Expressions, Equations, and Relationships

UNIT 4

Expressions, Equations, and Relationships Contents 6.7.A 6.7.A 6.7.A

6.7.C 6.7.A 6.7.D

6.9.A 6.10.A 6.10.A

6.9.A 6.10.A 6.10.A 6.9.B

MODULE 10

Generating Equivalent Numerical Expressions

Lesson 10.1 Lesson 10.2 Lesson 10.3

Exponents . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 269 Prime Factorization . . . . . . . . . . . . . . . . . . . . . . . . . . . . 275 Order of Operations . . . . . . . . . . . . . . . . . . . . . . . . . . . 281

MODULE 11

Generating Equivalent Algebraic Expressions

Lesson 11.1 Lesson 11.2 Lesson 11.3

Modeling Equivalent Expressions . . . . . . . . . . . . . . . . . 293 Evaluating Expressions . . . . . . . . . . . . . . . . . . . . . . . . . 301 Generating Equivalent Expressions. . . . . . . . . . . . . . . . 307

MODULE 12

Equations and Relationships

Lesson 12.1 Lesson 12.2 Lesson 12.3

Writing Equations to Represent Situations . . . . . . . . . . 321 Addition and Subtraction Equations . . . . . . . . . . . . . . . 327 Multiplication and Division Equations . . . . . . . . . . . . . 335

MODULE 13

Inequalities and Relationships

Writing Inequalities . . . . . . . . . . . . . . . . . . . . . Addition and Subtraction Inequalities . . . . . . Multiplication and Division Inequalities with Positive Numbers . . . . . . . . . . . . . . . . . . . . . . Lesson 13.4 Multiplication and Division Inequalities with Rational Numbers . . . . . . . . . . . . . . . . . . . . . . Lesson 13.1 Lesson 13.2 Lesson 13.3

MODULE 14

6.11 6.6.A 6.6.B 6.6.C

. . . . . . . 349 . . . . . . . 355 . . . . . . . 361 . . . . . . . 367

Relationships in Two Variables

Graphing on the Coordinate Plane . . . . . . . . . . . Independent and Dependent Variables in Tables and Graphs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Lesson 14.3 Writing Equations from Tables . . . . . . . . . . . . . . Lesson 14.4 Representing Algebraic Relationships in Tables and Graphs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Lesson 14.1 Lesson 14.2

. . . . . 379 . . . . . 385 . . . . . 393 . . . . . 399

UNIT 4

Unit Pacing Guide 45-Minute Classes Module 10 DAY 1

DAY 2

DAY 3

DAY 4

DAY 5

Lesson 10.1

Lesson 10.1

Lesson 10.2

Lesson 10.2

Lesson 10.3

DAY 6

DAY 7

Lesson 10.3

Ready to Go On? Texas Test Prep

Module 11 DAY 1

DAY 2

DAY 3

DAY 4

DAY 5

Lesson 11.1

Lesson 11.1

Lesson 11.2

Lesson 11.2

Lesson 11.3

DAY 6

DAY 7

Lesson 11.3

Ready to Go On? Texas Test Prep

Module 12 DAY 1

DAY 2

DAY 3

DAY 4

DAY 5

Lesson 12.1

Lesson 12.1

Lesson 12.2

Lesson 12.2

Lesson 12.3

DAY 6

DAY 7

Lesson 12.3

Ready to Go On? Texas Test Prep

Module 13 DAY 1

DAY 2

DAY 3

DAY 4

DAY 5

Lesson 13.1

Lesson 13.1

Lesson 13.2

Lesson 13.2

Lesson 13.3

DAY 6

DAY 7

DAY 8

DAY 9

Lesson 13.3

Lesson 13.4

Lesson 13.4

Ready to Go On? Texas Test Prep

DAY 1

DAY 2

DAY 3

DAY 4

DAY 5

Lesson 14.1

Lesson 14.1

Lesson 14.2

Lesson 14.2

Lesson 14.3

Module 14

DAY 6

DAY 7

DAY 8

DAY 9

Lesson 14.3

Lesson 14.4

Lesson 14.4

Ready to Go On? Texas Test Prep

Expressions, Equations, and Relationships

263B

Program Resources Plan

Engage and Explore

Online Teacher Edition Access a full suite of teaching resources online— plan, present, and manage classes, assignments, and activities.

Real-World Videos Engage students with interesting and relevant applications of the mathematical content of each module.

ePlanner Easily plan your classes, create and view assignments, and access all program resources with your online, customizable planning tool.

Animated Math Online interactive simulations, tools, and games help students actively learn and practice key concepts.

Professional Development Videos Author Juli Dixon models successful teaching practices and strategies in actual classroom settings.

QR Codes Scan with your smart phone to jump directly from your print book to online videos and other resources.

Explore Activities Students interactively explore new concepts using a variety of tools and approaches.

Teacher’s Edition Support students with point-of-use Questioning Strategies, teaching tips, resources for differentiated instruction, additional activities, and more.

LESSON

7.2 Rates ?

ESSENTIAL QUESTION How do you use rates to compare quantities?

6.4.D

EXPLORE ACTIVITY

LESSON LESSON

Rates

The student is expected to: Proportionality—6.4.D Give examples of rates as the comparison by division of two quantities having different attributes, including rates as quotients.

Mathematical Processes

?

Engage

Calculating Unit Rates

6.4.D Give examples of

rates as the comparison by division of two quantities having different attributes, including rates as quotients. Math On the Spot my.hrw.com

ESSENTIAL QUESTION How do you use rates to compare quantities?

ESSENTIAL QUESTION How do you use rates to compare quantities? Sample answer: You use division to compare two quantities with different units.

Yoga Classes This month’s special:

6 classes for \$90

Using Rates to Compare Prices A rate is a comparison of two quantities that have different units.

Motivate the Lesson

6 classes

Engage with the Whiteboard Have students fill in the table for each brand on the whiteboard. Ask students to write a rate for each brand, using the data from the last row of each table. Then have them compare that rate to the original rate in the problem statement. Help students to see that the two rates are equivalent.

B The cost of 2 cartons of milk is \$5.50. What is the unit price?

C An airplane makes a 2,748-mile flight in 6 hours. What is the airplane’s average rate of speed in miles per hour? 458 miles per hour Interactive Whiteboard Interactive example available online my.hrw.com

The unit price is \$2.75 per carton of milk.

3.84

÷2

÷2

8

1.92

4

0.96 0.48 0.24

÷2

÷2

2

÷2

1

B Brand A costs \$

0.24

÷5 ÷5

÷2

Ounces

Price (\$)

25

4.50

5

0.90 0.18

1

The first quantity in a unit rate can be less than 1.

÷5 ÷5

÷2

The ship travels 0.4 mile per minute.

They are equivalent. per ounce.

YOUR TURN 3. There are 156 players on 13 teams. How many players are on each

Analyze Relationships Describe another method to compare the costs.

team? Personal Math Trainer Online Assessment and Intervention

EXAMPLE 1

To compare the costs, Shana must compare prices for equal amounts of juice. How can she do this?

÷50

Analyze Relationships In all of these problems, how is the unit rate related to the rate given in the original problem?

B; it costs less per ounce.

Reflect 1.

1 carton

2 cartons

Reflect

÷2

0.18

\$5.50 \$2.75 ________ = _______

C A cruise ship travels 20 miles in 50 minutes. 20 miles = ________ 0.4 mile __________ How far does the ship travel per minute? 50 minutes 1 minute

2.

per ounce. Brand B costs \$

C Which brand is the better buy? Why?

Shana is at the grocery store comparing two brands of juice. Brand A costs \$3.84 for a 16-ounce bottle. Brand B costs \$4.50 for a 25-ounce bottle.

÷50

Brand B

Price (\$)

16

\$15 The unit rate _____ 1 class is the same as 15 ÷ 1 = \$15 per class.

÷2

Divide the cost of each bottle by the amount of juice; 3.84 4.50 A: ____ = 0.24; B: ____ = 0.18 16 25

Explain B Michael walks 30 meters in 20 seconds. How many meters does he walk per second? 1.5 meters per second

© Houghton Mifflin Harcourt Publishing Company

EXPLORE ACTIVITY

1 class

÷6

Brand A Ounces

107 miles . The rate is _______ 2 hours

÷6

\$15 \$90 ________ = ______

Gerald’s yoga classes cost \$15 per class.

A Complete the tables.

÷2

Chris drove 107 miles in two hours. You are comparing miles and hours.

To find the unit rate, divide both quantities in the rate by the same number so that the second quantity is 1:

107 miles . The rate is _______ 2 hours

Shana is at the grocery store comparing two brands of juice. Brand A costs \$3.84 for a 16-ounce bottle. Brand B costs \$4.50 for a 25-ounce bottle.

Explore

Connect Vocabulary

A rate is a comparison by division of two quantities that have different units.

6.4.D

\$90

Use the information in the problem to write a rate: _______ 6 classes

Chris drove 107 miles in two hours. You are comparing miles and hours.

Ask: Have you ever wanted to find out which was the best buy between two products when shopping or find out how fast you are walking or running? Begin the Explore Activity to find out how to compare quantities with different units.

To compare the costs, Shana must compare prices for equal amounts of juice. How can she do this?

ADDITIONAL EXAMPLE 1 A The cost of 3 candles is \$19.50. What is the unit price? \$6.50 per candle

EXAMPLE 1 A Gerald pays \$90 for 6 yoga classes. What is the cost per class?

6.4.D

EXPLORE ACTIVITY

6.1.G Display, explain, and justify mathematical ideas and arguments using precise mathematical language in written or oral communication.

Using Rates to Compare Prices

A unit rate is a rate in which the second quantity is one unit. When the first quantity in a unit rate is an amount of money, the unit rate is sometimes called a unit price or unit cost.

12

A Complete the tables. Brand A

photograph? \$

0.50

per photograph

my.hrw.com

ELL

Lesson 7.2

187

188

Unit 3

Remind students that a ratio is a comparison of two quantities expressed with the same units of measure, and a rate is a comparison of two quantities with different units of measure. A rate in which the second quantity is one unit is a unit rate.

Questioning Strategies

Mathematical Processes • How do you determine what number to divide by when finding a unit rate? Divide both quantities by the same number so that the second quantity is 1.

• How is finding a unit rate like simplifying a fraction? You find unit rates by dividing both quantities by the same number, just as you would to simplify a fraction.

PROFESSIONAL DEVELOPMENT Integrate Mathematical Processes This lesson provides an opportunity to address Mathematical Process TEKS 6.1.G, which calls for students to “display, explain, and justify mathematical ideas and arguments using precise mathematical language in written or oral communication.” In the Explore Activity, students

Math Background The words ratio and rate come from the Latin word ratus, which means “calculation.” A unit rate is a rate expressed in its simplest form, a to b, where a may or may not be a whole number and b is 1. The terms have only 1 as a common factor. Many real-world situations involve the use of rate

Brand B

Ounces

Price (\$)

÷2

16

3.84

÷2

8

1.92

4

0.96 0.48 0.24

players per team

4. A package of 36 photographs costs \$18. What is the cost per

lin Harcourt Publishing Company

Texas Essential Knowledge and Skills

Proportionality—

7.2 Rates

© Houghton Mifflin Harcourt Publishing Company

7.2

Proportionality— 6.4.D Give examples of rates as the comparison by division of two quantities having different attributes, including rates as quotients.

÷2 ÷2

2 1

B Brand A costs \$

0.24

Ounces ÷2 ÷2

÷5 ÷5

÷2

Price (\$)

25

4.50

5

0.90 0.18

1

÷2

per ounce. Brand B costs \$

0.18

per ounce.

B; it costs less per ounce.

÷5 ÷5

3 2 1

Teach

Assessment and Intervention

Math on the Spot video tutorials, featuring program authors Dr. Edward Burger and Martha Sandoval-Martinez, accompany every example in the textbook and give students step-by-step instructions and explanations of key math concepts.

The Personal Math Trainer provides online practice, homework, assessments, and intervention. Monitor student progress through reports and alerts. Create and customize assignments aligned to specific lessons or TEKS. • Practice – With dynamic items and assignments, students get unlimited practice on key concepts supported by guided examples, step-by-step solutions, and video tutorials. • Assessments – Choose from course assignments or customize your own based on course content, TEKS, difficulty levels, and more. • Homework – Students can complete online homework with a wide variety of problem types, including the ability to enter expressions, equations, and graphs. Let the system automatically grade homework, so you can focus where your students need help the most! • Intervention – Let the Personal Math Trainer automatically prescribe a targeted, personalized intervention path for your students.

Present engaging content on a multitude of devices, including tablets and interactive whiteboards. Continually monitor and assess student progress with integrated formative assessment.

Math Talk

Differentiated Instruction Print Resources Support all learners with Differentiated Instruction Resources, including • Leveled Practice and Problem Solving • Reteach • Reading Strategies • Success for English Learners • Challenge

Problem Solving

Calculating Unit Rates Math On the Spot my.hrw.com

You can solve rate

A unit rate is a rate in which the second quantity is one unit. When the first quantity in a unit rate is an amount of money, the unit rate is sometimes called a unit price or unit cost.

EXAMPLE 1

6.4.D

This month’s special:

6 classes for \$90

To find the unit rate, divide both quantities in the rate by the same number so that the second quantity is 1:

1 class

÷2

The unit price is \$2.75 per carton of milk.

112 camp _____ ers __ _____ _____ 8 campers per cabin = 14 cabins Method 2 Use equiv alent rates.

\$5.50 \$2.75 ________ = _______ 1 carton

2 cartons

The ship travels 0.4 mile per minute.

Divide to find the unit

rate.

÷2

Divide to find the numbe r of cabins.

16 camp ers _____ _____

112 campers _____ 2 cabins = 14 ______ cabins ×7

The camp needs 14

÷50

Animated Math

cabins.

my.hrw.com

Reflect Analyze Relationships In all of these problems, how is the unit rate related to the rate given in the original problem?

3. There are 156 players on 13 teams. How many players are on each players per team

4. A package of 36 photographs costs \$18. What is the cost per

in Harcourt Publishing Company

They are equivalent.

© Houghton Mifflin Harcourt Publishing Company

Reflect

Assessment Resources Tailor assessments to meet the needs of all your classes and students, including • Leveled Module Quizzes • Leveled Unit Tests • Unit Performance Tasks • Placement, Diagnostic, and Quarterly Benchmark Tests

×7

÷50

12

my.hrw.com

ers per cabin.

C A cruise ship travels 20 miles in 50 minutes. 20 miles = ________ 0.4 mile __________ How far does the ship travel per minute? 50 minutes 1 minute

team?

Math On the Spot

There are 8 camp

÷2

Personal

rate or by using equiv alent rates.

÷2

8 camp ers _____ ____ 2 cabins = 1 cabin

\$15 The unit rate _____ 1 class is the same as 15 ÷ 1 = \$15 per class.

Gerald’s yoga classes cost \$15 per class.

B The cost of 2 cartons of milk is \$5.50. What is the unit price?

2.

Prepare students with practice similar to the Texas assessment program at every module and unit.

16 camp ers _____ _____

÷6

\$90 \$15 ________ = ______ ÷6

The first quantity in a unit rate can be less than 1.

Texas Test Prep

6.4.D At a summer camp , the campers are divided into grou 16 campers and ps. Each group has 2 cabins. How many cabins are needed for 112 campers? Method 1 Find the unit rate. How many campers per cabin?

\$90

Use the information in the problem to write a rate: _______ 6 classes

6 classes

Raise the bar with homework and practice that incorporates higher-order thinking and mathematical processes in every lesson.

with Unit Rates

problems by using a unit

EXAMPLE 2

A Gerald pays \$90 for 6 yoga classes. What is the cost per class?

Yoga Classes

Response to Intervention

5.

What If? Suppose each group has 12 campers and 3 canoe unit rate of camp ers to canoes. s. Find the 12

÷ 3 = 4; there are

4 camper per cano

4 camp ers ____ e or ____ . 1 canoe

Petra jogs 3 miles in 27 minutes. At this rate, how long to jog 5 would

miles? 27 minu ____ tes____ ____ ÷3

9 minute __

it take her

Expressions, Equations, and Relationships

263D

Math Background Algebraic Expressions

TEKS 6.7.B, 6.7.C

LESSONS 11.1 to 11.3 An algebraic expression is a mathematical statement constructed from at least one variable. It may have one or more operation symbols and one or more numbers. The table shows examples and nonexamples of algebraic expressions. Algebraic Expressions Examples

x, 3y - __25 , -2xy, z 2 + 1

Nonexamples

7, 20 ÷ (4 + 1), 4x = 16

Note that an algebraic expression does not contain an equal sign. A mathematical statement that contains an equal sign is an equation. Some students may have trouble understanding that algebraic expressions can be represented in multiple equivalent forms, just like numbers and numerical expressions. Students should recognize that the following 5x . algebraic expressions are equivalent: x + 0, x, 1x, __ 5

Writing One-Variable Equations TEKS 6.7.B, 6.9.A

LESSON 12.1 Students must be able to translate English phrases and sentences into algebraic symbols. To avoid misunderstandings, there are conventions we all use when translating from words to math. The phrase “the difference of 3 and 7” translates to 3 - 7. An equation is a mathematical statement that two quantities are equal. An equation may involve only numbers, as in 6 + 5 = 11, or may have algebraic expressions, as in 2x = 6. When an equation contains one variable, the solution of the equation is a value of the variable that makes the equation true. For instance, x = 3 is the solution of 2x = 6 since 2(3) = 6 is a true statement. An equation such as 2x = 6 is sometimes called an open sentence. That is, it is neither true nor false until additional information (i.e., a value of x) is given. When x is replaced by

263E

a value that is a solution, the open sentence becomes a true statement. If the given value of x is not a solution, the open sentence becomes a false statement. Students should know that an equation may have no solutions, one solution, more than one solution, or infinitely many solutions. For example, the equation x + 5 = 2 + x + 3 has infinitely many solutions. Every real number is a solution of this equation. Such an equation is called an identity.

Solving One-Variable Equations TEKS 6.9.A, 6.9.B, 6.9.C, 6.10.A, 6.10.B

LESSONS 12.2 to 12.3 An equation is like a scale that is perfectly balanced. The quantities on both sides have exactly the same weight. When two quantities a and b cause a scale to balance, the same quantity c can be added to both sides of the scale while preserving the balance. Applying this idea to equations yields the Addition Property of Equality: If a = b, then a + c = b + c. Similarly, it is possible to subtract the same quantity c from both sides of the scale and preserve the balance. Applying this idea to equations gives the Subtraction Property of Equality: If a = b, then a – c = b – c. The Multiplication Property of Equality states that multiplying each side of an equation by the same nonzero number produces a new equation that has the same solutions as the original. In other words, if a = b and c ≠ 0, then ac = bc. (Strictly speaking, multiplying both sides by a constant c = 0 results in a true equation, 0 = 0, but this is not useful because we lose whatever information the original equation contained.) The Division Property of Equality states that dividing each side of an equation by the same nonzero number produces a new equation that has the same solutions as the original. That is, if a = b and c ≠ 0, then __ac = __bc .

Writing One-Variable Inequalities TEKS 6.9.A, 6.9.B, 6.9.C

LESSON 13.1 One of the essential skills of algebra is translating words into mathematics. This can be challenging for many students, especially those for whom English is a second language. Perhaps the most common error is the attempt to make a direct word-to-symbol translation that preserves the order of the words. Although this method works in many cases, it can cause problems. Consider the statement “5 less than a number n is greater than 2.” A student making the “word-order” error might translate this incorrectly as 5 - n > 2. The correct translation is n - 5 > 2. Students should realize that they can check their mathematical translations in much the same way that they can check a solution. In the above example, it is helpful to choose a specific value for n, such as 20, and ask whether “5 less than 20” is represented by 5 - 20.

Solving One-Variable Inequalities TEKS 6.9.B, 6.10.A, 6.10.B

LESSONS 13.2 to 13.4

The Addition and Subtraction Properties of Inequality state that the same quantity may be added to or subtracted from both sides of an inequality without changing the solution set. That is, if a > b, then a + c > b + c and a - c > b - c. Multiplying or dividing both sides of an inequality by a positive number also produces an inequality with the same solution set as the original inequality. In general terms, if a > b and c > 0, then ac > bc and __ac = __bc . When multiplying or dividing both sides of an inequality by a negative number, however, the inequality symbol must be reversed. Thus, if a > b and c < 0, then ac < bc and __ac = __bc . It is often helpful for students to check the solution of an inequality by substituting specific values for the variable. To check that n ≥ -10 is the solution of -18n ≤ 180, choose a value of n that is greater than or equal to -10, such as -2. In the original inequality, this gives -18(-2) ≤ 180 or 36 ≤ 180, which is a true inequality. Checking a single value can never guarantee that a solution to an inequality is correct, but it can help students catch some errors.

Solving an inequality in one variable is similar to solving an equation in one variable. The goal is to isolate the variable on one side of the inequality by writing a series of inequalities that have the same solution set.

Expressions, Equations, and Relationships

263F

UNIT 4

Expressions, Equations, and Relationships

MODULE MODULE

10 10

Generating Equivalent Numerical Expressions 6.7.A MODULE MODULE

11 11

Generating Equivalent Algebraic Expressions 6.7.A, 6.7.C, 6.7.D MODULE MODULE

12 12

Equations and Relationships 6.7.B, 6.9, 6.10

13 13 Inequalities and

MODULE MODULE

Relationships 6.9, 6.10

14 14 Relationships in Two

© Houghton Mifflin Harcourt Publishing Company • Image Credits: Andy Sotiriou/Photodisc/Getty Images

MODULE MODULE

Variables

6.6.A, 6.6.B, 6.6.C, 6.11

CAREERS IN MATH

Unit 4 Performance Task At the end of the unit, check out how botanists use math.

Botanist A botanist is a biologist who studies plants. Botanists use math to analyze data and create models of biological organisms and systems. They use these models to make predictions. They also use statistics to determine correlations. If you are interested in a career in botany, you should study these mathematical subjects: • Algebra • Trigonometry • Probability and Statistics • Calculus Research other careers that require the analysis of data and use of mathematical models.

Unit 4

263

UNIT 4

Careers in Math

Vocabulary

Botanist A botanist uses math to find correlations and to predict future results. You will learn more about using math to make predictions in the Performance Tasks at the end of the unit. For more information about careers in mathematics as well as various mathematics appreciation topics, visit the American Mathematical Society at www.ams.org

Preview

Use the puzzle to preview key vocabulary from this unit. Unscramble the circled letters within found words to answer the riddle at the bottom of the page.

T N E I C I F F E O C Z U S L

Vocabulary Preview Integrating the ELPS Use the puzzle to give students a preview of important concepts in this unit. Students may work individually, in pairs, or in groups. c.4.D Use prereading supports such as graphic organizers, illustrations, and pretaught topic-related vocabulary to enhance comprehension of written text.

O E K O F B S D E O R H B B F

Y S O F P N O X X W U X X F W

H S P J Z U X V P S T R F Q W

S V X S O A A X O S L H U O Z

D L U F E M F L N H O O B E T

F E C P E T U D E Z O P Y K V

P B D M U T A L N W P L H Y F

P C R Y I Z D N T T R Y J K O

O E A O N O C V I Y S C F P P

T R N Q V Q P J O D S U K H U

H V I H Z C Z S Q Q R X P C H

T N U G W C H A X E S O U S U

• A number that is multiplied by a variable in an algebraic expression. (Lesson 11-1) • A value of the variable that makes the equation true. (Lesson 12-1)

J N P C I J L V W B J Q O N U

J F N I C N P K B T A C E C B

coefficient

solution

• The point where the axes intersect to form the coordinate plane. (Lesson 14-1) origin • The part of an expression that is added or subtracted. (Lesson 11-1) term • The two number lines that intersect at right angles to form a coordinate plane. (Lesson 14-1) axes • Tells how many times the base is used in the product. (Lesson 10-1) exponent

Q: Unit Resources

my.hrw.com

Before Students understand: • operations with whole numbers, decimals, and fractions • order of operations • properties of operations: inverse, identity, commutative, associative, and distributive properties • graphs in the first quadrant

Go online to access all your unit resources.

In this Unit Students will learn about: • exponents • prime factorizations • numerical and algebraic expressions • equations and inequalities • the coordinate plane

Why did the paper rip when the student tried to stretch out the horizontal axis of his graph?

A: 264

© Houghton Mifflin Harcourt Publishing Company

• The numbers in an ordered pair. (Lesson 14-1) coordinates

Too much X – T

E N S

I

O N!

Vocabulary Preview

After Students will learn how to: • evaluate algebraic expressions with more than one variable • write two-step equations and inequalities to represent real-world problems and write a real-world problem to represent an equation or inequality • solve two-step equations and inequalities • graph linear equations in the form y = mx + b on the coordinate plane

Expressions, Equations, and Relationships

264

Generating Equivalent Numerical Expressions How can you generate equivalent numerical expressions and use them to solve real-world problems?

© Houghton Mifflin Harcourt Publishing Company • Image Credits: Vladimir Ivanovich Danilov / Shutterstock.com

?

ESSENTIAL QUESTION

Module 10

10

LESSON 10.1

Exponents 6.7.A

You can represent real-world problems with numerical expressions and simplify the expressions by applying rules relating to exponents, prime factorization, and order of operations.

LESSON 10.2

Prime Factorization 6.7.A

LESSON 10.3

Order of Operations 6.7.A

Real-World Video

my.hrw.com

my.hrw.com

265

MODULE

Assume that you post a video on the internet. Two of your friends view it, then two friends of each of those view it, and so on. The number of views is growing exponentially. Sometimes we say the video went viral.

my.hrw.com

Math On the Spot

Animated Math

Personal Math Trainer

Go digital with your write-in student edition, accessible on any device.

Scan with your smart phone to jump directly to the online edition, video tutor, and more.

Interactively explore key concepts to see how math works.

Get immediate feedback and help as you work through practice sets.

265

DO NOT EDIT--Changes must be made through "File info" CorrectionKey=B

Complete these exercises to review skills you will need for this chapter.

Use the assessment on this page to determine if students need intensive or strategic intervention for the module’s prerequisite skills.

Whole Number Operations

2 1

Response to Intervention

1. 992 × 16

Enrichment

my.hrw.com

2. 578 × 27

3 × 270 80 × 270 (3 × 270) + (80 × 270)

3. 839 × 65

15,606

15,872

Access Are You Ready? assessment online, and receive instant scoring, feedback, and customized intervention or enrichment.

Online Assessment and Intervention

← ← ←

Find the product.

Intervention

Personal Math Trainer

270 × 83 810 + 21,600 22,410

Online Assessment and Intervention

my.hrw.com

4. 367 × 23

54,535

8,441

Use Repeated Multiplication EXAMPLE

5×5× 5× 5

↓

25 × 5

Online and Print Resources

Multiply the first two factors.

Multiply the result by the next factor.

Multiply that result by the next factor.

125 × 5

Skills Intervention worksheets

Differentiated Instruction

• Skill 34 Whole Number Operations

• Challenge worksheets

• Skill 35 Use Repeated Multiplication

Extend the Math PRE-AP Lesson Activities in TE

Continue until there are no more factors to multiply.

625 Find the product.

PRE-AP

5. 7×7×7

6. 3×3×3×3

343

7. 6×6×6×6×6

7,776

81

8. 2×2×2×2×2×2

64

© Houghton Mifflin Harcourt Publishing Company

3

270 × 83

EXAMPLE

Personal Math Trainer

Division Facts

• Skill 38 Division Facts

EXAMPLE

54 ÷ 9 =

Think:

54 ÷ 9 = 6

So, 54 ÷ 9 = 6.

9 times what number equals 54? 9 × 6 = 54

Divide. 9. 20 ÷ 4

5

266

10. 21 ÷ 7

3

11. 42 ÷ 7

6

12. 56 ÷ 8

7

Unit 4

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PROFESSIONAL DEVELOPMENT VIDEO my.hrw.com

Author Juli Dixon models successful teaching practices as she explores equivalent numerical expressions in an actual sixth-grade classroom.

Online Teacher Edition Access a full suite of teaching resources online—plan, present, and manage classes and assignments.

Professional Development

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Interactive Answers and Solutions Customize answer keys to print or display in the classroom. Choose to include answers only or full solutions to all lesson exercises.

Interactive Whiteboards Engage students with interactive whiteboard-ready lessons and activities. Personal Math Trainer: Online Assessment and Intervention Assign automatically graded homework, quizzes, tests, and intervention activities. Prepare your students with updated, TEKS-aligned practice tests.

Generating Equivalent Numerical Expressions

266

Have students complete the activities on this page by working alone or with others.

Vocabulary Review Words ✔ factor (factor) factor tree (árbol de factores) ✔ integers (entero) ✔ numerical expression (expresión numérica) ✔ operations (operaciones) ✔ prime factorization (factorización prima) repeated multiplication (multiplicación repetida) simplified expression (expresión simplificada)

Visualize Vocabulary Use the ✔ words to complete the graphic. You may put more than one word in each box.

Visualize Vocabulary

Reviewing Factorization

The sequence diagram helps students review vocabulary associated with factorization and prepares them to work with exponents. After students complete the diagram, discuss the vocabulary as a class.

Factor tree

24

factor, prime factorization

integer

Understand Vocabulary Use the following explanation to help students learn the preview words. You may hear the terms power and exponent used in place of each other. However, they do not mean the same thing. An exponent is the number that is written beside and slightly above the base. It tells you how many times to use the base as a factor. A power is a number that is formed by repeated multiplication by the same factor (the base) and can be represented as the base with an exponent.

Integrating the ELPS Students can use these reading and note-taking strategies to help them organize and understand new concepts and vocabulary. c.4.D Use prereading supports such as graphic organizers, illustrations, and

pretaught topic-related vocabulary to enhance comprehension of written text.

8×3

integers, factors, operations, numerical expression

Preview Words base (base) exponent (exponente) order of operations (orden de las operaciones) power (potencia)

Understand Vocabulary Complete the sentences using the preview words.

1. A number that is formed by repeated multiplication by the same

power

factor is a

2. A rule for simplifying expressions is © Houghton Mifflin Harcourt Publishing Company

2×2×2×3

integers, factors, operations, numerical expression, prime factorization

3. The

base

.

order of operations

.

is a number that is multiplied. The number that

indicates how many times this number is used as a factor is the

exponent

.

Active Reading Three-Panel Flip Chart Before beginning the module, create a three-panel flip chart to help you organize what you learn. Label each flap with one of the lesson titles from this module. As you study each lesson, write important ideas like vocabulary, properties, and formulas under the appropriate flap.

Differentiated Instruction • Reading Strategies ELL Module 10

Grades 6–8 TEKS Before Students understand: • operations with whole numbers, decimals, and fractions • prime numbers • order of operations

267

Module 10

In this module Students will learn to: • generate equivalent numerical expressions using exponents • generate equivalent numerical expressions using prime factorization • simplify numerical expressions using the order of operations

After Students will connect: • order of operations and numerical expressions • numerical and algebraic expressions

267

MODULE 10

Unpacking the TEKS

Unpacking the TEKS Understanding the TEKS and the vocabulary terms in the TEKS will help you know exactly what you are expected to learn in this module.

6.7.A Generate equivalent numerical expressions using order of operations, including whole number exponents and prime factorization.

Texas Essential Knowledge and Skills Content Focal Areas

Key Vocabulary exponent (exponente) The number that indicates how many times the base is used as a factor. order of operations (orden de las operaciones) A rule for evaluating expressions: first perform the operations in parentheses, then compute powers and roots, then perform all multiplication and division from left to right, and then perform all addition and subtraction from left to right.

Expressions, equations, and relationships—6.7 The student applies mathematical process standards to develop concepts of expressions and equations.

Integrating the ELPS c.4.F Use visual and contextual support … to read grade-appropriate content area text … and develop vocabulary … to comprehend increasingly challenging language.

What It Means to You You will simplify numerical expressions using the order of operations. UNPACKING EXAMPLE 6.7.A

Ellen is playing a video game in which she captures frogs. There were 3 frogs onscreen, but the number of frogs doubled every minute when she went to get a snack. She returned after 4 minutes and captured 7 frogs. Write an expression for the number of frogs remaining. Simplify the expression. 3×2

number of frogs after 1 minute

3×2×2

number of frogs after 2 minutes

3×2×2×2

number of frogs after 3 minutes

3×2×2×2×2

number of frogs after 4 minutes

Since 3 and 2 are prime numbers, 3 × 2 × 2 × 2 × 2 is the prime factorization of the number of frogs remaining. 3 × 2 × 2 × 2 × 2 can be written with exponents as 3 × 24. The expression 3 × 24 – 7 is the number of frogs remaining after Ellen captured the 7 frogs. Use the order of operations to simplify 3 × 24 – 7.

Go online to see a complete unpacking of the .

3 × 24 – 7 = 3 × 16 – 7 = 48 – 7 = 41 41 frogs remain.

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© Houghton Mifflin Harcourt Publishing Company • Image Credits: Patrik Giardino/ Photodisc/Getty Images

Use the examples on this page to help students know exactly what they are expected to learn in this module.

Visit my.hrw.com to see all the unpacked. my.hrw.com

268

Lesson 10.1

Lesson 10.2

Unit 4

Lesson 10.3

6.7.A Generate equivalent numerical expressions using order of operations, including whole number exponents and prime factorization.

Generating Equivalent Numerical Expressions

268

LESSON

10.1 Exponents Texas Essential Knowledge and Skills The student is expected to: Expressions, equations, and relationships—6.7.A Generate equivalent numerical expressions using order of operations, including whole number exponents and prime factorization.

Mathematical Processes

Engage ESSENTIAL QUESTION How do you use exponents to represent numbers? Sample answer: You can use exponents to represent repeated multiplication. For example, in 5 × 5 × 5 × 5, the number 5 is multiplied 4 times, so you can represent it as 5 4.

Motivate the Lesson Ask: Have you ever heard the terms squared or cubed? Both of those expressions are used to describe exponents. Do you know what 3 squared means? Take a guess. Begin the Explore Activity to find out.

6.1.D Communicate mathematical ideas, reasoning, and their implications using multiple representations, including symbols, diagrams, graphs, and language as appropriate.

Explore EXPLORE ACTIVITY Engage with the Whiteboard Have students fill in the table on the whiteboard. Extend the table to include 8 hours, and have students fill in the table for 5, 6, 7, and 8 hours. Ask students if they could predict the total number of bacteria for hour 12 and hour 20. Discuss a rule that students could use to make those kinds of predictions.

Explain ADDITIONAL EXAMPLE 1 Use an exponent to write each expression. A 7 × 7 × 7 × 7 × 7 × 7 76 5 B __23 × __23 × __23 × __23 × __23 ( __23 )

Interactive Whiteboard Interactive example available online my.hrw.com

EXAMPLE 1 c.1.F ELL Students may know other definitions of the words raised and base. Point out that in math, raised means “to multiply by itself,” and base can mean “the foundation,” or that on which something is built. In 24, you build on the base 2 by multiplying it by itself 4 times.

Connect Vocabulary

Questioning Strategies

Mathematical Processes

4 • In B, why is the base __4 in parentheses in the power ( __4 ) ? The base is in parentheses to

5

5

show that the entire fraction is used for repeated multiplication, not just the numerator.

• Can a power have a base and an exponent that are the same number? Justify your answer. Yes; for example, 6 × 6 × 6 × 6 × 6 × 6 = 6 6.

YOUR TURN Avoid Common Errors Students may want to find the product for each expression. Review the direction line. They are not asked to simplify the expression, but to write it in exponential form.

Talk About It Check for Understanding Ask: What is the difference between the two numbers in a power? The first number is the base, which is the number that is multiplied. The second number is the exponent, which tells how many times the base is multiplied by itself.

269

Lesson 10.1

LESSON

Expressions, equations, and relationships—6.7.A Generate equivalent numerical expressions using… exponents.

10.1 Exponents

A number that is formed by repeated multiplication by the same factor is called a power. You can use an exponent and a base to write a power. For example, 73 means the product of three 7s:

Math On the Spot

73 = 7 × 7 × 7

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ESSENTIAL QUESTION How do you use exponents to represent numbers?

The base is the number that is multiplied. 6.7.A

EXPLORE ACTIVITY

Power

Identifying Repeated Multiplication A real-world problem may involve repeatedly multiplying a factor by itself. A scientist observed the hourly growth of bacteria and recorded his observations in a table.

© Houghton Mifflin Harcourt Publishing Company • Image Credits: D. Hurst/Alamy

Time (h) 0 1

6 squared, 6 to the power of 2, 6 raised to the 2nd power

73

7 cubed, 7 to the power of 3, 7 raised to the 3rd power

94

9 to the power of 4, 9 raised to 4th power

6.7.A

Use an exponent to write each expression. After 2 hours, there are 2 · 2 = ? bacteria.

2

2× 2=

3

2× 2× 2=

4

2× 2× 2× 2=

4

A 3×3×3×3×3

Math Talk

Find the base, or the number being multiplied. The base is 3.

Mathematical Processes

What is the value of a number raised to the power of 1?

8 16

When a number is raised to the power of 1, the value of the power is equal to the base.

A Complete the table. What pattern(s) do you see in the Total bacteria column?

Sample answer: Each number is 2 times the previous number. B Complete each statement. At 2 hours, the total is equal to the product of two 2s. At 3 hours, the total is equal to the product of

three

2s.

At 4 hours, the total is equal to the product of

four

2s.

Find the exponent by counting the number of 3s being multiplied. The exponent is 5. 3 × 3 × 3 × 3 × 3 = 35 5 factors of 3

B

4 _ _ × 45 × _45 × _45 5

Find the base, or the number being multiplied. The base is _45. Find the exponent by counting the number of times _45 appears in the expression. The exponent is 4.

( )

4 4 _ _ × 45 × _45 × _45 = _45 5 4 __

4 factors of

5

Reflect 1.

62

EXAMPLE 1

Total bacteria 1 2

The exponent tells how many times the base appears in the expression.

© Houghton Mifflin Harcourt Publishing Company

?

Using Exponents

Use exponents to write each expression.

Communicate Mathematical Ideas How is the time, in hours, related to the number of times 2 is used as a factor?

Personal Math Trainer

The number of hours is the number of times the factor

Online Assessment and Intervention

2 is repeated.

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Lesson 10.1

269

270

2. 4 × 4 × 4 4. _18 × _18

43

( )

1 2 _ 8

3. 6

61

5. 5 × 5 × 5 × 5 × 5 × 5

56

Unit 4

PROFESSIONAL DEVELOPMENT Integrate Mathematical Processes This lesson provides an opportunity to address Mathematical Process TEKS 6.1.D, which calls for students to “communicate mathematical ideas, …using multiple representations, including symbols, diagrams, graphs, and language as appropriate.” Students first use tables to identify patterns involving repeated multiplication. They then use exponents to rewrite expressions that involve repeated multiplication. Finally, students find the value of expressions that are written with exponents. This process helps students understand multiple ways to represent and use exponents.

Math Background The use of the terms squared and cubed is directly related to the measurements of area and volume. The area of a square with sides 5 units long is found by multiplying, 5 × 5, or 52, or 5 squared. The volume of a cube with sides 5 units long is found by multiplying, 5 × 5 × 5, or 53, or 5 cubed.

Exponents

270

EXAMPLE 2

Find the value of each power.

Avoid Common Errors

A 35 243

Since powers relate to multiplication, students may confuse powers with simple multiplication. After students evaluate each power, have them compare it to a simple multiplication problem to show that the two are not equal. For example, they find that 104 = 10,000, ask them to find 10 × 4 = 40. Since the two expressions have different answers, it should be clear that 104 and 10 × 4 are not equivalent.

0

B 17 1 C (-4)3 -64 D 0.62 0.36 Interactive Whiteboard Interactive example available online my.hrw.com

Questioning Strategies

Mathematical Processes • If a > b, which is greater: 2a or 2b ? Explain. 2a; If the exponent a is a bigger number than the exponent b, students find then the base 2 is used as a factor more times.

• If c > d, which is greater: c3 or d3? Explain. c3; The exponent with the greater base has to be greater if the exponent is the same. For example: 53 = 5 × 5 × 5 = 125, while 43 = 4 × 4 × 4 = 64. c.4.E ELL Encourage English learners to use the active reading strategies and the illustrated, bilingual glossary as they encounter new terms and concepts.

Integrating the ELPS

YOUR TURN Engage with the Whiteboard Have students rewrite each power as repeated multiplication. Seeing the power expressed as repeated multiplication can make it easier for students to find the correct value.

Elaborate Talk About It Summarize the Lesson Ask: How can you use an exponent to represent repeated multiplication? How can you find the value of a power? You can write a repeated multiplication, such as 23 = 2 × 2 × 2. To find the value of a power, rewrite the expression without using exponents by multiplying the base the number of times shown in the exponent—for example, 54 = 5 × 5 × 5 × 5 = 625.

GUIDED PRACTICE Engage with the Whiteboard For Exercise 1, have students complete the table on the whiteboard. Discuss different methods students may have for finding the value of each power, such as using parentheses to group the repeated multiplication or multiplying the value of the previous power by 5.

Avoid Common Errors Exercises 2–5 Remind students that their answers should be expressed as a power, not as the product of a repeated multiplication. Exercises 6–20 Some students may multiply a base by its exponent instead of using the base as a factor the number of times indicated by the exponent. Remind them that 43 means that 4 is used as a factor 3 times (4 × 4 × 4). Exercise 15 If students get the answer 8, remind them that the Property of Zero as an Exponent states that the value of any nonzero number raised to the power of 0 is 1.

271

Lesson 10.1

DO NOT EDIT--Changes must be made through “File info” CorrectionKey=A

DO NOT EDIT--Changes must be made through “File info” CorrectionKey=B

Guided Practice

Finding the Value of a Power

1. Complete the table. (Explore Activity)

To find the value of a power, remember that the exponent indicates how many times to use the base as a factor.

Exponential form Math On the Spot

Property of Zero as an Exponent The value of any nonzero number raised to the power of 0 is 1.

52

5×5

25

Example: 50 = 1

53

5×5×5

125

my.hrw.com

6.7.A

Find the value of each power.

My Notes

B 0.43

625 3,125

Use an exponent to write each expression. (Example 1)

63

107

3. 10 × 10 × 10 × 10 × 10 × 10 × 10

( _34 )5

( _79 )8

5. _79 × _79 × _79 × _79 × _79 × _79 × _79 × _79

Find the value of each power. (Example 2)

Evaluate: 0.4 = 0.4 × 0.4 × 0.4 = 0.064

2

1 __ 16

12. 0.82

0.64

9.

( )

3 0 _ 5

Identify the base and the exponent. The base is _35, and the exponent is 0.

Math Talk

Mathematical Processes

Evaluate.

3

=1

Is the value of 2 the same as the value of 32? Explain.

Any number raised to the power of 0 is 1.

D -112

23 = 2 · 2 · 2 = 8 and 32 = 3 · 3 = 9, so the values are not equivalent.

Identify the base and the exponent. The base is 11, and the exponent is 2. Evaluate. -112 = -(11 × 11) = -121

512

6. 83

3

The opposite of a positive number squared is a negative number.

( _14 )

15. 80 18.

? ?

( -2 )3

1 -8

2,401

7. 74 10.

( _13 )

1 __ 27

3

13. 0.53

0.125

16. 121

12

19.

( - _25 )

2

4 __ 25

8. 103 11.

( _67 )

2

14. 1.12 17.

( _12 )

0

20. -92

1,000 36 __ 49

1.21 1 -81

ESSENTIAL QUESTION CHECK-IN

21. How do you use an exponent to represent a number such as 16?

You use an exponent to write a number that can be written as a product of equal factors. 16 = 4 × 4 (or 2 × 2 × 2 × 2), so it can be written as 42 (or 24).

© Houghton Mifflin Harcourt Publishing Company

Identify the base and the exponent. The base is 0.4, and the exponent is 3.

0

5×5×5×5×5

4. _34 × _34 × _34 × _34 × _34

Evaluate: (-10)4 = -10 × (-10) × (-10) × (-10) = 10,000

( _35 )

5×5×5×5

55

3 factors of 6

Identify the base and the exponent. The base is -10, and the exponent is 4.

5

54

2. 6 × 6 × 6

A (-10)4

© Houghton Mifflin Harcourt Publishing Company

Simplified product

5

EXAMPL 2 EXAMPLE

C

Product

51

YOUR TURN Find the value of each power. 6. 34

81

7. (-1)9

-1

8.

( _25 )

3

8 ___ 125

Personal Math Trainer

9. -122-144

Online Assessment and Intervention

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Lesson 10.1

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22/10/12 10:51 PM

272

Unit 4

6_MTXESE051676_U4M10L1.indd 272

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DIFFERENTIATE INSTRUCTION Kinesthetic Experience

Critical Thinking

To help students remember the meaning of the base and the exponent in a power, have them use graph paper or square tiles to construct models of the squares of whole numbers 1–10. Label the models as shown below. A visual representation of a square number can help students remember that exponents represent repeated multiplication of the same factor.

Have students explore multiplication of numbers written as powers. Partners can work together to find the values of pairs of expressions such as these:

Differentiated Instruction includes: • Reading Strategies • Success for English Learners ELL • Reteach • Challenge PRE-AP

32 = 3 × 3

32 · 33 and 31 · 34 22 · 24 and 23 · 23 43 · 43 and 41 · 45

243 and 243; 64 and 64; 4,096 and 4,096; The values of both expressions in each pair are the same. The base is used the same number of times in each pair. The exponents in each pair have equal sums. You can add the exponents to multiply powers with the same base, for example, 2 2 · 24 = 26. Exponents

272

Personal Math Trainer Online Assessment and Intervention

Online homework assignment available

Evaluate GUIDED AND INDEPENDENT PRACTICE 6.7.A

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10.1 LESSON QUIZ 6.7.A Use exponents to write each expression. 1. __37 × __37 × __37

Concepts & Skills

Practice

Explore Activity Identifying Repeated Multiplication

Exercises 1, 38–44

Example 1 Using Exponents

Exercises 2–5, 22–37

Example 2 Finding the Value of a Power

Exercises 6–20, 38–44

2. 0.9 × 0.9 × 0.9 × 0.9 Find the value of each power.

Exercise

3. 74

Depth of Knowledge (D.O.K.)

Mathematical Processes

3 4. ( -__34 )

22–37

2 Skills/Concepts

1.C Select tools

Lesson Quiz available online

38–39

2 Skills/Concepts

1.A Everyday life

40

2 Skills/Concepts

1.D Multiple representations

41–42

2 Skills/Concepts

1.A Everyday life

43

3 Strategic Thinking

1.F Analyze relationships

44

2 Skills/Concepts

1.D Multiple representations

45

3 Strategic Thinking

1.G Explain and justify arguments

46–47

3 Strategic Thinking

1.F Analyze relationships

48

3 Strategic Thinking

1.G Explain and justify arguments

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Answers 3 1. ( __37 )

2. 0.94 3. 2,401 27 4. -__ 64

Additional Resources Differentiated Instruction includes: • Leveled Practice Worksheets

273

Lesson 10.1

Name

Class

Date

10.1 Independent Practice

Personal Math Trainer

6.7.A

my.hrw.com

Online Assessment and Intervention

Write the missing exponent.

2

22. 100 = 10

( )

1 = ___ 1 26. ____ 169 13

23. 8 = 2

2

3

24. 25 = 5

1

27. 14 = 14

28. 32 = 2

2

25. 27 = 3

5

1= 34. __ 9

(

1 _ 3

3

10

)

31. 256 =

4

4

()

64 = __ 8 29. ___ 81 9

2

35. 64 =

2

9 = 36. ___ 16

8

4

32. 16 =

33. 9 =

2

(

3 _ 4

)

Sample answer: 0.32 = 0.09; 0.3 > 0.09 44. Which power can you write to represent the volume of the cube shown? Write the power as an expression with a base and an exponent, and then find the volume of the cube.

1 in.3 ( _13 )3 = _13 × _13 × _13 = __ 27

3

FOCUS ON HIGHER ORDER THINKING

2

2

Work Area

The value of 1 raised to any power is 1. 1 multiplied by

3

2

37. 729 =

1 in. 3

45. Communicate Mathematical Ideas What is the value of 1 raised to the power of any exponent? What is the value of 0 raised to the power of any nonzero exponent? Explain.

Write the missing base. 30. 1,000 =

43. Write a power represented with a positive base and a positive exponent whose value is less than the base.

itself any number of times is 1. The value of 0 raised

3

9

to any power is 0. 0 multiplied by itself any number of times is still 0.

38. Hadley’s softball team has a phone tree in case a game is canceled. The coach calls 3 players. Then each of those players calls 3 players, and so on. How many players will be notified during the third round of calls?

46. Look for a Pattern Find the values of the powers in the following pattern: 101, 102, 103, 104… . Describe the pattern, and use it to evaluate 106 without using multiplication.

27 players

Sample answer: 10; 100; 1,000; 10,000… . Each term in

39. Tim is reading a book. On Monday he reads 3 pages. On each day after that, he reads triple the number of pages as the previous day. How many pages does he read on Thursday?

the pattern is a 1 followed by the same number of zeros as the exponent. 106 = 1,000,000

4

3 pages, or 81 pages 40. Which power can you write to represent the area of the square shown? Write the power as an expression with a base and an exponent, and then find the area of the square.

8.5 = 8.5 × 8.5 = 72.25 mm 2

47. Critical Thinking Some numbers can be written as powers of different bases. For example, 81 =  92 and 81 =  34. Write the number 64 using three different bases.

2

26, 43, and 82

41. Antonia is saving for a video game. On the first day, she saves two dollars in her piggy bank. Each day after that, she doubles the number of dollars she saved on the previous day. How many dollars does she save on the sixth day?

48. Justify Reasoning Oman said that it is impossible to raise a number to the power of 2 and get a negative value. Do you agree with Oman? Why or why not?

26 dollars, or \$64

Sample answer: Agree; because the product of two numbers with the same sign is always positive, and 0

42. A certain colony of bacteria triples in length every 10 minutes. Its length is now 1 millimeter. How long will it be in 40 minutes?

raised to the power of 2 is 0.

34 mm, or 81 mm

Lesson 10.1

EXTEND THE MATH

© Houghton Mifflin Harcourt Publishing Company

© Houghton Mifflin Harcourt Publishing Company

8.5 mm

PRE-AP

Activity Every integer can be written as the sum of square numbers. For example: Sum of 2 squares: 20 = 42 + 22 Sum of 3 squares: 24 = 42 + 22 + 22 Some integers, such as 22, can be written as the sum of square numbers more than one way. Sum of 3 squares: 22 = 32 + 32 + 22 Sum of 4 squares: 22 = 42 + 22 + 12 + 12

273

Activity available online

274

Unit 4

my.hrw.com

• Can you write the number 36 as the sum of squares in more than one way? Yes; 36 = 32 + 32 + 32 + 32 = 9 + 9 + 9 + 9; 36 = 42 + 42 + 22 = 16 + 16 + 4 • Write a number on one side of an index card and on the reverse write the number as a sum of squares. Challenge a classmate to write the number. For example: Write 62 as the sum of 3 squares.

12 + 52 + 62

• Write the integers 8, 13, and 18 as the sum of 2 squares. 8 = 22 + 22; 13 = 32 + 22; 18 = 32 + 32

Exponents

274

LESSON

10.2 Prime Factorization Texas Essential Knowledge and Skills The student is expected to: Expressions, equations, and relationships—6.7.A Generate equivalent numerical expressions using order of operations, including whole number exponents and prime factorization.

Mathematical Processes

Engage

ESSENTIAL QUESTION How do you write the prime factorization of a number? Sample answer: Use a factor tree or a ladder diagram to find the prime factorization of the number, then write the prime factorization using exponents.

Explore Motivate the Lesson

6.1.E Create and use representations to organize, record, and communicate mathematical ideas.

Ask students to name two numbers that can be multiplied to get a specific product. For example, you might ask them to name two numbers that can be multiplied to get 28 (1 and 28; 2 and 14; 4 and 7). Repeat the process using 36.

Explain ADDITIONAL EXAMPLE 1 Rayshawn is designing a mural. The mural must have an area of 42 square yards. What are the possible whole number lengths and widths for the mural? The possible lengths and widths are listed: Length (yd)

42

21

14

7

Width (yd)

1

2

3

6

EXAMPLE 1 Focus on Reasoning

Mathematical Processes Point out to students that you can tell when you have found all the factors of a number when the factor pairs start to repeat.

Questioning Strategies

Mathematical Processes • For any number, which numbers are always factors? 1 and the number itself. • Is it possible for a number to have all even factors? No; 1 is a factor for all numbers.

YOUR TURN Interactive Whiteboard Interactive example available online my.hrw.com

Avoid Common Errors Remind students that when they list the factors of a number they should always begin with 1 and end with the number itself.

EXPLORE ACTIVITY 1 Animated Math Prime Factorization Students use an interactive factor tree to find prime factors of composite numbers. my.hrw.com

Connect Vocabulary

ELL

Remind students that a prime number is a number with exactly 2 factors, 1 and itself, and a composite number is a number that has more than 2 factors.

Engage with the Whiteboard Have students make alternate factor trees for 240 on the whiteboard, next to the given factor tree. Have students start with the following pairs: 8 and 30; 24 and 10; and 12 and 20. Point out to students that while the order of the factors in a factor tree may differ, the prime factors of a number are always the same.

Questioning Strategies

Mathematical Processes • When choosing the first factor pair for the branches of the factor tree for 240, does one of the factors have to be a prime number? Explain. No. A factor tree can start with any factor pair.

