ____ units

h = ____ units

New Garden 5

How can you ﬁnd the area of this garden?

b2 = ____ units

Garden 5

b1 = ____ units

Explore 1

Area and Volume

282 | Area and Volume

Area:

Formula:

Area:

Formula:

New Garden 6

How can you ﬁnd the area of this garden?

Garden 6

How can you ﬁnd the area of this garden?

Explore 1

Area and Volume

Area:

Area and Volume | 283

Formula:

Area:

Formula:

New Garden 7

How can you decompose or arrange the shapes of this garden?

Garden 7

How can you ﬁnd the area of this garden?

Explore 1

Area and Volume

284 | Area and Volume

4. How did you determine the area formula for the trapezoids?

3. How did you determine the area formula for the triangles?

2. Why is the area formula for a parallelogram the same as the area formula for a rectangle?

1. How do you calculate the area of a rectangle?

Reflect

Explore 1

Area and Volume

Name: _______________________ Date: ___________

Finding the Area of Quadrilaterals

Area of garden:

What strategy did you use to ﬁnd the area?

Area and Volume | 285

Janice is landscaping her garden and only wants to have blanket ﬂowers in her garden. Find the area of her garden that will include blanket ﬂowers. (Each unit represents 1 square unit.)

Janice’s Garden

Decompose and rearrange the figures to find the area. Represent the decomposed and rearranged image in the grid. Label the base and height. Find the area, and describe the strategy you used to find the area. Represent how to find the area of each garden by using its area formula. Each grid square is 1 square unit.

Part I: Determining the Area of Quadrilaterals on Grids

Explore 2

Area and Volume

Tao’s Garden

286 | Area and Volume

Area of garden:

What strategy did you use to ﬁnd the area?

fountain

Tao is landscaping her garden and would like to have daisies planted around her fountain. Find the area of her garden that will include only daisies. (Each unit represents 1 square unit.)

Explore 2

Area and Volume

Tati’s Garden

Area of garden:

What strategy did you use to ﬁnd the area?

Area and Volume | 287

Tati is landscaping her garden and would like to have daisies planted. Find the area of her garden that will include daisies. (Each unit represents 1 square unit.)

Explore 2

Area and Volume

Pablo’s Garden

288 | Area and Volume

Area of garden:

What strategy did you use to ﬁnd the area?

statue statue

Pablo is landscaping his garden and has bought daffodils to be planted throughout the garden. Find the area of his garden that will include only daffodils. (Each unit represents 1 square unit.)

Explore 2

Area and Volume

Isabella’s Garden

Area and Volume

Area of garden:

What strategy did you use to ﬁnd the area?

Area and Volume | 289

Isabella is landscaping her garden and has bought roses. Find the area of her garden that will only include roses. (Each unit represents 1 square unit.)

Explore 2

Pergola

Inz’s Garden

290 | Area and Volume

Area of garden:

What strategy did you use to ﬁnd the area?

Inz is landscaping her garden and has bought roses. Find the area of her garden that will only include roses. (Each unit represents 1 square unit.)

Explore 2

Area and Volume

Marigolds

Solve:

Grass

6 ft.

Area: _______

Height: _______

Base: _______

Area and Volume | 291

There are new marigolds that are being added to the Biltmore Gardens. The gardeners are trying to ﬁgure out the area of the garden that will include the new ﬂowers.

Biltmore Gardens

Identify the base and height of the garden. Use the formula A = bh to ﬁnd the area of the 2-D ﬁgures.

Part II: Area of Quadrilaterals

Explore 2

9.5 ft.

Tuscan Villa Gardens

292 | Area and Volume

Solve:

1 ft. 2

Area: _______

Height: _______

Base2: _______

Base1: _______

The gardeners at Tuscan Villa Gardens would like to add sunﬂowers to the gardens. They are trying to determine the area that will include sunﬂowers.

Explore 2

Area and Volume

Daisies

8.5 ft.

Solve:

6.2 ft.

Sherman Oaks Gardens

Area and Volume

3.6 ft.

Lilacs

Height: ________

Area and Volume | 293

Area of garden with lilacs: _____________

Area of garden with daisies: ____________

Base: ________

Flower: __________

Base: ________

Flower: __________

The gardeners at Sherman Oaks Gardens would like to add daisies and lilacs to the gardens. They are trying to determine the area that will include daisies and lilacs.

Explore 2

4 ft.

294 | Area and Volume

3. How does the formula for the area of a rectangle help you understand how the area of a parallelogram and square are determined?

2. What is the relationship between the decomposed ﬁgure and its newly created ﬁgures?

1. What happens to the area of a parallelogram if the height doubles but the base stays the same?

Reflect

Explore 2

Area and Volume

Explore 3

Name: _______________________ Date: ___________

Finding the Area of Triangles

Part I: Determining the Area of Triangles on Grids Find the area of the 2-D figures to determine the area of the vegetable garden. Label the base and height. Find the area of the parallelogram and triangle. Represent how to find 1 the area of the triangular garden using the formula A = 2 bh bh. Describe the strategy you used to find the area. Each grid square is 1 square unit. Lettuce Garden

Area Parallelogram:

Triangle:

How did you ﬁnd the area of the garden that will include lettuce?

Area and Volume | 295

Area and Volume

Explore 3 Tomatoes and Radishes Garden

Area Parallelogram:

Triangle:

How did you ﬁnd the area of the garden that will include tomatoes or radishes?

Zucchini Garden

Area Parallelogram:

Triangle:

How did you ﬁnd the area of the garden that will include zucchini?

296 | Area and Volume

Area and Volume

Explore 3 Part II: Using the Formula for the Area of a Triangle

Use the area formula for triangles to ﬁnd the area of the 2-D ﬁgures to determine the area of the vegetable garden. Spinach Area

19.5 feet

Garden

16 feet How did you ﬁnd the area of the garden that will include spinach?

Peppers Garden

Area

17 feet 14 feet How did you ﬁnd the area of the garden that will include peppers?

Area and Volume | 297

Area and Volume

Explore 3 Carrots

Area

11 feet

Garden

24 feet How did you ﬁnd the area of the garden that will include carrots?

Peas Area

12 feet

Garden

27 feet How did you ﬁnd the area of the garden that will include peas?

298 | Area and Volume

Explore 3

Area and Volume

Reflect 1. How is the area formula for a parallelogram connected to the area formula for a triangle?

2. How many possible bases does a triangle have?

3. In the problem about the pea garden, how can we use the area of one shaded triangle to calculate the area of the entire shaded part of the garden?

Area and Volume | 299

Finding the Area of Composite Figures

Name: _______________________ Date: ___________

Area of composite ﬁgure:

Area and Volume | 301

_____ trapezoid(s)

_____ parallelogram(s)

Solve to ﬁnd the area of the garden.

_____ trapezoid(s)

_____ parallelogram(s)

_____ triangle(s)

_____ rectangle(s)

Garden B

Solve to ﬁnd the area of the garden.

_____ triangle(s)

_____ rectangle(s)

Garden A

Look at the composite figure of the gardens on each Garden Card. Write how many of each 2-D figure are used to find the total area of each composite figure. Use the area formulas to determine the area of each customer’s garden.

Explore 4

Area and Volume

302 | Area and Volume

Area of composite ﬁgure:

_____ trapezoid(s)

Area of composite ﬁgure:

_____ parallelogram(s)

_____ triangle(s)

Solve to ﬁnd the area of the garden.

_____ trapezoid(s)

_____ parallelogram(s)

_____ rectangle(s)

Garden D

Solve to ﬁnd the area of the garden.

_____ triangle(s)

Garden C

_____ rectangle(s)

Explore 4

Area and Volume

Area of composite ﬁgure:

Area and Volume | 303

_____ trapezoid(s)

Area of composite ﬁgure:

_____ parallelogram(s)

_____ triangle(s)

Solve to ﬁnd the area of the garden.

_____ trapezoid(s)

_____ parallelogram(s)

_____ rectangle(s)

Garden F

Solve to ﬁnd the area of the garden.

_____ triangle(s)

Garden E

_____ rectangle(s)

Explore 4

Area and Volume

304 | Area and Volume

Area of composite ﬁgure:

_____ trapezoid(s)

Area of composite ﬁgure:

_____ parallelogram(s)

_____ triangle(s)

Solve to ﬁnd the area of the garden.

_____ trapezoid(s)

_____ parallelogram(s)

_____ rectangle(s)

Garden H

Solve to ﬁnd the area of the garden.

_____ triangle(s)

Garden G

_____ rectangle(s)

Explore 4

Area and Volume

Area and Volume | 305

2. There are multiple ways you can decompose composite figures into figures that you know how to find the area of. Describe a strategy to solve one of the Garden Cards differently from the way you solved it.

1. What strategy can you use to ﬁnd the area of a composite ﬁgure?

Reflect

Explore 4

Area and Volume

Explore 5

Name: _______________________ Date: ___________

Volume of Rectangular Prisms Part I Read each Produce Carton Card. Build a model of each carton using linking cubes. Draw your model, and label the units using smile units. Record the length, width, and height of your model in both number of cubes and smile units. Produce Carton A

Cube Dimensions

Smile Unit Dimensions

Length = _______ cubes

Length = _______ smile units

Width = _______ cubes

Width = _______ smile units

Height = _______ cubes

Height = _______ smile units

Area and Volume | 307

Area and Volume

Explore 5 Produce Carton B

Cube Dimensions

Smile Unit Dimensions

Length = _______ cubes

Length = _______ smile units

Width = _______ cubes

Width = _______ smile units

Height = _______ cubes

Height = _______ smile units

Produce Carton C

Cube Dimensions

Smile Unit Dimensions

Length = _______ cubes

Length = _______ smile units

Width = _______ cubes

Width = _______ smile units

Height = _______ cubes

Height = _______ smile units

308 | Area and Volume

Area and Volume

Explore 5 Produce Carton D

Cube Dimensions

Smile Unit Dimensions

Length = _______ cubes

Length = _______ smile units

Width = _______ cubes

Width = _______ smile units

Height = _______ cubes

Height = _______ smile units

Find the volume of each produce carton in smile units using the formula V = lwh. Produce Carton A

Produce Carton B

Produce Carton C

Produce Carton D

Area and Volume | 309

Area and Volume

Explore 5 Part II

Look at the dimensions provided for each shipping box. Use this information to ﬁnd the volume in two ways. Lettuce Shipping Box Length = 2 1 feet 3

Width = 1 2 feet 3

Volume Formula 1 V=l×w×h

Height = 1 foot Volume Formula 2

Find the area of the base (B).

V = Bh

Tomato Shipping Box Length = 2 meters Volume Formula 1 V=l×w×h

Width = 1 4 meters

Height = 1 meter Volume Formula 2

Find the area of the base (B).

V = Bh

310 | Area and Volume

Area and Volume

Explore 5 Carrot Shipping Box 1

Length = 2 4 feet

Width = 4 4 feet

Volume Formula 1 V=l×w×h

Height = 3 feet Volume Formula 2

Find the area of the base (B).

V = Bh

Watermelon Shipping Box 1

Length = 18 2 inches Volume Formula 1 V=l×w×h

Width = 24 inches

Height = 5 5 inches Volume Formula 2

Find the area of the base (B).

V = Bh

Area and Volume | 311

Explore 5

Area and Volume

Reflect 1. How does the number of cubes you can put into the box relate to the length? How does it relate to the width? How does it relate to the height? How does it relate to the total cubes?

2. What do you notice about ﬁnding the volume of a rectangular prism using linking cubes versus using dimensions?

3. Explain how to multiply fractional measurements.

312 | Area and Volume

Surface Area

313

Surface Area

Explore 1

Name: _______________________ Date: ___________

Nets Part I: Nets Cut out and build each Pattern Card. Match each completed model to its corresponding Tent Model Card. Cut out the Student Journal Cutouts, match the cutouts to the Tent Model Card and 3-D model made from the Pattern Card, and glue the Student Journal Cutouts in the table. Model of Tent

Net of Tent

Surface Area | 315

Surface Area

Explore 1 Model of Tent

Net of Tent

316 | Surface Area

Surface Area

Explore 1 Model of Tent

Net of Tent

Reflect 1. What two-dimensional ﬁgures did you identify in the nets?

2. How are the tent model and its matching net similar?

3. How could recognizing nets be helpful not only in math but in everyday life?

4. What can each net, or pattern, tell us about the attributes of each tent?

Surface Area | 317

Surface Area

Explore 1 Part II: Attributes of Three-Dimensional Figures

Identify the three-dimensional ﬁgure for each tent model. Use markers to number and label the parts of each tent model image as follows: edges traced in green, vertices dotted with red, bases outlined in yellow, and faces numbered in blue. Record the number of each attribute, and identify the shape of each ﬁgure’s faces and bases.

3-D ﬁgure: ___________________________ Model of Tent _______

Number of edges:

318 | Surface Area

Number of vertices:

Number of curved surfaces:

Number of faces:

Number of bases:

Shape of faces:

Shape of bases:

Surface Area

Explore 1

3-D ﬁgure: ___________________________ Model of Tent _______

Number of edges:

Number of vertices:

Number of curved surfaces:

Number of faces:

Number of bases:

Shape of faces:

Shape of bases:

3-D ﬁgure: ___________________________ Model of Tent _______

Number of edges:

Number of vertices:

Number of curved surfaces:

Number of faces:

Number of bases:

Shape of faces:

Shape of bases:

Surface Area | 319

Surface Area

Explore 1

3-D ﬁgure: ___________________________ Model of Tent _______

Number of edges:

Number of vertices:

Number of curved surfaces:

Number of faces:

Number of bases:

Shape of faces:

Shape of bases:

3-D ﬁgure: ___________________________ Model of Tent _______

Number of edges:

320 | Surface Area

Number of vertices:

Number of curved surfaces:

Number of faces:

Number of bases:

Shape of faces:

Shape of bases:

Explore 1

Surface Area

Reflect 1. How many bases do prisms have?

2. How many bases do pyramids have?

3. Which three-dimensional ﬁgure has the most edges?

4. What is the difference between a triangular prism and a triangular pyramid?

Surface Area | 321

Surface Area

Explore 2

Name: _______________________ Date: ___________

Finding the Surface Area of 3-D Figures Look at the Nets for the net of each sample tent. Name the three-dimensional figure that can be made by assembling the net. Identify how many of each two-dimensional figure you see in the net. Use the measurements that are provided and find the missing measurements to calculate the area of each two-dimensional figure in the net. Use these areas to find the total surface area of the sample tent. Net A How many of each two-dimensional figure are in the net?

3-D Figure: ______________________ Area of each 2-D ﬁgure:

Total surface area of the sample tent:

Surface Area | 323

Surface Area

Explore 2 Net B How many of each two-dimensional ﬁgure are in the net?

3-D Figure: ______________________ Area of each 2-D ﬁgure:

Total surface area of the sample tent:

Surface area:

Net C How many of each two-dimensional ﬁgure are in the net?

3-D Figure: ______________________ Area of each 2-D ﬁgure:

Total surface area of the sample tent:

Surface area: 324 | Surface Area

Surface Area

Explore 2 Net D How many of each two-dimensional ﬁgure are in the net?

3-D Figure: ______________________ Area of each 2-D ﬁgure:

Total surface area of the sample tent:

Surface area:

Net E How many of each two-dimensional ﬁgure are in the net?

3-D Figure: ______________________ Area of each 2-D ﬁgure:

Total surface area of the sample tent:

Surface Area | 325

Explore 2

Surface Area

Reflect 1. How can you use a net of a three-dimensional ﬁgure to determine the ﬁgure’s surface area?

2. Describe a scenario in everyday life where knowing how to calculate surface area might be helpful.

326 | Surface Area

Represent and Interpret Data

327

Represent and Interpret Data

Explore 1

Name: _______________________ Date: ___________

Dot Plots Part I: Match Dot Plots and Explanations to Statistical Questions Read each statistical question, and determine which dot plot and explanation represent the possible data collected. Then, explain why you matched this dot plot and explanation to the statistical question. Question 1 Dot plot:

Explanation:

Explain why you chose this dot plot and explanation.

Question 2 Dot plot:

Explanation:

Explain why you chose this dot plot and explanation.

Question 3 Dot plot:

Explanation:

Explain why you chose this dot plot and explanation.

Represent and Interpret Data | 329

Represent and Interpret Data

Explore 1 Question 4 Dot plot:

Explanation:

Explain why you chose this dot plot and explanation.