275

Lesson 10.2

LESSON

Expressions, equations, and relationships— 6.7.A Generate equivalent numerical expressions using prime factorization.

10.2 Prime Factorization ESSENTIAL QUESTION

The prime factorization of 12 is 2 · 3 · 2 or 22 · 3.

Finding Factors of a Number

8 · 30, 10 · 24, 12 · 20, 15 · 16

Recall that area = length · width. For Ana’s garden, 24 ft2 = length · width.

3

2 · 3

2·2·2·2·3·5 Then write the prime factorization using exponents.

24 · 3 · 5

4

6

Math Talk

8 12 24

Mathematical Processes

Give an example of a whole number that has exactly two factors? What type of number has exactly two factors?

The factors of 24 are 1, 2, 3, 4, 6, 8, 12, 24. STEP 3

2 · 6

D Write the prime factorization of 240.

4 · 6 = 6 · 4, so you only list 4 · 6.

You can also use a diagram to show the factor pairs.

2

2 · 12

C Continue adding branches until the factors at the ends of the branches are prime numbers.

List the factors of 24 in pairs. List each pair only once.

1

240

5 · 48 2 · 24

B Choose any factor pair to begin the tree. If a number in this pair is prime, circle it. If a number in the pair can be written as a product of two factors, draw additional branches and write the factors.

Ana wants to build a rectangular garden with an area of 24 square feet. What are the possible whole number lengths and widths of the garden?

© Houghton Mifflin Harcourt Publishing Company • Image Credits: Brand X Pictures/ Getty Images

1 · 240, 2 · 120, 3 · 80, 4 · 60, 5 · 48, 6 · 40,

Math On the Spot

6.7.A

24 = 1 · 24 24 = 2 · 12 24 = 3 · 8 24 = 4 · 6

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Use exponents to show repeated factors.

A List the factor pairs of 240. my.hrw.com

EXAMPL 1 EXAMPLE

Animated Math

Use a factor tree to find the prime factorization of 240.

Whole numbers that are multiplied to find a product are called factors of that product. A number is divisible by its factors. For example, 4 and 2 are factors of 8 because 4 · 2 = 8, and 8 is divisible by 4 and by 2.

STEP 2

Finding the Prime Factorization of a Number The prime factorization of a number is the number written as the product of its prime factors. For example, the prime factors of 12 are 3, 2, and 2.

How do you write the prime factorization of a number?

STEP 1

6.7.A

The possible lengths and widths are: Length (ft)

24

12

8

6

Width (ft)

1

2

3

4

Reflect 5.

What If? What will the factor tree for 240 look like if you start the tree with a different factor pair? Check your prediction by creating another factor tree for 240 that starts with a different factor pair.

Sample answer: The intermediate steps on Sample answer: 13; its factors are 1 and 13; prime number

the factor tree will be different but the final prime factorization will be the same.

240

2 · 120 2 · 60 3 · 20 4 · 5 2 · 2

© Houghton Mifflin Harcourt Publishing Company

?

EXPLORE ACTIVITY 1

YOUR TURN List all the factors of each number. 1. 21 3. 42

1, 3, 7, 21 1, 2, 3, 6, 7, 14, 21, 42

Personal Math Trainer

2. 37

1, 37

4. 30

1, 2, 3, 5, 6, 10, 15, 30

Online Assessment and Intervention

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Lesson 10.2

275

276

Unit 4

PROFESSIONAL DEVELOPMENT Integrate Mathematical Processes This lesson provides an opportunity to address Mathematical Process TEKS 6.1.E, which calls for students to “create and use representations to organize, record, and communicate mathematical ideas.” Students use diagrams and factor trees to organize factor pairs of a number to find prime factorizations. They also use ladder diagrams to find prime factorizations, with the “ladder” as the means of recording and communicating the prime factorization.

Math Background Ancient Greeks started to study prime numbers circa 300 B.C.E. They observed that there were an infinite number of prime numbers and that there were irregular gaps between successive prime numbers. In 1984, Samuel Yates coined the term titanic prime. He used this term to refer to any prime number with 1,000 digits or more. When he first defined a titanic prime, only 110 of them were known. Today more than 110,000 titanic primes have been identified.

Prime Factorization

276

EXPLORE ACTIVITY 2 Engage with the Whiteboard Have students complete the ladder diagram starting with different combinations of prime factors. Point out to students that while the order in which they used the prime factors may differ, the prime factors of a number are always the same.

Focus on Modeling Mathematical Processes Students who find mental math difficult may find ladder diagrams to be challenging. Model other methods for dividing by 2, such as using long division or a calculator, to show that the ladder diagram is a useful organizational tool. Emphasize that they can use a combination of division methods when using ladder diagrams. Questioning Strategies

Mathematical Processes • How do you know when 2 is a factor of a number? How do you know when 2 is not a factor of a number? Even numbers have 2 as a factor. Odd numbers do not.

• Why do you have to divide by prime numbers when using the ladder diagram? The divisors on the left show the prime factorization, so all of them must be prime numbers.

Mathematical Processes Discuss with students ways to check that 2 · 3 · 3 · 3 is the prime factorization of 54. Students should understand that they can check their work two ways: by making sure that every number in the prime factorization is prime and by multiplying the expression 2 · 3 · 3 · 3 to verify that the product is 54.

Elaborate Talk About It Summarize the Lesson Ask: What is the prime factorization of a number, and how can you find the prime factorization of a number? The prime factorization of a number is an expression that shows the number as the product of its prime factors. You can use a factor tree or a ladder diagram to find the prime factorization of a number.

GUIDED PRACTICE Engage with the Whiteboard For Exercises 1–2, have students draw a diagram to list the factors of each number on the whiteboard. For Exercise 5, have students make several different factor trees on the whiteboard.

Avoid Common Errors Exercise 3 Remind students that you can tell when you have found all the factors of a number when the factor pairs start to repeat and that writing factor pairs in order makes it easier to check that all the factor pairs are listed. Exercises 4–7 Remind students that they can check their work by multiplying their answer for the prime factorization to make sure the product is the original number.

277

Lesson 10.2

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Guided Practice

6.7.A

Use a diagram to list the factor pairs of each number. (Example 1)

1. 18

A ladder diagram is another way to find the prime factorization of a number.

1

2

Use a ladder diagram to find the prime factorization of 132.

A Write 132 in the top “step” of the ladder. Choose a prime factor of 132 to write next to the step with 132. Choose 2. Divide 132 by 2 and write the quotient 66 in the next step of the ladder. B Now choose a prime factor of 66. Write the prime factor next to the step with 66. Divide 66 by that prime factor and write the quotient in the next step of the ladder.

2

4

13 26 52

1, 2, 4, 13, 26, 52

Length (ft)

72

36

24

18

12

9

Width (ft)

1

2

3

4

6

8

402

4. 402

201

E Write the prime factorization of 132 using exponents.

36

5. 36

3 · 12

· 2

4 · 3

3 · 67

22 · 3 · 11

2 · 2

2 · 3 · 67

Reflect 6. Complete a factor tree and a ladder diagram to find the prime factorization of 54.

© Houghton Mifflin Harcourt Publishing Company

1

Use a factor tree to find the prime factorization of each number. (Explore Activity 1)

to the left of the steps of the ladder.

22 · 32

Use a ladder diagram to find the prime factorization of each number. (Explore Activity 2)

54

6. 32

2 · 27

9 · 3

3 · 3

prime factorization: 2 · 33

2 32 2 16 2 8 2 4 2 2 1

7. 27

3 27 3 9 3 3 1 3 · 3 · 3 or 33

25

? ?

Communicate Mathematical Ideas If one person uses a ladder diagram and another uses a factor tree to write a prime factorization, will they get the same result? Explain.

ESSENTIAL QUESTION CHECK-IN

8. Tell how you know when you have found the prime factorization of a number.

Yes; there is only one unique prime factorization for

Sample answer: when all the factors are prime and their

every integer greater than 1.

product is the original number Lesson 10.2

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1. This is a list

2. 52

Complete the table with possible measurements of the stage.

The prime factors are 2, 2, 3, and 11. They are written

InCopy Notes

9 18

3. Karl needs to build a stage that has an area of 72 square feet. The length of the stage should be longer than the width. What are the possible whole number measurements for the length and width of the stage? (Example 1)

3 33 11 11 1

D What are the prime factors of 132? How can you tell from the ladder diagram?

7.

6

1, 2, 3, 6, 9, 18

2 132 2 66

C Keep choosing prime factors, dividing, and adding to the ladder until you get a quotient of 1.

2 54 3 27 3 9 3 3 1

3

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EXPLORE ACTIVITY 2

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InDesign Notes 1. This is a list

Kinesthetic Experience

Critical Thinking

Students can use a method called the sieve of Eratosthenes to help identify prime numbers. Start with a 10 × 10 grid showing the numbers 1 to 100. Cross out 1, because 1 is not a prime number. Circle 2 because it is prime. Then cross out all multiples of 2, because multiples of 2 are not prime. Circle the next prime number, 3, and cross out all multiples of 3. Repeat the process until all numbers are circled or crossed out. Students can refer to this chart when deciding whether a number is prime or composite.

Discuss with students how the prime factorization of a number can be used to find all the factors of a number, by using the Associative and Commutative properties. For example, the prime factorization of 30 is 2 · 3 · 5, which can be expressed as 2 · (3 · 5), (2 · 3) · 5, or (2 · 5) · 3.

Differentiated Instruction includes: • Reading Strategies • Success for English Learners ELL • Reteach • Challenge PRE-AP

When you multiply the numbers inside the parentheses, the expressions simplify to 2 · (15), (6) · 5, and (10) · 3.

These numbers along with the factor pair 1 · 30, give all the factors for 30: 1, 2, 3, 5, 6, 10, 15, 30.

Prime Factorization

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Personal Math Trainer Online Assessment and Intervention

Online homework assignment available

Evaluate GUIDED AND INDEPENDENT PRACTICE 6.7.A

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10.2 LESSON QUIZ 6.7.A 1. Find all the factors of 54. 2. Find the prime factorization of 54, and then write it using exponents.

Concepts & Skills

Practice

Example 1 Finding Factors of a Number

Exercises 1–3, 9–10

Explore Activity 1 Finding the Prime Factorization of a Number

Exercises 4–5, 12–15, 17–19

Explore Activity 2 Using a Ladder Diagram

Exercises 6–7, 16

3. Find all the factors of 60. 4. Find the prime factorization of 60, and then write it using exponents. 5. Chanasia has 30 beads. She wants to put them in boxes, so that each box will contain the same whole number of beads. Use factors to list all the different ways she can put the beads into boxes.

Exercise

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Answers 1. 1, 2, 3, 6, 9, 18, 27, 54 2. 2 × 3 × 3 × 3 = 2 · 3

3

3. 1, 2, 3, 4, 5, 6, 10, 12, 15, 20, 30, 60 4. 2 × 2 × 3 × 5 = 2 · 3 · 5

2 Skills/Concepts

1.D Multiple representations

10

2 Skills/Concepts

1.A Everyday life

11

3 Strategic Thinking

1.G Explain and justify arguments

2 Skills/Concepts

1.C Select tools

16

3 Strategic Thinking

1.G Explain and justify arguments

17

3 Strategic Thinking

1.C Select tools

18

3 Strategic Thinking

1.G Explain and justify arguments

19

3 Strategic Thinking

1.F Analyze relationships

20

3 Strategic Thinking

1.G Explain and justify arguments

21–22

3 Strategic Thinking

1.F Analyze relationships

2

5. 1 box with 30 beads 2 boxes with 15 beads each 3 boxes with 10 beads each 5 boxes with 6 beads each 6 boxes with 5 beads each 10 boxes with 3 beads each 15 boxes with 2 beads each 30 boxes with 1 bead each

279

Lesson 10.2

Mathematical Processes

9

12–15

Lesson Quiz available online

Depth of Knowledge (D.O.K.)

Additional Resources Differentiated Instruction includes: • Leveled Practice Worksheets

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Name

Class

Date

10.2 Independent Practice

18. In a game, you draw a card with three consecutive numbers on it. You can choose one of the numbers and find the sum of its prime factors. Then you can move that many spaces. You draw a card with the numbers 25, 26, 27. Which number should you choose if you want to move as many spaces as possible? Explain.

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6.7.A

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Online Assessment and Intervention

26; the prime factors of 25 are 5 and 5, the prime factors

9. Multiple Representations Use the grid to draw three different rectangles so that each has an area of 12 square units and they all have different widths. What are the dimensions of the rectangles?

of 26 are 2 and 13, and the prime factors of 27 are 3, 3, and 3. The sums are 10, 15, and 9. The greatest sum is 15,

1 × 12; 2 × 6; 3 × 4

so choose 26 to move 15 spaces. 19. Explain the Error When asked to write the prime factorization of the number 27, a student wrote 9 · 3. Explain the error and write the correct answer.

10. Brandon has 32 stamps. He wants to display the stamps in rows, with the same number of stamps in each row. How many different ways can he display the stamps? Explain.

9 is not a prime number; prime factorization of 27 = 33.

6 different ways; 1 row of 32 stamps; 2 rows of 16; 4 rows of 8; 32 rows of 1; 16 rows of 2; 8 rows of 4

FOCUS ON HIGHER ORDER THINKING

20. Communicate Mathematical Ideas Explain why it is possible to draw more than two different rectangles with an area of 36 square units, but it is not possible to draw more than two different rectangles with an area of 15 square units. The sides of the rectangles are whole numbers.

11. Communicate Mathematical Ideas How is finding the factors of a number different from finding the prime factorization of a number?

When you find the factors of a number, you find all

36 has five factor pairs, so five different rectangles can

factors, some of which are prime; when you find the

be drawn. 15 has only two factor pairs, so only two

prime factorization, you find only the prime factors.

14. 23

34 · 11

13. 504

23

15. 230

different rectangles can be drawn. 21. Critique Reasoning Alice wants to find all the prime factors of the number you get when you multiply 17 · 11 · 13 · 7. She thinks she has to use a calculator to perform all the multiplications and then find the prime factorization of the resulting number. Do you agree? Why or why not?

2 3 · 32 · 7 2 · 5 · 23

Disagree; the factors that are being multiplied are all prime numbers, so the prime factorization of the

16. The number 2 is chosen to begin a ladder diagram to find the prime factorization of 66. What other numbers could have been used to start the ladder diagram for 66? How does starting with a different number change the diagram?

number is 17 · 13 · 11 · 7. 22. Look for a Pattern Ryan wrote the prime factorizations shown below. If he continues this pattern, what prime factorization will he show for the number one million? What prime factorization will he show for one billion?

3 and 11 can be chosen because they are prime factors. The intermediate steps would be different, but the

10 = 5 · 2

prime factorization is the same.

100 = 52 · 22

17. Critical Thinking List five numbers that have 3, 5, and 7 as prime factors.

1,000 = 53 · 23

one million: 56 · 26; one billion: 59 · 29

Sample answer: 105, 315, 525, 735, 945 Lesson 10.2

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EXTEND THE MATH

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InDesign Notes

PRE-AP 1. This is a list

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Find the prime factorization of each number. 12. 891

Work Area

Unit 4

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InDesign Notes 1. This is a list

Activity In mathematics, a perfect number is a number that is equal to the sum of all its factors (excluding the number itself ). The number 6 is an example of a perfect number. The factors of 6 are 1, 2, 3, and 6. The sum of the factors excluding 6 is 1 + 2 + 3 = 6. • Find the next largest perfect number, and show why it is perfect. 28; the factors of 28 are 1, 2, 4, 7, 14, and 28, and 1 + 2 + 4 + 7 + 14 = 28. • A student claims that 128 is a perfect number. Prove or disprove the student’s claim. False; The factors of 128 are 1, 2, 4, 8, 16, 32, 64, and 128. Their sum, excluding the number itself, is 127. • Another student says that 496 is a perfect number. Prove or disprove the student’s claim. True; The factors of 496 are 1, 2, 4, 8, 16, 31, 62, 124, 248, and 496. Their sum, excluding the number itself, is 496.

Prime Factorization

280

LESSON

10.3 Order of Operations Texas Essential Knowledge and Skills The student is expected to: Expressions, equations, and relationships—6.7.A Generate equivalent numerical expressions using order of operations, including whole number exponents and prime factorization.

Mathematical Processes

Engage

ESSENTIAL QUESTION How do you use the order of operations to simplify expressions with exponents? Sample answer: Find the value of any expressions within parentheses first. Then evaluate all powers. Then multiply or divide in order from left to right, and finally, add or subtract in order from left to right.

Motivate the Lesson

Ask: Have you ever tried to simplify an expression such as 35 + 20(122 ÷ 9)? Try it. Need help? Begin the Explore Activity to find out how to use the order of operations.

6.1.C Select tools, including real objects, manipulatives, paper and pencil, and technology as appropriate, and techniques, including mental math, estimation, and number sense as appropriate, to solve problems.

Explore EXPLORE ACTIVITY Engage with the Whiteboard

Write the expression 2 + 3 × 4 on the whiteboard. Point out to students that this expression could have two different results (14 or 20) without guidelines to show which operation should be performed first. Show students that the correct solution is 14 based on the fact that 3 × 4 = 4 + 4 + 4. Thus the expression 2 + 3 × 4 can be written as 2 + 4 + 4 + 4, which equals 14. Now write the expressions 36 - 18 ÷ 6 and 7 + 24 ÷ 6 × 2 on the whiteboard and ask the students to solve them. After a few minutes, have students come up to the whiteboard and solve the equations, showing all the steps. Discuss the solutions with the class. Answers: 33 and 15

Explain ADDITIONAL EXAMPLE 1 Simplify each expression. A 30 - 3 × 23 6 B 128 ÷ (4 × 2)

2

32 C 40 - _____ (7 - 5)3

EXAMPLE 1 Avoid Common Errors Students may find it easier to perform operations from left to right as they appear in an expression, rather than use the order of operations. Remind students that using the order of operations correctly ensures that everyone who simplifies the same expression will get the same answer.

2

36

Interactive Whiteboard Interactive example available online my.hrw.com

Questioning Strategies

Mathematical Processes • Can you use the order of operations with expressions that have no parentheses? Explain. Yes. The order of operations tells the order in which operations should be performed but does not require that an expression include parentheses, exponents, or all the operations. • Are the expressions 3 + 5 · 2 and 3 + (5 · 2) equivalent? Explain. Yes. In both expressions 5 · 2 should be evaluated first, and then 3 should be added to the product.

281

Lesson 10.3

Check for Understanding Ask: In the expression 36 - 18 ÷ 2 + 6 × 1, which operation should you perform first? Explain. Division. This expression has no parentheses or exponents, so the first operation to perform is either multiplication or division from left to right.

LESSON

10.3 Order of Operations ?

ESSENTIAL QUESTION

EXPLORE ACTIVITY (cont’d)

Expressions, equations, and relationships— 6.7.A Generate equivalent numerical expressions using order of operations …

Reflect 1.

In C , why does it makes sense to write the values as powers? What is the pattern for the number of e-mails in each wave for Amy?

Sample answer: By writing the values as powers, you How do you use the order of operations to simplify expressions with exponents?

can see the exponent is equal to the wave number. The pattern would be 31, 32, 33, 34, and so on.

6.7.A

EXPLORE ACTIVITY

Exploring the Order of Operations

Simplifying Numerical Expressions

Order of Operations

A numerical expression is an expression involving numbers and operations. You can use the order of operations to simplify numerical expressions.

1. Perform operations in parentheses. Math On the Spot

2. Find the value of numbers with exponents.

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EXAMPLE 1

6.7.A

3. Multiply or divide from left to right.

My Notes

4. Add or subtract from left to right.

A 5 + 18 ÷ 32 5 + 18 ÷ 32 = 5 + 18 ÷ 9

Amy 1st wave

C Amy is just one of four friends initiating the first wave of e-mails. Write an expression for the total number of e-mails sent in the 2nd wave.

=5+2

Divide.

=7

B 4 × (9 ÷ 3)2

2nd wave

4 × (9 ÷ 3)2 = 4 × 32

A Use a diagram to model the situation for Amy. Each dot represents one e-mail. Complete the diagram to show the second wave. B Complete the table to show how many e-mails are sent in each wave of Amy’s diagram.

Wave

Number of emails

Power of 3

1st

3

31

2nd

9

32

4

3

Personal Math Trainer

Multiply 4 and 3/Find the value of 32

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36

Multiply.

2

Online Assessment and Intervention

=

= 36

(12 - 8) 42 = 8 +  __ 8 + _______ 2 2

D Identify the computation that should be done first to simplify the expression in C . Then simplify the expression.

9

Evaluate 32.

(12 - 8)2

2

The value of the expression is 4 ×

Perform operations inside parentheses.

=4×9

C 8 + _______ 2

number of people × number of emails in 2nd wave written as a power ×

Evaluate 32.

Perform operations inside parentheses.

16 = 8 + __ 2

Evaluate 42.

= 8+ 8

Divide.

= 16

© Houghton Mifflin Harcourt Publishing Company

Amy and three friends launch a new website. Each friend emails the web address to three new friends. These new friends forward the web address to three more friends. If no one receives the e-mail more than once, how many people will receive the web address in the second wave of e-mails?

© Houghton Mifflin Harcourt Publishing Company

Simplify each expression.

YOUR TURN Simplify each expression using the order of operations. 2. (3 - 1)4 + 3

19

3.

24 ÷ (3 × 22)

2

. Lesson 10.3

281

282

Unit 4

PROFESSIONAL DEVELOPMENT Integrate Mathematical Processes This lesson provides an opportunity to address Mathematical Process TEKS 6.1.C, which calls for students to “select tools, including…paper and pencil…and techniques, including mental math…and number sense…to solve problems.” Students work with pencil and paper using the order of operations to simplify expressions. Since use of the order of operations can involve many steps, students use mental math and number sense in situations such as finding the sum of two numbers with unlike signs and raising a negative number to a positive power.

Math Background A History of Mathematical Notations by Florian Cajori describes the history of various mathematical symbols. According to Cajori, parentheses and brackets have been used as grouping symbols since the sixteenth century. A work published in 1556, General trattato di numeri e misure by Niccolò Tartaglia, is one of the first works in which parentheses are used. Brackets have been found in a manuscript edition of Algebra, by Rafael Bombelli which dates back to 1550.

Order of Operations

282

ADDITIONAL EXAMPLE 2 Simplify each expression using the order of operations. A (-3)3 + 8(15 - 7) 32 B ____ +3×3 (-4)2

11

C 18 - (-3) + 3 × 2 3

37 51

EXAMPLE 2 Avoid Common Errors Point out to students that, now that they are working with integers, not every minus sign will indicate subtraction. Some minus signs are used to indicate negative numbers. Remind students to read equations carefully to distinguish between the two uses of the minus sign.

Engage with the Whiteboard

Interactive Whiteboard Interactive example available online

Cover up the blue text in each part and have students circle the operation to be performed for each step on the whiteboard. Ask students to explain their choices. Then discuss the choices with the class.

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Questioning Strategies

Mathematical Processes • In A, the order of operations says to perform operations in parentheses first. So why isn’t (-2)2 evaluated until the second step? When you are working with negative numbers, parentheses are used to separate addition, subtraction, multiplication, and division signs from negative numbers to avoid confusion. Since there is no operation in (-2)2, the first step is to subtract within the expression (3 - 9). (-3)2

• In B, can you evaluate the expression ____ the same as (-1)2? Explain. No. You cannot 3 cancel the 3s before you have evaluated (-3)2.

YOUR TURN Focus on Communication It may be helpful to review the rules for adding numbers with different signs before students try to work on their own. Remind them to subtract the absolute values of the numbers and then use the sign of the number with the greater absolute value in the difference.

Elaborate Talk About It Summarize the Lesson Ask: Why is it important to use the order of operations? The order of operations is important because correctly using the order or operations ensures that everyone who simplifies the same expression will get the same answer.

GUIDED PRACTICE Engage with the Whiteboard For Exercise 1, have students complete the diagram on the whiteboard. Then have them number the branches at each level to show the numbers of each type of fish that can be formed.

Avoid Common Errors Exercises 2–3 Remind students that when they work with an expression that has both multiplication and division, they should perform the operation that occurs first in the equation from left to right. Exercises 4–5 Remind students that when an expression inside parentheses has more than one operation, they need to perform those operations according to the order of operations.

283

Lesson 10.3

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Guided Practice

Using the Order of Operations with Integers

1. In a video game, a guppy that escapes a net turns into three goldfish. Each goldfish can turn into two betta fish. Each betta fish can turn into two angelfish. Complete the diagram and write the number of fish at each stage. Write and evaluate an expression for the number of angelfish that can be formed from one guppy. (Explore Activity)

You can use the order of operations to simplify expressions involving integers. Math On the Spot

EXAMPL 2 EXAMPLE

6.7.A

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1 guppy

Simplify each expression using the order of operations.

3 goldfish

A -4(3 - 9) + (-2)2 Perform operations inside parentheses.

B

= -4(-6) + 4

Evaluate (-2)2.

= 24 + 4

Multiply.

= 28

Divide.

= -18

=4+ =4+

© Houghton Mifflin Harcourt Publishing Company

2

3

)2 ÷ 3

9

=

12

angelfish

3

=2+

= -48 ÷ 3 + 1

Multiply.

= -16 + 1

=

Divide.

= -15

=

? ?

= = =

4. 2 + (-24 ÷  23) -9 = 2 + (-24 ÷

Evaluate (-2)3.

3. 36 ÷ 22 - 4 × 2 = 36 ÷

÷3

7

=

C 6 × (-2)3 ÷ 3 + 1 6 × (-2)3 ÷ 3 + 1 = 6 × (-8) ÷ 3 + 1

×2

Complete to simplify each expression. (Examples 1 and 2)

Evaluate (-3)2.

= -21 + 3

3 3 × 2 = 12 angelfish 2. 4 + (10 - 7)2 ÷ 3 = 4 + (

2

× 2 betta fish

2

(-3)2 -21 + _____ 3 (-3) -21 + _____ = -21 + _93 3

3

-1

-3

-10

8

)-9

9

9 1

4

-4 × 2

-4 × 2 -8

5. -42 × (-3 × 2 + 8) = -42 × ( -6 + 8)

2

= -42 ×

-9

=

-9

=

-16

-32

×

2

ESSENTIAL QUESTION CHECK-IN

6. How do you use the order of operations to simplify expressions with exponents?

Find the value of any expressions within parentheses first. Then evaluate all powers. Then multiply or divide from

left to right, and finally add or subtract from left to right.

Simplify each expression using the order of operations. 4.

-7 × (-4) ÷ 14 - 22

5.

-2

© Houghton Mifflin Harcourt Publishing Company

-4(3 - 9) + (-2)2 = -4(-6) + (-2)2

-5 (-3 + 1)3 - 3

37

Personal Math Trainer Online Assessment and Intervention

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Lesson 10.3

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Cooperative Learning

Cognitive Strategies

Have students work in groups to decide which operation signs to use to make the number sentences true. They may need to use operations more than once in each number sentence.

Students may be familiar with the abbreviation PEMDAS (or Please Excuse My Dear Aunt Sally) even before being introduced to the order of operations. But its abbreviation may give the impression that multiplication is always done before division and that addition is always done before subtraction. You may wish to present the mnemonic as P E M/D A/S

Differentiated Instruction includes: • Reading Strategies • Success for English Learners ELL • Reteach • Challenge PRE-AP

1. Operation signs: +, -, · 12

4

6

3

7 = 37

12 · 4 - 6 · 3 + 7 = 37

2. Operation signs: +, -, ÷ 18

2

24

12

4 = 22

18 + 2 - 24 ÷ 12 + 4 = 22

The slashes between the M and the D, and the A and the S, can help students remember that from left to right either operation can be performed first.

Order of Operations

284

Personal Math Trainer Online Assessment and Intervention

Online homework assignment available

Evaluate GUIDED AND INDEPENDENT PRACTICE 6.7.A

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10.3 LESSON QUIZ 6.7.A Simplify each expression using the order of operations. 1. 3 + (7 - 5)2 × 6

Concepts & Skills

Practice

Explore Activity Exploring the Order of Operations

Exercise 1

Example 1 Simplifying Numerical Expressions

Exercises 2–3, 7–12, 14–16

Example 2 Using the Order of Operations with Integers

Exercises 4–5, 13

25 2. ______ ×2 (-4 + 8)

3. 8 - 6 ÷ 2 + 3 × 5 4. 7 × 3 - 15 ÷ 5

Exercise

5. 8 + 2(-3 + 12 ÷ 2)2

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1.C Select tools

13

3 Strategic Thinking

1.F Analyze relationships

14

3 Strategic Thinking

1.G Explain and justify arguments

15–16

2 Skills/Concepts

1.A Everyday life

17–19

3 Strategic Thinking

1.F Analyze relationships

2. 16 3. 20

4. 18

Differentiated Instruction includes: • Leveled Practice Worksheets

5. 26

285

Lesson 10.3

Mathematical Processes

2 Skills/Concepts

7–12

Lesson Quiz available online

Depth of Knowledge (D.O.K.)

Class

Date

10.3 Independent Practice

Personal Math Trainer

6.7.A

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15. Ellen is playing a video game in which she captures butterflies. There are 3 butterflies onscreen, but the number of butterflies doubles every minute. After 4 minutes, she was able to capture 7 of the butterflies. a. Look for a Pattern Write an expression for the number of butterflies after 4 minutes. Use a power of 2 in your answer.

Online Assessment and Intervention

3 × 2 × 2 × 2 × 2 = 3 × 24

Simplify each expression using the order of operations. 7. 5 × 2 + 32

19

9. (11 - 8)2 - 2 × 6 2

9 11. 12 + __ 3

9

8. 15 - 7 × 2 + 23

-3

8 + 62 12. _____ +7×2 11

39

b. Write an expression for the number of butterflies remaining after Ellen captured the 7 butterflies. Simplify the expression.

14

10. 6 + 3(13 - 2) - 52

3 × 24 - 7 = 3 × 16 - 7 = 48 - 7 = 41; 41

18

butterflies remain.

13. Explain the Error Jay simplified the expression -3 × (3 + 12 ÷ 3) - 4. For his first step, he added 3 + 12 to get 15. What was Jay’s error? Find the correct answer.

16. Show how to write, evaluate and simplify an expression to represent and solve this problem: Jeff and his friend each text four classmates about a concert. Each classmate then texts four students from another school about the concert. If no one receives the message more than once, how many students from the other school receive a text about the concert?

Jay worked inside the parentheses first, but he should have performed the division 12 ÷ 3 = 4 first; -25.

2 × 42 = 32; 32 students receive a text.

14. Multistep A clothing store has the sign shown in the shop window. Pani sees the sign and wants to buy 3 shirts and 2 pairs of jeans. The cost of each shirt before the discount is \$12, and the cost of each pair of jeans is \$19 before the discount.

SALE

Today ONLY

\$3 off every purchase!

a. Write and simplify an expression to find the amount Pani pays if a \$3 discount is applied to her total.

3 × 12 + 2 × 19 - 3; \$71

17. Geometry The figure shown is a rectangle. The green shape in the figure is a square. The blue and white shapes are rectangles, and the area of the blue rectangle is 24 square inches. a. Write an expression for the area of the entire figure that includes an exponent. Then find the area.

b. Pani says she should get a \$3 discount on the price of each shirt and a \$3 discount on the price of each pair of jeans. Write and simplify an expression to find the amount she would pay if this is true. © Houghton Mifflin Harcourt Publishing Company

Work Area

FOCUS ON HIGHER ORDER THINKING

2 in. 6 in.

62 + 2 × 6 + 24 = 72 square inches

3 × (12 - 3) + 2 × (19 - 3); \$59

b. Find the dimensions of the entire figure.

c. Analyze Relationships Why are the amounts Pani pays in a and b different?

8 in. by 9 in.

In part a, the \$3 discount is applied 1 time; in b it is

18. Analyze Relationships Roberto’s teacher writes the following statement on the board: The cube of a number plus one more than the square of the number is equal to the opposite of the number. Show that the number is -1.

applied 5 times. d. If you were the shop owner, how would you change the sign? Explain.

Sample answer: If the shop owner wants to make more

(-1)3 + ((-1)2 + 1) = -1 + (1 + 1) = -1 + 2 = 1; 1 is

money, the sign should say “\$3 off your entire purchase.”

the opposite of -1.

If customers can take the discount off every item, a lot

19. Persevere in Problem Solving Use parentheses to make this statement true: 8 × 4 - 2 × 3 + 8 ÷ 2 = 25

more money is discounted from each purchase.

8 × 4 - (2 × 3 + 8) ÷ 2 Lesson 10.3

EXTEND THE MATH

PRE-AP

© Houghton Mifflin Harcourt Publishing Company • Image Credits: imagebroker/ Alamy

Name

285

286

Activity available online

Unit 4

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Activity The expression (4 × 4 - 4) × 4 uses exactly 4 fours. When simplified, its value is 48. • Write 10 expressions that use exactly 4 fours and that equal one of the numbers 0 to 9. Use what you know about the order of operations to write the expressions. You can use addition, subtraction, multiplication, division, parentheses, and exponents in the expressions. • Justify the expressions you have written by showing how to simplify them. Sample answers: 4+4-4-4=0

(4 + 4) ÷ (4 + 4) = 1

4÷4+4÷4=2

(4 + 4 + 4) ÷ 4 = 3

4 - (4 - 4) × 4 = 4

( __44 )

(4 + 4) 4 + _____ =6 4 (4 × 4) _____ +4=8 4

4

+4=5

4 + 4 - ( __44 ) = 7

4 + 4 + ( __44 ) = 9 Order of Operations

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MODULE QUIZ

Assess Mastery

10.1 Exponents

Use the assessment on this page to determine if students have mastered the concepts and standards covered in this module.

Find the value of each power.

5. 3

Response to Intervention

2 1

343

1. 73

( ) 2 _ 3

3

Enrichment

10. 120

Access Ready to Go On? assessment online, and receive instant scoring, feedback, and customized intervention or enrichment. Online and Print Resources Differentiated Instruction

Differentiated Instruction

• Reteach worksheets

• Challenge worksheets

• Reading Strategies • Success for English Learners ELL

ELL

6. ( -3

)5

-243

3.

( )

7. ( -2

)4

16

4.

( ) 1 _ 2

6

8. 1.42

1 __ 64

1.96

Additional Resources Assessment Resources includes: • Leveled Module Quizzes

1, 2, 3, 4, 5, 6, 8, 10, 12, 15, 20, 24, 30, 40, 60, 120

11. 58

2 × 29

12. 212

22 × 53

13. 2,800

2 4 × 52 × 7

14. 900

22 × 3 2 × 5 2

10.3 Order of Operations Simplify each expression using the order of operations. 15. ( 21 - 3 ) ÷ 32

PRE-AP

Extend the Math PRE-AP Lesson activities in TE

1, 2, 3, 4, 6, 8, 12, 16, 24, 32, 48, 96

Find the prime factorization of each number.

2

17. 17 + 15 ÷ 3 - 24 © Houghton Mifflin Harcourt Publishing Company

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81

2. 92

8 __ 27

49 __ 81

2

Find the factors of each number. 9. 96

Online Assessment and Intervention

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7 _ 9

10.2 Prime Factorization

Intervention

Personal Math Trainer

Personal Math Trainer Online Assessment and Intervention

16. 72 × ( 6 ÷ 3 )

6

98

18. ( 8 + 56 ) ÷ 4 - 32

19. The nature park has a pride of 7 adult lions and 4 cubs. The adults eat 6 pounds of meat each day and the cubs eat 3 pounds. Simplify 7 × 6 + 4 × 3 to find the amount of meat consumed each day by the lions.

7

54 pounds

ESSENTIAL QUESTION 20. How do you use numerical expressions to solve real-world problems?

Write an expression to model the situation. Simplify the expression using the order of operations. First perform operations in parentheses, then find the value of each power, multiply or divide from left to right, and finally add or subtract from left to right.

Module 10

Texas Essential Knowledge and Skills Lesson

Exercises

10.1

1–8

6.7.A

10.2

9–14

6.7.A

10.3

15–19

6.7.A

287

Module 10

TEKS

287

Personal Math Trainer

MODULE 10 MIXED REVIEW

Texas Test Prep

Texas Testing Tip Students can use logic to eliminate some or all of the answer choices. Item 5 Students can eliminate choices B, C, and D because they all have numbers that are not prime in the product expression. This leaves choice A as the correct answer.

Selected Response 1. Which expression has a value that is less than the base of that expression? A 2 5 B _ 6 C 32 3

( )

Item 8 Students can eliminate choices A and B because they have numbers that are not prime in the product expression.

Avoid Common Errors Item 3 Students sometimes will answer an order of operations question simply by completing the operations from left to right. Remind the students to perform operations in parentheses first. Item 7 Some students may see that 3.6 appears four times and then choose A. Remind them that multiplication is repeated addition and that exponents are needed to represent repeated multiplication.

2

2. After the game the coach bought 9 chicken meals for \$5 each and 15 burger meals for \$6 each. What percent of the total amount the coach spent was used for the chicken meals? 1 A 33 _% 3 B 45% 2 C 66 _% 3 D 90%

5. Which expression shows the prime factorization of 100? A 22 × 52

C

B 10 × 10

D 2 × 5 × 10

1010

A 21

C

B 23

D 27

25

7. Which expression is equivalent to 3.6 × 3.6 × 3.6 × 3.6? 34 × 64

A 3.6 × 4

C

B 36

D 3.64

3

8. Which expression gives the prime factorization of 80?

3. Which operation should you perform first when you simplify 75 - ( 8 + 45 ÷ 3 ) × 7? A addition

A 24 × 10

C

B 2×5×8

D 24 × 5

23 × 5

Gridded Response 9. Alison raised 10 to the 5th power. Then she divided this value by 100. What was the quotient?

B division

multiplication

D subtraction

4. At Tanika’s school, three people are chosen in the first round. Each of those people chooses 3 people in the second round, and so on. How many people are chosen in the sixth round? A 18

.

1

0

0

0

0

0

0

0

0

0

1

1

1

1

1

1

2

2

2

2

2

2

3

3

3

3

3

3

4

4

4

4

4

4

5

5

5

5

5

5

6

6

6

6

6

6

B 216

7

7

7

7

7

7

C

8

8

8

8

8

8

9

9

9

9

9

9

243

D 729

288

Online Assessment and Intervention

6. Which number has only two factors?

D 44

C

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© Houghton Mifflin Harcourt Publishing Company

Texas Test Prep

Unit 4

Texas Essential Knowledge and Skills Items

Mathematical Process TEKS

1

6.7.A

6.1.F

2

6.7.A

6.1.A

3*

6.4.E, 6.7.A

6.1.C

4

6.7.A

6.1.A, 6.1.F

5

6.7.A

6.1.E

6

6.7.A

6.1.F

7

6.7.A

6.1.E

8

6.7.A

6.1.E

9

6.7.A

6.1.C

* Item integrates mixed review concepts from previous modules or a previous course.

Generating Equivalent Numerical Expressions

288

Generating Equivalent Algebraic Expressions ?

MODULE

11

LESSON 11.1

Modeling Equivalent Expressions

You can model real-world problems with variable expressions, then use algebraic rules to solve the problems.

6.7.C

LESSON 11.2

ESSENTIAL QUESTION

Evaluating Expressions

How can you generate equivalent algebraic expressions and use them to solve real-world problems?

6.7.A

LESSON 11.3

Generating Equivalent Expressions 6.7.C, 6.7.D

© Houghton Mifflin Harcourt Publishing Company • Image Credits: ©Lloyd Sutton/ Alamy

Real-World Video

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289

Module 11

Carpenters use formulas to calculate a project’s materials supply. Sometimes formulas can be written in different forms. The perimeter of a rectangle can be written as P = 2(l + w) or P = 2l + 2w.

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Math On the Spot

Animated Math

Personal Math Trainer

Go digital with your write-in student edition, accessible on any device.

Scan with your smart phone to jump directly to the online edition, video tutor, and more.

Interactively explore key concepts to see how math works.

Get immediate feedback and help as you work through practice sets.

289

DO NOT EDIT--Changes must be made through "File info" CorrectionKey=B

Complete these exercises to review skills you will need for this chapter.

Use the assessment on this page to determine if students need intensive or strategic intervention for the module’s prerequisite skills.

Use of Parentheses

2 1

(6 + 4) × (3 + 8 + 1) = 10 × 12

Evaluate.

Intervention

Enrichment

1. 11 + (20 - 13)

2. (10 - 7) - (14 - 12)

3. (4 + 17) - (16 - 9)

4. (23 - 15) - (18 - 13)

5. 8 × (4 + 5 + 7)

6. (2 + 3) × (11 - 5)

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Differentiated Instruction

• Skill 50 Use of Parentheses

• Challenge worksheets

• Skill 53 Words for Operations • Skill 54 Evaluate Expressions

14

30

128

Words for Operations EXAMPLE

Online and Print Resources Skills Intervention worksheets

1

3

Access Are You Ready? assessment online, and receive instant scoring, feedback, and customized intervention or enrichment.

Online Assessment and Intervention

Online Assessment and Intervention

Do the operations inside parentheses first. Multiply.

= 120

Response to Intervention

18

Personal Math Trainer

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Write a numerical expression for the quotient of 20 and 5.

Think: Quotient means to divide.

20 ÷ 5

Write 20 divided by 5.

Write a numerical expression for the word expression. 7. the difference between 42 and 19

PRE-AP

Extend the Math PRE-AP Lesson Activities in TE

42 - 19

20 + 30

9. 30 more than 20

8. the product of 7 and 12 10. 100 decreased by 77

7 × 12

100 - 77

© Houghton Mifflin Harcourt Publishing Company

3

EXAMPLE

Personal Math Trainer

Evaluate Expressions EXAMPLE

Evaluate 2(5) – 32. 2(5) – 32 = 2(5) - 9 = 10 - 9 =1

Evaluate exponents. Multiply. Subtract.

Evaluate the expression. 11. 3(8) - 15

9 8

14. 4(2 + 3) - 12

290

59

12. 4(12) + 11 15. 9(14 - 5) - 42

39

13. 3(7) - 4(2)

13

16. 7(8) - 5(8)

16

Unit 4

6_MTXESE051676_U4MO11.indd 290

28/01/14 6:49 PM

InCopy Notes

PROFESSIONAL DEVELOPMENT VIDEO

InDesign Notes

1. This is a list Bold, Italic, Strickthrough.

1. This is a list

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Author Juli Dixon models successful teaching practices as she explores equivalent algebraic expressions in an actual sixth-grade classroom.

Online Teacher Edition Access a full suite of teaching resources online—plan, present, and manage classes and assignments.

Professional Development

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Interactive Answers and Solutions Customize answer keys to print or display in the classroom. Choose to include answers only or full solutions to all lesson exercises.

Interactive Whiteboards Engage students with interactive whiteboard-ready lessons and activities. Personal Math Trainer: Online Assessment and Intervention Assign automatically graded homework, quizzes, tests, and intervention activities. Prepare your students with updated, TEKS-aligned practice tests.

Generating Equivalent Algebraic Expressions

290

DO NOT EDIT--Changes must be made through "File info" CorrectionKey=B

Simplifying Expressions

operations, order of operations

Understand Vocabulary

Students can use these reading and note-taking strategies to help them organize and understand new concepts and vocabulary. c.4.D Use prereading supports such as graphic organizers, illustrations, and pretaught topic-related vocabulary to enhance comprehension of written text.

Preview Words

base, exponent

numerical expression

×, ÷, +, -

algebraic expression (expresión algebraica) coefficient (coeficiente) constant (constante) equivalent expression (expresión equivalente) evaluating (evaluar) like terms (términos semejantes) term (término, en una expresión) variable (variable)

23 2+1+3

Understand Vocabulary Complete the sentences using the preview words.

1. An expression that contains at least one variable is an

algebraic expression

.

2. A part of an expression that is added or subtracted is a © Houghton Mifflin Harcourt Publishing Company

Integrating the ELPS

base (base) exponent (exponente) numerical expression (expresión numérica) operations (operaciones) order of operations (orden de las operaciones)

Use the review words to complete the graphic. You may put more than one word in each oval.

The graphic organizer will help students to review concepts related to simplifying expressions. If time allows, discuss as a class the mnemonic Please Excuse My Dear Aunt Sally for order of operations (Parentheses, Exponent, Multiplication, Division, Addition, Subtraction).

Review Words

Visualize Vocabulary

Visualize Vocabulary

Use the following explanation to help students learn the preview words. Variable is an antonym of constant. Variable means “changeable”; something that is constant does not change. In math, a variable is a letter or symbol that represents a number. A letter is used because the number is unknown, and it may vary. A constant is a numeral, not a letter. For an expression to be an algebraic expression, it must contain at least one variable.

Vocabulary

3. A

constant

term

.

is a specific number whose value does not change.

Key-Term Fold Before beginning the module, create a key-term fold to help you learn the vocabulary in this module. Write the highlighted vocabulary words on one side of the flap. Write the definition for each word on the other side of the flap. Use the key-term fold to quiz yourself on the definitions used in this module.

Differentiated Instruction • Reading Strategies ELL

Module 11

6_MTXESE051676_U4MO11.indd 291

Grades 6–8 TEKS Before Students understand: • operations with whole numbers, decimals, and fractions • order of operations • properties of operations: inverse, identity, commutative, associative, and distributive properties

291

Module 11

28/01/14 6:49 PM

InCopy Notes

InDesign Notes

1. This is a list

1. This is a list

In this module Students will learn to: • determine if two expressions are equivalent using concrete models, pictorial models, and algebraic representations • evaluate algebraic expressions for the given value of a variable • generate equivalent expressions using the properties of operations: inverse, identity, commutative, associative, and distributive properties

291

After Students will connect: • numerical and algebraic expressions • variables and symbols to translate words into math

MODULE 11

Unpacking the TEKS

Unpacking the TEKS Understanding the TEKS and the vocabulary terms in the TEKS will help you know exactly what you are expected to learn in this module.

Use the examples on this page to help students know exactly what they are expected to learn in this module.

6.7.C

What It Means to You

Determine if two expressions are equivalent using concrete models, pictorial models, and algebraic representations.

Texas Essential Knowledge and Skills

You will use models to compare expressions. UNPACKING EXAMPLE 6.7.C

On a math quiz, Tina scored 3 points more than Yolanda. Juan scored 2 points more than Yolanda and earned 2 points as extra credit. Draw models for Tina's and Juan's scores. Use your models to decide whether they made the same score.

Key Vocabulary

Content Focal Areas

equivalent expressions (expresión equivalente) Expressions that have the same value for all values of the variables.

Expressions, equations, and relationships—6.7 The student applies mathematical process standards to develop concepts of expressions and equations.

y+3 Tina

y

3 y+2+2

Integrating the ELPS Juan

y

2

2

Tina and Juan did not make the same score because the models do not show equivalent expressions.

6.7.D Generate equivalent expressions using the properties of operations: inverse, identity, commutative, associative, and distributive properties.

Go online to see a complete unpacking of the .

What It Means to You You will use the properties of operations to find an equivalent expression. UNPACKING EXAMPLE 6.7.D

William earns \$13 an hour working at a movie theater. He worked h hours in concessions and three times as many hours at the ticket counter. Write and simplify an expression for the amount of money William earned.

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\$13 · hours at concessions + \$13 · hours at ticket counter Visit my.hrw.com to see all the unpacked.

© Houghton Mifflin Harcourt Publishing Company • Image Credits: Erik Dreyer/Getty Images

c.4.F Use visual and contextual support … to read grade-appropriate content area text … and develop vocabulary … to comprehend increasingly challenging language.

13h + 13(3h) 13h + 39h

Multiply 13 · 3h.

h(13 + 39)

Distributive Property

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292

Lesson 11.1

Lesson 11.2

Unit 4

Lesson 11.3

6.7.A Generate equivalent numerical expressions using order of operations, including whole number exponents and prime factorization. 6.7.C Determine if two expressions are equivalent using concrete models, pictorial models, and algebraic representations. 6.7.D Generate equivalent expressions using the properties of operations: inverse, identity, commutative, associative, and distributive properties.

Generating Equivalent Algebraic Expressions

292

LESSON

11.1 Modeling Equivalent Expressions Texas Essential Knowledge and Skills The student is expected to: Expressions, equations, and relationships—6.7.C Determine if two expressions are equivalent using concrete models, pictorial models, and algebraic expressions.

Engage

ESSENTIAL QUESTION How can you write algebraic expressions and use models to decide if expressions are equivalent? Sample answer: Write or model the variables, constants, and operations to represent each expression. Then compare the expressions or models.

Motivate the Lesson Ask: Is a quarter the same as 5 nickels? as 25 pennies? How can you describe something in different ways but not change its value? Begin the Explore Activity to find out.

Mathematical Processes 6.1.D Communicate mathematical ideas, reasoning, and their implications using multiple representations, including symbols, diagrams, graphs, and language as appropriate.

Explore EXPLORE ACTIVITY Focus on Modeling Mathematical Processes Point out to students that a balance scale can represent how two numbers or expressions compare in value. When the numbers or expressions on each side of the scale are equal in value, the scale is in balance. If the numbers or expressions are unequal, one side of the scale is higher than the other side.

Explain ADDITIONAL EXAMPLE 1 A Write each phrase as an algebraic expression. x less than 5 5 - x the product of z and 8 8z B Write a phrase for each algebraic expression. __z the quotient of z and 5 5

9 + y 9 more than y

EXAMPLE 1 Engage with the Whiteboard Write a mathematical operation on the whiteboard. Next to it, have students write an expression that uses the operation. Then ask them to describe their expression with words in several different ways and to list these on the whiteboard. For example, 6x can be described as 6 times x, the product of 6 and x, and 6 multiplied by x. Discuss with students the ways to describe each operation by looking at the list provided above Example 1.

Questioning Strategies

Interactive Whiteboard Interactive example available online my.hrw.com

Mathematical Processes • How do you identify the variable in an algebraic expression? Find a letter or symbol that represents an unknown. • Does it matter which letter you choose when writing an algebraic expression? No. You can choose any letter, but mathematicians often choose the last letters of the alphabet (e.g., x, y, and z) to represent variables. • How can you find the constant in an algebraic expression? Look for a specific number whose value does not change.

Connect Vocabulary

ELL

Explain to students that in the expression 8x + 15, the number 8 is a coefficient. A coefficient is the number multiplied by the variable, x. The number 15 in this expression is a constant.

293

Lesson 11.1

11.1 ?

Modeling Equivalent Expressions

ESSENTIAL QUESTION

Expressions, equations, and relationships— 6.7.C Determine if two expressions are equivalent using concrete models, pictorial models, and algebraic representations.

Writing Algebraic Expressions An algebraic expression is an expression that contains one or more variables and may also contain operation symbols, such as + or -. Math On the Spot my.hrw.com

How can you write algebraic expressions and use models to decide if expressions are equivalent?

A variable is a letter or symbol used to represent an unknown or unspecified number. The value of a variable may change. A constant is a specific number whose value does not change. constant

EXPLORE ACTIVITY

variable

6.7.C

Algebraic Expressions

Modeling Equivalent Expressions

150 + y

w+n

x

15

12 - 7

9 __ 16

Not Algebraic Expressions

Equivalent expressions are expressions that have the same value.

In algebraic expressions, multiplication and division are usually written without the symbols × and ÷. • Write 3 × n as 3n, 3 · n, or n · 3. • Write 3 ÷ n as _n3 . There are several different ways to describe expressions with words.

The scale shown to the right is balanced.

A Write an expression to represent the circles on the left side of the balance.

2+3

B The value of the expression on the left side is

5

.

Operation

C Write an expression to represent the circles on the right side of the balance.

1+4

D The value of the expression on the right side is

5

Words

.

added to plus sum more than

E Since the expressions have the same value, the expressions are

equivalent

subtracted from minus difference less than take away taken from

• • • •

Multiplication

Division

times multiplied by product groups of

• divided by • divided into • quotient

.

F What will happen if you remove a circle from the right side of the balance? © Houghton Mifflin Harcourt Publishing Company

Subtraction • • • • • •

EXAMPLE 1

The scale will no longer be balanced.

6.7.C

A Write each phrase as an algebraic expression.

G If you add a circle to the left side of the balance, what can you do to the right side to keep the scale in balance?

The sum of 7 and x

The algebraic expression is 7 + x.

Add a circle to the right side.

The quotient of z and 3

The operation is division.

The algebraic expression is _3z .

Reflect 1. What If? Suppose there were 2 + 5 circles on the right side of the balance and 3 on the left side of the balance. What can you do to balance the scale? Explain how the scale models equivalent expressions.

B Write a phrase for each expression. 11x

© Houghton Mifflin Harcourt Publishing Company

LESSON

The operation is multiplication.

Sample answer: Add 4 circles to the left side. 4 + 3 is equivalent

The product of 11 and x

to 2 + 5 because the value of both expressions is 7.

8-y

The operation is subtraction.

y less than 8 Lesson 11.1

293

294

Unit 4

PROFESSIONAL DEVELOPMENT Integrate Mathematical Processes This lesson provides an opportunity to address Mathematical Process TEKS 6.1.D, which calls for students to “communicate mathematical ideas … using multiple representations, including symbols, diagrams…and language as appropriate.” In this lesson, students use symbols, a model of a scale, and bar models to represent equivalent expressions. They employ these multiple representations to compare algebraic expressions and solve problems in real-world situations.

Math Background François Viète (1540–1603) was a lawyer in France who devoted his spare time to mathematics. In his book In artem analyticam isagoge, he introduced the idea of using vowels for variables and using consonants for constants. This was an important step toward modern algebra. Although Viète also used + and -, he had no symbol for equality. To write “equals,” he would use the Latin word aequatur. Viète is sometimes called the Father of Algebra.

Modeling Equivalent Expressions

294

YOUR TURN Avoid Common Errors ADDITIONAL EXAMPLE 2 Use a bar model to represent each expression. A 5+y

Talk About It Check for Understanding Ask: How can you tell if an expression is an algebraic expression? Look for a variable, a letter or symbol that represents an unknown. If the expression has a variable, it is an algebraic expression.

5+y

y

5

B __n4

Exercise 2 When students multiply a number by a variable, be sure that they write the number first: 4x, not x4. It’s easier to read and understand.

EXAMPLE 2

n

Questioning Strategies

Mathematical Processes • On the bar model for A, why is 7 + x written above the model, while 7 and x are written below? The total that the bar represents is 7 plus an unknown amount; the labels below show what part of the model is 7 and what part is x.

n 4

Interactive Whiteboard Interactive example available online my.hrw.com

• On the bar model for B, how do you know how many pieces to divide the bar into? The expression __z3 means “7 divided into 3 parts,” so you know you need to divide the bar into 3 equal parts.

Engage with the Whiteboard Have students change the constant in Example 2A and/or 2B. Then have students draw a model to represent the new expression. Ask volunteers to explain their models to the class, and then ask the class if the students are correct.

ADDITIONAL EXAMPLE 3 Amanda and Stuart began the week with the same amount of money. Amanda paid \$7 to go to the movies. Stuart spent \$4 on snacks and \$3 on a pen. Write algebraic expressions and draw bar models to represent the money each has left at the end of the week. Do Amanda and Stuart have the same amount of money left? n

n- 7

7 n

Exercise 7 Some students may try to model t - 2 as an addition equation. Remind students that for subtraction expressions, they must “take away” the 2 from the whole, not add to it.

EXAMPLE 3 Questioning Strategies

Mathematical Processes • How can you know which operation to use to solve the problem? Read the problem carefully. Katriana and Andrew “spent” or “took away” money from the money they started the day with. These words describe subtraction. • In Step 1, what do the labels on the model represent? The variable x represents the amount of money Katriana started with, the 5 represents the money she spent, and x - 5 represents the money she has left.

Focus on Reasoning 4

3

n- 4- 3

The models are equivalent, so Amanda and Stuart have the same amount of money left. Interactive Whiteboard Interactive example available online my.hrw.com

295

Lesson 11.1

Mathematical Processes Have students compare and contrast the models presented in the Explore Activity and Example 3. Encourage students to debate the advantages and disadvantages of each model, supporting their arguments with examples.

YOUR TURN Avoid Common Errors If students have difficulty with drawing the models, encourage them to circle the information for Tina in one color and for Juan in a different color. Then remind them to look at the list of ways to describe math operations to determine the operation they should use in the models.

Comparing Expressions Using Models

YOUR TURN Write each phrase as an algebraic expression. 2. n times 7

7n

3. 4 minus y

12

4. 13 added to x x

+ 13

Write a phrase for each expression. x 5. __

4-y

6. 10y 10

the quotient of x and 12

Personal Math Trainer Online Assessment and Intervention

Algebraic expressions are equivalent if they are equal for all values of the variable. For example, x + 2 and x + 1 + 1 are equivalent. Math On the Spot my.hrw.com

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EXAMPLE 3

6.7.C

Katriana and Andrew started the day with the same amount of money. Katriana spent 5 dollars on lunch. Andrew spent 3 dollars on lunch and 2 dollars on an afterschool snack.

multiplied by y

Do Katriana and Andrew have the same amount of money left?

EXAMPL 2 EXAMPLE

6.7.C

Math On the Spot

The variable represents the amount of money both Katriana and Andrew have at the beginning of the day.

Use a bar model to represent each expression.

A 7+x

7+x

x

7 B _3z

Divide z into 3 equal parts.

© Houghton Mifflin Harcourt Publishing Company

z

z 3

2

8. 4y

t

t- 2

y

x

3

2

x- 3- 2

Compare the models.

STEP 3

The models are equivalent, so the expressions are equivalent. Andrew and Katriana have the same amount of money left.

Mathematical Processes

9. On a math quiz, Tina scored 3 points more than Julia. Juan scored 2 points more than Julia and earned 2 points in extra credit. Write an expression and draw a bar model to represent Tina’s score and Juan’s score. Did Tina and Juan make the same grade on the quiz? Explain.

What two phrases can you use to describe the expression _4x ? What is different about the two phrases?

Draw a bar model to represent each expression. 7. t - 2

x−3−2

Math Talk

Write an algebraic expression to represent the money Andrew has left. Represent the expression with a model.

STEP 2

Sample answer: 4 divided by x; x divided into 4. The numerator, or dividend, comes first when using the term divided by. The denominator, or divisor, comes first when using the term divided into.

Combine 7 and x.

x-5

5

x−5

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© Houghton Mifflin Harcourt Publishing Company

Algebraic expressions can also be represented with models.

x

Write an algebraic expression to represent the money Katriana has left. Represent the expression with a model.

STEP 1

Modeling Algebraic Expressions

Juan: y + 2 + 2

Tina: y + 3

4y Personal Math Trainer

Personal Math Trainer

Online Assessment and Intervention

Online Assessment and Intervention

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Lesson 11.1

295

296

y

3

y

2

2

No; the expressions are not equivalent.

Unit 4

Modeling Equivalent Expressions

296

Elaborate Talk About It Summarize the Lesson Ask: How can you find out whether algebraic expressions are equivalent? Draw models of each expression and then compare them. If the models are equivalent, the expressions are equivalent.

GUIDED PRACTICE Engage with the Whiteboard For Exercise 1, have students write expressions that will keep the scale balanced next to the scale on the whiteboard. Emphasize to students that all the expressions are equivalent. For Exercise 4, have students circle the variable and the constant in y + 12. Then have them write three different algebraic expressions that are equivalent to y + 12 on the whiteboard. Sample answer: y + 3 + 9; y + 4 + 8; y + 5 + 7 For Exercise 7, have students complete the bar models for each city on the whiteboard and then explain their reasoning.

Avoid Common Errors Exercise 2 Remind students that the order of the variable and the constant is important in subtraction expressions, y - 5 is not the same as 5 - y. Exercise 3 When students multiply a number by a variable, be sure that they write the number first: 4x, not x4. It’s easier to read and understand. Exercise 6 Some students may try to model m ÷ 4 as an addition equation. Remind students that for division expressions, they must “divide” the whole into parts, not add to it.

297

Lesson 11.1

Guided Practice

Name

5+ 5

2p

3. The product of 2 and p

p 5. __ 10

11. Write an algebraic expression with two variables and one constant.

answers are given. 10 divided into p m

t

Tucson

© Houghton Mifflin Harcourt Publishing Company

2

3

4

14. n divided by 8

n __ 8

15. p multiplied by 4

4p b + 14

Tuesday

5

3

Wednesday

8

No; the models show that the temperature in Phoenix is 1 degree less than the temperature in Tucson.

18. a take away 16 19. k less than 24

24 - k

20. 3 groups of w

3w 1+q

21. the sum of 1 and q

13 __ z

22. the quotient of 13 and z

ESSENTIAL QUESTION CHECK-IN

45 + c

9. How can you use expressions and models to determine if expressions are equivalent?

Write a phrase in words for each algebraic expression. Sample answers given.

Write or model the variables, constants, and operations to represent each

24. m + 83

expression. Then compare the expressions or models.

25. 42s Lesson 11.1

297

298

32. Write an expression that represents Sarah’s total pay last week. Represent her hourly wage with w.

5w + 3w

90x a - 16

17. 90 times x

8. Are the expressions that represent the temperatures in the two cities equivalent? Justify your answer.

? ?

Monday

Noah

t

t- 4

Sarah

Write each phrase as an algebraic expression.

16. b plus 14

t- 2- 3

k less than 5

Read On Bookstore Work Schedule (hours)

x

Variable(s)

the quotient of h and 12

Sarah and Noah work at Read On Bookstore and get paid the same hourly wage. The table shows their work schedule for last week.

15

Constant(s)

the product of 11 and x

12

13. Identify the parts of the algebraic expression x + 15.

7. Represent the temperature in each city with an algebraic expression and a bar model.

28. 2 + g

31. 5 − k

12. What are the variables in the expression x + 8 − y?

m 4

t minus 29 g more than 2

h 30. __

x and y

At 6 p.m., the temperature in Phoenix, AZ, t, is the same as the temperature in Tucson, AZ. By 9 p.m., the temperature in Phoenix has dropped 2 degrees and in Tucson it has dropped 4 degrees. By 11 p.m., the temperature in Phoenix has dropped another 3 degrees. (Example 3)

27. t − 29

29. 11x

Sample answer: x + 24 − y

6. Draw a bar model to represent the expression m ÷ 4. (Example 2)

Phoenix

d

Online Assessment and Intervention

9 divided by d

26. __9

10. Write an algebraic expression with the constant 7 and the variable y.

Write a phrase for each algebraic expression. (Example 1) Sample 4. y + 12

my.hrw.com

33. Write an expression that represents Noah’s total pay last week. Represent his hourly wage with w.

8w 34. Are the expressions equivalent? Did Sarah and Noah earn the same amount last week? Use models to justify your answer.

Yes, Sarah and Noah got paid the same amount last week. Check

© Houghton Mifflin Harcourt Publishing Company

7+ 3

y−3

Personal Math Trainer

6.7.C

Write each phrase as an algebraic expression. (Example 1) 2. 3 less than y

Date

11.1 Independent Practice

1. Write an expression in the right side of the scale that will keep the scale balanced. (Explore Activity)

Class

students’ models.

83 added to m 42 times s

Unit 4

DIFFERENTIATE INSTRUCTION Cognitive Strategies

Cooperative Learning

Ask students for the meanings of the words variable and constant in a context such as the following: The air temperature in the desert was quite variable yesterday; it was cold overnight and warm during the day. The temperature at the equator was constant for 24 hours. Explain that the words have the same meaning in mathematics. A constant is a value that does not change, such as the number 5, and a variable is a symbol for a quantity that is not fixed, such as x.

Have students work in pairs to draw models of balance scales as shown in the Explore Activity. Instruct pairs to take turns writing simple expressions and drawing circles to represent them on the balance pans on each side of the scale. Then ask each student to write and illustrate two sets of equivalent expressions that balance when arranged on the scales. Invite pairs to explain how they chose their arrangements of circles.

Differentiated Instruction includes: • Reading Strategies • Success for English Learners ELL • Reteach • Challenge PRE-AP

Modeling Equivalent Expressions

298

Personal Math Trainer Online Assessment and Intervention

Evaluate GUIDED AND INDEPENDENT PRACTICE

Online homework assignment available

6.7.C

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11.1 LESSON QUIZ 6.7.C 1. Write each phrase as an algebraic expression: z times 5 6 plus n 2. Write a phrase for each algebraic expression: 6 8y __ m

Concepts & Skills

Practice

Explore Activity Modeling Equivalent Expressions

Exercise 1

Example 1 Writing Algebraic Expressions

Exercises 2–5, 14–33

Example 2 Modeling Algebraic Expressions

Exercise 6

Example 3 Comparing Expressions Using Models

Exercises 7–8, 34–37

3. Use a bar model to represent 4 + m. 4. Jan and Jackie check out the same number of library books. Jan turns in 4 books after 3 weeks. Jackie returns 2 books that week and 4 books later. Write algebraic expressions and draw bar models to represent the books Jan and Jackie have left. Do they have the same number of books left? Justify your answer. Lesson Quiz available online my.hrw.com

1. 5z, 6 + n 2. 8 multiplied by y, 6 divided by m 4+m

3.

Exercise

Depth of Knowledge (D.O.K.)

10–13

2 Skills/Concepts

1.F Analyze relationships

14–33

2 Skills/Concepts

1.C Select tools

34–36

3 Strategic Thinking

1.F Analyze relationships

37–40

2 Skills/Concepts

1.A Everyday life

41

3 Strategic Thinking

1.G Explain and justify arguments

42

3 Strategic Thinking

1.G Explain and justify arguments

43

3 Strategic Thinking

1.E Create and use representations

44

3 Strategic Thinking

1.A Everyday life

45

3 Strategic Thinking

1.F Analyze relationships

Additional Resources Differentiated Instruction includes: • Leveled Practice Worksheets m

4

4. No. Sample answer: The expressions are not equivalent. If b = the number of books each checked out, Jan’s books = b - 4 and Jackie’s books = b - 2 - 4 b

b-4

b-2-4

299

Lesson 11.1

4 b

2

4

Mathematical Processes

DO NOT EDIT--Changes must be made through “File info” CorrectionKey=A

DO NOT EDIT--Changes must be made through “File info” CorrectionKey=A

35. Critique Reasoning Lisa concluded that 3 · 2 and 32 are equivalent expressions. Is Lisa correct? Explain.

39. Abby baked 48 cookies and divided them evenly into bags. Let b represent the number of bags. Write an algebraic expression to represent the number of cookies in each bag.

No; 3 · 2 = 6 and 32 = 9, 6 ≠ 9; The expressions are not

40. Eli is driving at a speed of 55 miles per hour. Let h represent the number of hours that Eli drives at this speed. Write an algebraic expression to represent the number of miles that Eli travels during this time.

equal. 36. Multiple Representations How could you represent the expressions x - 5 and x - 3 - 3 on a scale like the one you used in the Explore Activity? Would the scale balance?

41. Represent Real-World Problems If the number of shoes in a closet is s, then how many pairs of shoes are in the closet? Explain.

x - 3 - 3 on the right side; no; you are removing 5 from

x on the left side and you are removing 6 from x on

_s ; there are half as many pairs of shoes as there are total 2

the right.

shoes. 42. Communicate Mathematical Ideas Is 12x an algebraic expression? Explain why or why not.

37. Multistep Will, Hector, and Lydia volunteered at the animal shelter in March and April. The table shows the number of hours Will and Hector volunteered in March. Let x represent the number of hours Lydia volunteered in March.

Sample answer: 12x is an algebraic expression because it contains a variable.

March Volunteering Will

3 hours

Hector

5 hours

43. Problem Solving Write an expression that has three terms, two different variables, and one constant.

Sample answer: 2x - 8y + 7.

a. Will’s volunteer hours in April were equal to his March volunteer hours plus Lydia’s March volunteer hours. Write an expression to represent Will’s volunteer hours in April.

44. Represent Real-World Problems Describe a situation that can be modeled by the expression x − 8.

Sample answer: Sam started the day with a pack of gum. During the day he gave out 8 pieces of gum.

b. Hector’s volunteer hours in April were equal to 2 hours less than his March volunteer hours plus Lydia’s March volunteer hours. Write an expression to represent Hector’s volunteer hours in April.

5-2+x

45. Critique Reasoning Ricardo says that the expression y + 4 is equivalent to the expression 1y + 4. Is he correct? Explain.

c. Did Will and Hector volunteer the same number of hours in April? Explain.

Sample answer: Yes; 1y is the product of 1 and y. Since 1 times any number is equal to the number, 1 · y = y. The

Yes; The expressions are equivalent.

expression y + 4 is equivalent to 1y + 4.

38. The town of Rayburn received 6 more inches of snow than the town of Greenville. Let g represent the amount of snow in Greenville. Write an algebraic expression to represent the amount of snow in Rayburn.

g+6

Lesson 11.1

6_MTXESE051676_U4M11L1.indd 299

1. This is a list

300

299

24/12/12 12:08 PM

EXTEND THE MATH

Unit 4

6_MTXESE051676_U4M11L1.indd 300

InCopy Notes

InDesign Notes

PRE-AP 1. This is a list

© Houghton Mifflin Harcourt Publishing Company

© Houghton Mifflin Harcourt Publishing Company

3+x

InCopy Notes

55h Work Area

FOCUS ON HIGHER ORDER THINKING

Represent x - 5 on the left side of the scale and

48 __ b

1. This is aonline list Bold, Italic, Strickthrough. my.hrw.com Activity available

10/25/12 3:05 PM

InDesign Notes 1. This is a list

Activity Each season the Ravens and the Hawks baseball teams play the same number of games. So far this year, the Ravens have played 3 games at home and 4 games on the road. The Hawks have played 5 games at home and 3 games on the road. Roberto drew these bar models and says that the Hawks have more games left to play in the season than the Ravens do. Is he correct? If not, what did he do incorrectly when he represented the given information? Ravens:

g

g–5–3

5

Hawks:

3

g

g–3–4

3

4

No, Roberto is not correct. He mixed up the data and mislabeled both models. 1) The tops of both bars should be labeled as g, the variable that stands for games left to play. 2) The Ravens model should have the labels g - 3 - 4, 3, and 4. 3) The Hawks model should have the labels g - 5 - 3, 5 and 3. Comparing the corrected models shows that the expressions are not equivalent: 7 < 8. So, the Ravens have one more game to play than the Hawks do. Modeling Equivalent Expressions

300

LESSON

11.2 Evaluating Expressions Texas Essential Knowledge and Skills The student is expected to:

Engage

ESSENTIAL QUESTION How can you use the order of operations to evaluate algebraic expressions? Sample answer: Substitute the given value for the variable in the expression and then use the order of operations to find the value of the resulting numerical expression.

Expressions, equations, and relationships—6.7.A Generate equivalent numerical expressions using order of operations, including whole number exponents and prime factorization.

Motivate the Lesson Ask: How can you evaluate an expression that includes an unknown value? Begin Example 1 to find out.

Mathematical Processes 6.1.G Display, explain, and justify mathematical ideas and arguments using precise mathematical language in written or oral communication.

Explore Engage with the Whiteboard

Write the expression 2(4 + x) - 5 on the whiteboard. Ask a student to evaluate the expression for x = 2. Then ask the student to explain his/her reasoning. Ask the class if the student’s work and reasoning are correct. If students have difficulty understanding which operation to perform first, review the order of operations.

ADDITIONAL EXAMPLE 1 Evaluate each expression for the given value of the variable. A b - 7; b = 16 28 B __ m; m = 4

9

EXAMPLE 1

7

C 0.2t; t = 1.6

Explain Focus on Math Connections

Mathematical Processes C and D involve an expression with a coefficient, a number that is multiplied by the variable. In the expression 0.5y, the coefficient is 0.5.

0.32

D 8s: s = __12 4 Interactive Whiteboard Interactive example available online my.hrw.com

Questioning Strategies

Mathematical Processes • How might substituting a negative value into one of the expressions affect its value? Substituting a negative value could change the sign of the answer and make it larger or smaller. In A, for example, if x = -15, the answer would be -24.

YOUR TURN ADDITIONAL EXAMPLE 2 Evaluate each expression for the given value of the variable. A 6(y - 6); y = 9 B 6y - 6; y = 9

18 48

C n - y + x; n = 5; y = 3; x = 4 D y - 2y; y = 7 2

E 4y - 3; y = -7

6

35

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Lesson 11.2

Check for Understanding Ask: How do you evaluate an expression for a given variable? Substitute the given value for the variable in the expression. Then perform the operations, using the order of operations to find the value of the expression.

EXAMPLE 2 Focus on Math Connections

-31

Interactive Whiteboard Interactive example available online

301

Mathematical Processes Remind students of the correct order of operations (parentheses, exponents, multiplication/ division, addition/subtraction) and the mnemonic device PEMDAS.

Questioning Strategies

Mathematical Processes • The answers to A and B are not the same, even though the expressions are very similar. Why? The parentheses in 4(x - 4) mean that you subtract first. There are no parentheses in 4x - 4, so you multiply first.

11.2 ?

Evaluating Expressions

ESSENTIAL QUESTION

Expressions, equations, and relationships—6.7.A Generate equivalent numerical expressions using order of operations, including whole number exponents and prime factorization.

Evaluate each expression for the given value of the variable.

Online Assessment and Intervention

Math On the Spot my.hrw.com

Math On the Spot my.hrw.com

A x - 9; x = 15

6

Subtract.

No; in w - x + y, you subtract x from w first. Then you add y to that result. In w - y + x, you subtract y from w first. Then you add x to that result.

When x = 15, x - 9 = 6. 16 B __ n ;n=8 16 __ 8

Substitute 8 for n.

2

Divide.

© Houghton Mifflin Harcourt Publishing Company

16 When n = 8, __ n = 2.

C 0.5y; y = 1.4 0.5(1.4)

Substitute 1.4 for y.

0.7

Multiply.

Math Talk

When y = 1.4, 0.5y = 0.7.

Mathematical Processes

Is w - x + y equivalent to w - y + x? Explain any difference in the order the math operations are performed.

D 6k; k = _13 6 . HINT: Think of 6 as __ 1

()

6 _13

1 for k. Substitute __ 3

2

Multiply.

m ; m = 18 3. __ 6

3

Expressions may have more than one operation or more than one variable. To evaluate these expressions, substitute the given value for each variable and then use the order of operations.

EXAMPLE 2

6.7.A

A 4(x - 4); x = 7

Evaluate each expression for the given value of the variable.

Substitute 15 for x.

4.7

Evaluate each expression for the given value of the variable.

6.7.A

15 - 9

2. 6.5 - n; n = 1.8

Using the Order of Operations

Evaluating Expressions

EXAMPL 1 EXAMPLE

32

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How can you use the order of operations to evaluate algebraic expressions?

Recall that an algebraic expression contains one or more variables. You can substitute a number for that variable and then find the value of the expression. This is called evaluating the expression.

1. 4x; x = 8

When k = _13, 6k = 2.

4(7 - 4)

Substitute 7 for x.

4(3)

Subtract inside the parentheses.

12

Multiply.

When x = 7, 4(x - 4) = 12.

B 4x - 4; x = 7 4(7) - 4

Substitute 7 for x.

28 - 4

Multiply.

24

Subtract.

When x = 7, 4x - 4 = 24.

C w - x + y; w = 6, x = 5, y = 3 (6) - (5) + (3)

Substitute 6 for w, 5 for x, and 3 for y.

1+3

Subtract.

4

When w = 6, x = 5, y = 3, w - x + y = 4.

D x2 - x; x = 9 (9)2 - (9)

Substitute 9 for each x.

81 - 9

Evaluate exponents.

72

© Houghton Mifflin Harcourt Publishing Company

LESSON

Subtract.

When x = 9, x - x = 72. 2

Lesson 11.2

301

302

Unit 4

PROFESSIONAL DEVELOPMENT Integrate Mathematical Processes This lesson provides an opportunity to address Mathematical Process TEKS 6.1.G, which calls for students to “display, explain, and justify mathematical ideas … using precise mathematical language in written … communication. ” In each Example and Exercise, students use mathematical ideas and language, including the order of operations, to evaluate algebraic expressions by substituting given values for variables. Students then evaluate real-world expressions such as formulas for finding surface area and volume and converting Celsius temperatures to Fahrenheit.

Math Background We translate (or write) words into algebraic expressions by using a consistent, universally understood system. This system has evolved over thousands of years. Archaeological records indicate that Babylonian mathematicians had developed prose-based algebra by 2000 B.C. The adoption of symbols to represent operations was also part of this evolution. The symbols + and - can be traced to Johann Widman (1498); the symbol · can be traced to Gottfried Leibniz (1698); and the symbol ÷ can be traced to Johann Heinrich Rahn (1659).

Evaluating Expressions

302

YOUR TURN Avoid Common Errors Exercises 7–9 Watch for students who substitute the wrong value for the variable. Caution students to be sure that they are substituting the correct value for each variable in expressions with more than one variable.

ADDITIONAL EXAMPLE 3 The expression 3.3m gives the number of feet in m (meters). Use the expression to find the number of feet that is equivalent to 400 meters. 1,320 feet Interactive Whiteboard Interactive example available online my.hrw.com

EXAMPLE 3 Focus on Math Connections

Mathematical Processes Remind students that when there is a coefficient in front of a variable, multiplication is indicated. Therefore, when they replace the variable with a value, they need to insert parentheses. In Step 2, for example, the expression 1.8c + 32 is written as 1.8(30) + 32 when substituting 30 for c.

Questioning Strategies

Mathematical Processes • How do you find the value of the variable c? Read the text of the problem carefully. The value of the variable is given in the last sentence.

• If the expression were written as 32 + 1.8c, would you still perform the multiplication first? Explain. Addition is associative, so changing the order of the terms does not change the expression in any way. You still need to do the multiplication before the addition. c.3.B ELL Be sure English learners understand the references to Celsius and Fahrenheit in Example 3. You may want to point out that both scales measure temperature but the Celsius Scale is part of the Metric System.

Integrating the ELPS

YOUR TURN Avoid Common Errors Exercise 10 Remind students that an exponent tells how many times to use the base as a factor, so x3 means x · x · x, not 3x.

Elaborate Talk About It Summarize the Lesson Ask: How can the order of operations help you evaluate algebraic expressions? When an expression contains more than one operation, the order of operations tells which operation to perform first.

GUIDED PRACTICE Engage with the Whiteboard For Exercises 7–8, have students circle all the important information provided, including key words that indicate operations, on the whiteboard. Then have them write an expression to represent each problem. Finally, have them complete the steps to evaluate each problem.

Avoid Common Errors Exercise 2 Remind students that when there is a coefficient in front of a variable, multiplication is indicated. Therefore, when they replace the variable with a value, they need to insert parentheses. Exercises 3, 5 Remind students that the fraction bar is another way to represent division.

303

Lesson 11.2

DO NOT EDIT--Changes must be made through “File info” CorrectionKey=B

DO NOT EDIT--Changes must be made through “File info” CorrectionKey=B

Guided Practice

YOUR TURN Evaluate each expression for n = 5. 5. 4 (n - 4) + 14

18

6. 6n + n2

18

8. bc + 5a

-9

9. a2 - (b + c)

5. _12 w + 2; w = _19

6.7.A

The expression 1.8c + 32 gives the temperature in degrees Fahrenheit for a given temperature in degrees Celsius c. Find the temperature in degrees Fahrenheit that is equivalent to 30 °C.

my.hrw.com

Find the value of c.

12(

Substitute the value into the expression.

1.8(30) + 32

Substitute 30 for c.

54 + 32

Multiply.

86

3

)+5=

36

+5=

The family spent \$41

\$6

Nonstudent tickets

\$12

Parking

\$5

41

to attend the game.

a. Write an expression that represents the perimeter of the rectangular tablecloth. Let l represent the length of the tablecloth and w represent its width. The expression would be

2w + 2l

.

b. Evaluate the expression P = 2w + 2l for l = 5 and w = 7.

2(

10. The expression 6x2 gives the surface area of a cube, and the expression x3 gives the volume of a cube, where x is the length of one side of the cube. Find the surface area and the volume of a cube with a side length of 2 m.

24

m2 ; V =

8

seconds

7

) + 2(

Stan bought

5

) = 14 +

24 feet

10

=

24

of trim to sew onto the tablecloth.

9. Essential Question Follow Up How do you know the correct order in which to evaluate algebraic expressions?

m3

11. The expression 60m gives the number of seconds in m minutes. How many seconds are there in 7 minutes?

420

Women’s Soccer Game Prices Student tickets

8. Stan wants to add trim all around the edge of a rectangular tablecloth that measures 5 feet long by 7 feet wide. The perimeter of the rectangular tablecloth is twice the length added to twice the width. How much trim does Stan need to buy? (Example 3)

86 °F is equivalent to 30 °C.

S=

50

b. Since there are three attendees, evaluate the expression 12x + 5 for x = 3.

1.8c + 32

© Houghton Mifflin Harcourt Publishing Company

6. 5(6.2 + z); z = 3.8

12x + 5 is an expression that represents the cost of one carful of nonstudent soccer fans.

c = 30 °C STEP 2

10.5

4. 9 + m; m = 1.5

1 2__ 18

a. Write an expression that represents the cost of one carful of nonstudent soccer fans. Use x as the number of people who rode in the car and attended the game.

Math On the Spot

6

2. 3a - b; a = 4, b = 6

7. The table shows the prices for games in Bella’s soccer league. Her parents and grandmother attended a soccer game. How much did they spend if they all went together in one car? (Example 3)

You can evaluate expressions to solve real-world problems.

STEP 1

2

3. _8t ; t = 4

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11

Evaluating Real-World Expressions EXAMPL 3 EXAMPLE

16

1. x - 7; x = 23

Online Assessment and Intervention

Evaluate each expression for a = 3, b = 4, and c = -6. 7. ab - c

Evaluate each expression for the given value(s) of the variable(s). (Examples 1 and 2)

Personal Math Trainer

55

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18

4. 3(n + 1)

Substitute for the variables and follow the order Personal Math Trainer

of operations that you would use for a numerical

Online Assessment and Intervention

expression.

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1. This is a list Bold, Italic, Strickthrough.

1. This is a list

Home Connection

Cooperative Learning

Have students record real-world math situations they experience at home, using both words and mathematical symbols.

Have students work in groups to solve a magic square. A magic square is an array of numbers in which each row, column, and diagonal has the same sum. Ask students if the array below is a magic square if x = 4; if x = 6; or if x = 0.

Differentiated Instruction includes: • Reading Strategies • Success for English Learners ELL • Reteach • Challenge PRE-AP

Sample answer: Mom works out for the same length of time each day. How long does she work out in a week? 7t, where t represents the length of time she works out each day.

x+7

x

x+2

0.5x + 6

3x - 5

3x

2x + 1 x+6

x+1

The array is a magic square if x = 4, but not if x = 6 or if x = 0.

Evaluating Expressions

304

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Online homework assignment available

Evaluate GUIDED AND INDEPENDENT PRACTICE 6.7.A

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11.2 LESSON QUIZ 6.7.A Evaluate each expression for the given value(s) of the variable(s). 1. a + 6; a = 2

Concepts & Skills

Practice

Example 1 Evaluating Expressions

Exercises 1–6

Example 2 Using the Order of Operations

Exercises 1–6, 13

Example 3 Evaluating Real-World Expressions

Exercises 7–8, 10–12, 14, 16

16 2. __ g ;g = 4

3. 7(m - 6); m = 8 4. 7m - 6; m = 8

Exercise

5. s - k + x; s = 7, k = 4, x = 6 6. The expression 4g gives the number of quarts in g gallons. How many quarts are there in 4 gallons? Lesson Quiz available online my.hrw.com

Answers 1. 8 3. 14 4. 50 5. 9 6. 16 quarts

Lesson 11.2

Mathematical Processes

2 Skills/Concepts

1.A Everyday life

13

3 Strategic Thinking

1.G Explain and justify arguments

14–16

3 Strategic Thinking

1.A Everyday life

17

3 Strategic Thinking

1.G Explain and justify arguments

18

3 Strategic Thinking

1.F Analyze relationships

10–12

Additional Resources Differentiated Instruction includes: • Leveled Practice Worksheets

2. 4

305

Depth of Knowledge (D.O.K.)

DO NOT EDIT--Changes must be made through “File info” CorrectionKey=B

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Class

Date

11.2 Independent Practice 6.7.A

Movie 16 Ticket Prices

x

0

1

2

3

4

5

6

\$8.75

6x - x2

0

5

8

9

8

5

0

Children

\$6.50

Seniors

\$6.50

The value of 6x - x2 increases

8.75a + 6.5c + 6.5s or 8.75a +

6.5(c + s)

b. The Andrews family bought 2 adult tickets, 3 children’s tickets, and 1 senior ticket. Evaluate your expression in part a to find the total cost of the tickets.

8.75(2) + 6.5(3) + 6.5(1) =

© Houghton Mifflin Harcourt Publishing Company

8.75(2) + 6.5(3 + 1) = \$43.50 c. The Spencer family bought 4 adult tickets and 2 children’s tickets. Did they spend the same as the Andrews family? Explain.

No; 8.75(4) + 6.5(2) = \$48.00

11. The area of a triangular sail is given by the expression _21 bh, where b is the length of the base and h is the height. What is the area of a triangular sail in a model sailboat when b = 12 inches and h = 7 inches?

42

in.2

V = 5,760 ft3

13. Look for a Pattern Evaluate the expression 6x - x2 for x = 0, 1, 2, 3, 4, 5, and 6. Use your results to fill in the table and describe any pattern that you see.

a. Write an expression for the total cost of tickets.

A=

Online Assessment and Intervention

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10. The table shows ticket prices at the Movie 16 theater. Let a represent the number of adult tickets, c the number of children’s tickets, and s the number of senior citizen tickets.

16. The volume of a pyramid with a square base is given by the expression _13s2h, where s is the length of a side of the base and h is the height. Find the volume of a pyramid with a square base of side length 24 feet and a height of 30 feet.

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17. Draw Conclusions Consider the expressions 3x(x - 2) + 2 and 2x2 + 3x - 12. a. Evaluate each expression for x = 2 and for x = 7. Based on your results, do you know whether the two expressions are equivalent? Explain.

from 0 at x = 0 to 9 at x = 3, then

decreases back to 0 at x = 6. Also,

For x = 2, each expression has a value of 2. For x = 7,

and 6, for x = 1 and 5, and for

suggest that the expressions may be equivalent but

the values are the same for x = 0

each expression has a value of 107. These results

x = 2 and 4.

do not prove that the expressions are equivalent.

14. The kinetic energy (in joules) of a moving object can be calculated from the expression _12mv2, where m is the mass of the object in kilograms and v is its speed in meters per second. Find the kinetic energy of a 0.145-kg baseball that is thrown at a speed of 40 meters per second. E=

116

b. Evaluate each expression for x = 1. Based on your results, do you know whether the two expressions are equivalent? Explain.

For x = 1, the 1st expression has a value of -1 and the 2nd expression has a value of -7. Because

the values are different, the expressions are not

joules

equivalent.

15. The area of a square is given by x2, where x is the length of one side. Mary’s original garden was in the shape of a square. She has decided to double the area of her garden. Write an expression that represents the area of Mary’s new garden. Evaluate the expression if the side length of Mary’s original garden was 8 feet.

18. Critique Reasoning Marjorie evaluated the expression 3x + 2 for x = 5 as shown: 3x + 2 = 35 + 2 = 37 What was Marjorie’s mistake? What is the correct value of 3x + 2 for x = 5?

2(x2); 2(64) = 128 square feet

3x means that 3 should be multiplied by the value

12. Ramon wants to balance his checking account. He has \$2,340 in the account. He writes a check for \$140. He deposits a check for \$268. How much does Ramon have left in his checking account?

of x; 17

\$2,468 Lesson 11.2

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Work Area

FOCUS ON HIGHER ORDER THINKING

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Name

EXTEND THE MATH

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Activity Harold has a globe that has a diameter of 10 inches. He builds a globe with a radius twice as long as his original globe. Use the formulas below to find the surface area and the volume of the new globe. Use π = 3.14. Explain your process and show your work. Surface Area (SA) of a sphere = 4πr2, where r is the radius of the sphere. Volume (V) of a sphere = __43 πr3, where r is the radius of the sphere. Sample answer: First, find the radius of the new globe. The diameter of the original globe is 10 inches, so its radius is 10 ÷ 2 = 5 inches. The radius of the new globe is twice as long: 5 inches × 2 = 10 inches. Then use r = 10 inches as the value to substitute for r in each formula. SA of new globe is 4πr2 ≈ 4(3.14)(10)2 ≈ 4(3.14)(100) ≈ 4(314) ≈ 1,256 in2; V of new globe is __43 πr3 ≈ __43 (3.14)(10)3 ≈ __43 (3.14)(1,000) ≈ __43 (3,140) ≈ 4,186 __23 in3

Evaluating Expressions

306

LESSON

11.3 Generating Equivalent Expressions Texas Essential Knowledge and Skills The student is expected to: Expressions, equations, and relationships—6.7.C Determine if two expressions are equivalent using concrete models, pictorial models, and algebraic representations.

Engage

ESSENTIAL QUESTION How can you identify and write equivalent expressions? Sample answer: Substitute the same value into each expression and compare the results, or simplify each expression to see if they are equivalent.

Motivate the Lesson

Ask: Are the expressions 3x + 8 and 2x + 14 equivalent expressions for x = 6? Begin the Explore Activity to find out.

Expressions, equations, and relationships—6.7.D Generate equivalent expressions using the properties of operations: inverse, identity, commutative, associative, and distributive properties.

Explore

Mathematical Processes

EXPLORE ACTIVITY 1

6.1.D Communicate mathematical ideas, reasoning, and their implications using multiple representations, including symbols, diagrams, graphs, and language as appropriate.

Mathematical Processes Focus on Critical Thinking Mathematical Processes Point out to students that even though expressions share an element, such as 5x, and include the same operation, they are not necessarily equivalent expressions. For example, x 2 and x 3 have the same base and look similar but they are not equivalent; x 2 = x · x and x 3 = x · x · x.

Explain EXPLORE ACTIVITY 2 Focus on Modeling Mathematical Processes Point out that the number and arrangement of the algebra tiles mirrors each expression. The first model shows three groups of 1 variable plus 2 ones, and the second model shows 3 variables plus 6 ones. Discuss with students why the two algebraic expressions are equivalent. Questioning Strategies

Mathematical Processes • Earlier, you used counters to make models. How are algebra tiles similar to counters? How are they different? Both counters and algebra tiles are used the same way, one tile or counter for each number being added. The difference is that algebra tiles have an x (variable) tile used to represent an unknown quantity, while counters represent +1 or -1 only. • Which two characteristics do you look for in the models to decide whether the two expressions are equivalent? The models of both expressions should have 1) an equal number of x tiles and 2) an equal number of +1-tiles or -1-tiles.

YOUR TURN Avoid Common Errors Remind students to arrange the tiles in the same groups and order as shown in the expressions. Then compare the number of x tiles and the number of +1-tiles or –1-tiles in each model to check whether the expressions are equivalent.

307

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Generating Equivalent Expressions

LESSON

11.3 ?

EXPLORE ACTIVITY 1 (cont’d)

Expressions, equations, and relationships— 6.7.D Generate equivalent expressions using the properties of operations: inverse, identity, commutative, associative, and distributive properties. Also 6.7.C

Reflect 1.

Evaluate the expressions for a different value of x; for

ESSENTIAL QUESTION

example, when x = 1, 2x = 2 and x2 = 1.

How can you identify and write equivalent expressions?

EXPLORE ACTIVITY 1

Error Analysis Lisa evaluated the expressions 2x and x2 for x = 2 and found that both expressions were equal to 4. Lisa concluded that 2x and x2 are equivalent expressions. How could you show Lisa that she is incorrect?

6.7.C

EXPLORE ACTIVITY 2

Identifying Equivalent Expressions One way to test whether two expressions might be equivalent is to evaluate them for the same value of the variable.

Modeling Equivalent Expressions

List B

5x + 65

5x + 1

5(x + 1)

5x + 5

1 + 5x

5(13 + x)

Algebra Tiles

You can also use models to determine if two expressions are equivalent. Algebra tiles are one way to model expressions.

Match the expressions in List A with their equivalent expressions in List B. List A

6.7.C

=1 = -1 =x

Determine if the expression 3(x + 2) is equivalent to 3x + 6.

A Model each expression using algebra tiles. 3(x + 2)

3x + 6

A Evaluate each of the expressions in the lists for x = 3.

© Houghton Mifflin Harcourt Publishing Company

5(3) + 65 = 5(3 + 1) = 1 + 5(3) =

List B 5(3) + 1 =

16

20

5(3) + 5 =

20

16

5(13 + 3) =

80

3

B The model for 3(x + 2) has The model for 3x + 6 has

80

3

6

x tiles and x tiles and

6

1 tiles. 1 tiles.

C Is the expression 3(x + 2) equivalent to 3x + 6? Explain.

Yes; each expression is represented by 3 x tiles and

B Which pair(s) of expressions have the same value for x = 3?

5(x + 1) and 5x + 5; 1 + 5x and 5x + 1; 5x + 65 and

6 1 tiles.

5(13 + x)

Reflect

C How could you further test whether the expressions in each pair are equivalent?

2.

Use algebra tiles to determine if 2(x - 3) is equivalent to 2x - 3. Explain your answer.

© Houghton Mifflin Harcourt Publishing Company

List A

No; 2(x - 3) is represented by 2 x tiles and 6 -1 tiles.

Sample answer: Evaluate for several other values of x.

2x - 3 is represented by 2 x tiles and 3 -1 tiles.

D Do you think the expressions in each pair are equivalent? Why or why not?

Sample answer: Yes; it appears that they will always have the same value. Lesson 11.3

307

308

Unit 4

PROFESSIONAL DEVELOPMENT Integrate Mathematical Processes This lesson provides an opportunity to address Mathematical Process TEKS 6.1.D, which calls for students to “communicate mathematical ideas… using multiple representations, including symbols, diagrams…and language as appropriate.” In this lesson’s Explore Activities and Examples, students use words and operational symbols as well as algebra tiles to identify, represent, and compare algebraic expressions and to generate equivalent expressions.

Math Background Algebraic expressions are equivalent if they simplify to the same value. The Commutative, Associative, Distributive, and Identity properties give rules about how to rewrite an expression without changing its value. x + (3x + 2) + (2x + 3) x + 3x + (2 + 2x) + 3 x + 3x + (2x + 2) + 3 (x + 3x + 2x) + (2 + 3) x(1 + 3 + 2) + (2 + 3) x(6) + (5) 6x + 5

Associative Commutative Associative Distributive Addition Commutative

Generating Equivalent Expressions

308

ADDITIONAL EXAMPLE 1 Use a property to write an expression that is equivalent to n × 5. Tell which property you used. 5 × n; Commutative Property of Multiplication Interactive Whiteboard Interactive example available online my.hrw.com

EXAMPLE 1 Questioning Strategies

Mathematical Processes • How does the Commutative Property of Addition allow you to rewrite an expression without changing its value? You can change the order of the terms in an addition expression. • How do you know which properties of operations may help you identify equivalent expressions? Look at the operational symbols that appear in the given expressions. Apply properties of operations that relate to those symbols.

Engage with the Whiteboard Have students circle the operational symbol in Example 1 and write an equivalent expression on the whiteboard. Then have students draw a model of each expression, using algebra tiles to show that the expressions are equivalent.

Focus on Communication

Mathematical Processes Make sure students understand that the properties of operations are rules about how to rewrite expressions by rearranging and combining terms without changing the value of the expression.

YOUR TURN Avoid Common Errors If students have difficulty determining which property to use, remind them to begin by identifying the operation used in the given expression. Then they should look at the list of properties to see which properties apply to that operation. Point out that a given expression may have more than one equivalent expression, as more than one property can be applied.

ADDITIONAL EXAMPLE 2 Use the properties of operations to determine if the expressions are equivalent. A 6 + y; __12 (12 + y) B 2(y - 3); 2y - 6

not equivalent equivalent

Interactive Whiteboard Interactive example available online my.hrw.com

Animated Math Equivalent Expressions Students explore equivalent expressions using an interactive model. my.hrw.com

309

Lesson 11.3

EXAMPLE 2 Questioning Strategies

Mathematical Processes • In A, how does the Distributive Property enable you to determine that the expressions are equivalent without evaluating them? The Distributive Property states that multiplying a number by a difference, as in 3(x - 2), is the same as multiplying each number in the difference and subtracting the products: (3)(x) - (3)(2) = 3x - 6.

YOUR TURN Engage with the Whiteboard For Exercises 5–6, have students use algebra tiles to draw a model of each expression on the whiteboard. Then have students explain whether the expressions are equivalent or not.

DO NOT EDIT--Changes must be made through “File info” CorrectionKey=B

DO NOT EDIT--Changes must be made through “File info” CorrectionKey=A

Identifying Equivalent Expressions Using Properties

Writing Equivalent Expressions Using Properties Properties of operations can be used to identify equivalent expressions. Examples 3+4=4+3

Commutative Property of Multiplication: When multiplying, changing the order of the numbers does not change the product.

2×4=4×2

Associative Property of Addition: When adding more than two numbers, the grouping of the numbers does not change the sum.

(3 + 4) + 5 = 3 + (4 + 5)

Associative Property of Multiplication: When multiplying more than two numbers, the grouping of the numbers does not change the product.

(2 × 4) × 3 = 2 × (4 × 3)

Identity Property of Multiplication: Multiplying a number by one does not change its value.

1×7=7

Inverse Property of Addition: The sum of a number and its opposite, or additive inverse, is zero.

- 3 + 3 = 0

© Houghton Mifflin Harcourt Publishing Company

EXAMPL 1 EXAMPLE

Math Talk

What property can you use to write an expression that is equivalent to 0 + c? What is the equivalent expression?

Sample answer: Represent 3 (x - 2) as three groups of x - 2 with each group showing one x tile and two - 1 tiles. To represent the Distributive Property, regroup the tiles to represent 3x - 6 by showing a group of three x tiles and a group of six - 1 tiles.

You can use the Commutative Property of Addition to write an equivalent expression: x + 3 = 3 + x

YOUR TURN For each expression, use a property to write an equivalent expression. Tell which property you used. Sample answers given.

a(bc); Associative Property of Multiplication

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Online Assessment and Intervention

Online Assessment and Intervention

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Distributive Property Commutative Property

YOUR TURN Use the properties of operations to determine if the expressions are equivalent. 5.

6x - 8; 2(3x - 5)

2(3x - 5) = 6x - 10;

6.

7.

2 - 2 + 5x; 5x

2 - 2 + 5x = 5x; equivalent

not equivalent

Jamal bought 2 packs of stickers and 8 individual stickers. Use x to represent the number of stickers in a pack of stickers and write an expression to represent the number of stickers Jamal bought. Is the expression equivalent to 2(4 + x)? Check your answer with algebra tile models.

Jamal bought 2x + 8 stickers. 2(4 + x) = 8 + 2x = 2x + 8; yes

Unit 4

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InCopy Notes

DIFFERENTIATE INSTRUCTION

Distributive Property

1 (4 + x) B 2 + x; __ 2 1x+ 2 1 (x + 4) = __ __ lk Ta th Ma 2 2 Mathematical Processes 1x = 2 + __ 2 Explain how you could use algebra tiles to 1 x. 2 + x does not equal 2 + __ represent the Distributive 2 Property in A. They are not equivalent expressions.

Mathematical Processes

6.7.D

(3 + 4)y; Distributive Property

3(x - 2) = 3x - 6

3(x 3 - 2) and 3x - 6 are equivalent expressions.

my.hrw.com my.h

The operation in the expression is addition.

4. 3y + 4y =

6.7.C

A 3(x - 2); 3x - 6

Use a property to write an expression that is equivalent to x + 3.

3. (ab)c =

EXAMPLE 2

1. This is a list Bold, Italic, Strickthrough.

10/25/12 8:35 PM

InDesign Notes 1. This is a list

Visual Cues

Cognitive Strategies

Point out to students that it is often helpful to use colored pencils to identify like terms before combining them.

A fun way for students to remember how to combine like terms is to name the variable part of like terms. For example, for the expression 2a + 5b + 4a, students can name variable a apples and variable b bananas. Thus, 2 apples + 5 bananas + 4 apples = 6 apples + 5 bananas = 6a + 5b.

Differentiated Instruction includes: • Reading Strategies • Success for English Learners ELL • Reteach • Challenge PRE-AP

Have students identify the like terms in the following expressions. 1. a + 2b + 2a + b + 2c 2b and b

Like terms: a and 2a,

© Houghton Mifflin Harcourt Publishing Company

9+0=9

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Animated Math

6(2 + 4) = 6(2) + 6(4) 8(5 - 3) = 8(5) - 8(3)

Identity Property of Addition: Adding zero to a number does not change its value.

Math On the Spot

my.hrw.com

Use the properties of operations to determine if the expressions are equivalent.

Properties of Operations Commutative Property of Addition: When adding, changing the order of the numbers does not change the sum.

Distributive Property: Multiplying a number by a sum or difference is the same as multiplying by each number in the sum or difference and then adding or subtracting.

Math On the Spot

2. 18 + 2d 3 + 5d + 3d 3 - 2d 2 Like terms: 2d 3 and 3d 3 3. 5x 3 + 3y + 7x 3 - 2y - 4x 2 Like terms: 5x 3 and 7x 3, 3y and 2y

Generating Equivalent Expressions

310

ADDITIONAL EXAMPLE 3 Combine like terms.

EXAMPLE 3 Connect Vocabulary

A 8y2 - 3y2 5y2 B 4m + 3(n + 7m)

25m + 3n

C x + 8y - 5y + 5x

6x + 3y

Interactive Whiteboard Interactive example available online my.hrw.com

ELL

Stress the use of correct mathematical terminology. The parts of an expression that are separated by + or - signs, such as 3x and 5x in the expression 3x + 5x, are called terms. Terms that have identical variable parts are like terms. In the expression 3x + 5x, 3x and 5x are like terms. The properties of operations allow you to rearrange and combine like terms.

Questioning Strategies

Mathematical Processes • In B, why do you have to apply the Distributive Property before adding like terms? Because of the order of operations, you need to multiply before you can add. • In C, why does y + 7y equal 8y? Because y is 1y, so 1y + 7y = (1 + 7)y = 8y.

YOUR TURN Avoid Common Errors Exercise 10 Students may neglect to add single variables. Remind them that b is 1b, so adding b to another b-term increases the coefficient by 1.

Elaborate Talk About It Summarize the Lesson Ask: How do properties of operations help you to write equivalent expressions? Properties of operations allow me to write an expression in different ways without changing its value. I can use the properties to form equivalent expressions by regrouping, reordering, and combining terms.

GUIDED PRACTICE Engage with the Whiteboard For Exercise 1, have students rewrite each expression on the whiteboard by substituting 5 for y and then simplifying the expressions. For Exercise 2, ask a student to circle the part of the model that shows that the two expressions are not equivalent on the whiteboard.

Avoid Common Errors Exercises 3–4 If students have difficulty determining which property to use, remind them to begin by identifying the operation used in the given expression. Then they should look at the list of properties to see which properties apply to that operation. Exercises 5–6 Remind students that when they apply the Distributive Property they must distribute the constant or variable that is outside the parentheses to each term that is inside the parentheses. Exercise 8 Students may neglect to add single variables. Remind them that -x is -1x, so adding -x to another x-term decreases the coefficient by 1.

311

Lesson 11.3

Generating Equivalent Expressions

Parts of an algebraic expression

coefficients like terms

12 + 3y2 + 4x + 2y2 + 4 12 + 3y2 + 4x + 2y2 + 4

Math On the Spot

Online Assessment and Intervention

my.hrw.com

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EXAMPL 3 EXAMPLE

4 + 4y = 4(y − 1) =

2

6x and 4x are like terms.

Subtract inside the parentheses.

= 2x2

Commutative Property of Multiplication

B 3a + 2(b + 5a)

© Houghton Mifflin Harcourt Publishing Company

4y + 1 =

Distributive Property

= x2(2)

= 3a + 2b + (2 · 5)a

Distributive Property

= 3a + 2b + 10a

Associative Property of Multiplication Multiply 2 and 5.

= 3a + 10a + 2b

= (3 + 10)a + 2b

Distributive Property

= 13a + 2b

Math Talk

Mathematical Processes

Write 2 terms that can be combined with 7y4.

y + 11x - 7x + 7y = y + 7y + 11x - 7x

List B

24 16 21

For each expression, use a property to write an equivalent expression. Tell which property you used. (Example 1) Sample 3. ab =

Distributive Property Commutative Property

ba

+

x-4

2(x - 2)

+

+

answers are given. 5(3x) - 5(2)

4. 5(3x − 2) =

Distributive Prop.

Commutative Prop. of Mult.

not equivalent

6. _12(6x − 2); 3 − x

not equivalent

Combine like terms. (Example 3) 7. 32y + 12y =

= 8y + 4x

4(y + 1) = 1 + 4y =

16 24 21

2. Determine if the expressions are equivalent by comparing the models. (Explore Activity 2) not equivalent

5. _12(4 − 2x); 2 - 2x

y and 7y are like terms; 11x and 7x are like terms. Commutative Property

= y(1 + 7) + x(11 - 7)

4y − 4 =

Use the properties of operations to determine if each pair of expressions is equivalent. (Example 2)

3a + 2(b + 5a) = 13a + 2b

C y + 11x - 7x + 7y

8m + 2 + 4n

2a5 + 5b

List A

A 6x2 - 4x2

3a + 2(b + 5a) = 3a + 2b + 2(5a)

11. 8m + 14 - 12 + 4n =

10. 4a5 - 2a5 + 4b + b =

1. Evaluate each of the expressions in the list for y = 5. Then, draw lines to match the expressions in List A with their equivalent expressions in List B. (Explore Activity 1)

Combine like terms.

6x2 - 4x2 = 2x2

10x2 - 4

Guided Practice

6.7.D

6x2 - 4x2 = x2 (6 - 4)

9. 6x2 + 4(x2 - 1) =

12 + 3y2 + 4x + 2y2 + 4

When an expression contains like terms, you can use properties to combine the like terms into a single term. This results in an expression that is equivalent to the original expression.

2

5y

8. 8y - 3y =

? ?

44y

8. 12 + 3x − x − 12 =

2x

ESSENTIAL QUESTION CHECK-IN

© Houghton Mifflin Harcourt Publishing Company

The parts of the expression that are separated by + or - signs Numbers that are multiplied by at least one variable Terms with the same variable(s) raised to the same power(s)

terms

Combine like terms.

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9. Describe two ways to write equivalent algebraic expressions.