Question 5 Dot plot:

Explanation:

Explain why you chose this dot plot and explanation.

Question 6 Dot plot:

Explanation:

Explain why you chose this dot plot and explanation.

330 | Represent and Interpret Data

Explore 1

Represent and Interpret Data

Part II: Describe Distribution of Dot Plots (spread and shape) The researchers used mathematical vocabulary to summarize each data set. Complete each explanation so the general public can interpret their observations. Number of Hours Sixth Graders Spent Watching TV in One Day When the shape of the data is symmetrical, it means–

When the spread has a small deviation from the mean, it means–

Number of Hours Seventh Graders Spent Watching TV in One Day When the shape of the data is asymmetrical and skewed right, it means–

When the spread has a large deviation from the mean, it means–

Number of Dogs Who Visited the Veterinarian Each Day of the Week When the shape of the data is asymmetrical and skewed left, it means–

When the spread has a large deviation from the mean, it means–

Represent and Interpret Data | 331

Explore 1

Represent and Interpret Data

Number of Cats Who Visited the Veterinarian Each Day of the Week When the shape of the data is symmetrical, it means–

When the spread has a small deviation from the mean, it means–

Number of Text Messages Each Boy in a Seventh-Grade Class Sent in a Month When the shape of the data is asymmetrical and skewed right, it means–

When the spread has a large deviation from the mean, it means–

Number of Text Messages Each Girl in a Seventh-Grade Class Sent in a Month When the shape of the data is asymmetrical and skewed left, it means–

When the spread has a large deviation from the mean, it means–

332 | Represent and Interpret Data

Explore 1

Represent and Interpret Data

Reflect 1. How would you describe the difference between the shape of data that is skewed left and the shape of data that is skewed right?

2. How would you describe the difference between a spread of data that has a small deviation and a spread of data that has a large deviation?

3. Describe how you would know if the data is a result of a statistical question.

Represent and Interpret Data | 333

Represent and Interpret Data

Explore 2

Name: _______________________ Date: ___________

Histograms Part I: Tori’s Points Tori’s points are listed below. Tori created a histogram of the points she earned during the mystery game. Use the list and histogram to answer the questions that follow.

10, 14, 17, 20, 14, 23, 19, 18, 18, 10, 10, 11, 22, 16, 15, 16, 20, 16, 22, 13, 13, 15, 19, 17, 21, 19, 16

Frequency

Tori’s Points

14 12 10 8 6 4 2 0

Tori’s Mystery Game Scores

10–12 13–15 16–18

19–21 22–24

Sum on spinners 1. Sort each of Tori’s individual point values into the following ranges. a. 10–12 b. 13–15 c. 16–18 d. 19–21 e. 22–24

Represent and Interpret Data | 335

Explore 2

Represent and Interpret Data

2. Which range of sums was landed on the most by Tori? Which range of sums was landed on the least?

3. How is the range of values with the most numbers in it related to the histogram?

4. What is the spread of the data?

5. What is the shape of the data?

Reflect 1. How can you determine the frequency of an interval on the histogram?

2. What does the tallest bar in the histogram represent?

3. How is data represented on the histogram when the interval has a frequency of zero?

336 | Represent and Interpret Data

Represent and Interpret Data

Explore 2 Part II: Jakob’s Points

Jakob took his turn at the mystery game. He recorded the following sums: 16, 16, 22, 22, 12, 22, 16, 23, 21, 13, 16, 16, 18, 25, 23, 21, 25, 14, 19, 18, 26, 21, 26, 18, 16, 21, 12, 23, 21, 21, 15, 17, 16, 18, 19, 12, 23, 21 1. Create a frequency table and a histogram of Jakob’s sums. Sums

Frequency

10 11 12 13 14 15 16 17 18 19 20 21 22 23

2. What was the lowest sum Jakob scored?

24 25 26

3. What was the highest sum Jakob scored?

Represent and Interpret Data | 337

Represent and Interpret Data

Explore 2 4. How many times were the sums in the range of 10–12?

5. How many times were the sums in the range of 28–30?

6. Which interval had the greatest number of sums?

7. What is the shape of the data?

8. What is the spread of the data?

Reflect 1. How can you determine the peak from a histogram?

2. What does each bar represent in the histogram?

3. What are some of the advantages of using a histogram?

338 | Represent and Interpret Data

Represent and Interpret Data

Explore 3

Name: _______________________ Date: ___________

Box Plots Part I Glue the Class A Box Plot Card in the top box, and answer the questions in the Predictions column about the box plot. Then, glue the Class A Data Card in the bottom box, and answer the questions in the Actual Answers column about the box plot and data.

Glue Class A Box Plot Card here.

Glue Class A Data Card here.

Predictions

Actual Answers

What is the shortest straw tower?

What is the tallest straw tower?

What is the median of the data?

Represent and Interpret Data | 339

Explore 3

Represent and Interpret Data

Part II Class C’s Straw Tower Challenge data is shown in the box plot and in the list below. Use the data to answer the questions that follow.

1. How many values are in class C’s data set?

2. What is the smallest value? Circle this number in the data set using a red colored pencil.

3. What is the largest value? Circle this number in the data set using a blue colored pencil.

4. What number is in the middle of the data? Circle this number in the data set using a green colored pencil.

5. What number does the box portion of the box plot start at? Circle this number in the data set using a purple colored pencil.

6. What number does the box portion of the box plot end at? Circle this number in the data set using an orange colored pencil.

7. Why does the box portion of the box plot start at 23 and end at 32?

340 | Represent and Interpret Data

Explore 3

Represent and Interpret Data

Reflect Use the graphic organizer to label where to ﬁnd each of the following pieces of information on the box plot. 1. Write the label “smallest value” in red at the location of the smallest value.

2. Write the label “largest value” in blue at the location of the largest value.

3. Write the label “middle value” in green at the location of the middle value.

4. Write the label “lower middle value” in purple at the location of the lower middle value.

5. Write the label “upper middle value” in orange at the location of the upper middle value.

Represent and Interpret Data | 341

Represent and Interpret Data

Explore 4

Name: _______________________ Date: ___________

Bar Graphs Part I Analyze the double bar graph, and answer the questions below. Favorite Movie

90 80

Lunch period 4

Lunch period 5

60 50 40 30 20 10 0

King

Fast Cars 9

Free Him

Terror of Thursday

Beware

1. Which movie had the most votes in period 4?

2. Which movie had the least votes in period 4?

3. Which movie had the most votes in period 5?

4. Which movie had the least votes in period 5?

5. Which movie should be chosen as the class favorite for 6th grade, and why?

Represent and Interpret Data | 343

Represent and Interpret Data

Explore 4 Part II

Use the data below to create a bar graph, and answer the questions.

Song Title

Period 4

Period 5

“Hover”

“Jump Up”

“Hot Wave”

“Moods”

“Great Habit”

90 80 70 60 50 40 30 20 10 0

“Hover”

344 | Represent and Interpret Data

“Jump Up”

“Hot Wave”

“Moods”

“Great Habit”

Explore 4

Represent and Interpret Data

1. Which song had the highest value from either period?

2. Which song had the fewest votes between both lunch periods?

3. Which song should be chosen as the class favorite, and why?

Reflect 1. When creating a bar graph, how do you decide the units needed for the y-axis?

2. How does displaying categorical data help you draw conclusions about the data?

Represent and Interpret Data | 345

Summarize Numerical Data

347

Summarize Numerical Data

Explore 1

Name: _______________________ Date: ___________

Mean and Median Part I: Mean as Balance Point Input the data from José’s Home Runs card in the table below. Use linking cubes to represent the data, and then record the data using a dot plot. Answer the following questions.

Title: ________________________________________

1. What do you notice about the distance between the points?

2. Determine the balance point of this data.

3. In series 6, José hit one home run. How many home runs would José have had to hit in series 7 to keep the balance point the same?

Summarize Numerical Data | 349

Explore 1

Summarize Numerical Data

Input the data from José’s Total Hits card in the table below. Represent the data using a dot plot. Answer the following questions.

Title: ________________________________________

1. How can you ﬁnd the balance point in a set of data?

2. What is the balance point for José’s total hits?

3. What does the balance point for this data represent?

350 | Summarize Numerical Data

Explore 1

Summarize Numerical Data

Reflect 1. How did your group equally distribute the number of cubes to make sure each stack of cubes got a fair share?