Use properties of operations or combine like terms

y + 11x - 7x + 7y = 8y + 4x

Lesson 11.3

311

312

Unit 4

Generating Equivalent Expressions

312

Personal Math Trainer Online Assessment and Intervention

Online homework assignment available

Evaluate GUIDED AND INDEPENDENT PRACTICE 6.7.C, 6.7.D

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Concepts & Skills

Practice

Explore Activity 1 Identifying Equivalent Expressions

Exercise 1

Explore Activity 2 Modeling Equivalent Expressions

Exercises 2, 14

Example 1 Writing Equivalent Expressions Using Properties

Exercises 3–4, 10–13, 27–29

Use the properties of operations to determine if the expressions are equivalent.

Example 2 Identifying Equivalent Expressions Using Properties

Exercises 5–6, 25

2. 3 + y; __13 (6 + y)

Example 3 Generating Equivalent Expressions

Exercises 7–8, 15–24, 26

11.3 LESSON QUIZ 6.7.D 1. Use one of the properties of operations to write an expression that is equivalent to 4 + m. Tell which property you used.

3. 4(y - 3); 4y - 12 Combine like terms. 4. 9y2 - 6y2

Exercise

5. 5c + 4(d+ 6c)

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1. m + 4; Commutative Property of Addition 2. not equivalent 3. equivalent 4. 3y2 5. 29c + 4d 6. 5x + 6y

313

Lesson 11.3

1.C Select tools

3 Strategic Thinking

1.G Explain and justify arguments

26–27

2 Skills/Concepts

1.A Everyday life

28–29

2 Skills/Concepts

1.D Multiple representations

30

3 Strategic Thinking

1.F Analyze relationships

31

3 Strategic Thinking

1.G Explain and justify arguments

32

3 Strategic Thinking

1.F Analyze relationships

25

Lesson Quiz available online

Mathematical Processes

2 Skills/Concepts

10–24

6. x + 10y - 4y + 4x

Depth of Knowledge (D.O.K.)

Additional Resources Differentiated Instruction includes: • Leveled Practice Worksheets

Class

Date

11.3 Independent Practice

Personal Math Trainer

6.7.D, 6.7.C

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27. Multiple Representations Use the information in the table to write and simplify an expression to find the total weight of the medals won by the top medal-winning nations in the 2012 London Olympic Games. The three types of medals have different weights.

Online Assessment and Intervention

2012 Summer Olympics

For each expression, use a property to write an equivalent expression. Tell which property you used. Sample answers given.

dc

10. cd =

12. 4(2x − 3) =

13 + x

11. x + 13 =

Commutative Prop. of Mult.

4(2x) - 4(3)

Distributive Prop.

Silver 29 27 17

113g + 73s + 71b

Write an expression for the perimeters of each given figure. Simplify the expressions.

14. Draw algebra tile models to prove that 4 + 8x and 4(2x + 1) are equivalent.

6x + 10 mm

28.

36.4 + 4x in.

29.

10.2 in.

3x - 1 mm 6 mm

4(2x + 1)

4 + 8x

x + 4 in. x + 4 in.

© Houghton Mifflin Harcourt Publishing Company

17. 6b + 7b − 10 =

13b - 10

16. 32y + 5y =

37y

18. 2x + 3x + 4 =

5x + 4 6a2 + 16

19. y + 4 + 3(y + 2) =

4y + 10

20. 7a2 − a2 + 16 =

21. 3y2 + 3(4y2 - 2) =

15y2 - 6

22. z2 + z + 4z3 + 4z2 =

23. 0.5(x4 - 3) + 12 =

0.5x + 10.5 4

FOCUS ON HIGHER ORDER THINKING

a. Write two equivalent expressions for

4z3 + 5z2 + z

the model.

The expressions become 6 - 9x and 3(2 - 3x).

Yes; Applying the Associative Property of Addition to 3x + 12 - 2x

31. Communicate Mathematical Ideas Write an example of an expression that cannot be simplified, and explain how you know that it cannot be simplified.

you get 3x - 2x + 12 which is equivalent to 1x + 12. Then apply

3x2 - 4x + 7; It does not have any like terms.

the Identity Property of Multiplication to get x + 12.

32. Problem Solving Write an expression that is equivalent to 8(2y + 4) that can be simplified.

26. William earns \$13 an hour working at a movie theater. Last week he worked h hours at the concession stand and three times as many hours at the ticket counter. Write and simplify an expression for the amount of money William earned last week.

13h + 13(3h) = 13h + 39h = 52h Lesson 11.3

PRE-AP

4 - 6x; 2(2 - 3x)

b. What If? Suppose a third row of tiles identical to the ones above is added to the model. How does that change the two expressions?

25. Justify Reasoning Is 3x + 12 - 2x equivalent to x + 12? Use two properties of operations to justify your answer.

EXTEND THE MATH

Work Area

30. Problem Solving Examine the algebra tile model.

4+p

24. _14 (16 + 4p) =

x + 4 in. 10.2 in.

Combine like terms.

2x4

x + 4 in.

6 mm 3x - 1 mm

15. 7x4 − 5x4 =

Bronze 29 23 19

(46 + 38 + 29)g + (29 + 27 + 17)s + (29 + 23 + 19)b;

(2 + a) + b

13. 2 + (a + b) =

Gold 46 38 29

United States China Great Britain

)PVHIUPO.JGGMJO)BSDPVSU1VCMJTIJOH\$PNQBOZt*NBHF\$SFEJUTª\$PNTUPDL +VQJUFSJNBHFT(FUUZ*NBHFT

Name

313

Activity available online

314

Unit 4

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Activity During a basketball game, Joelle scored 8 points on free throws. She also scored 2 points for each inside shot and 3 points for each outside shot she made. Joelle made n inside shots and s outside shots during the game. Write six equivalent expressions for the total number of points Joelle scored. Which properties of operations did you use to identify equivalent expressions? Sample answer: Commutative Property of Addition: 8 + 2n + 3s, 8 + 3s + 2n, 2n + 3s + 8, 2n + 8 + 3s; Distributive Property and Commutative Property of Addition: 2(4 + n) + 3s, 3s + 2(4 + n)

Generating Equivalent Expressions

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DO NOT EDIT--Changes must be made through “File info” CorrectionKey=B

Module Quiz

Assess Mastery

11.1  Modeling Equivalent Expressions

Personal Math Trainer

Write each phrase as an algebraic expression.

Use the assessment on this page to determine if students have mastered the concepts and standards covered in this module.

p __ 6

1. p divided by 6

185 + h

3. the sum of 185 and h 3

Response to Intervention

2 1

4 seasons.

4. the product of 16 and g

6. 8p; p = 9

4x

72

8. 4(d + 7); d = -2

20

7. 11 + r; r = 7 -60 9. ____ m ;m=5

of a triangle with a base of 6 and a height of 8?

Differentiated Instruction

Differentiated Instruction

11.3  Generating Equivalent Expressions

• Reteach worksheets

• Challenge worksheets

11. Draw lines to match the expressions in List A with their equivalent expressions in List B.

• Success for English Learners ELL

ELL

PRE-AP

Extend the Math PRE-AP Lesson Activities in TE

Additional Resources Assessment Resources includes: • Leveled Module Quizzes

18 -12

10. To find the area of a triangle, you can use the expression b × h ÷ 2, where b is the base of the triangle and h is its height. What is the area

Online and Print Resources

16g

Evaluate each expression for the given value of the variable.

© Houghton Mifflin Harcourt Publishing Company

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2. 65 less than j

11.2  Evaluating Expressions

Enrichment

Access Ready to Go On? assessment online, and receive instant scoring, feedback, and customized intervention or enrichment.

Online Assessment and Intervention

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j - 65

5. Let x represent the number of television show episodes that are taped in a season. Write an expression for the number of episodes taped in

Intervention

Personal Math Trainer

Online Assessment and Intervention

ESSENTIAL QUESTION

24 square units

List A 7x + 14 7 + 7x 7( x - 1 )

List B 7( 1 + x ) 7x - 7 7( x + 2 )

12. How can you determine if two algebraic expressions are equivalent?

Sample answer: Model the expressions with bar models or algebra tiles to determine if the expressions are equivalent; write equivalent expressions using properties of operations, order of operations, and combining like terms.

Module 11

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Texas Essential Knowledge and Skills Lesson

Exercises

11.1

1–5

6.7.C

11.2

6–10

6.7.C

11.3

11

6.7.D

315

Module 11

TEKS

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DO NOT EDIT--Changes must be made through “File info” CorrectionKey=B

Texas Test Prep

Texas Testing Tip  Students can circle or underline key words and phrases to identify important information. Item 1  Students should underline the word product. Product means the operation used is multiplication. Thus students can quickly see that choice C is the correct answer. Item 3  Students should underline the phrase divided them evenly between p pages. Since the operation used is division, ­students can eliminate choices A and B, and then check the order of division in the two remaining answer choices to reveal that choice D is the correct answer.

Selected Response 1. Which expression represents the product of 83 and x? A 83 + x

Item 5  Some students may forget that 7w means the product of 7 and the variable and mistakenly substitute 9 for w to make the number 79. Remind students that the variable in a term is multiplied by the coefficient. Item 6  Caution students to read the question carefully. Some students may not realize that this question is asking for how many pages Katie has left to read and may indicate that choice D is correct.

83x

A the product of r and 9 B the quotient of r and 9

9 less than r

3. Rhonda was organizing photos in a photo album. She took 60 photos and divided them evenly among p pages. Which algebraic expression represents the number of photos on each page? p

A p - 60

C

B 60 - p

60 D __ p​

​__ ​ 60

4. Using the algebraic expression 4n + 6, what is the greatest whole-number value of n that will give you a result less than 100? A 22

C

B 23

D 25

C

49

D 77

316

7. The expression 12(x + 4) represents the total cost of CDs Mei bought in April and May at \$12 each. Which property is applied to write the equivalent expression 12x + 48? A Associative Property of Addition

Commutative Property of Multiplication

C

D Distributive Property

Gridded Response 8. When traveling in Europe, Bailey converts the temperature given in degrees Celsius to a Fahrenheit temperature by using the expression 9x ÷ 5 + 32, where x is the Celsius temperature. Find the temperature in degrees Fahrenheit when it is 15 °C.

.

5

9

0

0

0

0

0

0

1

1

1

1

1

1

2

2

2

2

2

2

3

3

3

3

3

3

4

4

4

4

4

4

5

5

5

5

5

5

6

6

6

6

6

6

7

7

7

7

7

7

8

8

8

8

8

8

9

9

9

9

9

9

24

5. Evaluate 7w - 14 for w = 9. B 18

200

B Associative Property of Multiplication

D r more than 9

A 2

A 40

D 250

D 83 - x

Avoid Common Errors

6. Katie has read 32% of a book. If she has read 80 pages, how many more pages does Katie have left to read?

C

2. Which phrase describes the algebraic expression ​_9r​ ?

C

Online Assessment and Intervention

B 170

B 83 ÷ x

C

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Texas Test Prep

Personal Math Trainer

Module 11 MIXed ReVIeW

Unit 4

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Texas Essential Knowledge and Skills Items

Mathematical Process TEKS

1

6.7.C

6.1.D

2

6.7.C

6.1.D

3

6.7.C

6.1.A, 6.1.D

4

6.7.A

6.1.F

5

6.7.D

6*

6.5.A

6.1.A

7

6.7.D

6.1.A

8

6.7.D

6.1.A, 6.1.D

* Item integrates mixed review concepts from previous modules or a previous course.

Generating Equivalent Algebraic Expressions

316

Equations and Relationships ?

ESSENTIAL QUESTION How can you use equations and relationships to solve real-world problems?

MODULE

You can model real-world problems with equations, then use algebraic rules to solve the equations.

12

LESSON 12.1

Writing Equations to Represent Situations 6.7.B, 6.9.A, 6.10.B

LESSON 12.2

Addition and Subtraction Equations 6.9.B, 6.9.C, 6.10.A

LESSON 12.3

Multiplication and Division Equations

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6.9.B, 6.9.C, 6.10.A

Real-World Video

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317

Module 12

People often attempt to break World Records. To model how many seconds faster f an athlete's time a seconds must be to match a record time r seconds, write the equation f = a - r.

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Math On the Spot

Animated Math

Personal Math Trainer

Go digital with your write-in student edition, accessible on any device.

Scan with your smart phone to jump directly to the online edition, video tutor, and more.

Interactively explore key concepts to see how math works.

Get immediate feedback and help as you work through practice sets.

317

Are You Ready? Use the assessment on this page to determine if students need intensive or strategic intervention for the module’s prerequisite skills. 3 2 1

Personal Math Trainer

Complete these exercises to review skills you will need for this chapter.

Evaluate Expressions EXAMPLE

Response to Intervention

Online Assessment and Intervention

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Evaluate 8(3+2) - 52 8(3+2) - 52 = 8(5) - 52 = 8(5) - 25 = 40 - 25 = 15

Perform operations inside parentheses first. Evaluate exponents. Multiply. Subtract.

Evaluate the expression.

Online Assessment and Intervention

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Online and Print Resources Skills Intervention worksheets

Differentiated Instruction

• Skill 54 Evaluate Expressions

• Challenge worksheets

7. 2(8 + 3) + 42

38

9. 8(2 +1)2 - 42

56

39

4. 6(8 - 3) + 3(7 - 4)

1

5. 10(6 - 5) - 3(9 - 6)

64

2. 8(2 + 4) + 16

5

3. 3(14 - 7) - 16

Access Are You Ready? assessment online, and receive instant scoring, feedback, and customized intervention or enrichment.

Personal Math Trainer

29

1. 4(5 + 6) - 15

Enrichment

29

6. 7(4 + 5 + 2) - 6(3 + 5) 8. 7(14 - 8) - 62

6

Connect Words and Equations EXAMPLE

PRE-AP

• Skill 56 Connect Words and Extend the Math PRE-AP Equations Lesson Activities in TE

The product of a number and 4 is 32. The product of x and 4 is 32. Represent the unknown with a variable. Determine the operation. 4 × x is 32. 4 × x = 32. Determine the placement of the equal sign.

© Houghton Mifflin Harcourt Publishing Company

Intervention

Write an algebraic equation for the word sentence.

10. A number increased by 7.9 is 8.3.

x + 7.9 = 8.3

12. The quotient of a number and 8 is 4.

x÷8=4

14. The difference between 31 and a number is 7.

318

31 - x = 7

11. 17 is the sum of a number and 6.

17 = x + 6

13. 81 is three times a number.

81 = 3x 15. Eight less than a number is 19.

x - 8 = 19

Unit 4

PROFESSIONAL DEVELOPMENT VIDEO my.hrw.com

Author Juli Dixon models successful teaching practices as she explores equation concepts in an actual sixth-grade classroom.

Online Teacher Edition Access a full suite of teaching resources online—plan, present, and manage classes and assignments.

Professional Development

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Interactive Answers and Solutions Customize answer keys to print or display in the classroom. Choose to include answers only or full solutions to all lesson exercises.

Interactive Whiteboards Engage students with interactive whiteboard-ready lessons and activities. Personal Math Trainer: Online Assessment and Intervention Assign automatically graded homework, quizzes, tests, and intervention activities. Prepare your students with updated, TEKS-aligned practice tests.

Equations and Relationships

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DO NOT EDIT--Changes must be made through "File info" CorrectionKey=B

Visualize Vocabulary Use the ✔ words to complete the graphic.

Visualize Vocabulary The main idea web will help students review vocabulary and concepts related to algebraic expressions. Ask students to think of other algebraic expressions and their components and share them with the class, making sure to identify the variable, coefficient, terms, and constant in each expression.

4x + 5

Use the following explanations to help students learn the preview words. The words expression and equation are not synonyms. An algebraic expression is a mathematical statement that contains one or more variables. An equation is a mathematical statement stating that two expressions are equal. You evaluate an expression and solve an equation.

Students can use these reading and note-taking strategies to help them organize and understand new concepts and vocabulary. c.4.D Use prereading supports such as graphic organizers, illustrations, and pretaught topic-related vocabulary to enhance comprehension of written text.

Review Words ✔ algebraic expression (expresión algebraica) ✔ coefficient (coeficiente) ✔ constant (constante) evaluating (evaluar) like terms (términos semejantes) ✔ term (término, en una expresión) ✔ variable (variable)

Preview Words

5

4x and 5

constant

terms

Understand Vocabulary

equation (ecuación) equivalent expression (expresión equivalente) properties of operations (propiedades de las operaciones) solution (solución)

Match the term on the left to the correct expression on the right.

© Houghton Mifflin Harcourt Publishing Company

Integrating the ELPS

x

variable

algebraic expression

Understand Vocabulary

4

coefficient

Vocabulary

1. algebraic expression

A. A mathematical statement that two expressions are equal.

2. equation

B. A value of the variable that makes the statement true.

3. solution

C. A mathematical statement that includes one or more variables.

Booklet Before beginning the module, create a booklet to help you learn the concepts in this module. Write the main idea of each lesson on each page of the booklet. As you study each lesson, write important details that support the main idea, such as vocabulary and formulas. Refer to your finished booklet as you work on assignments and study for tests.

Differentiated Instruction • Reading Strategies ELL Module 12

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Grades 6–8 TEKS Before Students understand: • operations with rational numbers • properties of operations: inverse, identity, commutative, associative, and distributive properties

319

Module 12

In this module Students will learn to: • write one-variable, one-step equations to represent constraints or conditions within problems • model and solve one-variable, one-step equations that represent problems • write corresponding real-world problems given one-variable, one-step equations

319

After Students will learn how to: • write one-variable, two-step equations to represent real-world problems • write a real-world problem to represent a one-variable, two-step equation • solve one-variable, two-step equations

MODULE 12

Unpacking the TEKS

Unpacking the TEKS Understanding the TEKS and the vocabulary terms in the TEKS will help you know exactly what you are expected to learn in this module.

6.9.A Write one-variable, one-step equations and inequalities to represent constraints or conditions within problems.

Texas Essential Knowledge and Skills Content Focal Areas Expressions, equations, and relationships—6.7 The student applies mathematical process standards to develop concepts of expressions and equations.

Expressions, equations, and relationships—6.10 The student applies mathematical process standards to use equations and inequalities to solve problems.

You will learn to write an equation or inequality to represent a situation.

Key Vocabulary

UNPACKING EXAMPLE 6.9.A

equation (ecuación) A mathematical sentence that shows that two expressions are equivalent.

The Falcons won their football game with a score of 30 to 19. Kevin scored 12 points for the Falcons. Write an equation to determine how many points Kevin’s teammates scored.

inequality (desigualdad) A mathematical sentence that shows the relationship between quantities that are not equal.

Expressions, equations, and relationships—6.9 The student applies mathematical process standards to use equations and inequalities to represent situations.

What It Means to You

6.10.B Determine if the given value(s) make(s) one-variable, one-step equations or inequalities true.

Integrating the ELPS

Kevin’s points

+

Teammates’ points

=

Total points

12

+

t

=

30

What It Means to You You can substitute a given value for the variable in an equation or inequality to check if that value makes the equation or inequality true. UNPACKING EXAMPLE 6.10.B

Melanie bought 6 tickets to a play. She paid a total of \$156. Write an equation to determine whether each ticket cost \$26 or \$28.

content area text … and develop vocabulary … to comprehend increasingly challenging language.

Number of tickets bought 6

· ·

Price per ticket p

=

Total cost

=

156

Substitute 26 and 28 for p to see which equation is true.

Go online to see a complete unpacking of the .

Visit my.hrw.com to see all the unpacked.

6p = 156

6p = 156

? 156 6 · 26 =

? 156 6 · 28 =

? 156 ✓ 156 =

© Houghton Mifflin Harcourt Publishing Company • Image Credits: Robert Llewellyn/Corbis Super RF/Alamy Limited

Use the examples on this page to help students know exactly what they are expected to learn in this module.

? 156 ✗ 168 =

The cost of a ticket to the play was \$26. my.hrw.com

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320

Lesson 12.1

Lesson 12.2

Unit 4

Lesson 12.3

6.7.B Distinguish between expressions and equations verbally, numerically, and algebraically. 6.9.A Write one-variable, one-step equations and inequalities to represent constraints or conditions within problems. 6.9.B Represent solutions for one-variable, one-step equations and inequalities on number lines. 6.9.C Write corresponding real-world problems given one-variable, one-step equations or inequalities. 6.10.A Model and solve one-variable, one-step equations and inequalities that represent problems, including geometric concepts. 6.10.B Determine if the given value(s) make(s) one-variable, one-step equations or inequalities true.

Equations and Relationships

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LESSON

12.1 Writing Equations to Represent Situations Texas Essential Knowledge and Skills The student is expected to: Expressions, equations, and relationships—6.9.A Write one-variable, one-step equations and inequalities to represent constraints or conditions within problems. Expressions, equations, and relationships—6.7.B Distinguish between expressions and equations verbally, numerically, and algebraically. Expressions, equations, and relationships—6.10.B Determine if the given value(s) make(s) one-variable, one-step equations or inequalities true.

Mathematical Processes 6.1.D Communicate mathematical ideas, reasoning, and their implications using multiple representations, including symbols, diagrams, graphs, and language as appropriate.

ADDITIONAL EXAMPLE 1 Determine whether the given value is a solution of the equation. A x + 15 = 10; x = -5 m B __ = -11; m = 33 3

C 5n = 42; x = 7

yes no

no

Interactive Whiteboard Interactive example available online my.hrw.com

ADDITIONAL EXAMPLE 2 Eli is y years old. His 9-year-old cousin Jen is 4 years younger than he is. Write an equation to represent this situation. Sample answer: y - 4 = 9 Interactive Whiteboard Interactive example available online my.hrw.com

321

Lesson 12.1

Engage

ESSENTIAL QUESTION How do you write equations and determine whether a number is a solution of an equation? Write a statement that links two expressions with an equals sign. Substitute a number for the variable and simplify. If the final statement is true, the number is a solution.

Motivate the Lesson Ask: Ty wants to buy a video game. He has \$57, which is \$38 less than he needs. Does the game cost \$90 or \$95? Begin Example 1 to find out how to solve this problem.

Explore Engage with the Whiteboard

Write x + 4 and x + 4 = 9 on the whiteboard. Ask students how they differ. Explain the difference between an expression and an equation. Then have students model both sides of the equation, using algebra tiles on the whiteboard. Ask them what x must represent for the two sides to be equal. Tell them that x = 5 is the solution of the equation.

Explain EXAMPLE 1 Focus on Modeling Mathematical Processes Explain to students that an equation is like a balanced scale. Just as the weights on both sides of a balanced scale are exactly the same, the expressions on both sides of an equation represent exactly the same value. Show examples of equations on a balance scale. Questioning Strategies

Mathematical Processes • In A, why is x = 6 a solution of the equation? because the resulting statement, 15 = 15, is a true statement In B, why is y = -8 not a solution of the equation? because the resulting statement, -2 = -32, is a false statement

YOUR TURN Avoid Common Errors Watch for students who substitute the given value incorrectly. Caution students to doublecheck their work for accuracy.

EXAMPLE 2 Questioning Strategies

Mathematical Processes • Could you write another equation to represent the situation? Yes. I can write the equation p = Total points - Mark’s points or p = 46 - 17 because subtraction is the opposite of addition.

YOUR TURN Avoid Common Errors If students have difficulty writing equations to represent the word problems, encourage them to make a model, like the one in Example 2, to organize the information.

DO NOT EDIT--Changes must be made through “File info” CorrectionKey=A

12.1 ?

Writing Equations to Represent Situations

ESSENTIAL QUESTION

Expressions, equations, and relationships—6.9.A Write one-variable, one-step equations … to represent constraints or conditions within problems. Also 6.7.B, 6.10.B

How do you write equations and determine whether a number is a solution of an equation?

Determine whether the given value is a solution of the equation. 1. 11 = n + 6; n = 5

Online Assessment and Intervention

yes

5+4

5+4=9 a number plus 4 is 9.

n+4

Algebraic

Math On the Spot my.hrw.com

n+4=9

EXAMPL 1 EXAMPLE

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EXAMPLE 2

An equation relates two expressions using symbols for is or equals.

© Houghton Mifflin Harcourt Publishing Company

Mark’s points

17

Math Talk

6.10.B

Mathematical Processes

Use words to describe two expressions that represent the total points scored. What does the equation say about the expressions you wrote?

A x + 9 = 15; x = 6 ? 6 + 9 = 15 Substitute 6 for x. ? 15 = 15 Add. y __ = -32; y = -8 4 -8 ? ___ = -32 Substitute -8 for y. 4

=

Total points

+

p

=

46

HOME

PERIOD

GUEST

Math Talk: The sum of Mark’s points and his teammates’ points; the total points scored; the equation uses the symbol = to show that the two expressions are equal.

Write an equation to represent each situation.

Divide.

4. Marilyn has a fish tank that contains 38 fish. There are 9 goldfish and f other fish.

5. Juanita has 102 beads to make n necklaces. Each necklace will have 17 beads.

6. Craig is c years old. His 12-year-old sister Becky is 3 years younger than Craig.

7. Sonia rented ice skates for h hours. The rental fee was \$2 per hour and she paid a total of \$8.

f + 9 = 38

y

-8 is not a solution of the equation _4 = -32.

C 8x = 72; x = 9 ? 8(9) = 72 ? 72 = 72

+

Teammates’ points

6 is a solution of x + 9 = 15.

? -2 = -32

6.9.A

Mark scored 17 points for the home team in a basketball game. His teammates as a group scored p points. Write an equation to represent this situation.

Determine whether the given value is a solution of the equation.

B

You can represent some real-world situations with an equation. Making a model first can help you organize the information.

Math On the Spot

Equation

a number plus 4

Numerical

yes

Writing Equations to Represent Situations

An equation is a mathematical statement that two expressions are equal. An equation may or may not contain variables. For an equation that has a variable, a solution of the equation is a value of the variable that makes the equation true. An equation represents a relationship between two An expression represents values. a single value.

Words

no

36 3. __ x = 9; x = 4

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Determining Whether Values Are Solutions

Expression

2. y - 6 = 24; y = 18

Substitute 9 for x.

Personal Math Trainer

Multiply.

Online Assessment and Intervention

9 is a solution of 8x = 72.

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Lesson 12.1

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321

25/10/12 5:34 PM

322

c - 3 = 12

17n = 102

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LESSON

DO NOT EDIT--Changes must be made through “File info” CorrectionKey=B

2h = 8

Unit 4

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PROFESSIONAL DEVELOPMENT Integrate Mathematical Processes This lesson provides an opportunity to address Mathematical Process TEKS 6.1.D, which calls for students to “Communicate mathematical ideas, …using multiple representations, including symbols…and language as appropriate.” Students begin by distinguishing equations from expressions. Then they determine whether a given number is a solution of the equation. Next, students write equations that represent real-world situations expressed in words. Finally, students use substitution to check whether a value for a variable makes an equation true.

Math Background The equals sign, which first appeared about 450 years ago, is a relatively new math concept. The term equation comes from a Latin word meaning “to set equal.” There are, essentially, four types of equations: • True equation: 2 + 3 = 5 • False equation: 3 + 4 = 8 • Conditional equation: y - 6 = 7 (not true for all values) • Identity: 3n + 5n = 8n (true for all values)

Writing Equations to Represent Situations

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ADDITIONAL EXAMPLE 3 Suki paid \$132 for 6 DVDs. Write an equation to determine whether each DVD cost \$17 or \$22. Sample equation: 6n = \$132; \$22 Interactive Whiteboard Interactive example available online

EXAMPLE 3 Questioning Strategies

Mathematical Processes • Is the equation x - 47 = 18 true? Explain. The equation is true only when x = 65. An equation is a balanced mathematical statement, and only the correct solution will maintain the balance. • How can you check your answer by using a different mathematical operation? Add \$18 and \$47 to find the original amount on the card. You should get \$65.

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Connect Multiple Representations Mathematical Processes Have students explain why adding \$18 to \$47 to check that \$65 is the answer will yield the same result as subtracting \$47 from \$65 to see that \$18 is the money left over. Students should recognize that addition and subtraction are inverse operations. Animated Math Modeling Equations Students model equations using interactive algebra tiles. my.hrw.com

Focus on Math Connections Point out that an equals sign with a question mark above (≟) is used immediately after a variable has been substituted by a number. This symbol indicates that it is not yet known whether the equation is true or false.

YOUR TURN Avoid Common Errors If students have difficulty writing equations to represent the word problems, encourage them to make a model, like the one in Example 3, to organize the information. Then remind students to check that their solution makes the original equation true.

Elaborate Talk About It Summarize the Lesson Ask: How do you know when a number is a solution to a equation? When the number is substituted for the variable, it makes the equation true. An equation is true when the values of the expressions on opposite sides of the equals sign are the same.

GUIDED PRACTICE Engage with the Whiteboard For Exercises 1–2, have student fill in the boxes and determine whether the given values are solutions to the equations. For Exercise 13, have a student circle the important information in the problem statement on the whiteboard. Then have another student complete the model for the word equation and write the equation next to the model.

Avoid Common Errors Exercises 3–12 Watch for students who substitute the given value incorrectly. Caution students to double-check their work for accuracy. Exercises 14–16 If students have difficulty writing equations to represent the word problems, encourage them to make a model, like the one in Example 3, to organize the information. Then remind students to check that their solution makes the original equation true.

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Lesson 12.1

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Guided Practice

Writing an Equation and Checking Solutions

Determine whether the given value is a solution of the equation. (Example 1) Math On the Spot my.hrw.com

EXAMPL 3 EXAMPLE

6.10.B

Sarah used a gift card to buy \$47 worth of groceries. Now she has \$18 left on her gift card. Write an equation to determine whether Sarah had \$65 or \$59 on the gift card before buying groceries. STEP 1

Amount on card STEP 2

-

Amount spent

=

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STEP 3

=

Amount left on card

x

-

47

=

18

Substitute 65 and 59 for x to see which equation is true.

The amount spent and the amount left on the card are the known quantities. Substitute those values in the equation.

© Houghton Mifflin Harcourt Publishing Company

13

4 yes

7. 21 = m + 9; m = 11 9. d - 4 = 19; d = 15

no no

6. 2.5n = 45; n = 18

yes

8. 21 - h = 15; h = 6

yes

10. 5 + x = 47; x = 52

yes

yes

12. 5q = 31; q = 13

no no

Number of rooms

×

on each floor

=

Total number of rooms

8r = 256

14. In the school band, there are 5 trumpet players and f flute players. There are twice as many flute players as there are trumpet players. Write an equation to represent this situation. (Example 3)

15. Pedro bought 8 tickets to a basketball game. He paid a total of \$208. Write an equation to determine whether each ticket cost \$26 or \$28. (Example 3)

Sample equation: 8x = 208; \$26

8. What expressions are represented in the equation x - 47 = 18? How does the relationship represented in the equation help you determine if the equation is true?

16. The high temperature was 92°F. This was 24°F higher than the overnight low temperature. Write an equation to determine whether the low temperature was 62°F or 68°F. (Example 3)

x - 47 and 18; Subtract 47 from the given number and

Sample equation: x + 24 = 92; 68°F

if the difference is 18 the equation is true.

? ?

YOUR TURN 9. On Saturday morning, Owen earned \$24 raking leaves. By the end of the afternoon he had earned a total of \$62. Write an equation to determine whether Owen earned \$38 or \$31 on Saturday afternoon.

? =4

4. 17y = 85; y = 5

no

Number

Reflect

Personal Math Trainer

Substitute the number for the variable and simplify. If the final statement is true, the number is a solution.

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ESSENTIAL QUESTION CHECK-IN

17. Tell how you can determine whether a number is a solution of an equation.

Online Assessment and Intervention

Lesson 12.1

6_MTXESE051676_U4M12L1 323

52

? _____ = 4

of floors

x - 47 = 18 x - 47 = 18 ? ? 65 - 47 = 18 59 - 47 = 18 ? ? 18 = 18 12 = 18 The amount on Sarah’s gift card before she bought groceries was \$65.

Sample equation: x + 24 = 62; \$38

5

yes

13. Each floor of a hotel has r rooms. On 8 floors, there are a total of 256 rooms. Write an equation to represent this situation. (Example 2)

Let x be the amount on the card. Amount spent

? 23 =

n 2. __ = 4; n = 52 13

-9

11. w - 9 = 0; w = 9

Rewrite the equation using a variable for the unknown quantity and the given values for the known quantities.

-

14

3. 14 + x = 46; x = 32 Animated Math

Amount left on card

Amount on card

? 23 =

5. 25 = _5k ; k = 5

Identify the three quantities given in the problem.

no

1. 23 = x - 9; x = 14

© Houghton Mifflin Harcourt Publishing Company

You can substitute a given value for the variable in an equation to check if that value makes the equation true.

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DIFFERENTIATE INSTRUCTION Visual Cues

Critical Thinking

Display a set of pan balance scales. For each of the following equations, write the left side over the left scale and the right side over the right scale. Ask what the value of the variable must be for the scales to remain balanced.

Give students the following problem to solve.

1. x + 3 = 7 x = 4

Remind students that to compare quantities, they need to use the same units.

Differentiated Instruction includes: • Reading Strategies • Success for English Learners ELL • Reteach • Challenge PRE-AP

3. 4y = 28 y = 7

17 ≠ 350 ÷ 20; No, they do not have the same amount of money.

2. t - 5 = 3 t = 8 w 4. __ = 6 w = 12 2

Rebecca has 17 one-dollar bills. Courtney has 350 nickels. Do the two girls have the same amount of money?

Writing Equations to Represent Situations

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Personal Math Trainer Online Assessment and Intervention

Online homework assignment available

Evaluate GUIDED AND INDEPENDENT PRACTICE 6.9.A, 6.10.B

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12.1 LESSON QUIZ 6.9.A, 6.10.B Determine whether the given value is a solution of the equation. Write yes or no. 1. w – 7 = 20; w = 13

Concepts & Skills

Practice

Example 1 Determining Whether Values Are Solutions

Exercises 1–12

Example 2 Writing Equations to Represent Situations

Exercises 13, 18–21, 24–25

Example 3 Writing an Equation and Checking Solutions

Exercises 14–16, 26

2. 15t = -120; t = -8 y 3. __ = -2; y = 24 12

Exercise

4. -5 = x + 5; x = -10 5. In a choir there are 16 altos and s sopranos. There are twice as many sopranos as altos. Write an equation to represent this situation. 6. Jerome is one-third the age of his aunt, who is 51 years old. Write an equation to determine whether Jerome is 14 or 17.

Depth of Knowledge (D.O.K.) 2 Skills/Concepts

1.A Everyday life

22

3 Strategic Thinking

1.G Explain and justify arguments

23

2 Skills/Concepts

1.F Analyze relationships

24

3 Strategic Thinking

1.G Explain and justify arguments

25–26

3 Strategic Thinking

1.D Multiple representations

27–29

3 Strategic Thinking

1.G Explain and justify arguments

18–21

Lesson Quiz available online my.hrw.com

Answers 1. no 2. yes 3. no 4. yes 5. __2s = 16 6. 3x = 51; Jerome is 17 years old.

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Lesson 12.1

Mathematical Processes

Additional Resources Differentiated Instruction includes: • Leveled Practice Worksheets

Class

Date Personal Math Trainer

12.1 Independent Practice 6.7.B, 6.9.A, 6.10.B

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18. Andy is one-fourth as old as his grandfather, who is 76 years old. Write an equation to determine whether Andy is 19 or 22 years old.

4n = 76; 19 years old 19. A sleeping bag weighs 8 pounds. Your backpack and sleeping bag together weigh 31 pounds. Write an equation to determine whether the backpack without the sleeping bag weighs 25 or 23 pounds.

x + 8 = 31; 23 pounds

22. Write an equation that involves multiplication, contains a variable, and has a solution of 5. Can you write another equation that has the same solution and includes the same variable and numbers but uses division? If not, explain. If possible, write the equation.

© Houghton Mifflin Harcourt Publishing Company

29 32

a. Write an equation that relates Cindy’s age to her dad’s age when Cindy is 18.

Cindy’s Age 2 years old

36 years old

10 years old

?

18 years old

b. Determine if 42 is a solution to the equation. Show your work.

No; 42 - 26 = 16; 16 is not equal to 18.

represents one value such as

Cindy’s father will not be 42 when she is 18.

2 + 5 and an equation represents FOCUS ON HIGHER ORDER THINKING

the relationship between two equation is a statement that the

Yes, because 4 + f = 12 means there are 8 flute players,

equivalent. Each expression is

and 8 is twice 4.

24. Explain the Error The problem states that Ursula earns \$9 per hour. To write an expression that tells how much money Ursula earns for h hours, Joshua wrote _h9 . Sarah wrote 9h. Whose expression is correct and why?

Sarah is correct because the

x is the distance between

28. Problem Solving Ronald paid \$162 for 6 tickets to a basketball game. During the game he noticed that his friend paid \$130 for 5 tickets. The price of each ticket was \$26. Was Ronald overcharged? Justify your answer.

Yes, Ronald was overcharged because 6(26) = 156 and 156 < 162. 29. Communicate Mathematical Ideas Tariq said you can write an equation by setting an expression equal to itself. Would an equation like this be true? Explain.

earnings are the number of

b. Describe what your variable represents.

Work Area

27. Critical Thinking In the school band, there are 4 trumpet players and f flute players. The total number of trumpet and flute players is 12. Are there twice as many flute players as trumpet players? Explain.

two expressions, 2 + 5 and 7, are just part of that statement.

29 - 13 = x; 13 + x = 29

hours times the amount per

Yes, setting an expression equal to itself forms an

hour, or 9h.

equation that will always be true regardless of the value of the variable(s) or numbers in the expression.

Artaville and Greenville Lesson 12.1

EXTEND THE MATH

x - 26 = 18

values such as 2 + 5 = 7. An

a. Write two equations that state the relationship of the distances between Greenville, Artaville, and Jonesborough.

26. Multiple Representations The table shows ages of Cindy and her dad.

23. How are expressions and equations different? Explain using a numerical example.

x - 23 = 48; 71 students

Maybern

is 44 - 7 = x.

20 ÷ x = 4 or x = 20 ÷ 4

21. The table shows the distance between Greenville and nearby towns. The distance between Artaville and Greenville is 13 miles less than the distance between Greenville and Jonesborough. Distance between Greenville and Nearby Towns (miles)

Both equations are correct. Another correct equation

Sample answer: 4x = 20; Yes:

20. Halfway through a bus route, 23 students have been dropped off and 48 students remain on the bus. Write an equation to determine whether there are 61 or 71 students on the bus at the beginning of the route.

Jonesborough

Online Assessment and Intervention

25. Communicate Mathematical Ideas A dog weighs 44 pounds and the veterinarian thinks it needs to lose 7 pounds. Mikala wrote the equation x + 7 = 44 to represent the situation. Kirk wrote the equation 44 - x = 7. Which equation is correct? Can you write another equation that represents the situation?

© Houghton Mifflin Harcourt Publishing Company

Name

PRE-AP

325

Activity available online

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Unit 4

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Activity Following the release of the hit movie Attack of the Giant Muffins, muffin fever spread across the country. This resulted in a contest to choose the plot of the sequel Attack of the Giant Muffins, Part 2. The table shows the number of contest entries from some towns in one state.

Town Entries Hillville .....................................40 Dos Rios ..................................30 High Corn ..............................105 Moose .......................................n Ho-Hum ..................................25

Write an equation for each statement. 1. The sum of the number of entries from Hillville and Moose was 90. n + 40 = 90 2. The difference between the number of Moose entries and Ho-Hum entries was 25. n - 25 = 25 3. The total number of entries is 5 times the number of Moose entries. 5n = 250 Writing Equations to Represent Situations

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LESSON

12.2 Addition and Subtraction Equations Texas Essential Knowledge and Skills The student is expected to: Expressions, equations, and relationships—6.10.A Model and solve one-variable, one-step equations and inequalities that represent problems, including geometric concepts. Expressions, equations, and relationships—6.9.B Represent solutions for one-variable, one-step equations and inequalities on number lines. Expressions, equations, and relationships—6.9.C Write corresponding real-world problems given one-variable, one-step equations or inequalities.

Mathematical Processes 6.1.E Create and use representations to organize, record, and communicate mathematical ideas.

ADDITIONAL EXAMPLE 1 Solve the equation y + 8 = 22. Graph the solution on a number line. y = 14 10

15

20

Interactive Whiteboard Interactive example available online my.hrw.com

Engage

ESSENTIAL QUESTION How do you solve equations that contain addition and subtraction? Sample answer: Apply the inverse operation—subtraction for addition equations and addition for subtraction equations—to both sides of the equation.

Motivate the Lesson Ask: Does anyone have a puppy? Have you noticed how fast they grow? Begin the Explore Activity to find out how to model one puppy’s weight gain, using an equation and algebra tiles.

Explore EXPLORE ACTIVITY Focus on Modeling Mathematical Processes Remind students that an equation is like a balanced scale. If you increase or decrease the weights by the same amount on both sides, the scale will remain in balance. Emphasize that students must remove the same number of tiles from both sides of the mat.

Explain EXAMPLE 1 Focus on Math Connections

Mathematical Processes Remind students that subtraction is the inverse, or opposite, of addition. If an equation contains addition, solve it by subtracting from both sides to “undo” the addition.

Questioning Strategies

Mathematical Processes • Would you solve 15 + a = 26 differently from a + 15 = 26? no because addition is associative and the order of the variable and the added number does not change the process of subtracting 15 from both sides

• How can you use substitution to check an answer to an addition equation? Substitute the value for the variable into the original equation and simplify. If the result is a true statement, the value is the solution. • Why is the graph only a single point? because there is only one answer for the equation

YOUR TURN Avoid Common Errors Watch for students who perform the inverse operation of subtraction only on the side with the variable. Stress that to keep the equation “balanced” the same amount must be taken away from each side.

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12.2 ?

ESSENTIAL QUESTION

Expressions, equations, and relationships—6.10.A Model and solve one-variable, one-step equations ... that represent problems, including geometric concepts. Also 6.9.B, 6.9.C

How do you solve equations that contain addition or subtraction?

EXPLORE ACTIVITY

Using Subtraction to Solve Equations Removing the same number of tiles from each side of an equation mat models subtracting the same number from both sides of an equation. Math On the Spot my.hrw.com

You can subtract the same number from both sides of an equation, and the two sides will remain equal. When an equation contains addition, solve by subtracting the same number from both sides.

6.10.A

Modeling Equations

EXAMPLE 1

A puppy weighed 6 ounces at birth. After two weeks, the puppy weighed 14 ounces. How much weight did the puppy gain?

6

+

Weight gained

=

Weight after 2 weeks

+

x

=

14

© Houghton Mifflin Harcourt Publishing Company • Image Credits: ©Petra Wegner / Alamy

a + 15 = 26

Notice that the number 15 is added to a.

a + 15 = 26 - 15 -15 ___ a = 11

Subtract 15 from both sides of the equation.

Check: a + 15 = 26

To answer this question, you can solve the equation 6 + x = 14. Algebra tiles can model some equations. An equation mat represents the two sides of an equation. To solve the equation, remove the same number of tiles from both sides of the mat until the x tile is by itself on one side.

B How many 1 tiles must you remove on the left 6 side so that the x tile is by itself? Cross out these tiles on the equation mat.

1

1

1

1

1

1

x

1

1

1

1

1

1

1

1

1

1

1

1

1

1

Substitute 11 for a.

26 = 26

6+ x

5 6 7 8 9 10 11 12 13 14 15

14

Reflect 2.

C Whenever you remove tiles from one side of the mat, you must remove the same number of tiles from the other side of the mat. Cross out the tiles that should be removed on the right side of the mat. D How many tiles remain on the right side of the mat? This is the solution of the equation.

8

8

ounces.

variable is isolated, or alone, on one side of the equation.

Mathematical Processes

Why did you remove tiles from each side of your model?

You want to remove tiles equally from both The x tile is alone on one side of the mat. The solution is sides to keep the equation balanced. the number of small tiles on the other side. Communicate Mathematical Ideas How do you know when the model shows the final solution? How do you read the solution?

Lesson 12.2

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Communicate Mathematical Ideas How do you decide which number to subtract from both sides?

Subtract the number added to the variable so that the

Math Talk

Reflect 1.

? 11 + 15 = 26

Graph the solution on a number line.

A Model 6 + x = 14.

The puppy gained

6.10.A, 6.9.B

Solve the equation a + 15 = 26. Graph the solution on a number line.

Let x represent the number of ounces gained. Weight at birth

Subtraction Property of Equality

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Personal Math Trainer Online Assessment and Intervention

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3. Solve the equation 5 = w + 1.5. Graph the solution on a number line. w=

-5 -4 -3 -2 -1

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LESSON

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0 1 2 3 4 5

3.5

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PROFESSIONAL DEVELOPMENT Integrate Mathematical Processes This lesson provides an opportunity to address Mathematical Process TEKS 6.1.E, which calls for students to “create and use representations to organize, record, and communicate mathematical ideas.” Students use algebra tiles and number lines to model the solutions to one-step equations. They proceed to solve algebraic equations symbolically by using inverse operations to isolate the variable. They then apply the algebraic method to solve equations representing real-world situations.

Math Background In Elements, Book I, Euclid listed five axioms that he called “common notions.” 1. Things which are equal to the same thing are also equal to one another. 2. If equals be added to equals, the wholes are equal. 3. If equals be subtracted from equals, the remainders are equal. 4. Things which coincide with one another are equal to one another. 5. The whole is greater than the part.

328

Solve the equation y −13 = –12. Graph the solution on a number line. y=1 -5

0

5

Interactive Whiteboard Interactive example available online my.hrw.com

EXAMPLE 2 Focus on Math Connections

Mathematical Processes Remind students that addition and subtraction are inverse operations. If an equation contains subtraction, solve it by adding to both sides to “undo” the subtraction.

Questioning Strategies

Mathematical Processes • How is solving an equation containing subtraction similar to solving an equation containing addition? Solve both types of equations by using the inverse operation to get the variable by itself on one side of the equation. • Does it make sense that the solution is greater than 18? Explain. Yes. The equation indicates that subtracting 21 from some numbers gives an answer of 18, so the solution must be greater than 18.

Engage with the Whiteboard Cover up the sentences in blue next to each step of the solution and have students write a description of what is happening in each step. Then have the students graph the solution on the number line.

Focus on Modeling

Mathematical Processes Guide students to see that it makes sense to draw only the needed portion of the number line when graphing a solution, especially when the numbers are large.

YOUR TURN Focus on Math Connections

ADDITIONAL EXAMPLE 3 Write and solve an equation to find the measure of the unknown angle. z + 48° = 90°; z = 42°

z

Mathematical Processes Remind students that to add or subtract fractions with different denominators, it is necessary to rewrite the fractions with a common denominator. A fraction greater than 1 should be written as a mixed number for ease of graphing.

EXAMPLE 3 Questioning Strategies

Mathematical Processes • How do you know that x + 60 = 180 is the correct equation to use to find the measure of x? The two angles shown in the drawing are supplementary. • Does the sketch of the unknown angle x appear to be twice 60°? Explain. Yes. If I divide x into two halves, the two angles appear to have measures that are similar to the 60° angle.

48°

Interactive Whiteboard Interactive example available online my.hrw.com

Focus on Math Connections

Mathematical Processes Have students explain what will be true about the unknown measure of any angle if its supplement is less than 90°. Elicit that it will be an obtuse angle. Then ask the same question about angles whose supplements are greater than 90°.

Solve the equation n + 6.5 = 20. Then write a real-world problem that involves adding these two quantities. n = 13.5; Sample answer: Ivan needed 6.5 more points to take the lead with 20 points. How many points had he already scored? Interactive Whiteboard Interactive example available online my.hrw.com

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Lesson 12.2

Focus on Math Connections

Mathematical Processes Point out to students that the angle represented here is a right angle and that a right angle measures exactly 90°. Remind them that if the sum of the measures of two angles is 90°, the two angles are complementary angles. This information should help students to write and solve an equation to find the measure of the unknown angle.

EXAMPLE 4 Connect to Daily Life

Mathematical Processes Point out to students that the equation has decimals containing hundredths and that money is commonly represented as decimals containing hundredths. Thus, a good real-world application for this equation would be a situation involving money. Encourage students to brainstorm situations from everyday life that could make sense.

DO NOT EDIT--Changes must be made through “File info” CorrectionKey=B

DO NOT EDIT--Changes must be made through “File info” CorrectionKey=B

Solving Equations that Represent Geometric Concepts

When an equation contains subtraction, solve by adding the same number to both sides.

You can write equations to represent geometric relationships.

Math On the Spot

Math On the Spot

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Recall that a straight line has an angle measure of 180°. Two angles whose measures have a sum of 180° are called supplementary angles. Two angles whose measures have a sum of 90° are called complementary angles.

You can add the same number to both sides of an equation, and the two sides will remain equal.

EXAMPLE 3 EXAMPL 2 EXAMPLE

Find the measure of the unknown angle.

6.10.A, 6.9.B

STEP 1

Solve the equation y - 21 = 18. Graph the solution on a number line. y - 21 = 18

Notice that the number 21 is subtracted from y.

y - 21 = 18 + 21 +21 ___ y = 39

Add 21 to both sides of the equation. STEP 2

Subtract.

=

180°

Write a description to represent the model. Include a question for the unknown angle.

STEP 3

Write an equation.

STEP 4

Solve the equation.

x + 60 = 180

35 36 37 38 39 40 41 42 43 44 45

x+

Reflect © Houghton Mifflin Harcourt Publishing Company

60°

The sum of an unknown angle and a 60° angle is 180°. What is the measure of the unknown angle?

Substitute 39 for y.

Graph the solution on a number line.

4.

+

60°

x

60 = 180 -60 -60 _ _

Subtract 60 from each side.

Communicate Mathematical Ideas How do you know whether to add on both sides or subtract on both sides when solving an equation?

x

If the equation contains addition, subtract. If the

The unknown angle measures 120°.

=

120

The final answer includes units of degrees.

equation contains subtraction, add. YOUR TURN 6. Write and solve an equation to find the measure of the unknown angle.

YOUR TURN Solve the equation h - _12 = _43 .

5.

x + 65 = 90; x = 25°

-2

Graph the solution on a number line. h=

-1

0

1

5 _ , or 1_14 4

2 Personal Math Trainer

Personal Math Trainer

Online Assessment and Intervention

Online Assessment and Intervention

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x

© Houghton Mifflin Harcourt Publishing Company

18 = 18

Write the information in the boxes. Unknown angle

Check: y - 21 = 18 ? 39 - 21 = 18

6.10.A

65°

7. Write and solve an equation to find the measure of a complement of an angle that measures 42°.

x + 42 = 90; x = 48°

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DIFFERENTIATE INSTRUCTION Critical Thinking

Cognitive Strategies

Present the magic square shown below. Explain that the sum of any three numbers added across, down, or diagonally should have the same sum.

Have students work with a partner to create “mind-reading” puzzles that depend upon inverse operations to return to the original number. The following puzzle is an example. Pick a number.

Differentiated Instruction includes: • Reading Strategies • Success for English Learners ELL • Reteach • Challenge PRE-AP

Have students write and solve equations to fill in the table. The magic sum for this square is 465. 248

31

186

93

155

217

124

279

62

5. Subtract 4.

2. Subtract 1.

7. Subtract 7.

4. Subtract 5.

330

Questioning Strategies

Mathematical Processes • Using number sense, what can you determine about the value of x, given that the sum of two numbers is 25? The value of x will be less than 25.

• How else could you write the equation to solve for x? Explain. Sample answer: You could write x + 21.79 = 25; addition is commutative, so x + 21.79 = 21.79 + x. • How is the question in a real-world problem related to its equation? The question asks about the value of the variable.

Mathematical Processes Guide students to write problems that use sensible data, and have them explain, step by step, how to solve their equations. Challenge each student to create original word problems based on unique situations.

Elaborate Talk About It Summarize the Lesson Ask: How do you solve and check an equation containing only addition or only subtraction? You apply the inverse operation—subtraction for addition equations and addition for subtraction equations—to both sides of the equation.

GUIDED PRACTICE Engage with the Whiteboard For Exercise 1, have a student complete the verbal model on the whiteboard and then draw a model with algebra tiles below the verbal model. Ask students to provide an explanation of the solution.

Avoid Common Errors Exercises 2–6 Remind students to use the inverse operation of subtraction to undo addition or addition to undo subtraction. Exercise 7 Remind students to begin by identifying the angle they are using, as right, straight, or obtuse so that they will know what angle measure to use in their equation.

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Lesson 12.2

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DO NOT EDIT--Changes must be made through “File info” CorrectionKey=B

Guided Practice

Writing Real-World Problems for a Given Equation Math On the Spot

1. A total of 14 guests attended a birthday party. Three guests stayed after the party to help clean up. How many guests left when the party ended? (Explore Activity) a. Let x represent the number

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EXAMPL 4 EXAMPLE

b.

6.9.C

+

left at end of party

Write a real-world problem for the equation 21.79 + x = 25. Then solve the equation.

x

21.79 + x = 25 STEP 1

Number that

of guests who left when the party ended. Number that

stayed to clean 3

+

c. Draw algebra tiles to model the equation.

Examine each part of the equation.

11

Total at party

=

14

+ + +

+

friends left when the party ended.

=

x is the unknown or quantity we are looking for. 21.79 is added to x. = 25 means that after adding 21.79 and x, the result is 25. STEP 2

Write a real-world situation that involves adding two quantities. Joshua wants to buy his mother flowers and a card for Mother’s Day. Joshua has \$25 to spend and selects roses for \$21.79. How much can he spend on a card?