2. Is the balance point always toward the middle part of the graph?

3. How does adding or taking away a data point affect the balance point of the data?

Summarize Numerical Data | 351

Summarize Numerical Data

Explore 1 Part II: Finding the Center and Shape of Data

Input the data for Sammy’s Total Times at Bat in the frequency table below. Represent the data using a histogram. Then, answer the following questions. Interval

Frequency

Title: ________________________ 7

0–10

11–20

31–40 41–50

Frequency

21–30

5 4 3 2 1 0

Interval

1. Find the total number of at bats Sammy had for all 7 series. 2. If Sammy evenly distributed the number of times he was up to bat in each series, how many times would that be? 3. If Sammy had an at bat of 10 in series 8, how would the number of times he was up to bat change? Round to the nearest one. 4. Write the data set in a list in order from least to greatest. 5. Use your list in question 3 to determine the middle value in the data list. 6. If Sammy had an at bat of 10 in series 8, how would the middle value of the data list change? How would the histogram change?

352 | Summarize Numerical Data

Summarize Numerical Data

Explore 1

Input the data from Sammy’s Total Hits scenario card in the frequency table. Represent the data using a histogram. Answer the following questions.

Interval

Frequency

Title: ________________________ 7

1–4

5–8

13–16 17–20

Frequency

9–12

5 4 3 2 1 0

Interval

1. Find the total number of hits Sammy had in all 7 series of games. 2. If Sammy had the same number of hits in each series, how many hits would he have in one of these series? Round to the nearest one. 3. If Sammy eliminated his 18 hits in series 7 from the data, how would the number of hits in one of these series change? 4. Write the data set in a list in order from least to greatest. 5. Use your list in question 4 to determine the middle value in the data list. 6. If Sammy eliminated his 2 hits in series 7 from the data, how would the middle value change?

Summarize Numerical Data | 353

Explore 1

Summarize Numerical Data

Reflect 1. In Sammy’s Total Times at Bat scenario, what data point would you remove to make the mean and the median the same or closer in value?

2. How does the shape of the graph affect the mean and the middle value?

3. Is the mean or the median a better indicator of the center of data? Explain.

354 | Summarize Numerical Data

Summarize Numerical Data

Explore 2

Name: _______________________ Date: ___________

Range and Interquartile Range (IQR) Part I: Understanding Box Plots Using Measures of Center and Range Use the Baseball Team A card to answer the questions below. 1. Write the number of hits each player had from least to greatest.

2. What is the minimum number of hits by a single player?

3. What is the maximum number of hits by a single player?

4. What is the median number of hits?

5. What is the value of quartile 1 (Q1)?

6. What is the value of quartile 3 (Q3)?

Use the information above to draw a box plot representing the data.

Summarize Numerical Data | 355

Explore 2

Summarize Numerical Data

Use the information from team A’s hits and the box plot from the previous page to answer the questions below. 7. What is the range of the hits by each player?

8. How many values are below Q1?

9. What percentage of the data is below Q1?

10. How many values are above Q3?

11. What percentage of the data is above Q3?

12. What percentage of the data is inside the box?

13. How could we determine the range of hits by the middle 50% of batters?

356 | Summarize Numerical Data

Summarize Numerical Data

Explore 2

Part II: Understanding Box Plots Using Measures of Spread Use Baseball Team Cards B–D to ﬁnd the ﬁve-number summary for each team. Then, create a box plot for each team’s data. Team B List the number of hits from least to greatest below.

Minimum:

Q1:

Median:

Q3:

Maximum:

Draw a box plot to represent team B’s data.

1. What is the range of hits for team B?

2. What is team B’s IQR?

Summarize Numerical Data | 357

Summarize Numerical Data

Explore 2 Team C List the number of hits from least to greatest below.

Minimum:

Q1:

Median:

Q3:

Maximum:

Draw a box plot to represent team C’s data.

3. What is the range of hits for team C?

4. What is team C’s IQR?

358 | Summarize Numerical Data

Summarize Numerical Data

Explore 2 Team D List the number of hits from least to greatest below.

Minimum:

Q1:

Median:

Q3:

Maximum:

Draw a box plot to represent team D’s data.

5. What is the range of hits for team D?

6. What is team D’s IQR?

Summarize Numerical Data | 359

Explore 2

Summarize Numerical Data

Reflect 1. What does the median of a data set tell you? Explain how you located the median on a box plot, and explain what this value represents in relation to the context of the given situations.

2. What does the range of a data set tell you? Explain how you determined range, and explain what this value represents in relation to the context of the given situations.

3. What does the interquartile range of a data set tell you? Explain how you determined the IQR, and explain what this value represents in relation to the context of the given situations.

4. Which team has the lowest IQR? What does this tell you about the number of hits for the players on that team?

360 | Summarize Numerical Data

Summarize Numerical Data

Explore 3

Name: _______________________ Date: ___________

Mean Absolute Deviation The dot plot shows the ages of ten baseball players. Use the data to answer the questions below.

1. Describe the shape and spread of this data set.

2. What is the median of the data set?

3. What is the mean of this data set? Round this value to the nearest tenth.

4. Explain why the mean and median are not equal in value.

Summarize Numerical Data | 361

Summarize Numerical Data

Explore 3

5. Find the absolute value of the distance between each data point and the mean. Record each distance from the mean in the form of an equation in the table below.

Data Point

Distance from the Mean

32.8 – 22 = 10.8

32.8 – 27 = _____________

32.8 – 29 = _____________

29 32 34

34 – 32.8 = _____________

35 – 32.8 = _____________

35 39 46 6. Find the mean of the absolute value of the distances you found. Explain what this value means.

7. Why would we ﬁnd the absolute value of the distance between each point and the mean?

362 | Summarize Numerical Data

Explore 3

Summarize Numerical Data

Reflect 1. Why would the mean and median not be equal?

2. Describe the strategy you used to ﬁnd the mean absolute deviation.

3. What would a greater value represent when you ﬁnd the mean absolute deviation?

Summarize Numerical Data | 363

Summarize Numerical Data

Explore 4

Name: _______________________ Date: ___________

Comparing Different Representations of the Same Data Read the information on each Baseball Card. Use each set of graphs to answer the questions that follow.

RBIs 1. Describe the shape and spread of this data set.

2. What is the median of the data set? Which graph represents the median well?

3. What is the mean of this data set? Round this value to the nearest tenth. Which graph can you get the mean from?

4. Which graph makes it known that 6 is not a data point?

5. Which graph makes it easier to ﬁnd the interquartile range of RBIs?

Summarize Numerical Data | 365

Explore 4

Summarize Numerical Data

Total Hits 1. What is the range of total number of hits?

2. What is the median of the data set? Which graph represents the median well?

3. Describe the shape of this data set.

4. What is the mean of this data set? Round this value to the nearest tenth. Which graph can you get the mean from?

5. Which graph helps you see that there are many gaps in the data?

366 | Summarize Numerical Data

Explore 4

Summarize Numerical Data

Total Times at Bat 1. What is the range of the total number of at bats?

2. Which representation shows the distribution of data well?

3. What is the median of the data set? Which graph represents the median well?

4. Describe the shape of this data set.

5. Which graph shows the measure of center?

Summarize Numerical Data | 367

Summarize Numerical Data

Explore 4 Reflect 1. Describe what each graph is better at representing.

2. Which graph best represents gaps and clusters in data sets?

3. Which graph shows the median?

4. Which graph makes it easier to ﬁnd the interquartile range?

5. Can all graphs give you the same measures of data? Give one example.

368 | Summarize Numerical Data

Skills Quizzes

369

Add and Subtract Fractions

Skills Quiz

Name: _______________________ Date: ___________

Add and Subtract Fractions Directions: Solve each problem. Show or explain your mathematical thinking. 1. Write an equation to represent and solve the following scenario: 1 is taken away 2

3 from 4 .

2. Mr. Aldama has 2 of a cup of flour and pours out 4 of a cup of the flour. How much flour does he have left? Write an expression to represent what you did.

3. Write an expression to represent and solve the following scenario: 6 is removed from 5 . 3

Add and Subtract Fractions | 371

Add and Subtract Fractions

Skills Quiz

4. Write an equation to represent and solve the following scenario: 3 plus an 4

additional 8 .