© Houghton Mifflin Harcourt Publishing Company • Image Credits: ©Brand X Pictures/Getty Images

STEP 3

Math Talk

Mathematical Processes

How is the question in a real-world problem related to its equation?

Solve the equation. 21.79 + x = 25 -21.79 -21.79 __ __ x=

3.21

The final answer includes units of money in dollars.

Solve each equation. Graph the solution on a number line. (Examples 1 and 2) 2. 2 = x - 3

x=

-5 -4 -3 -2 -1

5

3. s + 12.5 = 14 -5 -4 -3 -2 -1

0 1 2 3 4 5

s=

1.5

0 1 2 3 4 5

Solve each equation. (Examples 1 and 2) 4. h + 6.9 = 11.4 h=

4.5

6. n + _12 = _74

5. 82 + p = 122

40

p=

n=

5 _ 4

7. Write and solve an equation to find the measure of the unknown angle. (Example 3)

Joshua can spend \$3.21 on a Mother’s Day card.

x + 45 = 180; x = 135°

Reflect 8.

+ + + + + + + + + + + + + +

45°

x

8. Write a real-world problem for the equation x - 75 = 200. Then solve the equation. (Example 4)

What If? How might the real-world problem change if the equation were x - 21.79 = 25 and Joshua still spent \$21.79 on roses?

Check students’ answers; x = 275

21.79 is subtracted from the unknown value, and the result would remain 25. The variable would be the amount Joshua started with and

? ?

25 would the amount of money available after buying roses.

ESSENTIAL QUESTION CHECK-IN

9. How do you solve equations that contain addition or subtraction?

You apply the inverse operation—subtraction for

YOUR TURN 9. Write a real-world problem for the equation x - 100 = 40. Then solve the equation.

Check students’ problems; x = 140

Personal Math Trainer

equations—to both sides of the equation.

Online Assessment and Intervention

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You can write a real-world problem for a given equation. Examine each number and mathematical operation in the equation.

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Personal Math Trainer Online Assessment and Intervention

Online homework assignment available

Evaluate GUIDED AND INDEPENDENT PRACTICE 6.9.B, 6.9.C, 6.10.A

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12.2 LESSON QUIZ 6.10.A, 6.9.C Solve each equation. 1. y + 8.3 = 12.7 2. w − 14 = 23 3. 18 + x = 6 4. t – 1.9 = 15.7 5. __12 = a - __45 6. Ellie spent \$88.79 at the computer store. She then had \$44.50 left to buy a cool hat. How much money did she originally have? Write and solve an equation to answer the question. 7. Write a real-world problem for the equation x + 12 = 35. Then solve the equation. Lesson Quiz available online my.hrw.com

2. w = 37 3. x = –12 4. t = 17.6 5. a = __74 6. x − \$88.79 = \$44.50; x = \$133.29 7. See students’ answers; x = 23

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Lesson 12.2

Concepts & Skills

Practice

Explore Activity Modeling Equations

Exercise 1

Example 1 Using Subtraction to Solve Equations

Exercises 2–6, 10, 12–14

Example 2 Using Addition to Solve Equations

Exercises 2–6, 11, 15–16

Example 3 Solving Equations that Represent Geometric Concepts

Exercise 7

Example 4 Writing Real-World Problems for a Given Equation

Exercises 8, 17

Exercise

Depth of Knowledge (D.O.K.)

Mathematical Processes

10–16

2 Skills/Concepts

1.A Everyday life

17–18

3 Strategic Thinking

1.F Analyze relationships

19

3 Strategic Thinking

1.A Everyday life

20

3 Strategic Thinking

1.G Explain and justify arguments

21

3 Strategic Thinking

1.F Analyze relationships

Additional Resources Differentiated Instruction includes: • Leveled Practice Worksheets

Name

Class

Date

12.2 Independent Practice 6.9.B, 6.9.C, 6.10.A

my.hrw.com

Write and solve an equation to answer each question.

Handy Dandy Grocery Regular price

Sample answer: m - 123.45 = 36.55; \$160

Sample answer: e + 8 = 31; 23

11. My sister is 14 years old. My brother says that his age minus twelve is equal to my sister’s age. How old is my brother?

Both a and r are equal to \$1.50, so the discount is the same. 20. Critical Thinking An orchestra has twice as many woodwind instruments as brass instruments. There are a total of 150 brass and woodwind instruments.

17. Represent Real-World Problems Write a real-world situation that can be represented by 15 + c = 17.50. Then solve the equation and describe what your answer represents for the problem situation.

Sample answer: x + 8.95 = 21.35; \$12.40

a. Write two different addition equations that describe this situation. Use w for woodwinds and b for brass.

w + b = 150; w = b + b = 2b

Check students’ answers. c = 2.50

13. The Acme Car Company sold 37 vehicles in June. How many compact cars were sold in June?

w = 100; b = 50

Number sold

SUV

8

Compact

?

18. Critique Reasoning Paula solved the equation 7 + x = 10 and got 17, but she is not certain if she got the correct answer. How could you explain Paula’s mistake to her?

Sample answer: x + 8 = 37;

21. Look for a Pattern Assume the following: a + 1 = 2, b + 10 = 20, c + 100 = 200, d + 1,000 = 2,000, ... a. Solve each equation for each variable.

a = 1; b = 10, c = 100, d = 1,000, ...

29 compact cars

b. What pattern do you notice between the variables?

of subtracting. She should have

14. Sandra wants to buy a new MP3 player that is on sale for \$95. She has saved \$73. How much more money does she need?

c. What would be the value of g if the pattern continues?

g = 1,000,000

the equation and found that

Sample answer: 73 + b = 95 or

x = 3.

95 - b = 73; \$22

Lesson 12.2

EXTEND THE MATH

Every variable is ten times the one before it.

subtracted 7 from both sides of

© Houghton Mifflin Harcourt Publishing Company

b. How many woodwinds and how many brass instruments satisfy the given information?

Acme Car Company - June Sales Type of car

\$3.99

b. Which fruit has a greater discount? Explain.

t = \$773

12. Kim bought a poster that cost \$8.95 and some colored pencils. The total cost was \$21.35. How much did the colored pencils cost?

\$2.99

5-pound bag of oranges

1.49 + a = 2.99; 2.49 + r = 3.99

Sample answer: 548 = t - 225;

Sample answer: 14 = b - 12; b = 26

5-pound bag of apples

a. Write an equation to find the discount for each situation using a for apples and r for oranges.

16. Brita withdrew \$225 from her bank account. After her withdrawal, there was \$548 left in Brita’s account. How much money did Brita have in her account before the withdrawal?

elephants

Work Area

19. Multistep Handy Dandy Grocery is having a sale this week. If you buy a 5-pound bag of apples for the regular price, you can get another bag for \$1.49. If you buy a 5-pound bag of oranges at the regular price, you can get another bag for \$2.49.

Online Assessment and Intervention

15. Ronald spent \$123.45 on school clothes. He counted his money and discovered that he had \$36.55 left. How much money did he originally have?

10. A wildlife reserve had 8 elephant calves born during the summer and now has 31 total elephants. How many elephants were in the reserve before summer began?

© Houghton Mifflin Harcourt Publishing Company

FOCUS ON HIGHER ORDER THINKING

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Activity available online

PRE-AP

Unit 4

my.hrw.com

Activity Use the code shown in the table below to send messages. A

B

C

D

E

F

G

H

I

J

K

L

M

1

2

3

4

5

6

7

8

9

10

11

12

13

N

O

P

Q

R

S

T

U

V

W

X

Y

Z

14

15

16

17

18

19

20

21

22

23

24

25

26

• Write a short secret message on a piece of paper. • Use the chart to find the number that matches each letter of your message. Write the number above each letter in your message. • To send your message in code, write a series of equations on a separate piece of paper. Use n as your variable and write the equations in the same order as the letters appear in your message. For example, for the letter G, n = 7. You might write n + 5 = 12. • Trade equations and decode classmates’ messages. Addition and Subtraction Equations

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LESSON

12.3 Multiplication and Division Equations Texas Essential Knowledge and Skills The student is expected to: Expressions, equations, and relationships—6.10.A Model and solve one-variable, one-step equations and inequalities that represent problems, including geometric concepts.

Engage

ESSENTIAL QUESTION How do you solve equations that contain multiplication or division? Sample answer: You apply the inverse operation—division for multiplication equations and multiplication for division equations—to both sides of the equation.

Motivate the Lesson Ask: How many of you like to bake? Can you imagine writing an equation to calculate the ingredients you need? Begin the Explore Activity to see an example for a cookie recipe.

Expressions, equations, and relationships—6.9.B Represent solutions for one-variable, one-step equations and inequalities on number lines. Expressions, equations, and relationships—6.9.C Write corresponding real-world problems given one-variable, one-step equations or inequalities.

Mathematical Processes

Explore EXPLORE ACTIVITY Engage with the Whiteboard For B, have a student circle the remaining two groups of the model on the whiteboard. Then have students fill in the answer for C and the final answer. Ask how the model helped them understand the problem.

6.1.C Select tools, including real objects, manipulatives, paper and pencil, and technology as appropriate, and techniques, including mental math, estimation, and number sense as appropriate, to solve problems.

Explain EXAMPLE 1 Focus on Math Connections

ADDITIONAL EXAMPLE 1 Solve each equation. Graph the solution on a number line.

-5

0

B -27 = -9z z = 3 -5

0

Questioning Strategies

Mathematical Processes • How will you know what number to divide both sides of an equation by in order to solve it? Divide both sides by the number that the variable is multiplied by.

A 6y = -24 y = -4 -10

Mathematical Processes Remind students that division is the inverse, or opposite, of multiplication. To solve an equation that contains multiplication, use division to “undo” the multiplication.

5

Interactive Whiteboard Interactive example available online my.hrw.com

• Why must you divide both sides of the equation by the same number? You do so to maintain the equality, or the balance, of the equation.

Focus on Critical Thinking Mathematical Processes Challenge students to describe how solving a multiplication equation is similar to solving an addition or subtraction equation. Students should indicate that the same inverse operation must be applied to both sides of the equation, leaving the variable isolated on one side of the equation.

YOUR TURN Avoid Common Errors Watch for students who multiply both sides of the equation when they should divide. Remind students to be certain they are using the inverse operation and to check their answers.

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12.3 ?

Multiplication and Division Equations

ESSENTIAL QUESTION

Expressions, equations, and relationships—6.10.A Model and solve one-variable, one-step equations . . . that represent problems, including geometric concepts. Also 6.9.B, 6.9.C

How do you solve equations that contain multiplication or division?

EXPLORE ACTIVITY

Using Division to Solve Equations Separating the tiles on both sides of an equation mat into an equal number of groups models dividing both sides of an equation by the same number. Math On the Spot my.hrw.com

You can divide both sides of an equation by the same nonzero number, and the two sides will remain equal. When an equation contains multiplication, solve by dividing both sides of the equation by the same nonzero number.

6.9.B, 6.9.C, 6.10.A

Modeling Equations

EXAMPLE 1

Deanna has a cookie recipe that requires 12 eggs to make 3 batches of cookies. How many eggs are needed per batch of cookies?

Number of eggs per batch

·

=

A 9a = 54 Total eggs

x

1 1 1 1

x

1 1 1 1

x

1 1 1 1

© Houghton Mifflin Harcourt Publishing Company

3x

4

Divide both sides of the equation by 9.

Check: 9a = 54 ? 9(6) = 54 54 = 54

12

18 = -3d

x

1 1 1 1

x

1 1 1 1

x

1 1 1 1

4

eggs are needed per batch of cookies.

Reflect 1.

Notice that 9 is multiplied by a.

9a __ __ = 54 9 9

0 1 2 3 4 5 6 7 8 9 10

Substitute 6 for a. Multiply on the left side.

B 18 = -3d

B There are 3 x-tiles, so draw circles to separate the tiles into 3 equal groups. One group has been circled for you.

C How many +1-tiles are in each group? This is the solution of the equation.

9a = 54 a=6

3 x 12 · = To answer this question, you can use algebra tiles to solve 3x = 12.

A Model 3x = 12.

6.10.A, 6.9.B

Solve each equation. Graph the solution on a number line.

Let x represent the number of eggs needed per batch. Number of batches

Division Property of Equality

Look for a Pattern Why does it make sense to arrange the 12 tiles in 3 rows of 4 instead of any other arrangement of 12 + 1-tiles, such as 2 rows of 6?

Notice that -3 is multiplied by d.

18 -3d ___ = ____ -3 -3

Math Talk

Mathematical Processes

Check: 18 = -3d ? 18 = -3(-6)

Why is the solution to the equation the number of tiles in each group?

18 = 18

The number of +1-tiles in each group equals one x-tile.

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

Substitute -6 for d. Multiply on the right side.

3 rows are evenly divisible by 3.

Divide both sides of the equation by -3.

-6 = d

© Houghton Mifflin Harcourt Publishing Company

LESSON

DO NOT EDIT--Changes must be made through “File info” CorrectionKey=B

Online Assessment and Intervention

Solve the equation 3x = -21. Graph the solution on a number line. 2. x =

-7

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

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PROFESSIONAL DEVELOPMENT Integrate Mathematical Processes This lesson provides an opportunity to address Mathematical Process TEKS 6.1.C, which calls for students to “select tools…and techniques, including mental math…and number sense to solve problems.” Students first use number sense to identify which inverse operation they should use to isolate the variable on one side of an equation. They then use pencil and paper to solve equations, to check their solutions by using substitution, and to graph the solutions on a number line. In Example 3, students use number sense to translate words to algebraic equations. Students should be encouraged to use mental math to check their answers.

Math Background Division is defined as multiplication by the reciprocal. To solve 3x = 15, for example, we could multiply both sides by __13 , which would be equivalent to dividing both sides by 3. While most students would rather divide by 3 than multiply by __13 , this alternate interpretation is useful when trying to solve equations such as __2 x = 4. Dividing by __2 is equivalent to multiplying 3 3 by __32 .

Multiplication and Division Equations

336

ADDITIONAL EXAMPLE 2 Solve each equation. Graph the solution on a number line. y A __7 = -2 y = -14

-20

-15

-10

15

20

y

B 5 = __3 y = 15 10

Interactive Whiteboard Interactive example available online my.hrw.com

EXAMPLE 2

Focus on Math Connections

Mathematical Processes Remind students that multiplication and division are inverse operations. To solve an equation that contains division, use multiplication to “undo” the division.

Questioning Strategies

Mathematical Processes • How is solving an equation containing division similar to solving an equation containing multiplication? You solve both by using the inverse operation on both sides to get the variable by itself. • Does it make sense that the solution is greater than 15? Yes. Since some number divided by 2 is 15, the unknown number will be greater than (double) the unknown number.

Engage with the Whiteboard Cover up the sentences in blue next to each step of the solution in B and have students write a description of what is happening in each step on the whiteboard. Then have the students graph the solution on the number line.

Avoid Common Errors Watch for students who divide both sides of the equation when they should multiply. Remind students to be certain that they are using the inverse operation and to check their answers.

ADDITIONAL EXAMPLE 3 The area of Danielle’s garden is one-twelfth the area of her entire yard. The area of the garden is 10 square feet. What is the area of the yard? Write and solve an equation to solve the y problem. 10 = __ ; 120 square feet 12 Interactive Whiteboard Interactive example available online my.hrw.com

EXAMPLE 3

Questioning Strategies

Mathematical Processes • Why does it make sense to express Juanita’s total scrapbooking time as the decimal 2.5? How else might you express the time? You can compute easily with the number 2.5. Another way is to express the time as the mixed number 2__12 or as the fraction __52 . • Is Julia’s scrapbooking speed best described as a ratio, a rate, or a unit rate? Explain. It is a unit rate because it not only compares quantities in different units, but it also has a denominator of 1. • Suppose Juanita works at her usual rate for 6 hours one weekend. How many pages can she expect to complete? about 54 pages

Focus on Math Connections

Mathematical Processes In solving the problem about Juanita’s scrapbooking, students need to subtract after they solve the division equation, to compare her rate last week to her usual rate. This Example not only reinforces the usefulness of the four-step problem-solving process but also prepares students for solving two-step equations—those containing more than one operation.

Integrating the ELPS

c.4.G ELL Encourage English learners to take notes on new terms or concepts and to write them in familiar language.

Focus on Reasoning

Mathematical Processes In this multistep problem, students first need to find the total number of cards Roberto started with. Have students describe why __5x = 9 is the correct equation for finding that number.

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DO NOT EDIT--Changes must be made through “File info” CorrectionKey=B

Using Multiplication to Solve Equations

Using Equations to Solve Problems You can use equations to solve real-world problems. Math On the Spot

Math On the Spot

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Multiplication Property of Equality

My Notes

You can multiply both sides of an equation by the same number, and the two sides will remain equal.

EXAMPLE 3

Problem Solving

Juanita is scrapbooking. She usually completes about 9 pages per hour. One night last week she completed pages 23 through 47 in 2.5 hours. Did she work at her average rate?

Solve each equation. Graph the solution on a number line.

© Houghton Mifflin Harcourt Publishing Company

Math Talk

0

10

20

30

40

50

Divide on the left side.

Notice that r is divided by the number 2. Multiply both sides of the equation by 2. 0 5 10 15 20 25 30 35 40 45 50

Check: 15 = __r 2 ? 30 15 = ___ 2 15 = 15

• Compare the number of pages Juanita can expect to complete with the number of pages she actually completed. Justify and Evaluate Solve

Both equations are solved by applying an inverse operation to both sides of the equation. The inverse operations applied are different.

Substitute 50 for x.

B 15 = __r 2 15 = __r 2 2 · 15 = 2 · __r 2 30 = r

• Solve an equation to find the number of pages Juanita can expect to complete.

How is solving a multiplication equation similar to solving a division equation? How are they different?

Multiply both sides of the equation by 5. -20 -10

Formulate a Plan

Mathematical Processes

Notice that x is divided by the number 5.

Substitute 30 for r.

TICKET TICKET

Identify the important information. • Worked for 2.5 hours • Starting page: 23 Ending page: 47 • Scrapbooking rate: 9 pages per hour

6.10.A, 6.9.B

x = 10 A __ 5 x = 10 __ 5 x = 5 · 10 5 · __ 5 x = 50 x = 10 Check: __ 5 ? 50 = ___ 10 5 10 = 10

Analyze Information

EXAMPL 2 EXAMPLE

6.9.C 6.9.C

Let n represent the number of pages Juanita can expect to complete in 2.5 hours if she works at her average rate of 9 pages per hour. Write an equation. n =9 ___ 2.5

n = 2.5 · 9 2.5 · ___ 2.5

Write the equation. Multiply both sides by 2.5.

n = 22.5 Juanita can expect to complete 22.5 pages in 2.5 hours. Juanita completed pages 23 through 47, a total of 25 pages. Because 25 > 22.5, she worked faster than her expected rate.

Divide on the right side.

Justify and Evaluate

You used an equation to find the number of pages Juanita could expect to complete in 2.5 hours if she worked at her average rate. You found that she could complete 22.5 pages.

Solve the equation __ = -1. Graph the solution on a number line. 9 y 3. _ = -1 9

y=

-9

Online Assessment and Intervention

The answer makes sense, because Juanita completed 25 pages in 2.5 hours, which is equivalent to a rate of 10 pages in 1 hour. Since 10 > 9, you know that she worked faster than her average rate.

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Since 22.5 pages is less than the 25 pages Juanita completed, she worked faster than her average rate.

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

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When an equation contains division, solve by multiplying both sides of the equation by the same number.

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DIFFERENTIATE INSTRUCTION Number Sense

Have students use a fraction bar to indicate division of both sides of the equation. This provides an easy visual check to compare the coefficient of the variable and the number chosen for dividing both sides of the equation.

Kinesthetic Experience

Give each group of students a set of nine cards numbered 1–9. On signal, each group picks a card at random and passes it to a different group. The number on this card becomes the divisor in a division equation. On a second signal, each group passes a different card to the same group, and this card becomes the quotient. Each group solves their equation, using the following form: (x ÷ first card) = (second card). The first group to solve their equation wins.

Differentiated Instruction includes: ••Reading Strategies ••Success for English Learners  ELL ••Reteach ••Challenge  PRE-AP

Multiplication and Division Equations

338

EXAMPLE 4 Focus on Reasoning

Mathematical Processes Encourage students to begin by analyzing the equation and using number sense to find situations from everyday life that could make sense. For example, if an equation contained decimals, a situation involving money would most likely be a good real-world situation.

Questioning Strategies

Mathematical Processes • Using number sense, what can you determine about the value of x given that the product of two numbers is 72? That the value of x will be less than 72.

• How is the question in a real-world problem related to its equation? The question asks about the value of the variable.

Mathematical Processes Guide students to write problems that use sensible data, and have them explain step by step how to solve their equations. Challenge each student to create original word problems based on unique situations.

Elaborate Talk About It Summarize the Lesson Ask: How do you solve equations that contain multiplication or division? You apply the inverse operation—division for multiplication equations and multiplication for division equations—to both sides of the equation.

GUIDED PRACTICE Engage with the Whiteboard For Exercise 1, have students complete the verbal model on the whiteboard and then draw a model with algebra tiles below the verbal model. Ask students to explain their reasoning.

Avoid Common Errors Exercises 2–3 Remind students to use the inverse operation of division to undo multiplication or multiplication to undo division. Exercise 4 Remind students that the formula for area of a rectangle is A = l · w.

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Guided Practice YOUR TURN 4. Roberto is dividing his baseball cards equally among himself, his brother, and his 3 friends. Roberto was left with 9 cards. How many cards did Roberto give away? Write and solve an equation to solve the problem.

Sample answer: _5x = 9; x = 45; Roberto gave away

Personal Math Trainer

1. Caroline ran 15 miles in 5 days. She ran the same distance each day. Write and solve an equation to determine the number of miles she ran each day. (Explore Activity)

b.

Number of

days

Math On the Spot my.hrw.com

8x = 72 Caroline ran

Examine each part of the equation. x is the unknown or quantity we are looking for.

2. x ÷ 3 = 3

= 72 means that after multiplying 8 and x, the result is 72.

x=

Write a real-world situation that involves multiplying two quantities. A hot air balloon flew at 8 miles per hour. How many hours did it take this balloon to travel 72 miles?

© Houghton Mifflin Harcourt Publishing Company

+ + + + +

Total number of miles

=

15

+ + + + +

+ + + + +

miles each day.

Solve each equation. Graph the solution on a number line. (Examples 1 and 2)

8 is multiplied by x.

3. 4x = -32

9

x=

0 1 2 3 4 5 6 7 8 9 10

Solve the equation.

-8

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

4. The area of the rectangle shown is 24 square inches. How much longer is its length than its width? (Example 3)

8x = 72 8x __ __ = 72 8 8

3

+ + + + +

=

6 in.

24 = 6w; w = 4; 6 - 4 = 2 inches longer

Divide both sides by 8.

? ?

x=9 The balloon traveled for 9 hours.

ESSENTIAL QUESTION CHECK-IN

5. How do you solve equations that contain multiplication or division?

You apply the inverse operation—division for multiplication equations and multiplication for division YOUR TURN 5. Write a real-world problem for the equation 11x = 385. Then solve the equation.

Check students’ problems; x = 35

equations—to both sides of the equation.

Personal Math Trainer Online Assessment and Intervention

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Lesson 12.3

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w

© Houghton Mifflin Harcourt Publishing Company

6.9.C

Write a real-world problem for the equation 8x = 72. Then solve the equation.

STEP 3

x

·

c. Draw algebra tiles to model the equation.

You can write a real-world problem for a given equation.

STEP 2

· Number of miles

run each day

5

Writing Real-World Problems

STEP 1

.

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45 - 9 = 36 cards.

EXAMPL 4 EXAMPLE

number of miles run each day

a. Let x represent the

Online Assessment and Intervention

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Multiplication and Division Equations

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Personal Math Trainer Online Assessment and Intervention

Online homework assignment available

Evaluate GUIDED AND INDEPENDENT PRACTICE 6.9.B, 6.10.A

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Concepts & Skills

Practice

Explore Activity Modeling Equations

Exercise 1

Example 1 Using Division to Solve Equations

Exercise 3

Example 2 Using Multiplication to Solve Equations

Exercise 2

Example 3 Using Equations to Solve Problems

Exercises 4, 6–12

5. The area of a rectangle is 48 square inches. The length is 8 inches. What is the measure of its width? Write and solve an equation.

Example 4 Writing Real-Word Problems

Exercises 13–14

Lesson Quiz available online

Exercise

12.3 LESSON QUIZ 6.10.A Solve each equation. 1. __4x = 12.5 2. 8w = -120 z 3. 6 = ___ 4.5

4. 20 = 2.5m

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Depth of Knowledge (D.O.K.)

Mathematical Processes

2 Skills/Concepts

1.A Everyday life

13

3 Strategic Thinking

1.C Select tools

1. x = 50

14

3 Strategic Thinking

1.F Analyze relationships

2. w = -15

15

3 Strategic Thinking

1.G Explain and justify arguments

16–17

3 Strategic Thinking

1.F Analyze relationships

3. z = 27

6–12

4. m = 8 5. Sample answer: 8x = 48 inches; x = 6 in.

Additional Resources Differentiated Instruction includes: • Leveled Practice Worksheets

341

Lesson 12.3

DO NOT EDIT--Changes must be made through “File info” CorrectionKey=A

DO NOT EDIT--Changes must be made through “File info” CorrectionKey=B

Name

Class

Date

12.3 Independent Practice 6.9.B, 6.9.C, 6.10.A

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Write and solve an equation to answer each question. Sample answers are given. 6. Jorge baked cookies for his math class’s end-of-year party. There are 28 people in Jorge’s math class including Jorge and his teacher. Jorge baked enough cookies for everyone to get 3 cookies each. How many cookies did Jorge bake?

14. Representing Real-World Problems Write and solve a problem involving money that can be solved with a division equation and has a solution of 1,350.

Personal Math Trainer

Sample answer: Marcy split her income from last week

Online Assessment and Intervention

equally between paying her student loans, rent, and

11. Dharmesh has a square garden with a perimeter of 132 feet. Is the area of the garden greater than 1,000 square feet?

savings. She put \$450 in savings. How much did Marcy earn last week? _3x = 450, x = 1,350; Marcy earned

S

\$1,350 last week. S

c __ = 3; 84 cookies 28

FOCUS ON HIGHER ORDER THINKING

4s = 132; s = 33 ft;

15. Communicate Mathematical Ideas Explain why 7 · _7x = x. How does this relate to solving division equations?

33 × 33 = 1,089 square feet

7. Sam divided a rectangle into 8 congruent rectangles that each have the area shown. What is the area of the rectangle before Sam divided it?

7x 7 · _7x = __ = 1x = x; 7 divided by 7 is 1. 1x is the same as x. 7

Yes, the area of the garden is greater than 1,000 square feet.

When you solve division equations, you multiply both

Area = 5 cm2

sides of the equation by the number that will give you 1x, or x.

12. Ingrid walked her dog and washed her car. The time she spent walking her dog was one-fourth the time it took her to wash her car. It took Ingrid 14 minutes to walk the dog. How long did it take Ingrid to wash her car?

8. Carmen participated in a read-a-thon. Mr. Cole pledged \$4.00 per book and gave Carmen \$44. How many books did Carmen read?

4k = 44; 11 books

16. Critical Thinking A number tripled and tripled again is 729. What is the number?

w __ = 14; 56 minutes 4

3(3x) = 729;

13. Representing Real-World Problems Write and solve a problem involving money that can be solved with a multiplication equation.

9. Lee drove 420 miles and used 15 gallons of gasoline. How many miles did Lee’s car travel per gallon of gasoline?

15m = 420; 28 mi/gal

10. On some days, Melvin commutes 3.5 hours per day to the city for business meetings. Last week he commuted for a total of 14 hours. How many days did he commute to the city?

9x = 729;

x = 81;

the number is 81.

17. Multistep Andre has 4 times as many model cars as Peter, and Peter has one-third as many model cars as Jade. Andre has 36 model cars. a. Write and solve an equation to find how many model cars Peter has.

4p = 36; p = 9; Peter has 9 model cars.

\$168 for babysitting over 6 weeks. If she earned the same

b. Using your answer from part a, write and solve an equation to find how many model cars Jade has.

amount each week, how much

1 _ j = 9; j = 27; Jade has 27 model cars. 3

did she earn for one week? 6x = 168; x = \$28.

© Houghton Mifflin Harcourt Publishing Company

a __ = 5; 40 square centimeters 8

© Houghton Mifflin Harcourt Publishing Company

Work Area

3.5d = 14; 4 days

Lesson 12.3

6_MTXESE051676_U4M12L3.indd 341

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26/10/12 12:03 AM

EXTEND THE MATH

PRE-AP

342

Unit 4

6_MTXESE051676_U4M12L3.indd 342

Activity available online

28/01/14 9:52 PM

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Activity Here is one way you can name 14, using four 7s and common arithmetic operations: 7×7 ____ +7 7

Use four 7s and the signs for common operations to name each given number. 1. 2 = __________ 2. 3 = __________ 3. 4 = __________ 4. 5 = __________

7+7+7

7+7

77 Sample solutions: 1. _77 + _77 2. _______ 3. __ - 7 4. 7 - _____ 7 7 7

Multiplication and Division Equations

342

MODULE QUIZ

Assess Mastery

12.1 Writing Equations to Represent Situations

Use the assessment on this page to determine if students have mastered the concepts and standards covered in this module.

Determine whether the given value is a solution of the equation.

3 1

no

1. p - 6 = 19; p = 13

yes

b 3. __ = 5; b = 60 12

Response to Intervention

2

Personal Math Trainer

5. 18 - h = 13; h = -5

no

Online Assessment and Intervention

my.hrw.com

2. 62 + j = 74; j = 12

yes

4. 7w = 87; w = 12

no

6. 6g = -86; g = -16

no

Write an equation to represent the situation. 7. The number of eggs in the refrigerator e decreased by 5 equals 18.

Intervention

e - 5 = 18

Enrichment

Access Ready to Go On? assessment online, and receive instant scoring, feedback, and customized intervention or enrichment.

Personal Math Trainer

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p + 17 = 29

12.2 Addition and Subtraction Equations Solve each equation.

Online and Print Resources

9. r - 38 = 9

Differentiated Instruction

Differentiated Instruction

• Reteach worksheets

• Challenge worksheets

• Reading Strategies • Success for English Learners ELL

ELL

11. n + 75 = 155

PRE-AP

Extend the Math PRE-AP Lesson Activities in TE

Additional Resources Assessment Resources includes: • Leveled Module Quizzes

r = 47 n = 80

10. h + 17 = 40

h = 23

12. q - 17 = 18

q = 35

12.3 Multiplication and Division Equations Solve each equation. © Houghton Mifflin Harcourt Publishing Company

Online Assessment and Intervention

8. The number of new photos p added to the 17 old photos equals 29.

13. 8z = 112 f 15. __ = 24 28

z = 14 f = 672

d 14. __ =7 14

16. 3a = 57

d = 98 a = 19

ESSENTIAL QUESTION 17. How can you solve problems involving equations that contain addition, subtraction, multiplication, or division?

Write an equation for the situation. Then apply the inverse operation to get the variable alone on one side of the equation.

Module 12

Texas Essential Knowledge and Skills Lesson

Exercises

12.1

1–8

6.7.B, 6.9.A, 6.10.B

12.2

9–12

6.9.B, 6.9.C, 6.10.A, 6.10.B

12.3

13–16

6.9.B, 6.9.C, 6.10.A, 6.10.B

343

Module 12

TEKS

343

Personal Math Trainer

MODULE 12 MIXED REVIEW

Texas Test Prep

Texas Test Prep

Item 8 Students can write the equation 17x = 680 to represent the situation and then start substituting answers in the equation to find which answer makes the sentence true. Choices A and B result in false equations, but choice C results in 680 = 680. Thus, choice C is the correct answer.

Selected Response 1. Kate has gone up to the chalkboard to do math problems 5 more times than Andre. Kate has gone up 11 times. Which equation represents this situation? A a - 11 = 5 B 5a = 11 C

Item 1 Some students will see the word times, automatically think of multiplication, and quickly pick choice B. Encourage them to read carefully to identify all key words or phrases such as more times, which indicates addition, not multiplication. Item 4 Some students may select choice A because they looked only at the endpoints under the arrow and didn’t pay attention to the direction in which the arrow was pointing. Remind them that they need to analyze art or diagrams carefully to fully understand them.

a - 5 = 11

2. For which equation is y = 7 a solution?

A 6x = 42

6 C _ x = 42

B 42 - x = 6

D 6 + x = 42

A t = 19

C

B t = 108

D t = 684

t = 120

8. The area of a rectangular deck is 680 square feet. The deck’s width is 17 feet. What is its length?

A 7y = 1 B y - 26 = -19 C

6. Jeordie spreads out a rectangular picnic blanket with an area of 42 square feet. Its width is 6 feet. Which equation could you use to find its length?

7. What is a solution to the equation 6t = 114?

D a + 5 = 11

Avoid Common Errors

Online Assessment and Intervention

y+7=0

y D _ = 14 2

A 17 feet

C

B 20 feet

D 51 feet

40 feet

3. Which is an equation? A 17 + x

C

20x = 200

B 45 ÷ x

D 90 - x

4. The number line below represents which equation?

9. Sylvia earns \$7 per hour at her afterschool job. One week she worked several hours and received a paycheck for \$91. Write and solve an equation to find the number of hours in which Sylvia would earn \$91.

.

1

3

0

0

0

0

0

0

1

1

1

1

1

1

2

2

2

2

2

2

3

3

3

3

3

3

3 + 7 = -4

4

4

4

4

4

4

D 3 - 7 = -4

5

5

5

5

5

5

6

6

6

6

6

6

7

7

7

7

7

7

8

8

8

8

8

8

9

9

9

9

9

9

-5 - 4 - 3 - 2 - 1

0 1 2 3 4 5

A -4 + 7 = 3 B -4 - 7 = 3 C

5. Becca hit 7 more home runs than Beverly. Becca hit 21 home runs. How many home runs did Beverly hit?

344

Gridded Response

A 3

C

B 14

D 28

© Houghton Mifflin Harcourt Publishing Company

Texas Testing Tip Instead of solving equations directly, students can work backward by substituting answers into given equations to see if they are correct. Item 7 Students can try substituting the answers in the equation in order. When they try choice A, they should find that the result is a true equation, indicating that this is the correct solution.

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21

Unit 4

Texas Essential Knowledge and Skills Items

Mathematical Process TEKS

1

6.9.A

6.1.A

2

6.10.B

6.1.C

3

6.7.B

6.1.F

4*

6.3.D, 6.9.A, 6.9.B

6.1.D

5

6.10.A

6.1.A, 6.1.F

6

6.10.A

6.1.A, 6.1.F

7

6.10.B

6.1.C

8

6.10.A

6.1.A, 6.1.F

9

6.10.A, 6.10.B

6.1.A, 6.1.F

* Item integrates mixed review concepts from previous modules or a previous course.

Equations and Relationships

344

Inequalities and Relationships ?

ESSENTIAL QUESTION How can you use inequalities and relationships to solve real-world problems?

MODULE

You can model real-world problems with inequalities, then use algebraic rules to solve the inequalities.

13

LESSON 13.1

Writing Inequalities 6.9.A, 6.9.B, 6.10.B

LESSON 13.2

Addition and Subtraction Inequalities 6.9.B, 6.9.C, 6.10

LESSON 13.3

Multiplication and Division Inequalities with Positive Numbers 6.9.B, 6.9.C, 6.10

LESSON 13.4

Multiplication and Division Inequalities with Rational Numbers

© Houghton Mifflin Harcourt Publishing Company

6.9.B, 6.10.A, 6.10.B

Real-World Video

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345

Module 13

Some rides at amusement parks indicate a minimum height required for riders. You can model all the heights that are allowed to get on the ride with an inequality.

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Math On the Spot

Animated Math

Personal Math Trainer

Go digital with your write-in student edition, accessible on any device.

Scan with your smart phone to jump directly to the online edition, video tutor, and more.

Interactively explore key concepts to see how math works.

Get immediate feedback and help as you work through practice sets.

345

Are You Ready? Use the assessment on this page to determine if students need intensive or strategic intervention for the module’s prerequisite skills. 2 1

Understand Integers EXAMPLE

Response to Intervention

-735 Write an integer to represent each situation.

Intervention

–75

Enrichment

Access Are You Ready? assessment online, and receive instant scoring, feedback, and customized intervention or enrichment.

Online Assessment and Intervention

my.hrw.com

Skills Intervention worksheets

Differentiated Instruction

• Skill 33 Understand Integers

• Challenge worksheets

• Skill 59 Solve Multiplication Equations

2. a football player’s 3. spending \$1,200 4. a climb of 2,400 gain of 9 yards on a flat screen feet 2,400 9 TV -1,200

Integer Operations EXAMPLE

Online and Print Resources

• Skill 47 Integer Operations

Online Assessment and Intervention

Decide whether the integer is positive or negative: into the ground → negative Write the integer.

A water well was drilled 735 feet into the ground.

1. a loss of \$75

Personal Math Trainer

my.hrw.com

3 × 8 = 24 -30 ÷ (-5) = 6

The product or quotient of two integers is positive if the signs of the integers are the same.

7 × (-4) = -28 -72 ÷ 9 = -8

The product or quotient of two integers is negative if the signs of the integers are different.

Find the product or quotient.

PRE-AP PRE-AP

Extend the Math Lesson Activities in TE

5. 6 × 9

54

9. 3 × (-7)

-21 10.

6. 15 ÷ (-5) -64 ÷ 8

-3 -8

7. -8 × 6

-48

11. -8 × (-2)

16

8. -100 ÷ (10) -10 12. 32 ÷ 2

16 © Houghton Mifflin Harcourt Publishing Company

3

Personal Math Trainer

Complete these exercises to review skills you will need for this chapter.

Solve Multiplication Equations 3 _ h = 15 4

EXAMPLE

4 _ _ · 3 h = 15 · _43 3 4 ·4 _____ h = 15 3

Write the equation. 3 . Multiply both sides by the h is multiplied by __ 4 4 , to isolate reciprocal, __ the variable. 3

h = 20

Simplify.

14. _35 n = 21

35

Solve. 13. 9p = 108

346

12

15. _47 k = 84

147

3 16. __   e = 24 20

160

Unit 4

PROFESSIONAL DEVELOPMENT VIDEO my.hrw.com

Author Juli Dixon models successful teaching practices as she explores inequality concepts in an actual sixth-grade classroom.

Online Teacher Edition Access a full suite of teaching resources online—plan, present, and manage classes and assignments.

Professional Development

my.hrw.com

Interactive Answers and Solutions Customize answer keys to print or display in the classroom. Choose to include answers only or full solutions to all lesson exercises.

Interactive Whiteboards Engage students with interactive whiteboard-ready lessons and activities. Personal Math Trainer: Online Assessment and Intervention Assign automatically graded homework, quizzes, tests, and intervention activities. Prepare your students with updated, TEKS-aligned practice tests.

Inequalities and Relationships

346

DO NOT EDIT--Changes must be made through "File info" CorrectionKey=B

Visualize Vocabulary Use the ✔ words to complete the graphic.

Visualize Vocabulary The graphic organizer helps students review vocabulary associated with evaluating expressions. If time allows, brainstorm other vocabulary that can be added to the chart.

>, <

4x + 4 = 12; x = 2

greater than, less than

solution

3x - 5

Understand Vocabulary

algebraic expression

Use the following explanation to help students learn the preview words. Inequalities are similar to equations in that they represent a statement relating two expressions with a symbol. Equations use an equal sign, while inequalities use an inequality symbol.

Students can use these reading and note-taking strategies to help them organize and understand new concepts and vocabulary. c.4.D Use prereading supports such as graphic organizers, illustrations, and

pretaught topic-related vocabulary to enhance comprehension of written text.

✔ algebraic expression (expresión algebraica) evaluating (evaluar) ✔ greater than (mayor que) ✔ less than (menor que) like terms (términos semejantes) ✔ numerical expression (expresión numérica) properties of operations (propiedades de las operaciones) ✔ solution (solución) term (término, en una expresión)

Preview Words Match the term on the left to the correct expression on the right.

© Houghton Mifflin Harcourt Publishing Company

Integrating the ELPS

6×4

numerical expression

Review Words

Understand Vocabulary

The solution of an inequality is a value or values that make the inequality true.

Evaluating Expressions

Vocabulary

1. solution of an inequality

A. A value or values that make the inequality true.

2. coefficient

B. A specific number whose value does not change.

3. constant

C. The number that is multiplied by the variable in an algebraic expression.

coefficient (coeficiente) constant (constante) solution of an inequality (solución de una desigualdad) variable (variable)

Two-Panel Flip Chart Create a two-panel flip chart to help you understand the concepts in this module. Label one flap “Adding and Subtracting Inequalities.” Label the other flap “Multiplying and Dividing Inequalities.” As you study each lesson, write important ideas under the appropriate flap.

Module 13

6_MTXESE051676_U4MO13.indd 347

Grades 6–8 TEKS Before Students understand: • operations with rational numbers • properties of operations: inverse, identity, commutative, associative, and distributive properties

347

Module 13

28/01/14 10:02 PM

InCopy Notes

InDesign Notes

1. This is a list

1. This is a list

In this module Students will learn how to: • write one-variable, one-step inequalities to represent constraints or conditions within problems • model and solve one-variable, one-step inequalities that represent problems • write corresponding real-world problems given one-variable, one-step inequalities

347

After Students will learn how to: • write one-variable, two-step inequalities to represent real-world problems • write a real-world problem to represent a one-variable, two-step inequality • solve one-variable, two-step inequalities

MODULE 13

Unpacking the TEKS

Unpacking the TEKS Understanding the TEKS and the vocabulary terms in the TEKS will help you know exactly what you are expected to learn in this module.

Use the examples on this page to help students know exactly what they are expected to learn in this module.

6.9.B Represent solutions for onevariable, one-step equations and inequalities on number lines.

Texas Essential Knowledge and Skills Content Focal Areas Expressions, equations, and relationships—6.9 The student applies mathematical process standards to use equations and inequalities to represent situations. Expressions, equations, and relationships—6.10 The student applies mathematical process standards to use equations and inequalities to solve problems.

What It Means to You You will learn to graph the solution of an inequality on a number line.

Key Vocabulary

UNPACKING EXAMPLE 6.9.B

equation (ecuación) A mathematical sentence that shows that two expressions are equivalent.

The temperature in a walk-in freezer must stay under 5 °C. Write and graph an inequality to represent this situation.

inequality (desigualdad) A mathematical sentence that shows the relationship between quantities that are not equal. solution of an inequality (solución de una desigualdad) A value or values that make the inequality true.

Write the inequality. Let t represent the temperature in the freezer. The temperature must be less than 5 °C. t<5 Graph the inequality. 0

5

10

6.10.A

c.4.F Use visual and contextual support … to read grade-appropriate content area text … and develop vocabulary … to comprehend increasingly challenging language.

Model and solve one-variable, one-step equations and inequalities that represent problems, including geometric concepts.

You can model and solve a one-variable, one-step inequality. UNPACKING EXAMPLE 6.10.A

Donny buys 3 binders and spends more than \$9. How much did he spend on each binder? Let x represent the cost of one binder.

Go online to see a complete unpacking of the .

Number of binders · Cost of a binder > Total cost of binders

x

>

Use algebra tiles to model 3x > 9 and solve the inequality. x>3 Donny spent more than \$3 on each binder.

+ + +

3 Visit my.hrw.com to see all the unpacked.

my.hrw.com my.hrw.com

348

What It Means to You

Lesson 13.1

Lesson 13.2

·

9

>

© Houghton Mifflin Harcourt Publishing Company • Image Credits: Image Source/Corbis

Integrating the ELPS

+ + + + + + + + +

Unit 4

Lesson 13.3

Lesson 13.4

6.9.A Write one-variable, one-step equations and inequalities to represent constraints or conditions within problems. 6.9.B Represent solutions for one-variable, one-step equations and inequalities on number lines. 6.9.C Write corresponding real-world problems given one-variable, one-step equations or inequalities. 6.10.A Model and solve one-variable, one-step equations and inequalities that represent problems, including geometric concepts. 6.10.B Determine if the given value(s) make(s) one-variable, one-step equations or inequalities true.

Inequalities and Relationships

348

LESSON

13.1 Writing Inequalities Texas Essential Knowledge and Skills The student is expected to: Expressions, equations, and relationships—6.9.A Write one-variable, one-step equations and inequalities to represent constraints or conditions within problems. Expressions, equations, and relationships—6.9.B

Engage

ESSENTIAL QUESTION How can you use inequalities to represent real-world constraints or conditions? Sample answer: You can choose a letter to represent the variable value in the situation and then use one of the inequality symbols to describe its range of values.

Motivate the Lesson Ask: If you know the lowest and highest temperatures recorded for yesterday, how could you describe yesterday’s temperature at any given time of day? For example, what could you say about the temperature at noon? Begin the Explore Activity to find out.

Represent solutions for one-variable, one-step equations and inequalities on number lines. Expressions, equations, and relationships—6.10.B Determine if the given value(s) make(s) one-variable, one-step equations or inequalities true.

Mathematical Processes 6.1.E Create and use representations to organize, record, and communicate mathematical ideas.

Explore EXPLORE ACTIVITY Engage with the Whiteboard

Have students graph -2 °F on the number line on the whiteboard and then graph -1 °F, 0 °F, 3 °F, 5 °F, and 6 °F in a different color on the same number line. Then have them write inequalities comparing each of the temperatures from B to -2 °F on the whiteboard. Students will see that all five temperatures are greater than -2 °F. Ask students to compare some of the numbers to the left of -2 as well, so that they use both the > and < symbols.

Explain ADDITIONAL EXAMPLE 1 Graph the solutions of each inequality. Check the solutions. A b ≥ -4 -5

0

5

5

Sample check: -3 > -5 Interactive Whiteboard Interactive example available online my.hrw.com

349

Lesson 13.1

Mathematical Processes Remind students that when a variable is less than a given number, all the values to the left of the given number on the number line make that inequality true. They are all solutions of the inequality. For example, if x < 4, then every number less than 4 is a solution. Mathematical Processes • What is the difference between the graph of y ≤ 4 and the graph of y < 4? In the first graph, 4 is included in the solution set. In the second graph, 4 is not included in the solution set.

B -3 > s 0

Focus on Communication

Questioning Strategies

Sample check: -1 ≥ -4

-5

EXAMPLE 1

• What is the difference between using a solid circle and an open circle? A solid circle includes the number at that point; an open circle does not. • How do you know when to use a solid circle or an open circle? A solid circle is used to represent ≤ or ≥ on a graph; an open circle is used to represent < or > on a graph. • How can you check that the graph of an inequality is correct? Pick a point on the shaded portion of the graph. Any point selected should make the inequality a true inequality.

LESSON

Expressions, equations, and relationships— 6.9.A Write … inequalities to represent constraints or conditions within problems. Also 6.9.B, 6.10.B.

13.1 Writing Inequalities ESSENTIAL QUESTION

Math On the Spot my.hrw.com

How can you use inequalities to represent real-world constraints or conditions?

EXAMPLE 1

Yes; yes; both numbers make the inequality true.

6.9.A

EXPLORE ACTIVITY

Using Inequalities to Describe Quantities You can use inequality symbols with variables to describe quantities that can have many values. Symbol

Meaning

Word Phrases

<

Is less than

Fewer than, below

>

Is greater than

More than, above

Is less than or equal to

At most, no more than

Is greater than or equal to

At least, no less than

A y ≤ -3

© Houghton Mifflin Harcourt Publishing Company

STEP 1

Draw a solid circle at -3 to show that -3 is a solution.

STEP 2

Shade the number line to the left of -3 to show that numbers less than -3 are solutions.

Mathematical Processes

Is -4 _14 a solution of y ≤ -3? Is -5.6?

-5 -4 -3 -2 -1

STEP 3

Draw an empty circle at 1 to show that 1 is not a solution.

STEP 2

Shade the number line to the right of 1 to show that numbers greater than 1 are solutions. -5 -4 -3 -2 -1

are located to the right of -2.

STEP 3

D How many other numbers have the same relationship to -2 as the temperatures in B ? Give some examples.

D

Use an open circle for an inequality that uses > or <.

0 1 2 3 4 5

Check your answer. Substitute 2 for m. 1<2 1 is less than 2, so 2 is a solution.

Reflect

infinitely many; any number greater than -2; sample answer: 1, 2, 8.5, 10

1.

on a number

How is x < 5 different from x ≤ 5?

For x < 5, 5 is not a solution and is not included in the graph. For x ≤ 5, 5 is a solution and is included

a ray extending to the right with its open endpoint at -2 >

-4 is less than -3, so -4 is a solution.

STEP 1

They are all greater than -2. All of the temperatures

D

-4 ≤ -3

C How do the temperatures in B compare to -2? How can you see this relationship on the number line?

Complete this inequality: x

0 1 2 3 4 5

B 1
0 1 2 3 4 5 6 7 8

F Let x represent all the possible answers to

Use a solid circle for an inequality that uses ≥ or ≤.

Choose a number that is on the shaded section of the number line, such as -4. Substitute -4 for y.

B The temperatures 0 °F, 3 °F, 6 °F, 5 °F, and -1 °F have also been recorded in Florida. Graph these temperatures on the number line.

E Suppose you could graph all of the possible answers to line. What would the graph look like?

6.9.B

Graph the solutions of each inequality. Check the solutions.

Math Talk

A The lowest temperature ever recorded in Florida was -2 °F. Graph this temperature on the number line. -8 -7 -6 -5 -4 -3 -2 -1

A solution of an inequality that contains a variable is any value of the variable that makes the inequality true. For example, 7 is a solution of  x > -2, since 7 > -2 is a true statement.

© Houghton Mifflin Harcourt Publishing Company

?

Graphing the Solutions of an Inequality

in the graph.

.

-2 Lesson 13.1

349

350

Unit 4

PROFESSIONAL DEVELOPMENT Integrate Mathematical Processes This lesson provides an opportunity to address Mathematical Process TEKS 6.1.E, which calls for students to “create and use representations to… communicate mathematical ideas.” In the Explore Activity and the Examples, students use number lines, word expressions, and mathematical symbols to express inequalities. They use graphs to represent inequalities and to determine if a given number is a solution.

Math Background Inequalities have a number of properties, including the Transitive Property of Inequality. This property states that for any real numbers a, b, c, if a > b and b > c, then a > c. if a < b and b < c, then a < c. if a > b and b = c, then a > c. if a < b and b = c, then a < c. This may seem like common sense, but the Transitive Property is not necessarily true in daily life. If Team A defeats Team B, and Team B defeats Team C, you can’t assume that Team A will defeat Team C. Writing Inequalities

350

ADDITIONAL EXAMPLE 2 A Write an inequality that represents the phrase “y minus 3 is less than or equal to -5.” Then graph the inequality. y - 3 ≤ -5 -5

0

5

B The temperature of the river is greater than -1 °C. Write and graph an inequality to represent this situation. t > -1 -5

0

5

Interactive Whiteboard Interactive example available online my.hrw.com

YOUR TURN Avoid Common Errors If students have trouble determining which side of the number line to shade, remind them that the inequality sign always points to the lesser of two numbers. Since t ≤ -4, they should shade the number line to the left of -4, because the numbers decrease to the left on a number line.

EXAMPLE 2 Focus on Reasoning

Mathematical Processes Point out to students that sometimes a problem may provide clues and facts that you must use to find a solution. Encourage them to use logical reasoning to solve this kind of problem. For example, in A, tell students that the first step is to identify key words or phrases that indicate operations or relationships. Then they can proceed to write an equation.

Questioning Strategies

Mathematical Processes • In A, how do you know which operation to use to write the inequality? The word sum indicates addition. • In A, how do you know which inequality symbol to use? The phrase greater than indicates the symbol >. • In B, how do you know which inequality symbol to use? The phrase keeps the temperature below 5 °C tells me that the temperature is less than 5, which indicates that the symbol < should be used.

Animated Math Modeling Inequalities Students model inequalities using interactive algebra tiles.

Exercise 3 Some students may read the problem quickly and use > instead of ≥. Encourage them to begin by underlining the key words or phrases before trying to graph the inequality.

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Elaborate Talk About It Summarize the Lesson Ask: How can you make sure you have graphed an inequality correctly? Test one of the values on the number line. If it makes the inequality true, the number line is correct.

GUIDED PRACTICE Engage with Whiteboard For Exercise 3, have students begin by underlining the key words or phrases on the whiteboard and then write an equation, using the same method shown in Example 2. Finally, ask students to graph the inequality on the number line.

Avoid Common Errors Exercises 1 and 2 Some students may shade in the wrong direction when they attempt to graph the solution set of an inequality, such as 1 ≤ x. Reading 1 ≤ x as “x is greater than or equal to 1” serves as a reminder to shade to the right. Exercise 4 Some students may have difficulty determining which inequality symbol to use. Encourage them to begin by underlining key words or phrases before graphing the inequality.

351

Lesson 13.1

Graph the solution of the inequality t ≤ -4. - 10 - 9 - 8 - 7 - 6 - 5 - 4 - 3 - 2 - 1

0 1 2 3 4 5 6 7 8 9 10

Personal Math Trainer

Personal Math Trainer

Online Assessment and Intervention

3.

1 + y ≥ 3; y = 1 is not a solution because 1 + 1 is not

Online Assessment and Intervention

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greater than or equal to 3.

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Writing Inequalities

Write an inequality that represents the phrase the sum of 1 and y is greater than or equal to 3 . Check to see if y = 1 is a solution.

Write and graph an inequality to represent each situation.

You can write an inequality to model the relationship between an algebraic expression and a number. You can also write inequalities to represent certain real-world situations.

The highest temperature in February was 6 °F.

5.

Each package must weigh more than 2 ounces.

Math On the Spot

6.9.A, 6.10.B

A Write an inequality that represents the phrase the sum of y and 2 is greater than 5. Draw a graph to represent the inequality. STEP 1

y+2

>

Animated Math

© Houghton Mifflin Harcourt Publishing Company

Guided Practice 1. Graph 1 ≤ x. Use the graph to determine which of these numbers are solutions of the inequality: -1, 3, 0, 1 (Explore Activity and Example 1)

Use an open circle at 3 and shade to the right of 3.

3, 1

0 1 2 3 4 5

Check your solution by substituting a number greater than 3, such as 4, into the original inequality. 4+2>5

0 1 2 3 4 5

-5 -4 -3 -2 -1

0 1 2 3 4 5

3. Write an inequality that represents the phrase “the sum of 4 and x is less than 6.” Draw a graph that represents the inequality, and check your solution. (Example 2)

-5 -4 -3 -2 -1

0 1 2 3 4 5

-5 -4 -3 -2 -1

0 1 2 3 4 5

4+x<6 4. During hibernation, a garter snake’s body temperature never goes below 3 °C. Write and graph an inequality that represents this situation. (Example 2)

6 is greater than 5, so 4 is a solution.

B To test the temperature rating of a coat, a scientist keeps the temperature below 5 °C. Write and graph an inequality to represent this situation.

Let t be temperature in °C; t ≥ 3

Write the inequality. Let t represent the temperature in the lab. t<5

STEP 2

-5 -4 -3 -2 -1

2. Graph -3 > z. Check the graph using substitution. (Example 1)

Substitute 4 for y.

6>5

STEP 1

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

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5

Graph the solution. For y + 2 to have a value greater than 5, y must be a number greater than 3.

-5 -4 -3 -2 -1

STEP 3

-2 -1

w>2

Write the inequality. The sum of y and 2 is greater than 5.

STEP 2

t≤6

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

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EXAMPL 2 EXAMPLE

4.

? ?

The temperature must be less than 5 °C.

Graph the inequality.

ESSENTIAL QUESTION CHECK-IN

© Houghton Mifflin Harcourt Publishing Company

2.

5. Write an inequality to represent this situation: Nina wants to take at least \$15 to the movies. How did you decide which inequality symbol to use?

d ≥ 15, where d represents dollars. “At least” means

0 1 2 3 4 5 6 7 8 9 10

she wants to take \$15 or more than \$15. Lesson 13.1

351

352

Unit 4

DIFFERENTIATE INSTRUCTION Critical Thinking

Cooperative Learning

Have students work together to consider absolute-value inequalities, such as | x | < 2.

Have students work in groups of 4. Have each group make a set of inequality symbol cards, a variable card, and 6 number cards (3 negative numbers and 3 positive numbers). Have the students take turns using the cards to create an inequality, such as this one:

Differentiated Instruction includes: • Reading Strategies • Success for English Learners ELL • Reteach • Challenge PRE-AP

First have students find numbers that make the inequality true. Then have them use the numbers to sketch a graph of what they think the solution should be. -5

-2

0

2

5

x

-4.5

Then have the groups record the inequalities and graph them on a number line.

Writing Inequalities

352

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Online homework assignment available

Evaluate GUIDED AND INDEPENDENT PRACTICE 6.9.A, 6.9.B, 6.10.B

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13.1 LESSON QUIZ 6.9.A Graph each inequality. 1. a ≤ -2 2. n < 4

Concepts & Skills

Practice

Explore Activity Using Inequalities to Describe Quantities

Exercise 1

Example 1 Graphing the Solutions of an Inequality

Exercises 1–2, 6–11, 16–19

Example 2 Writing Inequalities

Exercises 3–4, 12–15, 16–19

3. h > -1.5 4. t ≤ 3 Write an inequality that matches the number line model. Use x for the variable. 5. 6. 7. 8.

-10

-5

0

-5

0

5

-5

0

5

-5

0

5

Lesson Quiz available online my.hrw.com

2. 3. 4.

-5

0

5

-5

0

5

-5

0

5

-5

0

5

5. x ≤ -5 6. x > -1 7. x ≥ -0.5 8. x < 2 9. x ≥ 2.5

353

Depth of Knowledge (D.O.K.)

Lesson 13.1

Mathematical Processes

6

2 Skills/Concepts

1.F Analyze relationships

7–15

2 Skills/Concepts

1.D Multiple representations

16

2 Skills/Concepts

1.F Analyze relationships

17–19

2 Skills/Concepts

1.A Everyday life

20

3 Strategic Thinking

1.G Explain and justify arguments

21–22

3 Strategic Thinking

1.F Analyze relationships

9. The weight of a package is at least 2.5 pounds. Write an inequality to represent this situation.

1.

Exercise

Differentiated Instruction includes: • Leveled Practice Worksheets

DO NOT EDIT--Changes must be made through “File info” CorrectionKey=A

DO NOT EDIT--Changes must be made through “File info” CorrectionKey=B

Class

Name

Date

Write and graph an inequality to represent each situation.

13.1 Independent Practice

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6.9.A, 6.9.B, 6.10.B

10

Online Assessment and Intervention

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0.03, 0, 1.5, _12

-5

9. x ≥ -9 10. n > 2.5 11. -4 _12 >x

© Houghton Mifflin Harcourt Publishing Company

x>6

13.

x ≤ -3

14.

x < 1.5

15.

x ≥ -3.5

14

15

16

17

18

0

1

2

3

t < 3.5

-3

-2

-1

19

20

4

5

g > 150

50 100 150 200 250 300

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

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

0

1

2

3

4

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

0

1

2

20. Communicate Mathematical Ideas Explain how to graph the inequality 8 ≥ y.

Sample answer: Make a solid circle at 8 because of the

-5

-4

-3

-2

-1

0

1

2

2.5 3

4

5

-8

-7

-6

-5

-4

-3

-2

-1

0

1

2

-4 12

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

inequality symbol, greater than or equal to. Then shade in the numbers to the left of 8, which are the numbers that make y less than 8.

0 1 2 3 4 5 6 7 8 9 10

21. Represent Real-World Problems The number line shows an inequality. Describe a real-world situation that the inequality could represent.

0 1 2 3 4 5 6 7 8 9 10

Sample answer: Steve has more than \$2.75 in his wallet.

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

-5

-4

-3

-2

-1

0

-5

-4

-3

-2

-1

0

1

2

3

4

Lesson 13.1

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D x is less than or equal to -2 or x is greater than or equal to 3; -2 ≥ x or x ≥ 3

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

0 1 2 3 4 5 6 7 8 9 10

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

0 1 2 3 4 5 6 7 8 9 10

C

D

Unit 4

6_MTXESE051676_U4M13L1.indd 354

Activity available online

Activity The four graphs at right show constraints on A both ends of the graph. Challenge students to describe each graph in words and with an inequality statement. Tell them that graphs C and D describe the solutions for a single variable and that they should use the word “or” B to describe these situations.

C x is less than -2 or x is greater than 3; -2 > x or x>3

5

or fractional parts of students.

No; 46 is not greater than or equal to 48.

B x is greater than -2 and is less than or equal to 3; -2 < x ≤ 3

4

not make sense to have negative numbers of students

b. Can a child who is 46 inches tall ride the roller coaster? Explain.

A x is greater than -2 and less than 3: -2 < x < 3

3

3, and 4 as solutions is correct. In this example, it does

c ≥ 48

PRE-AP

2

Sample answer: The number line that shows only 0, 1, 2,

38 40 42 44 46 48 50 52 54 56 58

EXTEND THE MATH

1

22. Critique Reasoning Natasha is trying to represent the following situation with a number line model: There are fewer than 5 students in the cafeteria. She has come up with two possible representations, shown below. Which is the better representation, and why?

5

16. A child must be at least 48 inches tall to ride a roller coaster. a. Write and graph an inequality to represent this situation.

Work Area

FOCUS ON HIGHER ORDER THINKING

Write an inequality that matches the number line model. 12.

13

© Houghton Mifflin Harcourt Publishing Company

8. -7 < h

-4

0 -2 -1

12

19. The goal of the fundraiser is to make more than \$150.

Graph each inequality. 7. t ≤ 8

11

18. The temperature is less than 3.5 °F.

6. Which of the following numbers are solutions to x ≥ 0? -5, 0.03, -1, 0, 1.5, -6, _12

s ≥ 14.5

17. The stock is worth at least \$14.50.

28/01/14 10:12 PM

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-5

0

5

-5

0

5

-5

0

5

-5

0

5

Writing Inequalities

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LESSON

13.2 Addition and Subtraction Inequalities Texas Essential Knowledge and Skills The student is expected to: Expressions, equations, and relationships—6.10.A Model and solve one-variable, one-step inequalities that represent problems, including geometric concepts. Expressions, equations, and relationships—6.9.B Represent solutions for one-variable, one-step equations and inequalities on number lines. Expressions, equations, and relationships—6.9.C Write corresponding real-world problems given one-variable, one-step equations or inequalities. Expressions, equations, and relationships—6.10.B Determine if the given value(s) make(s) one-variable, one-step equations or inequalities true.

Engage

ESSENTIAL QUESTION How can you solve an inequality involving addition or subtraction? Sample answer: You can use the Addition and Subtraction Properties of Inequality: add or subtract the same amount from both sides to isolate the variable.

Motivate the Lesson Ask: You’ve used algebra tiles to model an equation. Do you think you can use algebra tiles to model an inequality? Take a guess. Begin the Explore Activity to find out how.

Explore EXPLORE ACTIVITY Engage with the Whiteboard

Have a student use algebra tiles to make a model of the equation x + 5 = 8 on the whiteboard. Then have another student make a model of the inequality x + 5 ≥ 8 on the whiteboard. Then have all students graph the results on number lines. Discuss with the students how the two models are similar and how they are different. Emphasize the differences.

Mathematical Processes 6.1.D Communicate mathematical ideas, reasoning, and their implications using multiple representations, including symbols, diagrams, graphs, and language as appropriate.

Explain EXAMPLE 1 Focus on Math Connection

ADDITIONAL EXAMPLE 1 Solve each inequality. Graph and check the solution. A x + 3 ≥ -2 -5

x ≥ -5 0

B -6 < y -5

5

Questioning Strategies

Mathematical Processes • In A, Step 1, why do you subtract 5 from both sides? The inequality has + 5 in it. To isolate the variable, you need to add the inverse, which is the same as subtracting 5.

y > -1

-5

0

Mathematical Processes Solving inequalities is very similar to solving equations. Remind students that they used the properties of equality and inverse operations to solve equations in an earlier module. Similarly, you can use the Addition and Subtraction Properties of Inequality and inverse operations to solve inequalities. Show how solving x + 5 = -12 is similar to solving x + 5 < -12.

5

Interactive Whiteboard Interactive example available online my.hrw.com

• In B, Step 2, why do you shade to the right when the inequality is ≤? Because the variable is on the right-hand side of the inequality, you need to read the inequality starting with the variable so that you will know how to shade the graph correctly. If you read the inequality 11 ≤ y as “y is greater than or equal to 11,” you will see that you need to shade to the right. • How can you check that the graph of an inequality is correct? Pick a point on the shaded portion of the graph. Any point selected should make the inequality a true inequality.

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Lesson 13.2

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LESSON

13.2 ?

DO NOT EDIT--Changes must be made through “File info” CorrectionKey=B

ESSENTIAL QUESTION

Expressions, equations, and relationships—6.10.A Model and solve one-variable, one-step… inequalities that represent problems. Also 6.9.B, 6.9.C, 6.10.B.

Math On the Spot my.hrw.com

How can you solve an inequality involving addition or subtraction?

EXPLORE ACTIVITY

Using Properties of Inequalities

6.10.A

EXAMPLE 1

Modeling One-Step Inequalities

A x + 5 < -12

On a day in January in Watertown, NY, the temperature was 5 °F at dawn. By noon it was at least 8 °F. By how many degrees did the temperature increase?

STEP 1

Increase in temperature

Solve the inequality. x + 5 < -12

A Let x represent the increase in temperature. Write an inequality. +

6.9.B, 6.10.B

Solve each inequality. Graph and check the solution.

You can use algebra tiles to model an inequality involving addition.

Temperature at dawn

Subtraction Property of Inequality

You can add the same number to You can subtract the same number both sides of an inequality and the from both sides of an inequality inequality will remain true. and the inequality will remain true.

8

5 ____ -5 ____ x < -17

Use the Subtraction Property of Inequality. Subtract 5 from both sides.

STEP 2

Graph the solution.

STEP 3

Check the solution. Substitute a solution from the shaded part of your number line into the original inequality. ? -18 + 5 < -12 Substitute -18 for x into x + 5 < -12

5

x

+

B The model shows 5 + x ≥ 8. How many tiles must you remove from each side to isolate x on one side of the inequality? Circle these tiles.

5

8

+ + + + + 5

+ +

x

≥ ≥

C What values of x make this inequality true? Graph the solution of the inequality on the number line. x≥

3

-5 -4 -3 -2 -1

8

Mathematical Processes

What would it tell you if the inequality is false when you check the solution?

No; Yes; It can only increase by an amount greater than or equal to 3.

0 1 2 3 4 5

Reflect 1.

Math Talk

+ + + + + + + +

Analyze Relationships How is solving the inequality 5 + x ≥ 8 like solving the equation 5 + x = 8? How is it different?

To solve both, you subtract 5 from both sides. But there is only one solution for the equation and

Math Talk

The number line is shaded in the wrong direction, and the number you chose is not a solution.

-13 < -12 STEP 1

8 ≤y - 3 + 3 + 3 _ _ 11 ≤ y

Lesson 13.2

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356

Use the Addition Property of Inequality. Add 3 to both sides. You can rewrite 11 ≤ y as y ≥ 11.

STEP 2

Graph the solution.

STEP 3

Check the solution. Substitute a solution from the shaded part of your number line into the original inequality.

infinitely many solutions for the inequality.

6_MTXESE051676_U4M13L2.indd 355

Solve the inequality.

Mathematical Processes

Could the temperature have increased by 2 degrees by noon? Could it have increased by 5 degrees? Explain.

The inequality is true.

B 8≤ y-3

5 6 7 8 9 10 11 12 13 14 15

? 8 ≤ 12 - 3

Substitute 12 for y in 8 ≤ y - 3

8≤ 9

The inequality is true.

© Houghton Mifflin Harcourt Publishing Company

© Houghton Mifflin Harcourt Publishing Company • Image Credits: ©Janusz Wrobel/ Alamy

-20 -19 -18 -17 -16 -15 -14 -13 -12 -11 -10

Unit 4

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PROFESSIONAL DEVELOPMENT Integrate Mathematical Processes This lesson provides an opportunity to address Mathematical Process TEKS 6.1.D, which calls for students to “communicate mathematical ideas… using multiple representations…as appropriate.” In the Explore Activity, students use algebra tiles and a number line to model solving a one-step inequality. In the Examples, students represent inequalities symbolically, graph them on a number line, and write word problems involving real-world situations that correspond to specific inequalities.

Math Background The Commutative and Associative properties hold for addition but not for subtraction, as shown below. Caution students not to use either property when simplifying inequalities that involve subtraction. Commutative Property: Subtraction: 5 - 3 ≠ 3 - 5 Associative Property: Subtraction: 5 - (3 - 1) ≟ (5 - 3) - 1 5-2≟2-1 3≠1

356

Mathematical Processes Exercise 3 Some students may shade in the wrong direction when they attempt to graph the solution set of an inequality with the variable on the right-hand side of the inequality symbol. Remind students that reading the inequality beginning with the variable will help them understand which side of the number line to shade.

ADDITIONAL EXAMPLE 2 Write a real-world problem that can be described by 14 < x + 7. Sample: A cook has 7 pounds of potatoes. He needs more than 14 pounds for dinner. What inequality describes the amount of potatoes the cook needs to buy? Interactive Whiteboard Interactive example available online my.hrw.com

EXAMPLE 2 Questioning Strategies

Mathematical Processes • What is the first step in writing a real-world problem to describe an inequality? First, analyze the numbers and relationships given in the inequality. Then think of real-world situations for which those numbers would make sense. • Are there any situations where the negative numbers on the number line would make sense? Yes. For example, it would make sense to use negative numbers for a situation involving changes in temperature.

Connect Vocabulary ELL

Mathematical Processes To help students understand the suggested comparison, relate the words “no more than 60 pounds” to the inequality symbol and the number 60 in the inequality. You may want to draw a simple sketch of a dog in a box on a weight scale to help students visualize the problem.

YOUR TURN Connect to Daily Life

Mathematical Processes Some students may need help conceiving inequality situations. You may want to make a list of various types of situations that students can use for inspiration. Invite the class to help you make the list.

Elaborate Talk About It Summarize the Lesson Ask: How does solving an addition inequality compare with solving an addition equation? The process is essentially the same: subtracting the same amount from each side to isolate the variable.

GUIDED PRACTICE Engage with the Whiteboard For Exercise 1, have a student circle the tiles that need to be removed from each side of the inequality on the whiteboard. Then ask the student to write the inequality and show how to “remove” the three tiles from the left-hand side of the inequality. Finally, have the student show how the solution matches the model. For Exercises 2–3, have students make models to represent the inequality on the whiteboard and explain how to solve it, using the same method that was used for Exercise 1.

Avoid Common Errors Exercises 2–5 Watch for students who incorrectly apply one of the Properties of Inequality. When this is the case, have the student circle the variable and the operation sign associated with it. Remind the student to apply the opposite operation to both sides of the inequality.

357

Lesson 13.2

DO NOT EDIT--Changes must be made through “File info” CorrectionKey=B

DO NOT EDIT--Changes must be made through “File info” CorrectionKey=B

Solve each inequality. Graph and check the solution. 2. y - 5 ≥ -7

y≥ - 2

-5 -4 -3 -2 -1

0 1 2

Personal Math Trainer

3. 21 > 12 + x

Online Assessment and Intervention

x< 9

3 4 5

0 1 2 3 4 5 6 7

Personal Math Trainer

5. Write a real-world problem that can be modeled by x - 13 > 20. Solve your problem and tell what values make sense for the situation.

Check students’ problems.

Online Assessment and Intervention

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8 9 10

Guided Practice

Interpreting Inequalities as Comparisons

1. Write the inequality shown on the model. Circle the tiles you would remove from each side and give the solution. (Explore Activity)

You can write a real-world problem for a given inequality. Examine each number and mathematical operation in the inequality.

EXAMPL 2 EXAMPLE

Inequality: Math On the Spot 6.9.C

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Write a real-world problem for the inequality 60 ≥ w + 5. Then solve the inequality. STEP 1

Solution:

3+x≤5

+ + +

x≤2

+

+ + + + +

Solve each inequality. Graph and check the solution. (Example 1)

Examine each part of the inequality.

2. x + 4 ≥ 9

x≥5

3. 5 > z - 3

z<8

w is the unknown quantity.

STEP 2

STEP 3

8 9 10

t>7

0 1 2 3 4 5 6 7

60 is greater than or equal to a number added to 5.

4. t + 5 > 12

Write a comparison that the inequality could describe. June’s dog will travel to a dog show in a pet carrier. The pet carrier weighs 5 pounds. The total weight of the pet carrier and the dog must be no more than 60 pounds. What inequality describes the weight of June’s dog?

6. Write a real-world problem that can be represented by the inequality y - 4 < 2. Solve the inequality and tell whether all values in the solution make sense for the situation. (Example 2)

0 1 2 3 4 5 6 7 8 9 10

0 1 2 3 4 5 6 7 8 9 10

Solve the inequality.

he had less than \$2 left on the card. How much money was on the

60 ≥ w + 5 -5 ____ 55 ≥ w

-5 ____

card before the purchase?; y < 6. Only values between 4 and 6 make sense, since these values result in positive dollar values.

June’s dog currently weighs ≤ 55 pounds.

? ?

Reflect

4. If you were to graph the solution, would all points on the graph make sense for the situation?

ESSENTIAL QUESTION CHECK-IN

7. Explain how to solve 7 + x ≥ 12. Tell what property of inequality you would use.

No; the dog’s weight cannot be 0 or a negative value,

Use the Subtraction Property of Inequality to subtract 7 from both sides of

so only positive numbers make sense.

the inequality. The answer is any number greater than or equal to 5. Lesson 13.2

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5. y - 4 < 2

8 9 10

y<6

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DIFFERENTIATE INSTRUCTION Cooperative Learning

Number Sense

Let students work in small groups to describe a situation from science that suggests inequalities. Topics may include comparing speeds, temperatures, weights, and so on. Have each group write an inequality for their situation and share the result with the class. Sample answers: The weight of sample A < 2.5 grams; the weight of sample A > the weight of sample B.

Start with the solution to an inequality, x > -3 Discuss with students that many different addition and subtraction inequalities have this same solution. Demonstrate that by “working backward” they can create an inequality that has this same solution. x + 4 > -3 + 4 x+4>1 Ask students to write three additional inequalities that have the solution x > -3. Then record them in a list for the class.

Differentiated Instruction includes: • Reading Strategies • Success for English Learners ELL • Reteach • Challenge PRE-AP

358

Personal Math Trainer Online Assessment and Intervention

Online homework assignment available

Evaluate GUIDED AND INDEPENDENT PRACTICE 6.10.A, 6.9.B, 6.9.C, 6.10.B

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13.2 LESSON QUIZ 6.10.A Solve each inequality. Graph and check the solution. 1. x + 20 ≥ -7

Concepts & Skills

Practice

Explore Activity Modeling One-Step Inequalities

Exercise 1

Example 1 Using Properties of Inequalities

Exercises 2–5, 8–11, 17

Example 2 Interpreting Inequalities as Comparisons

Exercises 6, 12–17

2. y - 30 ≤ 32 3. -16 < z -14 4. -2.5 > t + 3.5

Exercise

Write and solve an inequality for each situation. 5. Jeremy’s goal is to earn at least \$80 toward a week at soccer camp. He has earned \$32 so far. How much more does he need to earn? 6. The maximum weight an airline allows for a suitcase is 50 pounds. Ella’s suitcase weighs 8 pounds empty. a. If Ella meets the requirements, what could be the weight of the contents of her suitcase?

Depth of Knowledge (D.O.K.)

8–11

2 Skills/Concepts

1.D Multiple representations

12–16

2 Skills/Concepts

1.A Everyday life

17

3 Strategic Thinking

1.G Explain and justify arguments

18

3 Strategic Thinking

1.F Analyze relationships

19

3 Strategic Thinking

1.G Explain and justify arguments

20

3 Strategic Thinking

1.F Analyze relationships

Additional Resources Differentiated Instruction includes: • Leveled Practice Worksheets

b. Do all values in the solution make sense for the situation? Explain. Lesson Quiz available online my.hrw.com

-30

-25

-20

60

65

0

5

x ≥ -27 2.

55

y ≤ 62 3.

-5

z > -2 (or -2 < z) 4.

-10

-5

t < -6 (or -6 > t) 5. m + 32 ≥ 80; m ≥ 48

359

Lesson 13.2

0

Mathematical Processes

6. a. w + 8 ≤ 50; w ≤ 42; b. No. A negative value would make no sense, and 0 would mean that the suitcase is empty.

DO NOT EDIT--Changes must be made through “File info” CorrectionKey=A

DO NOT EDIT--Changes must be made through “File info” CorrectionKey=B

Name

Class

Date

13.2 Independent Practice

17. Multistep The table shows Marco’s checking account activity for the first week of June.

Personal Math Trainer

6.9.B, 6.9.C, 6.10.A, 6.10.B

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a. Marco wants his total deposits for the month of June to exceed \$1,500. Write and solve an inequality to find how much more he needs to deposit to meet this goal.

Online Assessment and Intervention

x > 50

y ≥ -10

-8

-7

-6

-5

-3

z ≤ -11

11. 15 ≥ z + 26 - 15 - 14 - 13 - 12 - 11 - 10 - 9

-4

-8

-7

-6

Yes; he has to spend less than \$356.68. He spent \$93.32 in the first week. \$93.32(3) = \$279.96, which is less than \$356.68.

-2

FOCUS ON HIGHER ORDER THINKING

18. Critique Reasoning Kim solved y - 8 ≤ 10 and got y ≤ 2. What might Kim have done wrong?

-5

Write an inequality to solve each problem.

Kim added 8 to the left side of the inequality and subtracted

12. The water level in the aquarium’s shark tank is always greater than 25 feet. If the water level decreased by 6 feet during cleaning, what was the water level before the cleaners took out any water?

8 from the right side of the inequality to isolate y. She should have added 8 to both sides of the inequality.

© Houghton Mifflin Harcourt Publishing Company

w - 6 > 25; w > 31 feet

19. Critical Thinking José solved the inequality 3 > x + 4 and got x < 1. Then, to check his solution, he substituted -2 into the original inequality to check his solution. Since his check worked, he believes that his answer is correct. Describe another check José could perform that will show his solution is not correct. Then explain how to solve the inequality.

13. Danny has at least \$15 more than his big brother. Danny’s big brother has \$72. How much money does Danny have?

72 ≤ d - 15; d ≥ 87 dollars

Substitute 0 for x in the original inequality. 0 is less than

14. The vet says that Ray’s puppy will grow to be at most 28 inches tall. Ray’s puppy is currently 1 foot tall. How much more will the puppy grow?

1 but is not a solution. To solve the inequality, subtract 4

12 + x ≤ 28; x ≤ 16 inches

from both sides to get x <-1.

15. Pierre’s parents ordered some pizzas for a party. 4.5 pizzas were eaten at the party. There were at least 5_12 whole pizzas left over. How many pizzas did Pierre’s parents order?

20. Look for a Pattern Solve x + 1 > 10, x + 11 > 20, and x + 21 > 30. Describe a pattern. Then use the pattern to predict the solution of x + 9,991 > 10,000.

p - 4.5 ≥ 5.5; p ≥ 10 pizzas

x > 9 for each inequality; in each case the number

16. To get a free meal at his favorite restaurant, Tom needs to spend \$50 or more at the restaurant. He has already spent \$30.25. How much more does Tom need to spent to get his free meal?

x + 30.25 ≥ 50

added to x is 9 less than the number on the right side of each inequality, so x > 9 is the solution.

x ≥ \$19.75

Lesson 13.2

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EXTEND THE MATH

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You may wish to present the model shown at the right for x - 2 ≤ 4. Remind students that x - 2 and x + (-2) are equivalent. Also remind them of the concept of zero pairs. Have students write an explanation of the way to use algebra tiles to solve subtraction inequalities and be prepared to demonstrate the method to other students.

Unit 4

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Activity available online

PRE-AP

Activity Challenge pairs of students to use what they know about adding and subtracting integers to model subtraction inequalities with algebra tiles.

Work Area

© Houghton Mifflin Harcourt Publishing Company

- 12 - 11 - 10 - 9

\$22.82

c. There are three weeks left in June. If Marco spends the same amount in each of these weeks that he spent during the first week, will he meet his goal of spending less than \$450 for the entire month? Justify your answer.

8 9 10

10. y - 5 ≥ -15

\$24.00

Purchase – Water bill

x + 93.32 < 450; x < 356.68; less than \$356.68

y≥8

0 1 2 3 4 5 6 7

\$46.50

Purchase – Movie Theatre

b. Marco wants his total purchases for the month to be less than \$450. Write and solve an inequality to find how much more he can spend and still meet this goal.

0 10 20 30 40 50 60 70 80 90 100

9. 193 + y ≥ 201

\$520.45

Purchase – Grocery Store

x + 520.45 > 1500; x > 979.55; more than \$979.55

Solve each inequality. Graph and check the solution. 8. x - 35 > 15

Deposit – Paycheck

28/01/14 11:00 PM

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x + (-2) ≤ 4

+

− −

+ +

+ +

x-2+2≤4+2

+

− −

+ +

+ +

+ +

+ +

+ +

+ +

+ +

x≤6

+

360

LESSON

13.3

Multiplication and Division Inequalities with Positive Numbers

Texas Essential Knowledge and Skills The student is expected to: Expressions, equations, and relationships—6.10.A Model and solve one-variable, one-step inequalities that represent problems, including geometric concepts. Expressions, equations, and relationships—6.9.B

Engage ESSENTIAL QUESTION How can you solve an inequality involving multiplication or division with positive numbers? Sample answer: You can use the Multiplication and Division Properties of Inequality: multiply or divide each side by the same positive amount to isolate the variable.

Motivate the Lesson

Ask: If you get a monthly allowance of \$36, how could you use the inequality 4x ≤ 36 to plan a weekly budget? Begin the Explore Activity to find out how to solve multiplication inequalities.

Represent solutions for one-variable, one-step equations and inequalities on number lines. Expressions, equations, and relationships—6.9.C Write corresponding real-world problems given one-variable, one-step equations or inequalities. Expressions, equations, and relationships—6.10.B Determine if the given value(s) make(s) one-variable, one-step equations or inequalities true.

Explore EXPLORE ACTIVITY Connect Multiple Representations Mathematical Processes In the activity, the real-world problem is represented first by an inequality in words, then as an inequality in symbols. The inequality is modeled and solved with algebra tiles. Finally, the solution is shown on a number line. Discuss with students how each representation leads to the next, and discuss the advantages and disadvantages of each type of representation.

Mathematical Processes 6.1.B Use a problem-solving model that incorporates analyzing given information, formulating a plan or strategy, determining a solution, justifying the solution, and evaluating the problem-solving process and the reasonableness of the solution.

ADDITIONAL EXAMPLE 1 Solve each inequality. Graph and check the solution. A 7x > 28

x>4

0 y B __5 ≤ 30

5

10

150

155

Interactive Whiteboard Interactive example available online my.hrw.com

361

Lesson 13.3

EXAMPLE 1 Focus on Math Connections

Mathematical Processes Present the equation 12x = 24. Encourage students to solve for x, x = 2. The solution to the equation is one point on the number line. The solution to the inequality, x < 2, is a section of the number line, in this case, everything to the left of 2.

Questioning Strategies

Mathematical Processes • In A, Step 1, why do you divide by 12? To isolate x, it is necessary to “remove” the 12. Since x is multiplied by 12, using the inverse operation, division, will “undo” the multiplication. • In A, Step 2, 24 ÷ 12 is 2. Why isn’t 2 a solution? Since the inequality sign is the less than sign, the solution is less than 2. • When graphing inequalities, how do you know when to use a solid circle or an open circle? A solid circle is used to represent ≤ or ≥ on a graph. An open circle is used to represent < or > on a graph.

y ≤ 150

145

Explain

Engage with the Whiteboard For A, have a volunteer make a model with algebra tiles on the whiteboard. Have the student solve the inequality, using the same method that was used in B of the Explore Activity. Then have students compare and contrast the model with the number line provided in the example.

DO NOT EDIT--Changes must be made through “File info” CorrectionKey=B

LESSON

13.3

DO NOT EDIT--Changes must be made through “File info” CorrectionKey=A

Multiplication and Division Inequalities with Positive Numbers

Expressions, equations, and relationships—6.10.A Model and solve one-variable, one-step inequalities that represent problems. Also 6.9.B, 6.9.C, 6.10.B.

Solving Inequalities Involving Multiplication and Division Math On the Spot

You can use properties of inequality to solve inequalities involving multiplication and division with positive integers.

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ESSENTIAL QUESTION

Multiplication and Division Properties of Inequality How can you solve an inequality involving multiplication or division with positive numbers?

• You can multiply both sides of an inequality by the same positive number and the inequality will remain true. • You can divide both sides of an inequality by the same positive number and the inequality will remain true.

6.10.A

EXPLORE ACTIVITY

Modeling One-Step Inequalities

EXAMPLE 1

You can use algebra tiles to solve inequalities that involve multiplying positive numbers.

Solve each inequality. Graph and check the solution.

A 12x < 24

Dominic is buying school supplies. He buys 3 binders and spends more than \$9. How much did he spend on each binder?

STEP 1

A Let x represent the cost of one binder. Write an inequality. Number of binders

3

Cost of a binder

B The model shows the inequality from There are

3

A

3

Mathematical Processes

9

Are all negative numbers solutions to 12x < 24? Explain.

.

+ + +

>

+ + + + + + + + +

Yes; because 12 times a negative number is a negative number, and all negative numbers are less than 24.

3

C What values make the inequality you wrote in true? Graph the solution of the inequality.

A

x>3

-5 -4 - 3 - 2 - 1

0 1 2 3 4 5

Reflect 1. Analyze Relationships Is 3.25 a solution of the inequality you wrote in A ? If so, does that solution make sense for the situation?

Yes, 3.25 is a solution. Dominic may have paid \$3.25 for each binder.

Divide both sides by 12.

x<2

Math Talk

equal groups.

How many units are in each group?

Solve the inequality. 12x __ ___ < 24 12 12

9

x-tiles, so draw circles

to separate the tiles into © Houghton Mifflin Harcourt Publishing Company

>

x

>

6.9.B, 6.10.B

Use an open circle to show that 2 is not a solution.

STEP 2

Graph the solution.

STEP 3

Check the solution by substituting a solution from the shaded part of the graph into the original inequality. ? Substitute 0 for x in the original inequality. 12(0) < 24

-5 -4 - 3 - 2 - 1

0 < 24

y B _3 ≥ 5

STEP 1

0 1 2 3 4 5

The inequality is true.

Solve the inequality. 3 ( _3 ) ≥ 3(5) y

Multiply both sides by 3.

y ≥ 15

Use a closed circle to show that 15 is a solution.

STEP 2

Graph the solution.

STEP 3

Check the solution by substituting a solution from the shaded part of the graph into the original inequality. 18 ? __ ≥5 Substitute 18 for x in the original inequality. 3

© Houghton Mifflin Harcourt Publishing Company

?

5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20

2. Represent Real-World Problems Rewrite the situation in represent the inequality 3x < 9.

A

to

Dominic is buying school supplies. He buys 3 binders and spends less than \$9. How much did he spend on each binder?

6≥5 Lesson 13.3

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InDesign Notes 1. This is a list

PROFESSIONAL DEVELOPMENT Integrate Mathematical Processes This lesson provides an opportunity to address Mathematical Process TEKS 6.1.B, which calls for students to “use a problem-solving model that incorporates analyzing given information, formulating a plan or strategy, determining a solution, justifying the solution, and evaluating the problem-solving process and the reasonableness of the solution.” Students directly apply each of these steps of the problem-solving model in Example 2, to determine the possible side lengths of a square flag.

The inequality is true.

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10/29/12 7:05 PM

InDesign Notes

1. This is a list Bold, Italic, Strickthrough.

1. This is a list

Math Background As with addition and subtraction, not all properties that hold for multiplication hold for division. For example, the Commutative and Associative properties hold for multiplication but not for division. Caution students to avoid rearranging or regrouping numbers that are being divided. Commutative Property: Division: 15 ÷ 3 ≠ 3 ÷ 15 Associative Property: Division: 50 ÷ (10 ÷ 5) ≟ (50 ÷ 10) ÷ 5 50 ÷ 2 ≟ 5 ÷ 5 25 ≠ 1

Multiplication and Division Inequalities with Positive Numbers

362

YOUR TURN Avoid Common Errors Exercise 4 Some students may divide both sides by 4 instead of multiplying. Rewriting the equation as z ÷ 4 = 11 may help these students see which operation they should use.

ADDITIONAL EXAMPLE 2 Kayla earned more than \$50 babysitting. Her mother paid her \$4 an hour to babysit her little brother. Write and solve an inequality to find the possible number of hours Kayla babysat. 4h > 50; h > 12.5; Kayla babysat more than 12.5 hours. Interactive Whiteboard Interactive example available online my.hrw.com

EXAMPLE 2 Focus on Reasoning

Mathematical Processes Point out to students that sometimes a problem may provide clues and facts that you must use to find a solution. Encourage students to begin by identifying the important information. They can underline or circle the information in the problem statement.

Questioning Strategies

Mathematical Processes • What information is given in the problem? Because the flag being made is a square, all the sides have the same length, and the perimeter needs to be 22 inches or longer. • How do you find the perimeter of a figure? Perimeter = sum of the lengths of the sides

Focus on Modeling Mathematical Processes Have students draw a square to represent the square flag. Then have students label each of the sides of the square with an x. Guide them to see that the four sides (x + x + x + x) are the same as the 4x shown in the inequality in the Solve step. c.2.I.4 ELL Encourage class discussion to develop the scenario in Reflect. English learners will benefit from hearing and participating in classroom discussions.

Integrating the ELPS

YOUR TURN Focus on Critical Thinking Mathematical Processes Point out to students that the unknown is the total weight of sand needed for 6 paperweights. Help students to see that the total weight divided by 6 is equivalent to the weight of a single paperweight.

Elaborate Talk About It Summarize the Lesson Ask: How do you solve a division inequality? Multiply both sides of the inequality by the same number to isolate the variable.

GUIDED PRACTICE Engage with the Whiteboard For Exercise 1, have students circle the groups of tiles on the whiteboard and then write the inequality and the solution below the model. Have students make a number line to show the solution. Ask the class to compare and contrast the model with the number line.

Avoid Common Errors Exercises 2–3 Watch for students who incorrectly apply one of the Properties of Inequality. Remind them that they should apply the opposite operation to both sides of the inequality. Exercise 4 If students have difficulty writing the inequality, encourage them to make a model to help them understand the situation better.

363

Lesson 13.3

DO NOT EDIT--Changes must be made through “File info” CorrectionKey=B

DO NOT EDIT--Changes must be made through “File info” CorrectionKey=A

YOUR TURN Solve each inequality. Graph and check the solution. 3. 5x ≥ 100 4. _4z < 11

Reflect

Personal Math Trainer

x ≥ 20

15 16 17 18 19 20 21 22 23 24 25

z < 44

40 41 42 43 44 45 46 47 48 49 50

5. Represent Real-World Problems Write and solve a real-world problem for the inequality 4x ≤ 60.

Online Assessment and Intervention

Sample problem: Cy is making a square flag. He wants

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the perimeter to be no more than 60 inches. What are the possible side lengths? x ≤ 15 inches

Solving Real-World Problems

You can use multiplication and division inequalities to model and solve real-world problems.

EXAMPL 2 EXAMPLE

Personal Math Trainer Math On the Spot

Problem Solving

6.10.A

Online Assessment and Intervention

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6. A paperweight must weigh less than 4 ounces. Brittany wants to make 6 paperweights using sand. Write and solve an inequality to find the possible weight of the sand she needs.

w __ < 4; w < 24; Brittany must use less than 24 oz of sand. 6

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Cy is making a square flag. He wants the perimeter to be at least 22 inches. Write and solve an inequality to find the possible side lengths.

Guided Practice 1. Write the inequality shown on the model. Circle groups of tiles to show the solution. Then write the solution. (Explore Activity)

Formulate a Plan

Inequality:

Write and solve a multiplication inequality. Use the fact that the perimeter of a square is 4 times its side length.

Solution:

© Houghton Mifflin Harcourt Publishing Company

Let x represent a side length.

4x __ __ ≥ 22 4 4

Divide both sides by 4.

x ≥ 5.5

2. 8y < 320

35 36 37 38 39 40 41 42 43 44 45

The side lengths must be greater than or equal to 5.5 in.

? ?

Check the solution by substituting a value in the solution set in the original inequality. Try x = 6.

30 31 32 33 34 35 36 37 38 39 40

ESSENTIAL QUESTION CHECK-IN

Use the Division Property of Inequality to divide both sides by 5 to get x < 8.

The statement is true.

Cy’s flag could have a side length of 6 inches.

To check, substitute a number from the solution into the original inequality. Lesson 13.3

364

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1. This is a list

r ≥ 33

5. Explain how to solve and check the solution to 5x < 40 using properties of inequalities.

Substitute 6 for x.

InCopy Notes

3. _3r ≥ 11

+ + + + + + + +

b __ ≥ 14; b ≥ 84; Karen had at least 84 books. 6

Justify and Evaluate

24 ≥ 22

y < 40

<

4. Karen divided her books and put them on 6 shelves. There were at least 14 books on each shelf. How many books did she have? Write and solve an inequality to represent this situation. (Example 2)

Cy’s flag should have a side length of 5.5 inches or more.

? 4(6) ≥ 22

+ +

Solve each inequality. Graph and check the solution. (Example 1)

Justify and Evaluate Solve

4x ≥ 22

2x < 8 x<4

© Houghton Mifflin Harcourt Publishing Company

Analyze Information

Find the possible lengths of 1 side of a square that has a perimeter of at least 22 inches.

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Unit 4

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InDesign Notes 1. This is a list

1. This is a list Bold, Italic, Strickthrough.

DIFFERENTIATE INSTRUCTION

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InDesign Notes 1. This is a list

Cooperative Learning

Critical Thinking

Have students work in pairs. Give each pair an index card. Have students write an inequality word problem on the card, then exchange problems with another pair and work together to solve the problems.

Let students work together to decide which of the following statements, a + c > b + c or ac > bc, are always true if a, b, and c are real numbers and a > b. If the statement is not always true, have students give an example to show that the statement is false.

Differentiated Instruction includes: • Reading Strategies • Success for English Learners ELL • Reteach • Challenge PRE-AP

If students have trouble writing word problems, suggest that they use the word problems in the Independent Practice as a template. They can change the names, numbers, and/or the direction of the inequality to create new problems.

a + c > b + c is always true; ac > bc is not always true. a = 2, b = 1, c = 3 2‧3<1‧3

6 < 3, false

Multiplication and Division Inequalities with Positive Numbers

364

Personal Math Trainer Online Assessment and Intervention

Online homework assignment available

Evaluate GUIDED AND INDEPENDENT PRACTICE 6.10.A, 6.9.B, 6.9.C, 6.10.B

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13.3 LESSON QUIZ 6.10.A Solve each inequality. Graph and check the solution. 1. 4x ≥ 12 y 2. __3 ≤ 6

Concepts & Skills

Practice

Explore Activity Modeling One-Step Inequalities

Exercise 1

Example 1 Solving Inequalities Involving Multiplication and Division with Positive Integers

Exercises 2–3, 10–11, 15–18

Example 2 Solving Real-World Problems

Exercises 4, 6–9, 12–14

3. 2p > 24 4. __8t < 20 Write and solve an inequality for each problem. 5. Kellin pays more than \$40 a day to rent a canoe. What amount might he pay to rent the canoe for a 7-day trip? 6. Amanda earns \$15 an hour. She needs at least \$90 to buy a computer desk. How many hours does she need to work to buy the desk?

Exercise

Depth of Knowledge (D.O.K.)

6–9

2 Skills/Concepts

1.B Problem-solving model

10–11

2 Skills/Concepts

1.D Multiple representations

12–14

2 Skills/Concepts

1.B Problem-solving model

15–18

2 Skills/Concepts

1.D Multiple representations

19–22

2 Skills/Concepts

1.A Everyday life

23–24

3 Strategic Thinking

1.F Analyze relationships

25

3 Strategic Thinking

1.B Problem-solving model

Lesson Quiz available online my.hrw.com

5

10

15

20

15

20

160

165

x≥3 2. 10

y ≤ 18 3. 10

p > 12 4. 155

t < 160 w 5. __ > 40; w > 280; more than \$280 7

6. 15h ≥ 90; h ≥ 6

365

Lesson 13.3

Mathematical Processes

Additional Resources Differentiated Instruction includes: • Leveled Practice Worksheets

DO NOT EDIT--Changes must be made through “File info” CorrectionKey=B

DO NOT EDIT--Changes must be made through “File info” CorrectionKey=B

Class

Date

The sign shows some prices at a produce stand.

13.3 Independent Practice 6.9.B, 6.9.C, 6.10.A, 6.10.B

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Write and solve an inequality for each problem.

22. The produce buyer for a local restaurant wants to buy more than 30 lb of onions. The produce buyer at a local hotel buys exactly 12 pounds of spinach. Who spends more at the produce stand? Explain.

numbers don’t make sense.

The restaurant produce buyer; he will spend more than

14. Multistep Lina bought 4 smoothies at a health food store. The bill was less than \$16.

\$37.50; the hotel’s produce buyer will spend \$36.

a. Write and solve an inequality to represent the cost of each smoothie.

8. In a litter of 7 kittens, each kitten weighs more than 3.5 ounces. Find the possible total weight of the litter. w __ 7 > 3.5; w > 24.5; more than

4s < 16; s < 4; the cost of each

_r 5

24. Represent Real-World Problems Write and solve a word problem that can be represented with 240 ≤ 2x.

Sample answer: Jake and his brother want to earn at

less than \$4.

least \$240. If each brother earns the same amount, how

c. Graph the values that make sense for this situation on the number line.

Solve each inequality. Graph and check the solution.

-1

x≤6

p 15. __ ≤ 30 13

0 1 2 3 4 5 6 7 8 9 10

t>0

16. 2t > 324 17. 12y ≥ 1

0 1 2 3 4 5

x 18. ___ < 11 9.5

much money does each brother have to earn?; x ≥ 120;

each brother must earn at least \$120.

0 1 2 3 4 5

Solve each inequality.

25. Persevere in Problem Solving A rectangular prism has a length of 13 inches and a width of _12 inch. The volume of the prism is at most 65 cubic inches. Find all possible heights of the prism. Show your work.

p ≤ 390

()

13 ∙ _12 ∙ h ≤ 65; 6.5h ≤ 65; h < 10; all heights greater

t > 162 1 y ≥ __ 12

than 0 inches but no more than 10 inches

x < 104.5 Lesson 13.3

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InCopy Notes

InDesign Notes

EXTEND THE MATH

Unit 4

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InCopy Notes

1. This is a list

Activity available online A Mystery Rectangle

There are only 3 possible rectangles.

Clue 2: Each dimension must be a whole number.

Perimeter

21

24

90

22

25

94

23

26

98

1. This is a list

my.hrw.com

Activity Challenge students to translate Clues 1, 3, and 4 into inequalities or equations. Direct them to use the clues together to find all the possible dimensions of the rectangle. List them in the table below.

Length

28/01/14 11:19 PM

InDesign Notes

1. This is a list Bold, Italic, Strickthrough.

PRE-AP

Width

is the

side and divided by 5 on the right side.

amount greater than \$0 and

of Will’s backyard is at least 11 feet.

2 __ . What 25

r ≤ 2; the student may have multiplied by 5 on the left

Each smoothie can cost any

15.5w ≥ 170.5; w ≥ 11; the width

_2 and 5

23. Critique Reasoning A student solves ≤ gets r ≤ correct solution? What mistake might the student have made?

b. What values make sense for this situation? Explain.

9. To cover his rectangular backyard, Will needs at least 170.5 square feet of sod. The length of Will’s yard is 15.5 feet. What are the possible widths of Will’s yard?

Work Area

FOCUS ON HIGHER ORDER THINKING

smoothie was less than \$4.

24.5 ounces.

© Houghton Mifflin Harcourt Publishing Company

No; 1.25x ≤ 3; x ≤ 2.4 so 2.4 pounds of onions are the most

Florence can buy. 2.4 < 2.5, so she cannot buy 2.5 pounds.

negative amount so negative

14x ≥ 84; x ≥ 6; at least 6 hours

1. This is a list

21. Florence wants to spend no more than \$3 on onions. Will she be able to buy 2.5 pounds of onions? Explain.

No; Steve would not pay a

7. Tamar needs to make at least \$84 at work on Tuesday to afford dinner and a movie on Wednesday night. She makes \$14 an hour at her job. How many hours does she need to work on Tuesday?

-5 -4 - 3 - 2 - 1

at most \$2.75

13. If you were to graph the solution for exercise 12, would all points on the graph make sense for the situation? Explain.

to 7 inches

11. _2t > 0

20. Gary has enough money to buy at most 5.5 pounds of potatoes. How much money does Gary have?

r __ < 32; r < 992; less than \$992 31

6s ≤ 42; s ≤ 7; less than or equal

10. 10x ≤ 60

3_13 pounds

Online Assessment and Intervention

12. Steve pays less than \$32 per day to rent his apartment. August has 31 days. What are the possible amounts Steve could pay for rent in August?

6. Geometry The perimeter of a regular hexagon is at most 42 inches. Find the possible side lengths of the hexagon.

Price per Pound Produce \$1.25 Onions \$0.99 Yellow Squash \$3.00 Spinach \$0.50 Potatoes

19. Tom has \$10. What is the greatest amount of spinach he can buy?

Personal Math Trainer

© Houghton Mifflin Harcourt Publishing Company

Name

Clue 1: A rectangle has a perimeter that is less than 100 units. 2w + 2l < 100 Clue 3: The width must be less than 25, but greater than 20. w > 20; w < 25 Clue 4: The length must be 3 units longer than the width. l=w+3

Multiplication and Division Inequalities with Positive Numbers

366

LESSON

13.4

Multiplication and Division Inequalities with Rational Numbers

Texas Essential Knowledge and Skills The student is expected to: Expressions, equations, and relationships—6.9.B Represent solutions for one-variable, one-step equations and inequalities on number lines. Expressions, equations, and relationships—6.10.A Model and solve one-variable, one-step equations and inequalities that represent problems, including geometric concepts. Expressions, equations, and relationships—6.10.B Determine if the given value(s) make(s) one-variable, one-step equations or inequalities true.

Mathematical Processes 6.1.F Analyze mathematical relationships to connect and communicate mathematical ideas.

Engage ESSENTIAL QUESTION How do you solve inequalities that involve multiplication and division of integers? Sample answer: You multiply or divide each side of the inequality to isolate the variable. If the integer is negative, you must reverse the direction of the inequality symbol.

Motivate the Lesson Ask: Have you ever had a friend ask you a question using double-negative language, such as “You don’t want to not go, do you?” Well, double negatives can change things around. Begin the Explore Activity and find out how multiplying or dividing both sides of an inequality by a negative number can change things.

Explore EXPLORE ACTIVITY Engage with the Whiteboard Have students draw number lines on the whiteboard to graph each example in A. First, have them graph the given inequality. Then have them graph the inequality after multiplying each side by the same number in a different color. For each example, ask if the original symbol still makes the inequality true after multiplying. If it doesn’t, ask what symbol would make the inequality true.

Explain ADDITIONAL EXAMPLE 1 Solve each inequality. Graph and check the solution. A -3x < 18 -10

B -__6 > -7 y

35

x > -6 -5

0

45

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Lesson 13.4

Questioning Strategies

Mathematical Processes

number you need to use to multiply both sides of the inequality

Interactive Whiteboard Interactive example available online

367

ELL c.1.F Remind students that the word reverse means to turn backward in position or direction. Connect this meaning to reversing the inequality, where the point of the symbol changes direction.

Connect Vocabulary

y • In B, why might you rewrite -__3 as ( -__13 )y before multiplying? to make it easier to see what

y < 42 40

EXAMPLE 1

• How can you check that the graph of an inequality is correct? Pick a point on the shaded portion of the graph. Any point selected should make the inequality a true inequality.

DO NOT EDIT--Changes must be made through "File info" CorrectionKey=A

LESSON

13.4

DO NOT EDIT--Changes must be made through "File info" CorrectionKey=B

Multiplication and Division Inequalities with Rational Numbers

Expressions, equations, and relationships—6.9.B Represent solutions for one-step inequalities on number lines. Also 6.10.A, 6.10.B

Multiplication and Division Properties of Inequality Math On the Spot

Recall that you can multiply or divide both sides of an inequality by the same positive number, and the statement will still be true.

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ESSENTIAL QUESTION

Multiplication and Division Properties of Inequality How do you solve inequalities that involve multiplication and division of integers?

EXPLORE ACTIVITY

• If you multiply or divide both sides of an inequality by the same negative number, you must reverse the inequality symbol for the statement to still be true.

6.10.A

Investigating Inequality Symbols

EXAMPLE 1

You have seen that multiplying or dividing both sides of an inequality by the same positive number results in an equivalent inequality. How does multiplying or dividing both sides by the same negative number affect an inequality?

Solve each inequality. Graph and check the solution.

My Notes

A Complete the tables. New inequality

New inequality is true or false?

3<4

2

true

2 ≥ -3

6< 8

3

© Houghton Mifflin Harcourt Publishing Company

5>2

-1

-8 > -10

-8

6 ≥ -9

-5 > -2

false

x < -13

Divide each side by:

New inequality

New inequality is true or false?

4<8

4

1< 2

true

4 ≤ -3

false

4 ≥ -5

3

-16 ≤ 12

-4

15 > 5

-5

-3 > -1

Divide both sides by -4. Reverse the inequality symbol.

-4x 52 ____ < ___ -4 -4

false

64 > 80

Solve the inequality. -4x > 52

true

Inequality

12 ≥ -15

A -4x > 52 STEP 1

Multiply each side by:

Inequality

6.9.B, 6.10.B

STEP 2

Graph the solution.

STEP 3

Check your answer using substitution. ? Substitute -15 for x in -4x > 52. -4(-15) > 52 60 > 52

y

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

-8

The statement is true.

B - _3 < -5

true

STEP 1

Solve the inequality. y

- _3 < -5

false

( )

y -3 - _3 > -3(-5)

Multiply both sides by -3. Reverse the inequality symbol.

y > 15

B What do you notice when you multiply or divide both sides of an inequality by the same negative number?

The inequality is no longer true. C How could you make each of the multiplication and division inequalities that were not true into true statements?

STEP 2

Graph the solution.

STEP 3

Check your answer using substitution. ? 18 < - ___ -5 3 -6 < -5

Reverse the inequality symbol.