5. What is the total when 8 is added to 4 ? Write an expression to represent what you did.

6. Margaret read 3 of her book and then read an additional 6 of her book. How much of her book did she read in all? Write an expression to represent what you did.

372 | Add and Subtract Fractions

Skills Quiz

Add and Subtract Fractions

Use strategies to determine a common denominator, and solve. 7.

3 4

5 6

12 5

+ 31 2

41 2

– 23 = 4

10. 8

– 21 = 6

Add and Subtract Fractions | 373

Multiplication and Division Problem Solving Using Fractions

Skills Quiz

Name: _______________________ Date: ___________

Multiplication and Division Problem Solving Using Fractions Directions: Solve each problem. Show or explain your mathematical thinking.

1. Last week, you ordered 15 pouches of a strawberry blend for smoothies. Yesterday, 3 of the order arrived. Create a model to help determine the equation to 5

use and solve for the number of pouches you received.

2. Determine the mathematical expression for the model shown.

Multiplication and Division Problem Solving Using Fractions | 375

Multiplication and Division Problem Solving Using Fractions

Skills Quiz 3

3. Model 4 × 4 on the number line.

2 4

1 4

3 4

3 2 1 14 1 24 1 34 2 2 14 2 24 2 34 3 3 14 3 4 3 4 4

4. Model 3 ÷ 3 on the number line. 4

1 8

2 8

1 4

3 8

4 8

2 4

5 8

6 8

3 4

7 8

5. Model 7 ÷ 1 on the tape diagram. 10

376 | Multiplication and Division Problem Solving Using Fractions

Multiplication and Division Problem Solving Using Fractions

Skills Quiz

6. Which expression represents the model shown below?

1 Whole

1 2

2 2 ÷ 4

12 ÷ 2 2

22 ÷ 4

2 2 × 2

7. Write an equation to represent the model shown below.

2 3

2 3 5 6

Multiplication and Division Problem Solving Using Fractions | 377

Skills Quiz

Multiplication and Division Problem Solving Using Fractions

8. Determine an equation to represent the following scenario: 8 chocolate bars are split into thirds. A.

1 ×8=2 2 3 3

8 ÷ 3 = 24

1 1 ÷ 8 = 24 3

8× 1 =22

9. Calculate 12 ÷ 8 .

10. Calculate 12 ÷ 1 10 .

378 | Multiplication and Division Problem Solving Using Fractions

Add and Subtract Decimals

Skills Quiz

Name: _______________________ Date: ___________

Add and Subtract Decimals Directions: Solve each problem. Show or explain your mathematical thinking.

1. A peanut butter bar costs $2.35. A chocolate bar costs $0.15 less. How much does a chocolate bar cost? A.

$0.85

$2.20

$2.50

$2.17

2. Janice runs a lemonade stand. A cup of lemonade costs $0.89. She also sells sugar cookies for $2.79. Determine how much each order below would cost. a. 3 cups of lemonade

b. 1 cup of lemonade and 2 sugar cookies

3. Janice decides to increase the price of a cup of lemonade so it is only $1.80 less than the sugar cookies listed at $2.79. What is the new cost? How much did she raise the price of a cup of lemonade from the price listed in question 2?

Add and Subtract Decimals | 379

Skills Quiz

Add and Subtract Decimals

4. Which of the following expressions is the same as 15.8 – 0.452? A.

15.888 – 0.452

15.8 – 0.452

15.800 – 0.452

5. Alison had 0.5 of a cup of yogurt for making strawberry smoothies. Each smoothie requires 0.125 of a cup of yogurt. She made 2 smoothies for her friends. How much yogurt was left?

380 | Add and Subtract Decimals

Skills Quiz

Add and Subtract Decimals

6. Which of the following statements is true? A.

The sum of 0.75 and 0.89 is less than 2.

The sum of 1.25 and 0.81 is less than 2.

The sum of 0.75 and 0.89 is greater than 2.

The sum of 0.7 and 1.25 is greater than 2.

7. Marissa is making 3 different types of cookies. The first cookie calls for 2.25 cups of flour. The second cookie calls for 1.25 cups of flour, and the last type of cookie calls for 3.7 cups of flour. How much flour does she need for all 3 types of cookies? A.

6.12 cups of ﬂour

4.7 cups of ﬂour

7.2 cups of ﬂour

3.87 cups of ﬂour

8. Is the difference of 5.24 and 1.89 greater than or less than the difference of 7.42 and 3.94?

Add and Subtract Decimals | 381

Add and Subtract Decimals

Skills Quiz

9. Finn paid $9.80 for a shirt on clearance. The original price was $15.28. How much money did Finn save by buying the shirt on clearance?

10. Anika walked 2.5 miles on Monday and Wednesday. On Tuesday, she walked 0.42 miles less than she walked on Monday. On both Thursday and Friday, Anika walked 1.15 miles more than she walked on Monday. What is the total distance she walked over all 5 days? A.

8.23 miles

14.38 miles

10.73 miles

15.1 miles

382 | Add and Subtract Decimals

Multiply and Divide Decimals

Skills Quiz

Name: _______________________ Date: ___________

Multiply and Divide Decimals Directions: Solve each problem. Show or explain your mathematical thinking.

1. The dimensions of a rectangular quilt are 7.5 yards by 5.25 yards. What is the total area of the quilt? A.

3.9375 square yards

39.375 square yards

393.75 square yards

3,937.5 square yards

2. Ateeba buys 5.5 pounds of organic cheese from the grocery store. Each pound is priced at $4.96. How much does Ateeba spend on organic cheese?

3. Yanette works as a library assistant and earns $9.75 per hour. How much will she get paid if she works 38.5 hours in one week?

Multiply and Divide Decimals | 383

Multiply and Divide Decimals

Skills Quiz

4. Rohan is solving the problem 4.21 × 1.35 using an area model as shown below. He knows he has made a mistake, but he cannot ﬁgure out where exactly. Identify the location of Rohan’s mistake. 1

+ 0.3

+ 0.05

1.2

0.2

= 5.4

+ 0.2

0.2

0.6

0.01

= 0.81

+ 0.01

0.01

0.003

0.0005

= 0.0135

The total sum from the area model is 6.2235. A.

He multiplied 0.3 and 0.2 incorrectly.

He multiplied 0.05 and 0.01 incorrectly.

He broke apart the two decimal numbers using incorrect place values.

He added all of the terms inside of the area model incorrectly.

5. Namrata earns $8.50 per hour babysitting her neighbor’s son. How many hours must Namrata babysit to earn $408?

384 | Multiply and Divide Decimals

Skills Quiz

Multiply and Divide Decimals

6. Match each of the expressions below with its correct representation. Connect each expression and representation by drawing a line. A.

8÷4

0.8 ÷ 4

0.08 ÷ 4

7. Mrs. Hawkins deposits $915.60 into her checking account to spend on her grandchildren. She decides to write a check for each of her 3 grandchildren. The checks are all for equal amounts. How much did each of her grandchildren receive?

Multiply and Divide Decimals | 385

Multiply and Divide Decimals

Skills Quiz

8. Your school hosted a bake sale to raise money for a dance, and you were put in charge of selling chocolate chip cookies. The cookies were sold for $1.50 each. At the end of the day, you calculated that you had earned $88.50 in sales. How many chocolate chip cookies did you sell?

9. When multiplying 20.81 by 0.083, how many decimal places are in the product? A.

10. When solving 85.500 ÷ 1.25 using the standard algorithm, how many times must you move the decimal to make the divisor a whole number? A.

386 | Multiply and Divide Decimals

Integers

Skills Quiz

Name: _______________________ Date: ___________

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

1. A diver can swim up to 130 feet below sea level. Write a negative integer to represent a depth that a diver could possibly dive and explain your reasoning.

2. It is 8°F in International Falls, Minnesota. In Utqiagvik, Alaska, it is −6°F. Which city’s temperature is closer to zero? Explain your reasoning.

Integers | 387

Skills Quiz

Integers

3. Reggie earned $15 after mowing the lawn for his grandparents. He owes his brother $20. Choose the answer that best describes how much money Reggie will have after he pays his debt. A.

Reggie will have $5 left over.

Reggie will have zero dollars left over.

Reggie will be $5 in debt.

Reggie will have $35 total.

4. Determine whether each situation describes a positive or negative value. Write + for a positive value or − for a negative value. a.

A credit of $10

A loss of 3 points

A weight gain of 5 pounds

A deposit of $5

A depth of 10 feet below sea level

A temperature increase of 5°F

388 | Integers

Integers

Skills Quiz 5. Select all of the true statements. A.

Opposite integers are equidistant from zero.

Opposite integers sum to zero.

Zero is its own opposite.

Opposite integers greater than or less than zero have opposite signs.

6. Describe the relationship between −7 and the opposite of −7.

7. What is the distance between |–7| and –|4|? Explain your reasoning.

Integers | 389

Integers

Skills Quiz 8. Use absolute value to describe the elevation of a diver at −288 meters.

9. Which answer describes a bank account with less than −$405? A.

The absolute value of the bank account is −$405.

The bank account has more than $405 of debt.

The bank account has exactly $405 of debt.

The bank account has $405.

10. Select all of the true statements. A.

Zero is its own opposite.

Zero is neither negative nor positive.

Zero is an integer.

Zero is both positive and negative.

390 | Integers

Rational Numbers

Skills Quiz

Name: _______________________ Date: ___________

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

1. Order the rational numbers below from least to greatest. −2.0, –3.5, −4.25, 3.5, −2.5, −2.75

2. Order the rational numbers below from least to greatest. 1

|–3|, –3 2 , |2.5|, –1.05, −1.5, 1.8, –2.2, –2.2,_ _8_

3. Choose the correct representation for the following statement: −4.2 degrees Fahrenheit is greater than −12.6 degrees Fahrenheit. A.

−4.2 < −12.6

−12.6 > −4.2

−4.2 > −12.6

−12.6 ≥ −4.2

Rational Numbers | 391

Rational Numbers

Skills Quiz

4. Choose the correct representation for the following statement: The absolute value of −2 is less than the absolute value of 10. A.

−2 < −10

−10 > −2

2 > −10

2 < 10

5. Order the rational numbers below from greatest to least. 1 − 1 , 1 ,− 3 ,−12 3, 1 2 ,– 2 2 8 2 __

6. Choose the correct representation for the following statement: 20 feet below sea level is less than 3 feet above sea level. A.

3 < −20

20 < −3

−20 < 3

−3 > 20

392 | Rational Numbers

Rational Numbers

Skills Quiz 7. Select all of the true statements. A.

A temperature of −5°F is warmer than a temperature of −4°F.

A bank account balance less than −$25 describes a debt greater than $25.

An elevation of −5 ft. is farther below sea level than an elevation of −2 ft.

A score of −5 points is less than a score of −6 points.

8. Select all of the numbers that are located below −2 on a vertical number line. A.

–2 1

–1.8

–3.3

9. Find and position each rational number on the number line. A=– 2 3

B = |−1|

C= 1 3

D = the opposite of 1 1 3

Rational Numbers | 393

Skills Quiz

Rational Numbers

10. Select all of the numbers located to the left of 5 on a horizontal number line. A.

5.5

4 3

394 | Rational Numbers

Equivalent Numerical Expressions

Skills Quiz

Name: _______________________ Date: ___________

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

1. Find the prime factorization of 84. Write 84 as a product of its prime factors.

2. What are the ﬁrst two common multiples of 4 and 5?

3. Find the greatest common factor of 52 and 68.

Equivalent Numerical Expressions | 395

Equivalent Numerical Expressions

Skills Quiz 4. Find the least common multiple of 6 and 8.

5. Create an area model and equivalent expression for 33 + 99.

6. What is the value of (0.5)3?

7. Evaluate the following expression. (2 + 32) + (4.5 − 1.5)

396 | Equivalent Numerical Expressions

Equivalent Numerical Expressions

Skills Quiz

8. Use the pattern in the table to determine the last digit of 28.

Power of 2

Result

Last Digit

128

9. Find the value of the following expressions, and describe the relationship between the powers of ten. 102

103

104

Equivalent Numerical Expressions | 397

Equivalent Numerical Expressions

Skills Quiz 1

10. Evaluate 9(7 + 8) × ( 3 )3.

398 | Equivalent Numerical Expressions

Algebraic Expressions

Skills Quiz

Name: _______________________ Date: ___________

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

1. Choose the expression that represents the following scenario: Add 8 to a number, and then multiply by 6. A. k + 8 × 6 B. 6(k k + 8) C. 8k k+6 D. 6k k−8

2. Choose the expression that represents the following scenario: 9 subtracted from 1.5 times a number. A. 1.5 − 9 + x B. 1.5 − 9x C. 1.5 × 9 + x D. 1.5x x−9

Algebraic Expressions | 399

Algebraic Expressions

Skills Quiz 3. How many terms are in the following expression: 3 ∙ 2 + 7y y + –10? A. 5 B. 6 C. 2 D. 3

4. What is the largest coefﬁcient in the following expression: 4x x + 8y + 7z? A. 8 B. 4 C. 4x D. 7

5. Select all of the expressions equivalent to 6(2x + 3y). A. 12x x + 3y B. 12x x + 18y C. 6(2x) + 3y D. 6(2x) + 6(3y)

400 | Algebraic Expressions

Algebraic Expressions

Skills Quiz x

6. Select all of the expressions equivalent to 5 . A. x – 4 x 5

1 B. 5 x

C. 5x 1

D. x – 5

7. Evaluate the expression 4(c c – 1) – 3c, where c = 7.

8. Describe the expression 2(3 + 9).

Algebraic Expressions | 401

Algebraic Expressions

Skills Quiz 9. Prove that the two expressions below are equivalent. 5(2b + c)

and

10b + 5c

10. Write an expression to represent the perimeter of this ﬁgure. Then, ﬁnd the perimeter when m = 12 and n = 3.

n m

402 | Algebraic Expressions

Skills Quiz

Equations and Inequalities

Name: _______________________ Date: ___________

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

1. Kaden needs to grill 55 hot dogs for his family barbecue. He has already grilled 33 hot dogs. Choose the equation that represents the number of hot dogs, d, he still needs to grill. A.

55 + 33 = d

d − 33 = 55

d + 33 = 55

55 + d = 33

2. Kim has p pens. Jill has 3 times as many pens as Kim. Write and solve an equation to represent the number of pens that Kim and Jill have combined.

3. Solve the equation 0.5x x = 10.

Equations and Inequalities | 403

Equations and Inequalities

Skills Quiz 4. Solve the equation x – 8.4 = 10.6.

5. Write an equation for the model shown below. Solve for the value of x.

x 3.4

3.4

6. Jonah needs to sell more than 15 T-shirts for his school’s fundraiser. Write an inequality that represents the number of T-shirts, t, he needs to sell.

7. Maria is in line for a roller-coaster ride at the state fair. The height requirement for the ride is at least 60 inches and no more than 84 inches. Write an inequality that represents the height requirement, h, of the roller-coaster ride.

404 | Equations and Inequalities

Equations and Inequalities

Skills Quiz

8. Consider the inequality x ≤ 3. Determine if each value makes a true or false statement. a.

−1

−100

9. Explain the meaning of the graphed solution for the inequality p < –7 in your own words.

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

9 10

Equations and Inequalities | 405

Skills Quiz

Equations and Inequalities

10. Explain the meaning of the graphed solution for the inequality m ≥ 1.5 in your own words.

406 | Equations and Inequalities

Ratios, Rates, and Unit Rates

Skills Quiz

Name: _______________________ Date: ___________

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

1. Which statement relates to the ratio 2 : 3? A.

For every 6 cats there are 4 dogs.

The ratio of boys to girls in the class is 20-to-30.

For a cake recipe, every 2 cups of ﬂour needs 6 cups of sugar.

The ratio of legs to tails is 4-to-1 for every cow on a farm.

2. Which statement relates to the ratio 9 : 1? A.

A plant grows 1 inch every 9 days.

Amanda buys 2 betta ﬁsh for every 18 goldﬁsh in her aquarium.

Roberto earns $9 per hour.

The ratio of lemons to water for a lemonade recipe is 1-to-9.