Lesson 13.4

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10 11 12 13 14 15 16 17 18 19 20

© Houghton Mifflin Harcourt Publishing Company

?

y Substitute 18 for y in - __ < -5. 3 The inequality is true.

Unit 4

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PROFESSIONAL DEVELOPMENT Integrate Mathematical Processes This lesson provides an opportunity to address Mathematical Process TEKS 6.1.F, which calls for students to “analyze mathematical relationships to connect and communicate mathematical ideas.” In the Explore Activity, students use tables and numerical inequalities to explore the effects of multiplying or dividing an inequality by a negative number. Students examine examples and counter-examples, leading to the conclusion that the inequality symbol must be reversed when multiplying or dividing both sides of the inequality by a negative number.

Math Background The solution set also can be described by using interval notation. In this notation, parentheses indicate that the endpoints are not included in the solution set. Brackets indicate that the endpoints are included in the solution set. Suppose a and b are any real numbers: Inequality

Interval notation

a
(a, b)

a
(a, b] [a, b) [a, b]

Multiplication and Division Inequalities with Rational Numbers

368

YOUR TURN Avoid Common Errors Exercise 2 Some students may divide both sides by -6 instead of multiplying. Rewriting the equation as 7 ≥ t ÷ (-6) may help these students see which operation they should use.

ADDITIONAL EXAMPLE 2 During the month, Jenna uses the dining commons at the college for dinner. Each dinner costs \$12. Each time she uses her dining commons card, her balance changes by -12. Last month the balance change was an amount greater than or equal to -324. How many times did she use her dining commons card? -12n ≥ -324; n ≤ 27; Jenna used the dining commons card 27 or fewer times. Interactive Whiteboard Interactive example available online my.hrw.com

EXAMPLE 2 Focus on Reasoning

Mathematical Processes Point out to students that sometimes a problem may provide clues and facts that must be used to find a solution. Encourage students to begin by identifying the important information. They can underline or circle the information in the problem statement.

Questioning Strategies

Mathematical Processes • Why is “40 feet below sea level” written as a negative number? Because the words “below sea level” indicate a negative direction, down. • What is the unknown in this situation? The unknown is the time it took the submersible to reach its final elevation of more than 40 feet below sea level.

Focus on Modeling Mathematical Processes It may help students to understand the problem better if they draw a vertical number line to represent this situation. They can label the descent in intervals of -5 feet for each second.

YOUR TURN Focus on Modeling Mathematical Processes To help students visualize the \$35 being deducted each month, draw a vertical number line from 0 to -315. Show the decreases in multiples of -35 for each month. Help students understand that what they need to find is the number of months it takes to go no farther than -315.

Elaborate Talk About It Summarize the Lesson Ask: How do you know when to reverse the inequality symbol when solving an inequality and when not to reverse it? If you multiply or divide both sides by a negative number, you must reverse the inequality symbol. If you multiply or divide by a positive number, the inequality symbol stays the same.

GUIDED PRACTICE Engage with the Whiteboard For Exercises 1–2, have students solve and graph each inequality on the whiteboard, showing all their work. Then have students explain how they knew which operation to use to solve each inequality.

Avoid Common Errors Exercises 3–5 Watch for students who may forget to reverse the inequality symbol when multiplying or dividing by a negative number. Remind them that they should always check their solution. Exercise 5 Remind students that “not colder than -80 °C” means that the temperature is at least -80 °C or warmer. Warmer temperatures are above -80° on a thermometer or to the right of -80 on a number line.

369

Lesson 13.4

DO NOT EDIT--Changes must be made through "File info" CorrectionKey=B

DO NOT EDIT--Changes must be made through "File info" CorrectionKey=B

Solve each inequality. Graph and check the solution. 1. -10y < 60 2. 7 ≥ - __t 6

y > -6

t ≥ -42

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

0 1

-47 -46 -45 -44 -43 -42 -41 -40

Personal Math Trainer Online Assessment and Intervention

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3. Every month, \$35 is withdrawn from Tom’s savings account to pay for his gym membership. He has enough savings to withdraw no more than \$315. For how many months can Tony pay for his gym membership?

Personal Math Trainer Online Assessment and Intervention

m ≤ 9; Tom can pay for no more than 9 months of his

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gym membership using this account.

Solving a Real-World Problem

EXAMPL 2 EXAMPLE

Problem Solving

Guided Practice Math On the Spot my.hrw.com

6.10.A

z ≤ -3

1. -7z ≥ 21

A marine submersible descends more than 40 feet below sea level. As it descends from sea level, the change in elevation is -5 feet per second. For how many seconds does it descend?

2. -__t > 5 4

t < -20

t >5 4. -___

Rewrite the question as a statement. • Find the number of seconds that the submersible decends below sea level.

10

-10 -9 - 8 - 7 - 6 - 5 - 4 - 3 - 2 - 1 -50 -40 -30 -20 -10

x < -6

3. 11x < -66

Analyze Information

© Houghton Mifflin Harcourt Publishing Company • Image Credits: ©Jeffrey L. Rotman/Peter Arnold Inc/Getty Images

Solve each inequality. Graph and check the solution. (Explore Activity and Example 1)

t < -50

0

10

20

30

40

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

0

50

0 1 2

-100 -90 -80 -70 -60 -50 -40 -30 -20 -10

0

5. For a scientific experiment, a physicist must make sure that the temperature of a metal does not get colder than -80 °C. The metal begins the experiment at 0 °C and is cooled at a steady rate of -4 °C per hour. How long can the experiment run? (Example 2)

List the important information: • The final elevation is greater than 40 feet below sea level or < -40 feet. • The rate of descent is -5 feet per second.

a. Let t represent time in hours. Write an inequality. Use the fact that the rate of change in temperature times the number of seconds equals the final temperature.

Formulate a Plan

Write and solve an inequality. Use this fact:

-4 · t ≥ -80

Rate of change in elevation × Time in seconds = Total change in elevation

b. Solve the inequality in part a. How long will it take the physicist to change the temperature of the metal?

20 or fewer hours

Justify and Evaluate Solve

c. The physicist has to repeat the experiment if the metal gets cooler than -80 °C. How many hours would the physicist have to cool the metal for this to happen?

more than 20 hours

-5t < -40 -5t > ____ -40 ____ -5 -5 t> 8

Rate of change × Time < Maximum elevation Divide both sides by -5. Reverse the inequality symbol.

? ?

The submersible descends for more than 8 seconds.

6. Suppose you are solving an inequality. Under what circumstances do you reverse the inequality symbol?

Justify and Evaluate

when you divide or multiply both sides by a negative number

Check your answer by substituting a value greater than 8 seconds in the original inequality. ? Substitute 9 for t in the inequality -5t < -40. -5(9) < -40   -45 < -40

ESSENTIAL QUESTION CHECK-IN

© Houghton Mifflin Harcourt Publishing Company

Although elevations below sea level are represented by negative numbers, we often use absolute value to describe these elevations. For example, -50 feet relative to sea level might be described as 50 feet below sea level.

The statement is true. Lesson 13.4

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370

Unit 4

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DIFFERENTIATE INSTRUCTION Cooperative Learning

Modeling

Have students work in pairs to explain to each other how to solve multiplication and division inequalities. Have one student explain how to solve an inequality that involves multiplying or dividing by a positive number, and have the other student explain how to solve an inequality that involves multiplying or dividing by a negative number. Then have each pair work with specific inequalities. Invite several pairs of students to share their explanations with the class.

Draw a number line with 1 and 3 graphed.

Differentiated Instruction includes: • Reading Strategies • Success for English Learners ELL • Reteach • Challenge PRE-AP

Sample inequalities for a pair of students:

Use similar steps to show that dividing both sides of a numerical inequality by -1 will change the sign as well.

-3x ≥ 60 and __6x < -4

Ask if it is correct to write 1 < 3. Yes

Multiply both numbers by -1 and graph the products. Ask if it is correct to write -1 < -3. No

Why? -1 is farther to the right than -3. Ask if it is correct to write -1 > -3. Yes

What changed? the direction of the inequality sign

Multiplication and Division Inequalities with Rational Numbers

370

Personal Math Trainer Online Assessment and Intervention

Online homework assignment available

Evaluate GUIDED AND INDEPENDENT PRACTICE 6.9.B, 6.10.A, 6.10.B

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13.4 LESSON QUIZ 6.9.B Solve each inequality. Graph and check the solution. 1. -__3x > -7 2. 10z ≤ -20 3. -6t ≥ 54

Concepts & Skills

Practice

Explore Activity Investigating Inequalities Involving Multiplication and Division of Integers

Exercises 1–4, 7–12, 17

Example 1 Multiplication and Division Properties of Inequality

Exercises 1–4, 7–12, 18–24

Example 2 Solving a Real-World Problem

Exercises 5, 13–16

4. -__2r < 7 5. Melissa’s rent check is lost in the mail. There is a -\$9 late fee charged to Melissa’s account for every day the rent is late. Her account started out at \$0, and after the late fees showed a balance that was less than -\$36. How many days late was the check? Lesson Quiz available online my.hrw.com

Answers 1. x < 21 15

20

25

0

5

-10

-5

-15

-10

3. t ≤ -9 -15

4. r > -14 -20

5. -9d < -36; d > 4; The check was more than 4 days late.

371

Depth of Knowledge (D.O.K.)

Lesson 13.4

Mathematical Processes

2 Skills/Concepts

1.D Multiple representations

3 Strategic Thinking

1.G Explain and justify arguments

14–16

2 Skills/Concepts

1.A Everyday life

17

2 Skills/Concepts

1.F Analyze relationships

18–23

2 Skills/Concepts

1.C Select tools

24

2 Skills/Concepts

1.C Select tools

25

3 Strategic Thinking

1.F Analyze relationships

26

3 Strategic Thinking

1.E Create and use representations

7–12 13

Additional Resources Differentiated Instruction includes: • Leveled Practice Worksheets

2. z ≤ -2 -5

Exercise

Name

Class

Date

Solve each inequality.

13.4 Independent Practice 6.9.B, 6.10.A, 6.10.B

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Solve each inequality. Graph and check your solution. q q≤7 7. - __ ≥ -1 7 0 1 2 3 4 5 6 7 8 9 10

x > -5

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

y 9. 0.5 ≤ __ 8

11. -12 > 2x

© Houghton Mifflin Harcourt Publishing Company

0

2

x>4 1 x < __ 20

21. 0.4 < -x

x ≥ 161 x < -0.4

x ≤ -30 23. - ___ 0.8

x ≥ 24

24. Use the order of operations to simplify the left side of the inequality below. What values of x make the inequality a true statement? - _12 (32 + 7)x > 32

- _12 (32 + 7)x > 32; - _12 (9 + 7)x > 32; - _12(16)x > 32; -8x > 32; x < -4. The solution would be all values less than -4.

FOCUS ON HIGHER ORDER THINKING

Work Area

25. Counterexamples John says that if one side of an inequality is 0, you don’t have to reverse the inequality symbol when you multiply or divide both sides by a negative number. Find an inequality that you can use to disprove John’s statement. Explain your thinking.

10 seconds or more

x < -6

If you divide both sides of -7z ≥ 0 by -7, you get z ≥ 0.

0 1 2

x≥3

x ≤ -0.5 12. - __ 6 -5 -4 - 3 - 2 - 1

14. A veterinarian tells Max that his cat should lose no more than 30 ounces. The veterinarian suggests that the cat should lose 7 ounces or less per week. What is the shortest time in weeks and days it would take Max’s cat to lose the 30 ounces?

15. The elevation of an underwater cave is -120 feet relative to sea level. A submarine descends to the cave. The submarine’s rate of change in elevation is no greater than -12 feet per second. How long will it take to reach the cave?

r < -6

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

8

1 22. 4x < __ 5

0

y≥4

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

Online Assessment and Intervention

4 weeks and 2 days

0 1 2 3 4 5 6 7 8 9 10

10. 36 < -6r

x < - __ 1 20. - __

x ≤ -23 19. - __ 7

0 1 2 3 4 5

13. Multistep Parav is playing a game in which he flips a counter that can land on either a -6 or a 6. He adds the point values of all the flips to find his total score. To win, he needs to get a score less than -48. a. Assuming Parav only gets -6s when he flips the counter, how many times does he have to flip the counter?

-6x < -48; x > 8

16. The temperature of a freezer is never greater than -2 °C. Yesterday the temperature was -10 °C, but it increased at a steady rate of 1.5 °C per hour. How long in hours and minutes did the temperature increase inside the freezer?

This is incorrect because if you choose a value from the possible solutions, such as z = 1, and substitute it into the original equation, you get -7 ≥ 0, which is not true.

less than 5 hours and 20 minutes 26. Communicate Mathematical Thinking Van thinks that the answer to -3x < 12 is x < -4. How would you convince him that his answer is incorrect?

17. Explain the Error A student's solution to the inequality -6x > 42 was x > -7. What error did the student make in the solution? What is the correct answer?

b. Suppose Parav flips the counter and gets five 6s and twelve -6s when he plays the game. Does he win? Explain.

No; his score is 5(6) +

I would remind him of the properties of inequality that we learned. I would show him that if you substitute -5 into the original inequality, the statement is not true.

The student did not reverse the inequality sign. The answer

Therefore his answer is not correct. Then I would show

should be x < -7.

him that the several solutions from x > -4 do hold true.

© Houghton Mifflin Harcourt Publishing Company

8. -12x < 60

x ≤ -9

18. 18 ≤ -2x

Personal Math Trainer

12(-6) = -42, which is not less than -48. Lesson 13.4

EXTEND THE MATH

PRE-AP

Activity Present the statements at the right. Challenge students to make the statements true by filling in the blanks with <, >, or =. For each choice, students should give an example to demonstrate that the statement is true. 1. > example: a = 3, b = 4, c = -2 (3)(-2) > (4)(-2); -6 > -8 2. < example: x = -5, y = -3, z = 2 (-5)(2) < (-3)(2); -10 < -6 3. = example: n = -3, m = 2, p = 0 (-3)(0) = (2)(0); 0 = 0

371

372

Activity available online

Unit 4

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Algebraically Speaking 1. If a < b and c < 0, then ac

bc.

2. If x < y and z > 0, then xz

yz.

3. If n < m and p = 0, then np

mp.

4. If f < 0 and g < 0, then fg

0.

4. > example: f = -6, g = -2 (-6)(-2) > 0; 12 > 0

Multiplication and Division Inequalities with Rational Numbers

372

DO NOT EDIT--Changes must be made through “File info” CorrectionKey=B

MODULE QUIZ

Assess Mastery

13.1 Writing Inequalities

Personal Math Trainer Online Assessment and Intervention

Write an inequality to represent each situation, then graph the solutions.

Use the assessment on this page to determine if students have mastered the concepts and standards covered in this module.

1. There are fewer than 8 gallons of gas in the tank. 0 1 2 3 4 5 6 7 8 9 10

3

Response to Intervention

2 1

0 1 2 3 4 5 6 7 8 9 10

Enrichment

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

Access Ready to Go On? assessment online, and receive instant scoring, feedback, and customized intervention or enrichment.

Personal Math Trainer

v ≤ -4

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Solve each inequality. Graph the solution. 4. c - 28 > -32

Online and Print Resources Differentiated Instruction

Differentiated Instruction

• Reteach worksheets

• Challenge worksheets

• Reading Strategies • Success for English Learners ELL

ELL

0 1 2 3 4 5 6 7 8 9 10

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

5. 0

v + 17 ≤ 20

0 1 2 3 4 5 6 7 8 9 10

6. Today’s high temperature of 80 °F is at least 16 ° warmer than yesterday’s high temperature. What was yesterday’s high temperature? 80

PRE-AP

Extend the Math PRE-AP Lesson activities in TE

Additional Resources Assessment Resources includes: • Leveled Module Quizzes

≥ 16 + T; T ≤ 64 °F

13.3, 13.4 Multiplication and Division Inequalities © Houghton Mifflin Harcourt Publishing Company

Online Assessment and Intervention

p≥3

2. There are at least 3 pieces of gum left in the pack.

3. The valley was at least 4 feet below sea level.

Intervention

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f<8

Solve each inequality. Graph the solution. 8. __a2 < 4

7. 7f ≤ 35 0 1 2 3 4 5 6 7 8 9 10

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

0 1 2 3 4 5 6 7 8 9 10 k <3 10. ___ -3

9. -25g ≥ 150 0

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

0

Module 13

6_MTXESE051676_U4M13RT.indd 373

Texas Essential Knowledge and Skills Lesson

Exercises

13.1

1–3

6.9.A, 6.9.B, 6.10.B

13.2

4–6

6.9.B, 6.9.C, 6.10.A, 6.10.B

13.3

7–10

6.9.B, 6.9.C, 6.10.A, 6.10.B

13.4

7–10

6.9.B, 6.10.A, 6.10.B

373

Module 13

TEKS

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MODULE 13 MIXED REVIEW

Texas Test Prep

Texas Test Prep

Item 6 Students can look at each graph and test the ending point to see if it makes the inequality true. Then, using the ≤ symbol, students can identify that choice B is the correct answer.

Selected Response 1. Em saves at least 20% of what she earns each week. If she earns \$140 each week for 4 weeks, which inequality describes the total amount she saves? A t > 112 B t ≥ 112 C

t < 28

D t ≤ 28

2. Which number line represents the inequality r > 6?

Avoid Common Errors

A

Item 1 If students do not read the item carefully, they may find the amount saved per week rather than the total amount saved. Remind them to read all items fully and carefully. Item 7 When students divide \$150 by \$12, they will get 12.5. Many students might insert this answer, not realizing that the question asks for full weeks, which requires the answer to be rounded up to 13. Remind students to read the questions carefully and be sure to understand what is being asked.

0 1 2 3 4 5 6 7 8 9 10

5. The number line below represents the solution to which inequality? 0 1 2 3 4 5 6 7 8 9 10 m A __ > 2.2 4

C

m __ > 2.5 3

B 2m < 17.6

D 5m > 40

6. Which number line shows the solution to w - 2 ≤ 8? A

0 1 2 3 4 5 6 7 8 9 10

B

0 1 2 3 4 5 6 7 8 9 10

C

0 1 2 3 4 5 6 7 8 9 10

B

0 1 2 3 4 5 6 7 8 9 10

Online Assessment and Intervention

D

0 1 2 3 4 5 6 7 8 9 10

C

0 1 2 3 4 5 6 7 8 9 10

Gridded Response

D

0 1 2 3 4 5 6 7 8 9 10

3. For which inequality below is z = 3 a solution? A z+5≥9

7. Hank needs to save at least \$150 to ride the bus to his grandparent’s home. If he saves \$12 a week, what is the least number of weeks he needs to save?

B z+5>9

3

0

0

0

0

0

0

D z+5<8

1

1

1

1

1

1

2

2

2

2

2

2

3

3

3

3

3

3

4

4

4

4

4

4

5

5

5

5

5

5

6

6

6

6

6

6

7

7

7

7

7

7

8

8

8

8

8

8

9

9

9

9

9

9

4. What is the solution to the inequality −6x < −18? A x>3 B x<3 C

x≥3

D x≤3

374

.

1

z+5≤8

C

© Houghton Mifflin Harcourt Publishing Company

Texas Testing Tip Students can use the given solutions to work backwards and find a solution. Item 3 Students are given the solution of 3 in the main problem. Instead of solving every problem to see which would give 3 as a true solution, the students can substitute 3 into each inequality. The choice with the true statement is the correct answer.

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Unit 4

Texas Essential Knowledge and Skills Items

Mathematical Process TEKS

1

6.10.A

6.1.A, 6.1.F

2

6.9.B

6.1.D, 6.1.E

3

6.10.B

6.1.D

4

6.10.A

6.1.D

5

6.9.B, 6.10.A

6.1.D

6

6.9.B, 6.10.A

6.1.D, 6.1.E

7*

6.3.D, 6.9.A, 6.10.A

6.1.A

* Item integrates mixed review concepts from previous modules or a previous course.

Inequalities and Relationships

374

Relationships in Two Variables ?

ESSENTIAL QUESTION How can you use relationships in two variables to solve real-world problems?

MODULE

You can use tables, graphs, and equations in two variables to model real-world problems, then use algebraic methods to solve the problems.

14

LESSON 14.1

Graphing on the Coordinate Plane 6.11

LESSON 14.2

Independent and Dependent Variables in Tables and Graphs 6.6.A, 6.6.C

LESSON 14.3

Writing Equations from Tables 6.6.B, 6.6.C

LESSON 14.4

© Houghton Mifflin Harcourt Publishing Company • Image Credits: Blickwinkel/Alamy

Representing Algebraic Relationships in Tables and Graphs 6.6.A, 6.6.B, 6.6.C

Real-World Video

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375

Module 14

A two-variable equation can represent an animal’s distance over time. A graph can display the relationship between the variables. You can graph two or more animals’ data to visually compare them.

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Math On the Spot

Animated Math

Personal Math Trainer

Go digital with your write-in student edition, accessible on any device.

Scan with your smart phone to jump directly to the online edition, video tutor, and more.

Interactively explore key concepts to see how math works.

Get immediate feedback and help as you work through practice sets.

375

Are You Ready? Use the assessment on this page to determine if students need intensive or strategic intervention for the module’s prerequisite skills. 2 1

Multiplication Facts EXAMPLE

Response to Intervention

1. 7 × 6

Enrichment

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5.

Skills Intervention worksheets

Differentiated Instruction

• Skill 36 Multiplication Facts

• Challenge worksheets

42

2. 10 × 9

90

3. 13 × 12

6.

x

1

2

3

4

y

7

14

21

28

156

4. 8 × 9

72

x

1

2

3

4

y

7

8

9

10

y is 7 times x.

Online and Print Resources

• Skill 69 Graph Ordered Pairs (First Quadrant)

Use a related fact you know. 7 × 7 = 49 Think: 8 × 7 = (7 × 7) + 7 = 49 + 7 = 56

8×7=

Write the rule for each table.

Access Are You Ready? assessment online, and receive instant scoring, feedback, and customized intervention or enrichment.

Online Assessment and Intervention

Online Assessment and Intervention

Multiply.

Intervention

Personal Math Trainer

my.hrw.com

7.

y is 6 more than x.

x

1

2

3

4

y

-5

-10

-15

-20

8.

x

0

4

8

12

y

0

2

4

6

y is -5 times x.

PRE-AP

y is one-half x.

Extend the Math PRE-AP Lesson Activities in TE

EXAMPLE

10 8

B

D

6 4

A

Start at the origin. Move 9 units right. Then move 5 units up. Graph point A(9, 5).

C

2 O

© Houghton Mifflin Harcourt Publishing Company

3

Personal Math Trainer

Complete these exercises to review skills you will need for this chapter.

E 2

4

6

8 10

Graph each point on the coordinate grid above.

9. B (0, 8)

376

10. C (2, 3)

11. D (6, 7)

12. E (5, 0)

Unit 4

PROFESSIONAL DEVELOPMENT VIDEO my.hrw.com

Author Juli Dixon models successful teaching practices as she explores graphing in the coordinate plan in an actual sixth-grade classroom.

Online Teacher Edition Access a full suite of teaching resources online—plan, present, and manage classes and assignments.

Professional Development

my.hrw.com

Interactive Answers and Solutions Customize answer keys to print or display in the classroom. Choose to include answers only or full solutions to all lesson exercises.

Interactive Whiteboards Engage students with interactive whiteboard-ready lessons and activities. Personal Math Trainer: Online Assessment and Intervention Assign automatically graded homework, quizzes, tests, and intervention activities. Prepare your students with updated, TEKS-aligned practice tests.

Relationships in Two Variables

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DO NOT EDIT--Changes must be made through "File info" CorrectionKey=B

Parts of the Algebraic Expression 14 + 3x Definition

Understand Vocabulary Use the following explanations to help students learn the preview words. A coordinate plane is formed by two number lines that intersect at right angles. Coordinate planes are used in geographic maps and for locating images on computer screens.

Students can use these reading and note-taking strategies to help them organize and understand new concepts and vocabulary.

Mathematical Representation

Review Word

A specific number whose value does not change

14

constant

A number that is multiplied by a variable in an algebraic expression

3

coefficient

A letter or symbol used to represent an unknown

x

variable

Preview Words

Understand Vocabulary Complete the sentences using the preview words.

1. The numbers in an ordered pair are © Houghton Mifflin Harcourt Publishing Company

Integrating the ELPS

✔ coefficient (coeficiente) ✔ constant (constante) equation (ecuación) negative number (número negativo) positive number (número positivo) scale (escala) ✔ variable (variable)

Use the ✔ words to complete the chart.

The chart helps students review vocabulary associated with algebraic expressions. Write additional expressions on the board and have students identify the parts of each expression.

Review Words

Visualize Vocabulary

Visualize Vocabulary

The two lines that make a coordinate plane are called the axes. The x-axis is the horizontal number line that runs left to right on the coordinate plane. The y-axis is the vertical line that runs up and down on the coordinate plane.

Vocabulary

coordinate plane 2. A lines that intersect at right angles.

coordinates

.

is formed by two number

axes (ejes) coordinate plane (plano cartesiano) coordinates (coordenadas) dependent variable (variable dependiente) independent variable (variable independiente) ordered pair (par ordenado) origin (origen) quadrants (cuadrantes) x-axis (eje x) x-coordinate (coordenada x) y-axis (eje y) y-coordinate (coordenada y)

c.4.D Use prereading supports such as graphic organizers, illustrations, and pretaught topic-related vocabulary to enhance comprehension of written text.

Layered Book Before beginning the module, create a layered book to help you learn the concepts in this module. Label each flap with lesson titles from this module. As you study each lesson, write important ideas such as vocabulary and formulas under the appropriate flap. Refer to your finished layered book as you work on exercises from this module.

Differentiated Instruction • Reading Strategies ELL

Module 14

6_MTXESE051676_U4MO14.indd 377

29/01/14 12:23 AM

Grades 6–8 TEKS Before Students understand: • how to recognize the difference between additive and multiplicative numerical patterns given in a table or graph • how to graph a relationship on a number line • how to identify and locate ordered pairs of whole numbers in the first quadrant

377

Module 14

In this module Students will learn to: • identify independent and dependent quantities from tables and graphs • write an equation that represents the relationship between independent and dependent quantities from a table • represent a given situation using verbal descriptions, tables, graphs, and equations in the form y = kx or y=x+b • graph points in all four quadrants using ordered pairs of rational numbers

377

After Students will connect: • tables and verbal descriptions with a linear relationship • graphs and equations with a linear relationship • ordered pairs with an equation

MODULE 14

Unpacking the TEKS

Unpacking the TEKS Understanding the TEKS and the vocabulary terms in the TEKS will help you know exactly what you are expected to learn in this module.

Use the examples on this page to help students know exactly what they are expected to learn in this module.

6.6.B Write an equation that represents the relationship between independent and dependent quantities from a table.

Texas Essential Knowledge and Skills Content Focal Areas

Key Vocabulary equation (ecuación) A mathematical sentence that shows that two expressions are equivalent.

Expressions, equations, and relationships—6.6 The student applies mathematical process standards to use multiple representations to describe algebraic relationships.

What It Means to You You will learn to write an equation that represents the relationship in a table. UNPACKING EXAMPLE 6.6.B

Emily has a dog-walking service. She charges a daily fee of \$7 to walk a dog twice a day. Create a table that shows how much Emily earns for walking 1, 6, 10, and 15 dogs. Write an equation that represents the situation. Dogs walked

1

6

10

15

Earnings (\$)

7

42

70

105

Earnings is 7 times the number of dogs walked. Let the variable e represent earnings and the variable d represent the number of dogs walked.

Measurement and data—6.11 The student applies mathematical process standards to use coordinate geometry to identify locations on a plane.

e=7×d

6.6.C Represent a given situation using verbal descriptions, tables, graphs, and equations in the form y = kx or y = x + b.

Go online to see a complete unpacking of the .

What It Means to You You can use words, a table, a graph, or an equation to model the same mathematical relationship.

Key Vocabulary

UNPACKING EXAMPLE 6.6.C

coordinate plane (plano cartesiano) A plane formed by the intersection of a horizontal number line called the x-axis and a vertical number line called the y-axis.

The equation y = 4x represents the total cost y for x games of miniature golf. Make a table of values and a graph for this situation.

Visit my.hrw.com to see all the unpacked.

my.hrw.com

Number of games, x Total cost (\$), y

1

2

3

4

4

8

12

16

y

Total cost (\$)

c.4.F Use visual and contextual support … to read grade-appropriate content area text … and develop vocabulary … to comprehend increasingly challenging language.

© Houghton Mifflin Harcourt Publishing Company • Image Credits: PhotoDisc/Getty Images

Integrating the ELPS

20 16 12 8 4 x O

2

4

6

8

Number of games my.hrw.com

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Lesson 14.1

Lesson 14.2

Unit 4

Lesson 14.3

Lesson 14.4

6.6.A Identify independent and dependent quantities from tables and graphs. 6.6.B Write an equation that represents the relationship between independent and dependent quantities from a table. 6.6.C Represent a given situation using verbal descriptions, tables, graphs, and equations in the form y = kx or y = x + b. 6.11 Graph points in all four quadrants using ordered pairs of rational numbers.

Relationships in Two Variables

378

LESSON

14.1 Graphing on the Coordinate Plane Engage

Texas Essential Knowledge and Skills

ESSENTIAL QUESTION

The student is expected to:

How do you locate and name points in the coordinate plane? Sample answer: Points in the coordinate plane are located and named by their locations from the origin along the x-axis first, followed by the y-axis. The order of the coordinates is important.

Measurement and data—6.11 Graph points in all four quadrants using ordered pairs of rational numbers.

Motivate the Lesson Ask: Have you ever tried to find a city or town by its location on a map grid? Maps are like a coordinate plane. Begin Example 1 to find out how to locate a point on a coordinate plane.

Mathematical Processes 6.1.E Create and use representations to organize, record, and communicate mathematical ideas.

Explore

Engage with the Whiteboard

Identify the coordinates of each point. Name the quadrant where each point is located.

To introduce students to the four-quadrant coordinate plane, sketch a simple “treasure map” with a coordinate plane on the whiteboard. Mark a point for “Start” at the origin and a point for “Treasure” in Quadrant I. Ask students to draw a path to the treasure using the grid lines and then to describe the path in words, such as, “Walk east 3 steps. Then walk north 5 steps.” Repeat several times with new coordinates for the “Treasure.”

y 2

B

x -4

O

-2

2

4

A

-2

EXAMPLE 1 Focus on Communication

Mathematical Processes Point out to students that coordinates describe a location in relation to the origin, so it is important to always start at the origin when identifying the coordinates of a point.

Point A: (3, -2), Quadrant IV; Point B: (-4, 1), Quadrant II Interactive Whiteboard Interactive example available online my.hrw.com

Students may give incorrect coordinates for a point because they transposed the x- and y-coordinates. Remind students that the x-coordinate is the first number in an ordered pair.

y

EXAMPLE 2

2

-2

R 2

x

4

P

Interactive Whiteboard Interactive example available online my.hrw.com

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Lesson 14.1

ELL c.2.C Some students may have difficulty remembering what the x- and y-coordinates mean in an ordered pair. Encourage students to think of plotting points as physical movements, run and jump. The x-coordinate tells how far to run to the right or left, and the y-coordinate tells how far to jump up or down. So, when plotting points, students should always “run before they jump.”

Connect Vocabulary

O

-2

Mathematical Processes • Is point (2, 3) the same as point (3, 2)? Explain. No, they are not the same point. Point (2, 3) lies 2 units to the right of the origin and 3 units up, while point (3, 2) lies 3 units to the right and 2 units up.

Avoid Common Errors

P(0, -2), Q(-4, 1.5), R(3.5, 0)

-4

Questioning Strategies

ADDITIONAL EXAMPLE 2 Graph and label each point on the coordinate plane.

Q

Explain

Questioning Strategies

Mathematical Processes • Describe how graphing the point (0, -3) is similar to graphing the point (-3, 0). How is it different? Sample answer: They are similar because you start at the origin and move three units to graph each point. They are different because in (0, -3), you move down from the origin. In (-3, 0), you move left from the origin.

14.1 ?

Graphing on the Coordinate Plane

Measurement and data—6.11 Graph points in all four quadrants using ordered pairs of rational numbers.

Reflect 1.

If both coordinates of a point are negative, in which quadrant is the

2.

Describe the coordinates of all points in Quadrant I.

point located?

ESSENTIAL QUESTION

Both coordinates are positive.

How do you locate and name points in the coordinate plane? 3.

Naming Points in the Coordinate Plane

The horizontal axis is called the x-axis.

The vertical axis is called the y-axis.

The point where the axes intersect is called the origin.

The two axes divide the coordinate plane into four quadrants.

left of the origin. The x-coordinate in (3, 5) is 3 which

y

The two number lines are called the axes.

Math On the Spot

2

x-axis x

-6

-4

-2

O

2 4 Origin

will be to the right of the origin.

my.hrw.com

4

y-axis

Personal Math Trainer

6

Online Assessment and Intervention

-4

my.hrw.com

4.

© Houghton Mifflin Harcourt Publishing Company

EXAMPL 1 EXAMPLE

Point A is 1 unit left of the origin, and 5 units down. It has x-coordinate -1 and y-coordinate -5, written (-1, -5). It is located in Quadrant III. Point B is 2 units right of the origin, and 3 units up. It has x-coordinate 2 and y-coordinate 3, written (2, 3). It is located in Quadrant I.

Math On the Spot my.hrw.com

2

(-1, -3); III

x -4

-2

O

2

4

-2 H -4

G

EXAMPLE 2

6.11 y 4

Point A is 5 units left and 2 units up from the origin.

x O

H

F

2

Graph and label each point on the coordinate plane. A(-5, 2), B(3, 1.5), C(0, -3)

B

2 -2

F

(-2, 4); II (3, 2); I

4

Points that are located on the axes are not located in any quadrant. Points on the x-axis have a y-coordinate of 0, and points on the y-axis have an x-coordinate of 0.

y

-4

(4, -4); IV

E

Graphing Points in the Coordinate Plane

6.11

4

G E

5.

The numbers in an ordered pair are called coordinates. The first number is the x-coordinate and the second number is the y-coordinate.

y

Identify the coordinates of each point. Name the quadrant where each point is located.

-6

An ordered pair is a pair of numbers that gives the location of a point on a coordinate plane. The first number tells how far to the right (positive) or left (negative) the point is located from the origin. The second number tells how far up (positive) or down (negative) the point is located from the origin.

Identify the coordinates of each point. Name the quadrant where each point is located.

Communicate Mathematical Ideas Explain why (-3, 5) represents a different location than (3, 5).

The x-coordinate in (-3, 5) is -3 which will be to the

A coordinate plane is formed by two number lines that intersect at right angles. The point of intersection is 0 on each number line. •

III

© Houghton Mifflin Harcourt Publishing Company

LESSON

4

Point B is 3 units right and 1.5 units up from the origin. Graph the point halfway between (3, 1) and (3, 2).

-2 -4 A

Point C is 3 units down from the origin. Graph the point on the y-axis. Lesson 14.1

379

380

A

2

B x

-4

-2

O -2

2

4

C

-4

Unit 4

PROFESSIONAL DEVELOPMENT Integrate Mathematical Processes This lesson provides an opportunity to address Mathematical Process TEKS 6.1.E, which calls for students to “create and use representations to organize, record, and communicate mathematical ideas.” Students use coordinate planes to locate points. Then students solve a real-world problem on a coordinate plane in which the scale on each axis represents a real-world situation. In this way, students are able to connect a coordinate plane to the real world.

Math Background The concept of the rectangular coordinate system is generally credited to French mathematician and philosopher René Descartes and, therefore, is sometimes referred to as the Cartesian plane. Every point on the plane can be located because all real numbers, not just integers, are used. The points represented by integer coordinates are sometimes called lattice points.

Graphing on the Coordinate Plane

380

YOUR TURN Avoid Common Errors Students may graph the points incorrectly by using the x- and y-coordinates in the wrong order. Remind students to run before they jump.

EXAMPLE 3

The graph shows the location of a fountain, a slide, and a sandbox in a park. The scale on each axis represents yards. Give the coordinates for the sandbox and use them to describe the location of the sandbox relative to the fountain.

Focus on Math Connections

Mathematical Processes Point out to students that the coordinate plane also indicates directions. The x-axis points east (to the right) and west (to the left) and the y-axis points north (up) and south (down). For Example 3, have students add the correct north, south, east, and west labels to the axes of the coordinate plane.

Questioning Strategies

Mathematical Processes • Describe the direction you would go from Gary’s house to Jen’s house. I would travel east to go from Gary’s house to Jen’s house.

y

• What would the coordinates of Gary’s house be if he lived 30 miles directly east of Jen? Explain. (35, 15); Jen lives at (5, 15), so 30 miles directly east would be (35, 15).

20

10

x

Fountain -20

Slide

O

-10

-10

10

20

Sandbox

-20

The sandbox is at (10, -15). This is 10 yards east and 15 yards south of the fountain. Interactive Whiteboard Interactive example available online my.hrw.com

YOUR TURN Focus on Math Connections

Mathematical Processes Show students how to translate directions to movements by using the axes. Remind students that “20 miles south” is on the y-axis below the origin and that “20 miles west” is on the x-axis to the left of the origin. Both movements are in a negative direction, so the coordinates of Ted’s home are (-20, -20). Since Ned lives “50 miles directly north of Ted’s house,” only the y-coordinate changes. Because north is a positive direction, -20 + 50 = 30. So, Ned’s home is located at (-20, 30).

Elaborate Talk About It Summarize the Lesson Have students complete a graphic organizer that shows the number of each quadrant and the signs of the coordinates in each quadrant. y

Q __ II Q __ I - __) + (__, + __) + (__,

x

O

III IV Q __ Q __ - __) - (__, + __) (__,

GUIDED PRACTICE Engage with the Whiteboard For Exercises 1–2, have students fill in the blanks by identifying the coordinates of each point on the whiteboard. Then have them name the quadrant where each point is located. Also ask students to describe how they would graph each point.

Avoid Common Errors Exercises 3–4, 6–7 Students may graph the points incorrectly by using the x- and y-coordinates in the wrong order. Remind students to run before they jump.

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Lesson 14.1

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DO NOT EDIT--Changes must be made through “File info” CorrectionKey=A

6.

P(-4, 2)

7.

Q(3, 2.5)

8.

R(-4.5, -5)

9. 10.

S(4, -5)

4

P -4

2

T

Online Assessment and Intervention

x

-2

O

2

left

1. Point A is 5 units

my.hrw.com

4

1 unit

-2

up

S

2

2. Point B is

T(-2.5, 0)

and

3

.

2

A -4

(2, -3)

IV

-2

D

O

2

x

4

-2 B

-4

.

Each grid square is _12 unit on a side.

y

Gary

Jen

-20 -10

x

City O 10

Each grid square is 5 miles on a side.

-10

Gary’s house is at (-25, 15), which is 25 miles west and 15 miles north of the city.

-20

20

N E

W

( (

S

)

-1

O -1 -2

1

2

B

The first number, the x-coordinate, tells how many units to the right or left the point is located from the origin. The second number, the y-coordinate, tells how many units up or down the point is located from the origin.

? ?

How are north, south, east, and west represented on the graph in Example 3?

ESSENTIAL QUESTION CHECK-IN

9. Give the coordinates of a point that could be in each of the four quadrants, a point on the x-axis, and an point on the y-axis.

Use the graph in the Example. Ted lives 20 miles south and 20 miles west of the city represented on the graph in Example 3. His brother Ned lives 50 miles north of Ted’s house. Give the coordinates of each brother’s house.

Personal Math Trainer

(5, -2). x-axis: (-3, 0); y-axis: (0, 5).

Online Assessment and Intervention

my.hrw.com

Lesson 14.1

6_MTXESE051676_U4M14L1.indd 381

x -2

Mathematical Processes

Jen’s house is located 6 grid squares to the right of Gary’s house. Since each grid square is 5 miles on a side, her house is 6 · 5 = 30 miles east of Gary’s.

Ted (-20, -20), Ned (-20, 30)

2 1

8. Vocabulary Describe how an ordered pair represents a point on a coordinate plane. Include the terms x-coordinate, y-coordinate, and origin in your answer.

Math Talk

B Describe the location of Jen’s house relative to Gary’s house.

)

6. Plot point A at -_12 , 2 . 7. Plot point B at 2 _12 , -2 .

North and south are the positive and negative directions along the y-axis; east and west are the positive and negative directions on the x-axis.

10

y

A

5. Describe the scale of the graph.

6.11

20

4. Point D at (5, 0)

For 5–7, use the coordinate plane shown. (Example 3)

Math On the Spot my.hrw.com

EXAMPL 3 EXAMPLE

© Houghton Mifflin Harcourt Publishing Company

II

units right of the origin

3. Point C at (-3.5, 3)

The scale of an axis is the number of units that each grid line represents. So far, the graphs in this lesson have a scale of 1 unit, but graphs frequently use other units.

11.

Graph and label each point on the coordinate plane above. (Example 2)

A Use the scale to describe Gary’s location relative to the city.

(-5, 1)

4

C

units down from the origin.

Its coordinates are

The graph shows the location of a city. It also shows the location of Gary’s and Jen’s houses. The scale on each axis represents miles.

y

of the origin and

from the origin.

Its coordinates are

-4

R

Identify the coordinates of each point in the coordinate plane. Name the quadrant where each point is located. (Example 1)

Personal Math Trainer

Q

© Houghton Mifflin Harcourt Publishing Company

Graph and label each point on the coordinate plane.

Guided Practice

y

381

29/01/14 12:30 AM

382

Unit 4

6_MTXESE051676_U4M14L1.indd 382

10/30/12 9:55 AM

DIFFERENTIATE INSTRUCTION Curriculum Integration

Cooperative Learning

Have students draw coordinate grid lines on maps of Texas. Instruct students to draw the x- and y-axes through the state capital and the other lines at __12 -inch increments above, below, to the left, and to the right of the axes. Have the students label the grid lines, beginning with the axes, with the appropriate numbers and give coordinates for various cities and towns on the map.

Have students work in three teams to play coordinate tic-tac-toe. Use a coordinate plane that is 5 units from the origin in all directions. One player on each team alternates calling out the coordinates of a point. Another player on each team locates the point and marks it on the coordinate plane. The first team to place three marks in an uninterrupted row horizontally, vertically, or diagonally wins the round.

Differentiated Instruction includes: • Reading Strategies • Success for English Learners ELL • Reteach • Challenge PRE-AP

Graphing on the Coordinate Plane

382

Personal Math Trainer Online Assessment and Intervention

Evaluate GUIDED AND INDEPENDENT PRACTICE

Online homework assignment available

6.11

my.hrw.com

14.1 LESSON QUIZ 6.11 Use the coordinate plane shown. Each unit represents 1 city block. y Henry

Concepts & Skills

Practice

Example 1 Naming Points in the Coordinate Plane

Exercises 1–2, 10–13

Example 2 Graphing Points in the Coordinate Plane

Exercises 3–4, 10–13

Example 3 Reading Scales on Axes

Exercises 5–7, 14–15

4 2

x -4

-2

O

Emma -2 Ice cream -4 shop

2

4

Library

1. Write the ordered pairs that represent Henry and the library. 2. Describe Henry’s location relative to the library. 3. Henry wants to meet his friend Emma at an ice cream shop before they go to the library. The ice cream shop is 7 blocks west of the library. Plot and label a point representing the ice cream shop. What are the coordinates of the point? 4. Emma describes her current location: “I’m directly west of the library, halfway to the ice cream shop.” Plot and label a point representing Emma’s location. What are the coordinates of the point? Lesson Quiz available online my.hrw.com

1. Henry (-3, 4), Library (2, -3) 2. Henry is 5 blocks west and 7 blocks north of the library. 3. (-5, -3) 4. (-1.5, -3)

383

Lesson 14.1

Exercise

Depth of Knowledge (D.O.K.)

Mathematical Processes

10

2 Skills/Concepts

1.A Everyday life

11

3 Strategic Thinking

1.F Analyze relationships

12–13

2 Skills/Concepts

1.A Everyday life

14–15

2 Skills/Concepts

1.D Multiple representations

16

3 Strategic Thinking

1.F Analyze relationships

17

3 Strategic Thinking

1.G Explain and justify arguments

18–19

3 Strategic Thinking

1.F Analyze relationships

Additional Resources Differentiated Instruction includes: • Leveled Practice Worksheets

Class

Date

14.1 Independent Practice 6.11

my.hrw.com

17. Critical Thinking Choose scales for the coordinate plane shown so that you can graph the points J(2, 40), K(3, 10), L(3, -40), M(-4, 50), and N(-5, -50). Explain why you chose the scale for each axis.

Online Assessment and Intervention

For 10–13, use the coordinate plane shown. Each unit represents 1 kilometer. 10. Write the ordered pairs that represent the location of Sam and the theater.

FOCUS ON HIGHER ORDER THINKING

Personal Math Trainer

Theater

Sam: (4, 2); Theater: (-3, 5)

4

11. Describe Sam’s location relative to the theater.

Sam is 3 km south and 7 km east of the

Beth -4

-2

theater.

Sam

2 2

S

(-3, -4)

-40

L

down (in a negative direction). 19. Represent Real-World Problems Zach graphs some ordered pairs in the coordinate plane. The x-values of the ordered pairs represent the number of hours since noon, and the y-values represent the temperature at that time.

y V

© Houghton Mifflin Harcourt Publishing Company

N

4

direction) along the x-axis. Then count 12 grid squares

For 14–15, use the coordinate plane shown. 14. Find the coordinates of points T, U, and V.

U 1.0

T (0.75, -1.0); U (0.75, 1.25); V (-0.75, 1.25)

0.5

Quadrants I and IV; time is always positive, but

x -1.0 -0.5 O

0.5

temperatures can be positive or negative. In

1.0

-0.5

W

16. Explain the Error Janine tells her friend that ordered pairs that have an x-coordinate of 0 lie on the x-axis. She uses the origin as an example. Describe Janine’s error. Use a counterexample to explain why Janine’s statement is false.

each grid square. On the

2

Count 18 grid squares to the right (in a positive

(-3, 0.5)

W(-0.75, -1.0)

K

18. Communicate Mathematical Ideas Edgar wants to plot the ordered pair (1.8, -1.2) on a coordinate plane. On each axis, one grid square equals 0.1. Starting at the origin, how can Edgar find (1.8, -1.2)?

13. Beth describes her current location: “I’m directly south of the theater, halfway to the restaurant.” Plot and label a point representing Beth’s location. What are the coordinates of the point?

15. Points T, U, and V are the vertices of a rectangle. Point W is the fourth vertex. Plot point W and give its coordinates.

I used a scale of 1 unit for

-4 -2 -20

O

y-coordinates ranged from -50 to 50.

E

W

Restaurant

20

The x-coordinates ranged from -5 to 3, and the

N

-4

J

units for each grid square.

4

-2

12. Sam wants to meet his friend Beth at a restaurant before they go to the theater. The restaurant is 9 km south of the theater. Plot and label a point representing the restaurant. What are the coordinates of the point?

40

y-axis I used a scale of 10

x O

M

x

y

Work Area y

quadrants I and IV, the x-coordinate is always positive,

T

-1.0

but the y-coordinate can be positive or negative.

b. In what part of the world and at what time of year might Zach collect data so that the points he plots are in Quadrant IV?

Janine is describing points that lie on the y-axis. Ordered pairs that lie on the x-axis have a y-coordinate of 0. The origin lies on the x- and y-axis. Any

Sample answer: in a region with a cold climate during

other point with an x-coordinate of 0 lies on the y-axis such as (0, 3).

the winter

Lesson 14.1

EXTEND THE MATH

PRE-AP

383

Activity available online

© Houghton Mifflin Harcourt Publishing Company

Name

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Activity Plot the points for each set of ordered pairs below. Then connect the points in the order shown to reveal a figure. Name the figure and find its area. Set 1: (2, 5), (2, -1), (-3, -1), (-3, 5) Set 2: (-4, -3), (6, -3), (6, 4) Set 3: (1, 3), (-4, 3), (-4, -2), (1, -2) Write the coordinates for another set of points that form a figure. Find its area. Then challenge a classmate to draw the figure and find its area. Set 1: rectangle, A = 30 square units Set 2: triangle, A = 35 square units Set 3: square, A = 25 square units

Graphing on the Coordinate Plane

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LESSON

14.2

Independent and Dependent Variables in Tables and Graphs

Texas Essential Knowledge and Skills The student is expected to: Expressions, equations, and relationships—6.6.A Identify independent and dependent quantities from tables and graphs. Expressions, equations, and relationships—6.6.C Represent a given situation using verbal descriptions, tables, graphs, and equations in the form y = kx or y = x + b.

Mathematical Processes 6.1.D Communicate mathematical ideas, reasoning, and their implications using multiple representations, including symbols, diagrams, graphs, and language as appropriate.

Engage ESSENTIAL QUESTION How can you identify independent and dependent quantities from tables and graphs? Sample answer: The dependent variable is the quantity that depends on the other variable. On a graph, the independent variable is shown on the horizontal axis and the dependent variable is shown on the vertical axis.

Motivate the Lesson Ask: What is the relationship between the amount of time a person works and the amount of money that person earns? Begin the Explore Activity to find out what independent and dependent quantities are and how to recognize them.

Explore EXPLORE ACTIVITY 1 Connect to Vocabulary

ELL

Have students describe the meaning of the following phrases: Sample answers are given. • independently wealthy doesn’t need to work for money • Independence Day day of freedom • working independently doesn’t need help to do a job • dependent child needs a parent • insulin-dependent needs insulin daily • dependent clause cannot stand alone in a sentence Then ask students to define independent and dependent variables. An independent variable stands alone and isn’t changed by the other variables. A dependent variable depends on or is changed by another variable.

Explain EXPLORE ACTIVITY 2 Connect Multiple Representations Mathematical Processes Have students complete this table, which represents the situation about the art teacher and clay, to reinforce that both a table and a graph can represent this relationship. Clay bought by teacher (lb)

0

10

20

30

Clay available for classes (lb)

20

30

40

50

Engage with the Whiteboard Ask a student volunteer to locate the point on the graph that shows the 50 pounds of clay that is available for the art class. Have the volunteer draw a line from the y-axis to the point and a line from the point to the x-axis. Have students repeat this process for several more values.

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LESSON

14.2 ?

Independent and Dependent Variables in Tables and Graphs

ESSENTIAL QUESTION

EXPLORE ACTIVITY (cont’d)

Expressions, equations, and relationships—6.6.A Identify independent and dependent quantities from tables and graphs. Also 6.6.C.

Reflect

1. Analyze Relationships Describe how the value of the independent variable is related to the value of the dependent variable. Is the relationship additive or multiplicative?

The value of y is always 50 times the value of x; multiplicative. 2. What are the units of the independent variable and of the dependent variable?

How can you identify independent and dependent quantities from tables and graphs?

independent variable: hours; dependent variable: miles.

6.6.A

3. A rate is used in the equation. What is the rate?

Identifying Independent and Dependent Quantities from a Table

50 miles per hour

Many real-world situations involve two variable quantities in which one quantity depends on the other. The quantity that depends on the other quantity is called the dependent variable, and the quantity it depends on is called the independent variable.

EXPLORE ACTIVITY 2

A freight train moves at a constant speed. The distance y in miles that the train has traveled after x hours is shown in the table. 0

1

2

3

Distance y (mi)

0

50

100

150

Identifying Independent and Dependent Variables from a Graph In Explore Activity 1, you used a table to represent a relationship between an independent variable (time) and a dependent variable (distance). You can also use a graph to show a relationship of this sort.

A What are the two quantities in this situation?

time and distance

An art teacher has 20 pounds of clay but wants to buy more clay for her class. The amount of clay x purchased by the teacher and the amount of clay y available for the class are shown on the graph.

© Houghton Mifflin Harcourt Publishing Company

Which of these quantities depends on the other?

Distance depends on time. time, x

What is the independent variable?

A If the teacher buys 10 more pounds of clay, how many

distance, y

What is the dependent variable?

B How far does the train travel each hour?

pounds will be available for the art class?

50 miles

=

Distance traveled per hour

=

50

↓ y

·

30 lb; find the point on the graph with a

Time (hours)

y-coordinate of 50. Then find the x-coordinate of this point, which is 30.

x

Lesson 14.2

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Clay Used in Art Class

lb

How can you use the graph to find this information?

↓ ·

30

If the art class has a total of 50 pounds of clay available, how many pounds of clay did the teacher buy?

The relationship between the distance traveled by the train and the time in hours can be represented by an equation in two variables. Distance traveled (miles)

© Houghton Mifflin Harcourt Publishing Company • Image Credits: © Hill Street Studios/Corbis

Time x (h)

6.6.A

y

Clay available for classes (Ib)

EXPLORE ACTIVITY 1

80 60 40 20 O

x 20 40 60 80

Clay bought by teacher (Ib)

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PROFESSIONAL DEVELOPMENT Integrate Mathematical Processes This lesson provides an opportunity to address Mathematical Process TEKS 6.1.D, which calls for students to “communicate mathematical ideas… using multiple representations…as appropriate.” Students use tables, graphs, equations, and language to describe and model relationships between independent and dependent variables. In this way, students use multiple representations to model real-world situations involving independent and dependent variables.

Math Background Although the term function is not mentioned in this lesson, the tables in the lesson represent functions. A function is a rule that relates two quantities so that each input value corresponds to one output value exactly. When y is a function of x, x is called the independent variable and y is called the dependent variable. Whenever a value is assigned to x, a value is automatically assigned to y by an applicable rule or correspondence.