Ratios, Rates, and Unit Rates | 407

Ratios, Rates, and Unit Rates

Skills Quiz

3. The double line diagram shows the cost of oranges. How much does one pound of oranges cost?

4. The ratio of blueberries to mufﬁns is shown in the table below. Complete the missing value. Blueberries

Mufﬁns

408 | Ratios, Rates, and Unit Rates

60 8

Skills Quiz

Ratios, Rates, and Unit Rates

5. A plant grows 2 2 centimeters every 5 weeks. At this rate, how many centimeters does the plant grow in 12 weeks?

Ratios, Rates, and Unit Rates | 409

Ratios, Rates, and Unit Rates

Skills Quiz

6. The ratio of blue paint to yellow paint in a green paint mixture is shown below. Select all of the true statements.

Blue Yellow A.

For every 2 parts of yellow, use one part of blue.

2 parts of blue are used to make 6 cups of green paint.

The ratio of blue paint to yellow paint is 2 : 1.

The ratio of yellow paint to blue paint is 4-to-8.

410 | Ratios, Rates, and Unit Rates

Ratios, Rates, and Unit Rates

Skills Quiz

7. For every 3 hours of babysitting, Rhea earns $27. How much will she earn in 9 hours?

8. The graph below shows the relationship between the number of doughnuts purchased and their cost. y

Cost ($)

Number of Doughnuts Complete each statement below. a.

The ratio of doughnuts to cost is 8-to- ________.

The unit price per doughnut is ________.

For every 4 dollars spent, ________ doughnuts are purchased.

The ________ depends on the ____________________ purchased.

Ratios, Rates, and Unit Rates | 411

Ratios, Rates, and Unit Rates

Skills Quiz

9. Four T-shirts cost a total of $52 at the school spirit-wear store. At this rate, how much will 5 shirts cost?

10. The table below shows Henry’s reading rate. What is the missing value?

412 | Ratios, Rates, and Unit Rates

Hours

Chapters

Percents

Skills Quiz

Name: _______________________ Date: ___________

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

1. Find 20% of $70.

% Value

0 10

100

2. Given that 27 is 30% of a number, x, ﬁnd the value of x.

Percents | 413

Percents

Skills Quiz 3. What percentage of the hundreds grid has been shaded?

4. 100 pennies represent 100% of a dollar. What total percentage of a dollar is one dime and one nickel?

414 | Percents

Skills Quiz

Percents

5. Select all of the true equations. A.

0.3 = 30%

3 = 30% 6

1 = 20% 5

1.2 = 12%

6. There are 60 seconds in 1 minute. How many seconds make up 75% of a minute?

Percents | 415

Percents

Skills Quiz

7. A comic book has 16 pages. A short story has 400% of the number of pages in the comic book. How many pages does the short story have? A.

400

8. What is 20% of $60? Show or explain your reasoning.

9. During a musical, the curtain closed after 66 minutes had passed, signaling an intermission. At that point, the performance was 75% complete. How many minutes were in the entire performance? A.

88 minutes

198 minutes

91 minutes

83 minutes

416 | Percents

Skills Quiz

Percents

10. Charles completed 5 of a novel. What percentage of the novel does he have left to complete? A.

80%

20%

45%

55%

Percents | 417

Measurement Conversions

Skills Quiz

Name: _______________________ Date: ___________

Measurement Conversions Directions: Solve each problem. Show or explain your mathematical thinking. 1. There are 60 minutes in one hour. How many minutes are in 1 1 hours? 4

2. Francisco drove 537.6 kilometers on his road trip across Texas. One mile is approximately 1.6 kilometers. Approximately how many miles did Francisco drive?

3. Complete the table to determine how many feet are in 7 yards.

Yard(s)

Feet

Measurement Conversions | 419

Measurement Conversions

Skills Quiz

4. Convert 200 gallons to liters. (1 gallon is approximately 3.8 liters.)

5. If 1 inch is approximately 2.5 centimeters, approximately how many inches is 80 centimeters?

6. Find the missing values on the double number line below.

420 | Measurement Conversions

Measurement Conversions

Skills Quiz

7. Complete the missing values in the table showing the ratio of days to hours.

Day(s)

Hours

24 72

8. The ratio of centimeters to meters is 100 : 1. Complete each set of equivalent ratios.

100 : 1 = ____ : 4 = ____ : 8

9. Solve the proportion below. For every 8 ounces, there is 1 cup. Ounces Cups

8 1

64 x

Measurement Conversions | 421

Measurement Conversions

Skills Quiz

10. There are 4 ounces in 1 of a pound. How many ounces are in 2 pounds? 4

422 | Measurement Conversions

Coordinate Planes

Skills Quiz

Name: _______________________ Date: ___________

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

1. Select the answer choice that best describes the two points on the graph.

4 3 2 1 x –5

–4

–3

–2

–1

–1 –2 –3 –4 –5

The points are reﬂected over the y-axis.

Both points have the same x value of −5.

The points have opposite x values of 5 and −5.

The points have opposite y values of 5 and −5.

Coordinate Planes | 423

Coordinate Planes

Skills Quiz 1

2. Describe the relationship between the following points: A (–1 2 , 4) and B (1 2 , 4).

3. Describe the relationship between the following points: A (8, −6) and B (8, 6).

4. Select all of the points located on the x-axis. A.

(−3, 0)

(0, 0)

(3, 0)

(0, 3)

424 | Coordinate Planes

Skills Quiz

Coordinate Planes

5. Point M is plotted in quadrant ΙΙΙ on a coordinate plane.

Part A. Point N is a reﬂection of point M over the y-axis. Plot point N.

Part B. Describe the relationship between point N and point M, the x-axis, the y-axis, and the origin.

Coordinate Planes | 425

Coordinate Planes

Skills Quiz 6. Which point is located 5 units to the right and 3 units down? A.

(3, −5)

(5, 3)

(5, −3)

(−5, 3)

7. Point A is located at (2, 5). Select all of the points that are reﬂections of point A. A.

(−5, −2)

(−2, 5)

(2, −5)

(5, 2)

8. What is the location of the origin on a coordinate plane?

426 | Coordinate Planes

Skills Quiz

Coordinate Planes

9. Identify in which quadrant (Ι, ΙΙ, ΙΙΙ, or ΙV) each point below is located. (1, 3) _______ (−3, −2) _______ (−5, 1) _______ (3, −2) _______

10. Select all of the points that are not located in a quadrant. A.

(0, 0)

(3, 0)

(0, 3)

(3, 3)

Coordinate Planes | 427

Coordinate Plane Problem Solving

Skills Quiz

Name: _______________________ Date: ___________

Coordinate Plane Problem Solving Directions: Solve each problem. Show or explain your mathematical thinking.

1. How many units from the x-axis is the point (3, –4) located? A.

2. Find the area of a square with the vertices below. (−1, 2), (−1, 4), (1, 2), (1, 4)

Coordinate Plane Problem Solving | 429

Coordinate Plane Problem Solving

Skills Quiz

3. A polygon has vertices of (2, 2), (2, 0), (−1, 2), and (−1, 0). Part A. Draw the polygon on a coordinate plane.

Part B. What kind of polygon is this? How do you know?

4. The base of a triangle on a coordinate plane is drawn from (−5, 3) to (2, 3). What is the measurement of the base?

430 | Coordinate Plane Problem Solving

Coordinate Plane Problem Solving

Skills Quiz

5. A polygon has vertices of (−6, 1), (−4, 3), (−1, 2), and (−4, −1). Part A. Draw the polygon on a coordinate plane.

Part B. What kind of polygon is this? How do you know?

6. What is the distance between point A (5, −2) and point B (5, 3)?

7. What is the distance between point A (−3, 1) and point B (6, 1)?

Coordinate Plane Problem Solving | 431

Coordinate Plane Problem Solving

Skills Quiz

8. Draw a polygon with vertices A, B, C, and D each located in a different quadrant.

List your vertices as ordered pairs. A=

432 | Coordinate Plane Problem Solving

Coordinate Plane Problem Solving

Skills Quiz

9. What is the distance between point A (−3, 4) and point B (−3, −2)? A.

2 units

6 units

7 units

5 units

10. Choose the answer that best describes the relationship between the origin and a point at (2, −2). A.

The absolute value of the x value is −2 units from the origin.

The absolute value of the y value is 2 units from the origin.

The absolute value of the y value is −2 units from the origin.

The absolute value of the x value is −2 1 units from the origin.

Coordinate Plane Problem Solving | 433

Area and Volume

Skills Quiz

Name: _______________________ Date: ___________

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

1. Find the total area of the composite ﬁgure.

6 inches

10 inches

5 ft.

4 ft.

8 ft.

2. Find the total area of the composite ﬁgure.

5 ft.

12 ft.

Area and Volume | 435

Area and Volume

Skills Quiz 3. Find the area of the dark shaded region.

10 yd.

2 yd.

14 yd.

15.5 meters

4. Select the formula to ﬁnd the area of the green shaded region.

18 meters

A = bh

A=b+h

A = 2bh

A = 1 bh 2

436 | Area and Volume

Skills Quiz

Area and Volume

5. Find the area of the composite ﬁgure.

6. What is the volume of a cube with side lengths of 1.5 feet? A.

3.5 ft.²

4.5 ft.³

3.375 ft.³

2.25 ft.³

7. What is the volume of a rectangular prism with a length of 3.5 yards, a width of 4 yards, and a height of 6 yards? A.

56 yd.³

84 yd.³

252 yd.²

13.5 yd.³

Area and Volume | 437

Area and Volume

Skills Quiz

8. Write the equation to ﬁnd the volume of a rectangular prism with a height of 8 centimeter cubes. A.

V = 8h

B=8

V = 8B

V=8

9. If the volume of a prism is 56.25 in.3, ﬁnd the width.

10. The prism has a total volume of 162 cm3. How tall is the prism?

B = 27cm2

438 | Area and Volume

Surface Area

Skills Quiz

Name: _______________________ Date: ___________

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

1. The cube has side lengths of 2 centimeters. Find the surface area.

2 cm 2 cm

2. Use the net of the rectangular prism to determine the surface area of the prism.

Surface Area | 439

Surface Area

Skills Quiz 3. Circle the net that corresponds with the featured triangular prism.

4. Calculate the surface area of the prism shown below.

1.5 in. 8 in. 6 in.

440 | Surface Area

Surface Area

Skills Quiz Use the net below to answer questions 5 and 6. 4 4 4 6

4 6

4 4

4 6

4 4

5. What type of 3-D ﬁgure is formed when the net is folded? Explain your reasoning.

6. What is the surface area of the prism? A.

96 square units

128 square units

96 cubic units

60 square units

7. What is the surface area of a rectangular prism that measures 2 cm by 5.25 cm by 6.5 cm?

Surface Area | 441

Surface Area

Skills Quiz 8. Will this net form a 3-D ﬁgure? Explain your reasoning.

4 in.

9. A prism has equilateral triangle bases. The base of the prism is pictured below.

3 in. The height of the entire prism is 2.5 inches. Find the surface area of the entire prism.

442 | Surface Area

Surface Area

Skills Quiz

10. The net of a right triangular prism is shown below. What is the surface area of the prism?

5 mm

Surface Area | 443

Represent and Interpret Data

Skills Quiz

Name: _______________________ Date: ___________

Represent and Interpret Data Directions: Solve each problem. Show or explain your mathematical thinking.

For questions 1 and 2, refer to the box plot below.

100

Math Quiz Grades

1. Choose all of the true statements that apply to the box plot. A. The data is not symmetrical. B. Quartile 1 represents 50% of all of the data. C. The center is approximately 90. D. The data is skewed right.

Represent and Interpret Data | 445

Represent and Interpret Data

Skills Quiz 2. Write a ﬁve-number summary for the distribution of data.

Minimum Lower Quartile (Q1) Median Upper Quartile (Q3) Maximum

3. Use the dot plot to determine the mean of the data. Explain what that represents.

Hours spent practicing dance or sports each week A. The mean is 3. This represents the largest number of students who practiced each week. B. The mean is 10. This represents the most hours spent practicing each week. C. The mean is 3.4. This represents the average number of hours a student practices each week. D. The mean is 10. This represents the range of hours spent practicing each week.

446 | Represent and Interpret Data

Represent and Interpret Data

Skills Quiz

Macie surveyed her classmates about the number of electronic devices in their households. The data 1, 1, 2, 2, 2, 3, 3, 3, 3, 3, 4, 4, 6 was collected. Use this data to answer questions 4–7.

4. Represent Macie’s data on a dot plot.

5. Is the distribution of data symmetrical or skewed? Explain your reasoning.

6. What is the typical number of electronic devices that Macie’s classmates have? Explain your reasoning.

Represent and Interpret Data | 447

Represent and Interpret Data

Skills Quiz

7. If the data point of 6 were deleted, how would that change the mean of the data?

8. A data set with 150 data points is represented on a dot plot. How does the range of the data set relate to the spread of the data set? A. A larger range will result in a larger spread of data. B. A smaller range will result in a larger spread of data. C. The range does not have any effect on the spread of data. D. A larger range will result in a smaller spread of data.

448 | Represent and Interpret Data

Represent and Interpret Data

Skills Quiz

9. Based on the bar graph below, how many total student representatives are there in student council? Representatives in Student Council

Number of responses

6 5

5 4

4 3

3 2

2 1 0

6th grade

7th grade

8th grade

Staff

A. 6 B. 11 C. 14 D. 150

Represent and Interpret Data | 449

Represent and Interpret Data

Skills Quiz 10. Refer to the data in ﬁgure 1 to complete parts A and B. Figure 1 Athletic Girls’ Mile Run Times (minutes) 4.5 7 7.5 4.75 5 5.5 8 7.5 6.5 6.5 6 5.5 4.25 5 7.25 450 | Represent and Interpret Data

Represent and Interpret Data

Skills Quiz Part A. Complete the grouped frequency table for this data. Times in Minutes

Frequency

Between 4.2 and 5.1 Between 5.2 and 6.1 Between 6.2 and 7.1 Between 7.2 and 8.1

Part B. Create a histogram using your answers in the table in part A.

Represent and Interpret Data | 451

Summarize Numerical Data

Skills Quiz

Name: _______________________ Date: ___________

Summarize Numerical Data Directions: Solve each problem. Show or explain your mathematical thinking.

1. Jillian collected data to ﬁnd the arm span of each student in her class. She wants to display the data so that each student can see how their individual measurement compared with the others in the class. Should she choose to represent the data using a dot plot, box plot, or histogram? Explain your reasoning.

2. Anna is analyzing this data set: {10, 12, 7, 9, 8, 14, 13, 2}. What will happen if she adds 20 to the set? A.

The mean will increase.

The median will be the same as the mean.

The mean will decrease.

The mean will stay the same.

Summarize Numerical Data | 453

Summarize Numerical Data

Skills Quiz

A science class is weighing seeds for an experiment. Refer to group A’s data to answer questions 3–5. Group A’s Data Seed

Weight in Grams

1.5

2.5

3. Create a dot plot to represent this data.

454 | Summarize Numerical Data

Skills Quiz

Summarize Numerical Data

4. Calculate the mean and median of group A’s data. Describe how your answers relate to the dot plot you created.

5. Is there an outlier in this data set? Explain your reasoning.

Summarize Numerical Data | 455

Summarize Numerical Data

Skills Quiz

6. The table below shows the absolute value of the distance between each data point and the mean. Determine the mean absolute deviation of the data, and explain what this value means.

456 | Summarize Numerical Data

Data Point

Distance from Mean

32.8 – 22 = 11.8

32.8 – 27 = 5.8

32.8 – 29 = 3.8

32.8 – 32 = 0.8

34 – 32.8 = 1.2

35 – 32.8 = 2.2

39 – 32.8 = 6.2

46 – 32.8 = 13.2

Summarize Numerical Data

Skills Quiz

Sara and her friends were playing basketball. The points they scored in the ﬁrst half of the game are shown in the table below. Use this data to answer questions 7 and 8.

7. Create a box plot to represent this data.

8. Calculate the IQR.

Summarize Numerical Data | 457

Skills Quiz

Summarize Numerical Data

Janiesha’s last 8 test scores were 77, 77, 82, 85, 98, 75, 82, and 80. Use this data set to answer questions 9 and 10.

9. Calculate the IQR.

10. Use the IQR to explain whether or not Janiesha is a consistent test taker and if there are any striking deviations from the overall pattern.

458 | Summarize Numerical Data

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

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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

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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

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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

462

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

463

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

464

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

465

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

466

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

467

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

468

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

469

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 470

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

471

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

472

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

473

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

474

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

475

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

476

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

477

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

478

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

479

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 480

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

481

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

482

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

483

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 484

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

485

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°

486

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

487

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

488

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) 489

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 490

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

491

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

492

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

493

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 494

Workspace

495

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