Independent and Dependent Variables in Tables and Graphs

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EXPLORE ACTIVITY 2 CONTINUED Questioning Strategies

Mathematical Processes • Why does the graph show only Quadrant I? Negative amounts do not make sense in this situation, so the values and the graph are limited to positive x- and y-values. • Why does the graph start at (0, 20)? The art teacher had 20 pounds of clay to start with.

• As the x-value is increasing, what is happening to the y-value? The y-value is also increasing.

ADDITIONAL EXAMPLE 1 A The table below shows a relationship between two variables, x and y. Describe a possible situation the table could represent. Describe the independent and dependent variables in this situation. Independent variable, x

1

2

Dependent variable, y

8

16 24 32

3

4

Sample answer: The table could represent the amount a person earns at a rate of \$8 per hour. The independent variable, x, is the number of hours the person works. The dependent variable, y, is the total earnings. B The graph below shows a relationship between two variables, x and y. Describe a possible situation the graph could represent. Describe the independent and dependent variables. y 6

2

x 2

4

6

Sample answer: The graph could represent the progress of a rock climber, starting at a 2-foot height and continuing at a pace of 1 foot every second. The independent variable is the number of seconds, and the dependent variable is the total number of feet climbed after x seconds. Interactive Whiteboard Interactive example available online my.hrw.com

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Engage with the Whiteboard Have a volunteer sketch a graph of the relationship shown in the table in A. Have a second volunteer make a table of the relationship shown in the graph in B. This will help students to see that both a table and a graph can represent the same relationship.

Connect Multiple Representations Mathematical Processes Point out to students that each of these situations can be represented by a verbal description, a table, a graph, or an equation. Questioning Strategies • How can the relationship in A be represented by an equation? The table begins with a y-value of 10, so the y-value always will be 10 units greater than the x-value. Then, as x increases by 1, y also increases by 1, resulting in the equation y = x + 10. • How would you describe the relationship in A? Explain. The relationship is an additive relationship because the value of y is always 10 units greater than the value of x. • How can the relationship in B be represented by an equation? The graph begins at the origin, so both variables begin at 0. Then, as x increases by 1, y increases by 12, resulting in the equation y = 12x. • How could you check that the equation is correct for either A or B? Pick a point from either the table or the graph and substitute it into the equation. The result should be a true equation.

Focus on Reasoning

Mathematical Processes Ask students to identify the independent and dependent quantities in the following situations. • A veterinarian must weigh an animal before determining the amount of medication it needs. independent quantity, weight of animal, dependent quantity: amount of medication

4

O

EXAMPLE 1

Lesson 14.2

• A company charges \$10 per hour to rent a jackhammer. independent quantity: time, dependent quantity: cost c.4.D ELL Encourage English learners to use the active reading strategies presented at the beginning of the module.

Integrating the ELPS

DO NOT EDIT--Changes must be made through “File info” CorrectionKey=B

DO NOT EDIT--Changes must be made through “File info” CorrectionKey=A

Describing Relationships Between Independent and Dependent Variables

B What are the two quantities in this situation?

the clay bought by the teacher and the amount of clay available to the class Math On the Spot

Which of these quantities depends on the other?

my.hrw.com

The amount of clay available to the class depends on the amount of clay bought by the teacher.

Thinking about how one quantity depends on another helps you identify which quantity is the independent variable and which quantity is the dependent variable. In a graph, the independent variable is usually shown on the horizontal axis and the dependent variable on the vertical axis.

EXAMPLE 1

clay bought by teacher

A The table shows a relationship between two variables, x and y. Describe a possible situation the table could represent. Describe the independent and dependent variables in the situation.

clay available for the class

What is the dependent variable?

C The relationship between the amount of clay purchased by the teacher and the amount of clay available to the class can be represented by an equation in two variables. Amount of clay available (pounds) ↓ y

=

Amount of clay Current amount + purchased (pounds) of clay (pounds) ↓

=

+

20

x

2

3

11

12

13

B The graph shows a relationship between two variables. Describe a possible situation that the graph could represent. Describe the independent and dependent variables.

Reflect

© Houghton Mifflin Harcourt Publishing Company

1

10

The independent variable, x, is the number of days she has been adding money to her savings. The dependent variable, y, is her savings after x days.

value of x.

y 36 24 12

As x increases by 1, y increases by 12. The relationship O is multiplicative. The value of y is always 12 times the value of x.

4. In this situation, the same units are used for the independent and dependent variables. How is this different from the situation involving the train in the first Explore?

x 2

4

6

The graph could represent the number of eggs in cartons that each hold 12 eggs.

The other situation involves two different units

The independent variable, x, is the number of cartons. The dependent variable, y, is the total number of eggs.

(miles and hours). 5. Analyze Relationships Tell whether the relationship between the independent variable and the dependent variable is a multiplicative or an additive relationship.

Reflect 7. What are other possible situations that the table and graph in Example 1 could represent?

Sample answer: Table: Paul has 10 DVDs and buys more. Independent: number of DVDs he buys; dependent:

6. What are the units of the independent variable, and what are the units of the dependent variable? ; dependent variable:

0

Dependent variable, y

The table could represent Jina’s savings if she starts with \$10 and adds \$1 to her savings every day.

The value of y is always 20 units greater than the

pounds

Independent variable, x

As x increases by 1, y increases by 1. The relationship is additive. The value of y is always 10 units greater than the value of x.

D Describe in words how the value of the independent variable is related to the value of the dependent variable.

independent variable:

6.6.A

© Houghton Mifflin Harcourt Publishing Company

What is the independent variable?

number he has after he buys x DVDs. Graph: 12 photos

pounds

fit on each page of a yearbook. Independent: number of pages; dependent: total number of photos on x pages. Lesson 14.2

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DIFFERENTIATE INSTRUCTION Curriculum Integration

Cooperative Learning

Music: The notes you hear played by a musical instrument are an example of a dependent relationship. For example, a clarinet’s pitch at a particular moment depends on the number of holes covered by the musician. A harp’s pitch depends on the length of the string being plucked.

One way to remember which is the independent variable and which is the dependent variable is to use the names of the two variables in a sentence that makes sense. For example:

Differentiated Instruction includes: • Reading Strategies • Success for English Learners ELL • Reteach • Challenge PRE-AP

Dollars Earned depends on Hours Worked, but Hours Worked does not depend on Dollars Earned. So, Dollars Earned must be the dependent variable and Hours Worked must be the independent variable.

Independent and Dependent Variables in Tables and Graphs

388

YOUR TURN Avoid Common Errors If students have difficulty distinguishing between independent and dependent variables, remind them that the independent variable causes a change in the dependent variable, while the dependent variable could not cause a change in the independent variable.

Elaborate Talk About It Summarize the Lesson Ask: How do you know which is the dependent variable and which is the independent variable in a table or graph? In a table, the independent variable usually is represented by the variable x. The dependent variable usually is represented by the variable y. On a graph, the independent variable usually is shown on the horizontal axis and the dependent variable on the vertical axis.

GUIDED PRACTICE Engage with the Whiteboard For Exercises 1–2, have a student sketch the graph to represent the table on the whiteboard. Have the student explain how to know which quantity should be represented by the x-axis and which by the y-axis. For Exercise 3, have a student volunteer label the axes in the graph to represent the real-world situation suggested by the student.

Avoid Common Errors Exercise 3 If students have difficulty determining whether a relationship is additive or multiplicative, remind them that in a multiplicative relationship the graph will pass through the origin, but in an additive relationship the graph will not pass through the origin. Exercise 4 Remind students that if the independent variable is on the horizontal axis of a graph, the dependent variable is on the vertical axis of the graph.

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Lesson 14.2

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DO NOT EDIT--Changes must be made through “File info” CorrectionKey=B

Guided Practice

8.

x

0

1

2

3

y

15

16

17

18

Personal Math Trainer Online Assessment and Intervention

1. A boat rental shop rents paddleboats for a fee plus an additional cost per hour. The cost of renting for different numbers of hours is shown in the table.

0

1

2

3

Cost (\$)

10

11

12

13

What is the independent variable, and what is the dependent variable? How do you know? (Explore Activity 1)

my.hrw.com

Sample answer: Bridget’s grandmother gave her a

Time is the independent variable and cost is the

collection of 15 perfume bottles. Bridget adds one

dependent variable, because cost depends on the

bottle per week to the collection. The independent

number of hours rented.

variable is the number of weeks. The dependent variable

2. A car travels at a constant rate of 60 miles per hour. (Explore Activity 1)

is the number of perfume bottles in her collection. The

a. Complete the table.

value of y is always 15 units greater than the value of x. 9.

x

0

1

2

3

4

y

0

16

32

48

64

1

2

3

0

60

120

180

dependent variable. c. Describe how the value of the dependent variable is related to the value of the independent variable.

The value of y is always 60 times the value of x.

profit per T-shirt. The independent variable is the

Use the graph to answer the questions.

number of T-shirts he sells, and the dependent variable

3. Describe in words how the value of the dependent variable is related to the value of the independent variable. (Explore Activity 2)

is his profit in dollars. The value of y is always 16 times

The dependent variable is 5 times the value of the

the value of x.

independent variable.

y 18

4. Describe a real-world situation that the graph could represent. (Example 1)

12 6

O

0

Distance y (mi)

Time is the independent variable and distance is the

that are printed with funny slogans. He makes a \$16

10.

Time x (h)

b. What is the independent variable, and what is the dependent variable?

Sample answer: Colin created a website to sell T-shirts

© Houghton Mifflin Harcourt Publishing Company

Time (hours)

4

20 10

O

x 2

4

6

Sample answer: The graph could represent the total

x 2

y 30

cost y of buying x carnival tickets for \$5 each.

6

Sample answer: Tickets to the school musical cost \$3

? ?

each. The independent variable is the number of tickets

ESSENTIAL QUESTION CHECK-IN

5. How can you identify the dependent and independent variables in a real-world situation modeled by a graph?

purchased, and the dependent variable is the total cost. The value of y is always 3 times the value of x.

© Houghton Mifflin Harcourt Publishing Company

Describe a real-world situation that the variables could represent. Describe the relationship between the independent and dependent variables.

Sample answer: The dependent variable is the quantity that depends on the other variable. On a graph, the independent variable is usually shown on the horizontal axis and the dependent variable on the vertical axis. Lesson 14.2

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Independent and Dependent Variables in Tables and Graphs

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Personal Math Trainer Online Assessment and Intervention

Online homework assignment available

Evaluate GUIDED AND INDEPENDENT PRACTICE 6.6.A, 6.6.C

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14.2 LESSON QUIZ 6.6.A The graph below shows the relationship between the number of tickets Lisa is ordering for a raffle and the cost. Use the graph to answer questions 1–3.

Total Cost (including postage)

Lisa’s Ticket Order

Concepts & Skills

Practice

Explore Activity 1 Identifying Independent and Dependent Quantities from a Table

Exercises 1–2, 7

Explore Activity 2 Identifying Independent and Dependent Variables from a Graph

Exercises 3, 6, 8

Example 1 Describing Relationships Between Independent and Dependent Variables

Exercises 4, 6

y 12

Exercise

8

Depth of Knowledge (D.O.K.)

Mathematical Processes

6

2 Skills/Concepts

1.A Everyday life

7

3 Strategic Thinking

1.F Analyze relationships

Number of Tickets

8–9

3 Strategic Thinking

1.G Explain and justify arguments

1. What are the dependent and independent variables?

10

3 Strategic Thinking

1.F Analyze relationships

4

x O

2

4

6

2. Is the relationship between the two variables additive or multiplicative? 3. Describe the relationship between the two quantities in words.

Additional Resources Differentiated Instruction includes: • Leveled Practice Worksheets

Use the table for question 4. x

0

1

2

3

4

y

0

7

14

21

28

4. Describe a possible situation that can be represented by the table. Identify the dependent and independent variables in this situation. Lesson Quiz available online my.hrw.com

Answers 1. Independent variable, x, is the number of tickets bought; dependent variable, y, is the total cost. 2. It is an additive relationship. 3. The total cost is the number of tickets bought plus \$5 for postage for the tickets.

391

Lesson 14.2

4. Sample answer: Parking at the airport costs \$7 per day. Independent variable, x, is the number of days a vehicle is parked; dependent variable, y, is the total cost for parking.

DO NOT EDIT--Changes must be made through “File info” CorrectionKey=B

Class

Date

14.2 Independent Practice

8. Ty borrowed \$500 from his parents. The graph shows how much he owes them each month if he pays back a certain amount each month.

Personal Math Trainer

6.6.A, 6.6.C

a. How many hours did the soccer team practice before the season began?

6 hours b. What are the two quantities in this situation?

hours practiced during the season and total

Total practice time for year (hours)

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6. The graph shows the relationship between the hours a soccer team practiced after the season started and their total practice time for the year.

a. Describe the relationship between the number of months and the amount Ty owes. Identify an independent and dependent variable and explain your thinking.

Online Assessment and Intervention

y 10

Ty starts out owing \$500 and every

8 6 4 2

400 300 200 100

O

2

8 10

b. How long will it take Ty to pay back his parents?

10 months FOCUS ON HIGHER ORDER THINKING

d. Analyze Relationships Describe the relationship between the quantities in words.

Work Area

9. Error Analysis A discount store has a special: 8 cans of juice for a dollar. A shopper decides that since the number of cans purchased is 8 times the number of dollars spent, the cost is the independent variable and the number of cans is the dependent variable. Do you agree? Explain.

Total practice time for the year is 6 hours more than practice time during the season.

Sample answer: I disagree because the amount

e. Is the relationship between the variables additive or multiplicative? Explain.

a shopper pays depends on the number of cans

Additive; the total practice increases by 1 hour as the

purchased. So, the number of cans is the independent

practice time during the season increases by 1 hour.

variable, and cost is the dependent variable.

b. What is the independent variable?

6

owes.

dependent: total practice time for year

a. What is the dependent variable?

4

Months

months; dependent variable: amount he

4

independent: hours practiced during season;

7. Multistep Teresa is buying glitter markers to put in gift bags. The table shows the relationship between the number of gift bags and the number of glitter markers she needs to buy.

2

by \$50; independent variable: number

x

O

c. What are the dependent and independent variables?

© Houghton Mifflin Harcourt Publishing Company

500

month the amount he owes decreases

Practice time during the season (hours)

practice time for year

Ty’s Loan Payments

0

1

2

3

Number of markers, y

0

5

10

15

10. Analyze Relationships Provide an example of a real-world relationship where there is no clear independent or dependent variable. Explain.

Sample answer: Andrea is 4 years older than Lisa. You

number of markers number of gift bags

could say that Andrea’s age depends on Lisa’s because you can add 4 to Lisa’s age. You can also say that Lisa’s

c. Describe the relationship between the quantities in words.

© Houghton Mifflin Harcourt Publishing Company

Name

Amount Ty owes (dollars)

DO NOT EDIT--Changes must be made through “File info” CorrectionKey=A

age depends on Andrea’s age because you can subtract

The number of glitter markers is 5 times the number of gift bags.

4 from Andrea’s age.

d. Is the relationship additive or multiplicative? Explain.

The relationship is multiplicative because y increases by a factor of 5 as x increases by 1. Lesson 14.2

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EXTEND THE MATH

3/11/13 9:40 AM

PRE-AP

Introduce students to independent and dependent variables in situations that involve decimals or fractions. For example: Gina is charged \$0.15 for each text message that she sends. 1. What is the independent variable? a a. number of texts sent b. charge per text c. total amount charged for texting

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Activity available online 2. What is the dependent variable?

29/01/14 12:52 AM

my.hrw.com

c

a. number of texts sent b. charge per text c. total amount charged for texting 3. Write an equation that expresses the situation. Let b be the amount of Gina’s bill. Let s be the number of texts sent. b = 0.15s

Independent and Dependent Variables in Tables and Graphs

392

LESSON

14.3 Writing Equations from Tables Texas Essential Knowledge and Skills The student is expected to: Expressions, equations, and relationships—6.6.B Write an equation that represents the relationship between independent and dependent quantities from a table. Also 6.6.C

Engage

ESSENTIAL QUESTION How can you use an equation to show a relationship between two variables? Use a table to find the relationship between the two variables. Use that relationship to write an equation.

Motivate the Lesson Ask students to imagine a hot dog stand that charges \$3 per hot dog. How much would 4 hot dogs cost? 30 hot dogs? Begin the Explore Activity to find out how to write an equation that will help you predict the total cost for any number of hot dogs.

Mathematical Processes 6.1.B Use a problem-solving model that incorporates analyzing given information, formulating a plan or strategy, determining a solution, justifying the solution, and evaluating the problem-solving process and the reasonableness of the solution.

Explore EXPLORE ACTIVITY Focus on Patterns

Mathematical Processes Point out to students that to write an equation from the data in the table, they need to look for a pattern in the data. First, they should look for changes in both the input values and the output values. Then they need to see how the changes are related. For example:

8=8·1 16 = 8 · 2 24 = 8 · 3

Write an equation that expresses y in terms of x. A x

1

2

3

4

5

y

3

6

9

12

15

x

2

4

6

8

10

y

7

9

11

13

15

So, the pattern is y = 8 · x, where x is the number of dogs walked and y is the amount of money earned.

y = 3x B

y=x+5

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Animated Math Writing Equations from Tables Students generate patterns with an interactive model, record the data in a table, and write equations to represent the pattern. my.hrw.com

Lesson 14.3

EXAMPLE 1 Focus on Reasoning

Interactive Whiteboard Interactive example available online

393

Explain Mathematical Processes In A, point out to students that the y-value is always less than the x-value. Therefore, the operation in the equation must be subtraction, division, or multiplication by a factor that is less than 1. In B, point out to students that the y-value is always more than the x-value. Therefore, the operation in the equation must be addition or multiplication by a factor that is greater than 1.

Questioning Strategies

Mathematical Processes • For A, how can you write an equation that expresses x in terms of y? I compared the x- and y-values and found that each x-value is twice the corresponding y-value, which gives me the equation x = 2y.

YOUR TURN Engage with the Whiteboard For Exercises 2–5, have students write a pattern on the whiteboard for each table. Then have them use the pattern to write an equation to represent each table. Ask students to explain their reasoning.

Writing Equations from Tables

14.3 ?

ESSENTIAL QUESTION

Expressions, equations, and relationships—6.6.B Write an equation that represents the relationship between independent and dependent quantities from a table. Also 6.6.C.

Writing an Equation Based on a Table The relationship between two variables where one variable depends on the other can be represented in a table or by an equation. An equation expresses the dependent variable in terms of the independent variable.

Math On the Spot

When there is no real-world situation to consider, we usually say x is the independent variable and y is the dependent variable. The value of y depends on the value of x.

my.hrw.com

How can you use an equation to show a relationship between two variables?

EXAMPLE 1

6.6.B, 6.6.C

EXPLORE ACTIVITY

Write an equation that expresses y in terms of x.

Writing an Equation to Represent a Real-World Relationship

Animated Math

A

my.hrw.com

Many real-world situations involve two variable quantities in which one quantity depends on the other. This type of relationship can be represented by a table. You can also use an equation to model the relationship.

1

2

3

5

10

20

Earnings

\$8

\$16

\$24

\$40

\$80

\$160

For 1 dog, Amanda earns 1 · 8 = \$8. For 2 dogs, she earns 2 · 8 = \$16.

© Houghton Mifflin Harcourt Publishing Company

3

4

5

1

1.5

2

2.5

Compare the x- and y-values to find a pattern.

Use the pattern to write an equation expressing y in terms of x. y = 0.5x

B

x

2

4

6

8

10

y

5

7

9

11

13

Compare the x- and y-values to find a pattern.

STEP 1

Each y-value is 3 more than the corresponding x-value.

Use the pattern to write an equation expressing y in terms of x.

STEP 2

Math Talk

Mathematical Processes

number of dogs walked.

y=x+3

How can you check that your equations are correct?

for each dog she walks.

C Write an equation that relates the number of dogs Amanda walks to the amount she earns. Let e represent earnings and d represent dogs.

For each table, write an equation that expresses y in terms of x. 2.

e=8·d D Use your equation to complete the table for 5, 10, and 20 walked dogs. E Amanda’s earnings depend on the number of dogs walked

x

12

11

10

y

10

9

8

3.

x

10

12

14

y

25

30

35

y=x-2

y = 2.5x

. 4.

Reflect

Personal Math Trainer

1. What If? If Amanda changed the amount earned per dog to \$11, what equation could you write to model the relationship between number of dogs walked and earnings?

2

STEP 2

Substitute each value of x in the equation. If the equation is correct, the result is the corresponding y-value.

Each earnings amount is 8 times the corresponding 8

1 0.5

1 Each y-value is __ , or 0.5 times, the corresponding x-value. 2

A For each column, compare the number of dogs walked and earnings. What is the pattern?

B Based on the pattern, Amanda earns \$

x y STEP 1

The table shows how much Amanda earns for walking 1, 2, or 3 dogs. Use the table to determine how much Amanda earns per dog. Then write an equation that models the relationship between number of dogs walked and earnings. Use your equation to complete the table. Dogs walked

6.6.B, 6.6.C

Online Assessment and Intervention

e = 11 · d

x

5

4

3

y

10

9

8

y=x+5

5.

x

0

1

2

y

0

2

4

© Houghton Mifflin Harcourt Publishing Company

LESSON

y = 2x

my.hrw.com

Lesson 14.3

393

394

Unit 4

PROFESSIONAL DEVELOPMENT Integrate Mathematical Processes This lesson provides an opportunity to address Mathematical Process TEKS 6.1.B, which calls for students to “use a problem-solving model that incorporates analyzing given information, formulating a plan or strategy, determining a solution, justifying the solution, and evaluating the problem-solving process and the reasonableness of the solution.” Each of these steps is explicitly used in solving a real-world problem in this lesson.

Math Background A rule that relates the x- and y-values in a table also can be called a relation. A relation can describe a function if, for each x-value (input), there is only one y-value (output). There are several different ways to describe the variables of a function: Independent Variable

Dependent Variable

x-value

y-value

Domain

Range

Input

Output

x

f(x)

Writing Equations from Tables

394

ADDITIONAL EXAMPLE 2 Meredith is playing a video game. She earns the same number of points for each alien she captures. She earned 750 points for capturing 5 aliens and 1,350 points for capturing 9 aliens. Write an equation to represent the relationship. Then solve the equation to find how many points Meredith will earn if she captures 27 aliens. p = 150a, where p represents the number of points and a represents the number of aliens captured; 4,050 points Interactive Whiteboard Interactive example available online my.hrw.com

EXAMPLE 2 Focus on Reasoning

Mathematical Processes Point out to students that sometimes a problem may provide clues and facts that you must use to find a solution. Encourage students to begin by identifying the important information. They can underline or circle the information in the problem statement.

Engage with the Whiteboard Have students extend the table on the whiteboard, continuing with sale prices of \$400 through \$1,200 in increments of \$100. Then have students find the donation amount for these sale prices. After they have completed the table, ask students to identify any patterns in the table.

Questioning Strategies • What is the independent variable in this situation? the dependent variable? The independent variable is the sale price of a painting. The dependent variable is the amount donated to charity. • How else could you solve this problem? Write and solve a proportion to find the amount 50 75 x x of the donation if a painting sells for \$1,200: ___ = ____ or ___ = ____ . 200 1,200 300 1,200

YOUR TURN Avoid Common Errors If students have difficulty identifying the independent and dependent variables, remind them to begin by using the given information to make a table and then look for a pattern.

Elaborate Talk About It Summarize the Lesson Ask: How can you use a table to write an equation that represents the relationship in the table? In the table, find the relationship between the independent and dependent variables. Then write the equation that represents the relationship.

GUIDED PRACTICE Engage with the Whiteboard For Exercises 1–2, have students write a pattern on the whiteboard for each table. Then have students use the pattern to write an equation to represent each table. Ask students to explain their reasoning. For Exercise 5, have students fill in the missing information in the table on the whiteboard. Then have them identify the pattern and write an equation.

Avoid Common Errors Exercises 1–4 Some students may write an equation that expresses x in terms of y instead of y in terms of x. Remind them that the form of the equation should be y = kx or y = x + b. Exercise 5 If students have difficulty identifying the independent and dependent variables, remind them to begin by using the given information to make a table and then look for a pattern.

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DO NOT EDIT--Changes must be made through “File info” CorrectionKey=B

DO NOT EDIT--Changes must be made through “File info” CorrectionKey=B

Using Tables and Equations to Solve Problems Problem Solving

EXAMPL 2 EXAMPLE

6. When Ryan is 10, his brother Kyle is 15. When Ryan is 16, Kyle will be 21. When Ryan is 21, Kyle will be 26. Complete the table for Ryan and Kyle. Write and solve an equation to find Kyle’s age when Ryan is 52.

Personal Math Trainer

You can use tables and equations to solve real-world problems. 6.6.B, 6.6.C

Math On the Spot

Online Assessment and Intervention

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my.hrw.com

A certain percent of the sale price of paintings at a gallery will be donated to charity. The donation will be \$50 if a painting sells for \$200. The donation will be \$75 if a painting sells for \$300. Find the amount of the donation if a painting sells for \$1,200.

Ryan

10

16

21

Kyle

15

21

26

k = r + 5; 57 years old

Analyze Information

Guided Practice

You know the donation amount when the sale price of a painting is \$200 and \$300. You need to find the donation amount if a painting sells for \$1,200.

Write an equation to express y in terms of x. (Explore Activity, Example 1) 1.

Formulate a Plan

You can make a table to help you determine the relationship between sale price and donation amount. Then you can write an equation that models the relationship. Use the equation to find the unknown donation amount. 3. 200

300

Donation amount (\$)

50

75

50 ÷ 2 50 25 ___ = ______ = ___ = 25% 200 200 ÷ 2 100

75 ÷ 3 75 25 ___ = _______ = ___ = 25% 300 300 ÷ 3 100

Write an equation. Let p represent the sale price of the painting. Let d represent the donation amount to charity. © Houghton Mifflin Harcourt Publishing Company

The donation amount is equal to 25% of the sale price. d = 0.25 · p Find the donation amount when the sale price is \$1,200.

One way to determine the relationship between sale price and donation amount is to find the percent.

d = 0.25 · 1,200

p is the independent variable; its value does not depend on any other value. d is the dependent variable; its value depends on the price of the painting.

26

36

2.

x

0

1

2

3

y

0

4

8

12

y = 4x

x

4

6

8

10

y

7

9

11

13

4.

1

2

5

1.35

2.70

6.75

Number of songs = n; Cost =

? ?

x

12

24

36

48

y

2

4

6

8

y = _6x

1.35n \$33.75

.

ESSENTIAL QUESTION CHECK-IN

6. Explain how to use a table to write an equation that represents the relationship in the table.

Justify and Evaluate

16

The total cost of 25 songs is

Substitute values from the table for p and d to check that they are solutions of the equation d = 0.25 · p. Then check your answer of \$300 by substituting for d and solving for p.

6

Total cost (\$)

Simplify to find the donation amount.

d = 0.25 · p 300 = 0.25 · p p = 1,200

y

Substitute \$1,200 for the sale price of the painting.

d = 0.25 · p d = 0.25 · 300 d = 75

40

y=x+3

When the sale price is \$1,200, the donation to charity is \$300.

d = 0.25 · p d = 0.25 · 200 d = 50

30

5. Jameson downloaded one digital song for \$1.35, two digital songs for \$2.70, and 5 digital songs for \$6.75. Complete the table. Write and solve an equation to find the cost to download 25 digital songs. (Example 2)

d = 0.25 · p

d = 300

20

© Houghton Mifflin Harcourt Publishing Company

Sale price (\$)

10

y=x-4

Justify and Evaluate Solve

Make a table.

x

Compare the x- and y-values to find a pattern. Use the pattern to write an equation expressing y in terms of x.

✓ Lesson 14.3

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Unit 4

6_MTXESE051676_U4M14L3.indd 396

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DIFFERENTIATE INSTRUCTION Curriculum Integration

Cognitive Strategies

Discuss the relationship between Celsius temperature and Kelvin temperature. Show students the following table and ask them to write an equation to convert from degrees Celsius to degrees Kelvin.

Some students may find it helpful to include a “Process” column in a table to help them identify patterns. Have students complete the table below.

Differentiated Instruction includes: • Reading Strategies • Success for English Learners ELL • Reteach • Challenge PRE-AP

Celsius (°C)

Kelvin (°K)

-100

173

K = C + 273

x

Process

y

-3

-2 = -3 + 1

-2

0 = -1 + 1

-2

-50

223

-1

0

273

0

50

323

1

100

373

2

-1 = -2 + 1 1=0+1 2=1+1 3=2+1

-1 0 1 2 3

Each value of y is one more than the value of x. Writing Equations from Tables

396

Personal Math Trainer Online Assessment and Intervention

Online homework assignment available my.hrw.com

Evaluate GUIDED AND INDEPENDENT PRACTICE 6.6.B, 6.6.C

Concepts & Skills

Practice

Explore Activity Writing an Equation to Represent a Real-World Relationship

Exercises 1–4, 8, 11

Write an equation that expresses y in terms of x.

Example 1 Writing an Equation Based on a Table

Exercises 1–4, 9–10

1.

Example 2 Using Tables and Equations to Solve Problems

Exercises 5, 11

14.3 LESSON QUIZ 6.6.B

x

1

2

3

4

5

y

5

10 15 20 25

x

10 20 30 40 50

y

7

2. 17 27 37 47

3. Jaime bought 2 puzzles for \$5.00 and 3 puzzles for \$7.50. Write and solve an equation to find the cost of 15 puzzles. 4. A submarine descends to -100 feet in 2 minutes and -250 feet in 5 minutes. Write and solve an equation to find the depth of the submarine in 8 minutes. Lesson Quiz available online my.hrw.com

2. y = x - 3

3. c = 2.50p; \$37.50

4. d = -50m; -400 feet

397

Lesson 14.3

Exercise

Depth of Knowledge (D.O.K.)

Mathematical Processes

7

3 Strategic Thinking

1.F Analyze relationships

8

2 Skills/Concepts

1.A Everyday life

9–10

3 Strategic Thinking

1.F Analyze relationships

11

3 Strategic Thinking

1.A Everyday life

12

3 Strategic Thinking

1.G Explain and justify arguments

13–14

3 Strategic Thinking

1.F Analyze relationships

15

3 Strategic Thinking

1.G Explain and justify arguments

Additional Resources Differentiated Instruction includes: • Leveled Practice Worksheets

DO NOT EDIT--Changes must be made through “File info” CorrectionKey=B

DO NOT EDIT--Changes must be made through “File info” CorrectionKey=A

Name

Class

Date

14.3 Independent Practice

12. Communicate Mathematical Ideas For every hour that Noah studies, his test score goes up 3 points. Explain which is the independent variable and which is the dependent variable. Write an equation modeling the relationship between hours studied h and the increase in Noah’s test score s.

Personal Math Trainer

6.6.B, 6.6.C

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Online Assessment and Intervention

Independent: hours studied; its value does not depend on

7. Vocabulary What does it mean for an equation to express y in terms of x?

another variable; dependent: test score; its value depends on the number of hours that Noah studies; s = 3h

The variable y is on one side of the equation. The expression on the other side of the equation shows the relationship

FOCUS ON HIGHER ORDER THINKING

between x and y.

13. Make a Conjecture Compare the y-values in the table to the corresponding x-values. Determine whether there is an additive relationship or a multiplicative relationship between x and y. If possible, write an equation modeling the relationship. If not, explain why.

8. The length of a rectangle is 2 inches more than twice its width. Write an equation relating the length l of the rectangle to its width w.

l = 2w + 2

9. Look for a Pattern Compare the y-values in the table to the corresponding x-values. What pattern do you see? How is this pattern used to write an equation that represents the relationship between the x- and y-values? 20

24

28

32

y

5

6

7

8

© Houghton Mifflin Harcourt Publishing Company

2

4

6

8

8

16

24

32

3

5

7

3

6

8

21

y-values and corresponding x-values. 14. Represent Real-World Problems Describe a real-world situation in which there is an additive or multiplicative relationship between two quantities. Make a table that includes at least three pairs of values. Then write an equation that models the relationship between the quantities.

10. Explain the Error A student modeled the relationship in the table with the equation x = 4y. Explain the student’s error. Write an equation that correctly models the relationship. x

1

Not possible; there is no consistent pattern between the

The y-value is _14 of the x-value. Write an equation that relates y to _14 of x.

y

x y

Sample answer: The distance Yasmine traveled in miles is equal to 50 times the number of hours she drove; d = 50 × t (multiplicative relationship). Time (h)

The student switched the variables; y = 4x

2

Distance (mi) 100

3

4

5

150

200

250

15. Critical Thinking Georgia knows that there is either an additive or multiplicative relationship between x and y. She only knows a single pair of data values. Explain whether Georgia has enough information to write an equation that models the relationship between x and y.

11. Multistep Marvin earns \$8.25 per hour at his summer job. He wants to buy a video game system that costs \$206.25. a. Write an equation to model the relationship between number of hours worked h and amount earned e.

No; with only one pair of values, Georgia cannot tell

e = 8.25h

© Houghton Mifflin Harcourt Publishing Company

x

Work Area

whether the relationship is additive or multiplicative, so

b. Solve your equation to find the number of hours Marvin needs to work in order to afford the video game system.

she cannot write an equation for the relationship.

206.25 = 8.25h; 25 = h; 25 hours

Lesson 14.3

6_MTXESE051676_U4M14L3.indd 397

10/30/12 12:23 PM

InCopy Notes 1. This is a list

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Unit 4

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InDesign Notes

EXTEND THE MATH

1. This is a list

PRE-AP

Activity available online

my.hrw.com

Activity To introduce the idea of relationships that are not purely additive or multiplicative, have students find the y-values in the following tables. Remind them to use the order of operations. Have them compare and contrast these relationships with additive and multiplicative relationships. 1. y = 2x + 1 3. y = __12 x + 3 x

0

1

2

3

x

0

2

4

6

y

1

3

5

7

y

3

4

5

6

4. y = __13 x -1

2. y = 3x - 2 x

1

2

3

4

x

3

6

9

12

y

1

4

7

10

y

0

1

2

3

Writing Equations from Tables

398

LESSON

14.4

Representing Algebraic Relationships in Tables and Graphs

Texas Essential Knowledge and Skills The student is expected to: Expressions, equations, and relationships—6.6.C Represent a given situation using verbal descriptions, tables, graphs, and equations in the form y = kx or y = x + b. Expressions, equations, and relationships—6.6.A

Engage ESSENTIAL QUESTION How can you use verbal descriptions, tables, and graphs to represent algebraic relationships? Sample answer: You can make a table from the verbal description and then make a graph from the ordered pairs in the table. From a graph, you can make a table and write an equation.

Motivate the Lesson Ask: Have you ever thought about running in a marathon? Do you know how many kilometers you could run in an hour? in two hours? Begin Explore Activity 1 to find out how to make a table and a graph to estimate how far you could run in a given period of time.

Identify independent and dependent quantities from tables and graphs. Expressions, equations, and relationships—6.6.B Write an equation that represents the relationship between independent and dependent quantities from a table.

Mathematical Processes 6.1.D Communicate mathematical ideas, reasoning, and their implications using multiple representations, including symbols, diagrams, graphs, and language as appropriate.

Explore EXPLORE ACTIVITY 1 Connect Multiple Representations

Mathematical Processes Point out to students that the ordered pairs from each table are used to make the graphs. However, after the lines are drawn on the graphs, they represent a more complete picture of the relationship than the tables do because all positive real numbers, not just integers, are included in the graph.

Explain EXPLORE ACTIVITY 2 Focus on Math Connections

Mathematical Processes Point out to students that when finding an equation from a graph, it is easier to first make a table of values from the graph. Then they can look for a pattern for the equation.

Questioning Strategies

Mathematical Processes • Is the relationship additive or multiplicative? Explain how you know. The relationship is additive, because the line drawn through the points does not go through the origin. • Explain how you can find the entrance fee for the museum from the graph. The starting point of the graph is (0, 5). This ordered pair represents the cost of Cherise’s expenses at the museum without any purchases at the gift shop, so it represents the entrance fee, \$5.

c.4.C ELL Be sure English learners understand the context in Explore Activity 2. You may want to discuss the terms “museum”, “souvenir”, and “entrance fee” before starting the activity.

Integrating the ELPS

Engage with the Whiteboard Ask a student volunteer to complete the table and identify the pattern. Then have the student write the equation to represent the total amount spent at the museum gift shop. Finally, discuss with the class what the independent and dependent variables are.

399

Lesson 14.4

Representing Algebraic Relationships in Tables and Graphs

?

ESSENTIAL QUESTION

How can you use verbal descriptions, tables, and graphs to represent algebraic relationships?

Writing an Equation from a Graph Cherise pays the entrance fee to visit a museum, then buys souvenirs at the gift shop. The graph shows the relationship between the total amount she spends at the museum and the amount she spends at the gift shop. Write an equation to represent the relationship.

A Read the ordered pairs from the graph. Use them to complete a table comparing total spent y to amount spent at the gift shop x.

6.6.C

EXPLORE ACTIVITY 1

Representing Algebraic Relationships Angie’s walking speed is 5 kilometers per hour, and May’s is 4 kilometers per hour. Use tables and graphs to show how the distance each girl walks is related to time.

A For each girl, make a table comparing time and distance. 0

1

2

3

4

Angie’s distance (km)

0

5

10

15

20

Time (h)

0

1

2

3

4

May’s distance (km)

0

4

8

12

16

For every hour May walks, she travels 4 km.

Distance (km)

Distance (km)

© Houghton Mifflin Harcourt Publishing Company

20 16 12 8 4 1

2

3

4

Time (h)

5

10

15

20

Total amount (\$)

5

10

15

20

25

y 32

shop amount.

C Write an equation that expresses the total amount y in terms of the gift shop amount x.

28 24 20 16 12 8 4 x

O

4

8 12 16 20 24

Reflect Math Talk

20

Mathematical Processes

16

2. Identify the dependent and independent quantities in this situation.

Dependent: total amount spent; independent: amount

Why does it make sense to connect the points in each graph?

12 8

4 x

x O

5

y= x+5

May

y

0

The total amount is 5 more than the gift

B For each girl, make a graph showing her distance y as it depends on time x. Plot points from the table and connect them with a line. Angie

B What is the pattern in the table? For every hour Angie walks, she travels 5 km.

Time (h)

y

6.6.C, 6.6.B

EXPLORE ACTIVITY 2

O

1

2

3

4

5

Time (h)

Reflect 1. Analyze Relationships How can you use the tables to determine which girl is walking faster? How can you use the graphs?

The girls can walk for fractional parts of an hour and can travel fractional parts of a kilometer.

3. Draw a line through the points in the graph. Find the point that represents Cherise spending \$25 at the gift shop. Use this point to find the total she would spend if she spent \$25 at the gift shop. Then use your equation from C to verify your answer.

© Houghton Mifflin Harcourt Publishing Company • Image Credits: © Thinkstock/ Corbis

14.4

Expressions, equations, and relationships—6.6.C Represent a given situation using tables, graphs, and equations…. Also 6.6.A, 6.6.B

Total amount (\$)

LESSON

(25, 30); \$30; 30 = 25 + 5; 30 = 30

Compare the distances walked after the same amount of time; compare the steepness of the lines. Lesson 14.4

399

400

Unit 4

PROFESSIONAL DEVELOPMENT Integrate Mathematical Processes This lesson provides an opportunity to address Mathematical Process TEKS 6.1.D, which calls for students to “communicate mathematical ideas...using multiple representations…as appropriate.” Students use verbal descriptions to make tables and draw graphs that represent real-life situations. They also represent information from graphs by using tables and equations; and represent equations by using tables and graphs.

Math Background The equations in this lesson are linear equations. A linear equation is an equation whose solutions fall on a line on a coordinate plane. All solutions of a particular linear equation fall on the line, and all the points on the line are solutions of the equation. Linear equations have constant slope. In the slope-intercept form, y = mx + b, m is the slope and b is the y-intercept. Linear equations fit into the general category of polynomial equations. Linear equations are called first-degree equations, because the greatest power of x is 1.

Representing Algebraic Relationships in Tables and Graphs

400

ADDITIONAL EXAMPLE 1 Graph each equation.

A y=x+2

Check for Understanding Ask: Can the points on the graphs only have whole number coordinates? Explain your answer. No, the coordinates can be any pair of rational numbers that satisfies the equation. For example, (0.5, 1.5) is on the graph of y = x + 1.

y 6 4 2

x

O

EXAMPLE 1

2

4

6

B y = 3x

Questioning Strategies

Mathematical Processes • By looking at the graph, how can you tell if the relationship in A is additive or multiplicative? The relationship is additive because the line drawn through the points does not go through the origin. • By looking at the graph, how can you tell if the relationship in B is additive or multiplicative? The relationship is multiplicative because the line drawn through the points goes through the origin.

y 6 4 2 O

x 2

4

6

Interactive Whiteboard Interactive example available online

Students may plot the points in the table but forget to draw a line connecting the points. Remind them that they must connect the points with a line for the graph to be correctly drawn.

my.hrw.com

Elaborate Talk About It Summarize the Lesson Ask: How can you use tables and graphs to represent algebraic relationships? You can make a table from the verbal description and then make a graph from the ordered pairs in the table. From a graph, you can make a table and write an equation.

GUIDED PRACTICE Engage with the Whiteboard For Exercises 1–2, have students complete the table and graph the points on the coordinate grid on the whiteboard. Then have another student identify the pattern and write the equation.

Avoid Common Errors Exercises 1–2 Students may plot the points in the table but forget to draw a line connecting the points. Remind them that they must connect the points with a line for the graph to be drawn correctly.

401

Lesson 14.4

Graphing an Equation

An ordered pair (x, y) that makes an equation like y = x + 1 true is called a solution of the equation. The graph of an equation represents all the ordered pairs that are solutions.

EXAMPL 1 EXAMPLE

4. Graph y = x + 2.5.

Personal Math Trainer Math On the Spot

Online Assessment and Intervention

my.hrw.com

my.hrw.com

6.6.C

Graph each equation.

y

x

x + 2.5 = y

(x, y)

0 1 2 3

0 + 2.5 = 2.5 1 + 2.5 = 3.5 2 + 2.5 = 4.5 3 + 2.5 = 5.5

(0, 2.5) (1, 3.5) (2, 4.5) (3, 5.5)

10 8 6 4 2 x

A y=x+1

O

STEP 1

Make a table of values. Choose some values for x and use the equation to find the corresponding values for y.

STEP 2

Plot the ordered pairs from the table. Mathematical Processes

Is the ordered pair (3.5, 4.5) a solution of the equation y = x + 1? Explain.

y

x

x+1=y

(x, y)

1+1=2

(1, 2)

8

10

2

2+1=3

(2, 3)

6

3

3+1=4

(3, 4)

4

4

4+1=5

(4, 5)

2

5

5+1=6

(5, 6)

O

x 1

2

3

4

5

B y = 2x

© Houghton Mifflin Harcourt Publishing Company

3

4

5

STEP 1

Make a table of values. Choose some values for x and use the equation to find the corresponding values for y.

STEP 2

Plot the ordered pairs from the table.

STEP 3

Draw a line through the plotted points to represent all of the ordered pair solutions of the equation.

Frank mows lawns in the summer to earn extra money. He can mow 3 lawns every hour he works. (Explore Activity 1 and Explore Activity 2) 1. Make a table to show the relationship between the number of hours Frank works, x, and the number of lawns he mows, y. Graph the relationship and write an equation.

Yes; (3.5, 4.5) is on the graph. You can also substitute for the variables in the equation to check.

Hours worked

Lawns mowed

0

0 3 6 9

1

2 3

y

10 8 6 4 2

y = 3x

x O

1

2

3

4

5

Hours worked

Graph y = 1.5x. (Example 1)

y 5

2. Make a table to show the relationship. x y

y

x

2x = y

(x, y)

1

2×1=2

(1, 2)

2

2×2=4

(2, 4)

6

3

2×3=6

(3, 6)

4

4

2×4=8

(4, 8)

2

5

2 × 5 = 10

(5, 10)

O

2 1

10

3. Plot the points and draw a line through them.

8

? ?

x 1

2

3

4

4 3

0 1 2 3 0 1.5 3 4.5

x O

1

2

3

4

5

ESSENTIAL QUESTION CHECK-IN

© Houghton Mifflin Harcourt Publishing Company

Draw a line through the plotted points to represent all of the ordered pair solutions of the equation.

1

2

Guided Practice Math Talk

Lawns mowed

STEP 3

1

4. How can a table represent an algebraic relationship between two variables?

5

It shows pairs of values that satisfy the relationship.

Lesson 14.4

401

402

Unit 4

DIFFERENTIATE INSTRUCTION Cooperative Learning

Modeling

Have students work in pairs to write an equation with two variables. Each equation should be an additive equation. Collect students’ equations and randomly redistribute them. Have the students make tables for the equations and write solutions of the equations as ordered pairs. Then have the students graph the equations.

Draw an equilateral triangle and a square, each with a side length of 6 inches, on the chalkboard. Have students find the perimeter of each. Ask students to come up with a formula for the perimeter of an equilateral triangle and a square. Then have students make a table and a graph for each formula.

Differentiated Instruction includes: • Reading Strategies • Success for English Learners ELL • Reteach • Challenge PRE-AP

Representing Algebraic Relationships in Tables and Graphs

402

Personal Math Trainer Online Assessment and Intervention

Evaluate GUIDED AND INDEPENDENT PRACTICE

Online homework assignment available

6.6.A, 6.6.C

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14.4 LESSON QUIZ 6.6.C The graph shows the number of bracelets Olivia can make in an hour. Number of Bracelets

y

Concepts & Skills

Practice

Explore Activity 1 Representing Algebraic Relationships

Exercise 1

Explore Activity 2 Writing an Equation from a Graph

Exercises 1, 5–7

Example 1 Graphing an Equation

Exercises 2–3, 8–9, 11

8 6

Exercise

4

5–6

2

7 2

4

1.D Multiple representations

3 Strategic Thinking

1.F Analyze relationships

8–9

2 Skills/Concepts

1.D Multiple representations

10

3 Strategic Thinking

1.G Explain and justify arguments

11

3 Strategic Thinking

1.F Analyze relationships

12

3 Strategic Thinking

1.G Explain and justify arguments

13–14

3 Strategic Thinking

1.F Analyze relationships

6

Hours

1. Read the ordered pairs from the graph to make a table. 2. Write an equation to model the relationship. The equation y = x + 2 represents the total cost of doing x loads of laundry at a laundromat, including buying a box of detergent.

Additional Resources Differentiated Instruction includes: • Leveled Practice Worksheets

3. Make a table that represents the relationship between number of loads and total cost. 4. Make a graph showing the relationship. Lesson Quiz available online my.hrw.com

0

1

2

Number of bracelets

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8

2. y = 4x 3.

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4. Total Cost

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Mathematical Processes

2 Skills/Concepts

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Depth of Knowledge (D.O.K.)

6 4 2

x O

2

4

6

DO NOT EDIT--Changes must be made through “File info” CorrectionKey=B

Class

Date

14.4 Independent Practice

Personal Math Trainer

6.6.A, 6.6.B, 6.6.C

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11. Multistep The equation y = 9x represents the total cost y for x movie tickets. a. Make a table and a graph to represent the relationship between x and y.

Online Assessment and Intervention

Students at Mills Middle School are required to work a certain number of community service hours. Students may work additional hours beyond the requirement.

5

10

15

20

Total hours

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50

Total (h)

0

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Total cost (\$), y

9

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y 50 40 30 20 10

of tickets; the total cost depends on how many

20

O

2

Dependent: total cost; independent: number

40

10

6. Write an equation that expresses the total hours in terms of the additional hours.

1

b. Critical Thinking In this situation, which quantity is dependent and which is independent? Justify your answer.

y

5. Read the ordered pairs from the graph to make a table.

Number of tickets, x

2

3

4

5

Number of tickets

c. Multiple Representations Eight friends want to go see a movie. Would you prefer to use an equation, a table, or a graph to find the cost of 8 movie tickets? Explain how you would use your chosen method to find the cost.

7. Analyze Relationships How many community service hours are students required to work? Explain.

Sample answer: an equation; substitute 8 for x in

20 hours; when 0 additional hours are worked,

y = 9x to get y = 9(8) = \$72.

the total is 20 hours.

FOCUS ON HIGHER ORDER THINKING

Work Area

12. Critical Thinking Think about graphing the equations y = 5x and y = x + 500. Which line would be steeper? Why?

Beth is using a map. Let x represent a distance in centimeters on the map. To find an actual distance y in kilometers, Beth uses the equation y = 8x.

The graph of y = 5x would be steeper because y

8. Make a table comparing a distance on the map to the actual distance.

increases more rapidly for each value of x.

Map distance (cm)

1

2

3

4

5

Actual distance (km)

8

16

24

32

40

9. Make a graph that compares the map distance to the actual distance. 10. Critical Thinking The actual distance between Town A and Town B is 64 kilometers. What is the distance on Beth’s map? Did you use the graph or the equation to find the answer? Why?

8 cm; sample answer: I used the equation because the scales on the graph don’t extend far enough.

Actual distance (km)

© Houghton Mifflin Harcourt Publishing Company

x 1

tickets were purchased.

x 10 20 30 40 50

y = x + 20

O

13. Persevere in Problem Solving Marcus plotted the points (0, 0), (6, 2), (18, 6), and (21, 7) on a graph. He wrote an equation for the relationship. Find another ordered pair that could be a solution of Marcus’s equation. Justify your answer.

Sample answer: (30, 10); every y-value is _13 of the x-value.

y 50

So, 10 = _13 (30).

40 30 20 10 O

x 1

2

3

4

5

14. Error Analysis The cost of a personal pizza is \$4. A drink costs \$1. Anna wrote the equation y = 4x + 1 to represent the relationship between total cost y of buying x meals that include one personal pizza and one drink. Describe Anna’s error and write the correct equation.

Map distance (cm)

© Houghton Mifflin Harcourt Publishing Company

Name

Total cost (\$)

DO NOT EDIT--Changes must be made through “File info” CorrectionKey=B

Anna’s equation does not show that every meal includes both a pizza and a drink; the correct equation is y = 5x.

Lesson 14.4

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EXTEND THE MATH

PRE-AP

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Activity available online

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Activity Ask students if they know that you can find the approximate temperature by listening to crickets chirp? The chirp rate of a cricket varies by temperature. The hotter it is, the more chirps per minute. The temperature in °F can be found by multiplying the number of chirps in one minute by __14 and adding 40. Write the equation that represents this situation. Then make a table of values to find the temperature for 20, 40, 60, 80, and 100 chirps per minute. t = __14 c + 40, where t is the Fahrenheit temperature and c is the number of chirps in a minute c

20

40

60

80

100

t

45

50

55

60

65

Representing Algebraic Relationships in Tables and Graphs

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MODULE QUIZ

Assess Mastery

14.1 Graphing on the Coordinate Plane

Use the assessment on this page to determine if students have mastered the concepts and standards covered in this module.

Graph each point on the coordinate plane.

3

Response to Intervention

2 1

Differentiated Instruction

Differentiated Instruction

• Reteach worksheets

• Challenge worksheets

• Success for English Learners ELL

ELL

Additional Resources Assessment Resources includes: • Leveled Module Quizzes

E

2 -6

6. F(-4, 6)

-2O

D

x 2

6

C

-6

14.3 Writing Equations from Tables Write an equation that represents the data in the table. 8.

x

3

5

8

10

y

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35

56

70

9.

x

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10

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y

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32

y = x + 12

y = 7x

14.4 Representing Algebraic Relationships in Tables and Graphs

PRE-AP

Extend the Math PRE-AP Lesson activities in TE

A

independent: number of packages; dependent: total cost

Online and Print Resources

2. B(3, 5) 4. D(-3, -5)

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B

6

7. Jon buys packages of pens for \$5 each. Identify the independent and dependent variables in the situation.

Enrichment

Graph each equation. © Houghton Mifflin Harcourt Publishing Company

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1. A(-2, 4)

14.2 Independent and Dependent Variables in Tables and Graphs

Access Ready to Go On? assessment online, and receive instant scoring, feedback, and customized intervention or enrichment.

Online Assessment and Intervention

y

F

3. C(6, -4) 5. E(7, 2)

Intervention

Personal Math Trainer

Personal Math Trainer Online Assessment and Intervention

10. y = x + 3

11. y = 5x

8

40

6

30

4

20

2

O

10 2

4

6

8

O

2

4

6

8

ESSENTIAL QUESTION 12. How can you write an equation in two variables to solve a problem?

Decide which variable depends on the other. Use a table to find the relationship between the variables and write an equation.

Module 14

Texas Essential Knowledge and Skills Lesson

Exercises

14.1

1–6

6.11

14.2

7

6.6.A, 6.6.C

14.3

8–9

6.6.B, 6.6.C

14.4

10–11

6.6.A, 6.6.B, 6.6.C

405

Module 14

TEKS

405

Personal Math Trainer

MODULE 14 MIXED REVIEW

Texas Test Prep

Texas Testing Tip Some items are called context-based items, which means the student has to examine each answer choice in order to determine the correct answer. Item 2 If students don’t remember that for every point in quadrant II the x-coordinate is a negative number and the y-coordinate is a positive number, they may need to plot each point to see that choice C is the correct answer.

Selected Response 1. What are the coordinates of point G on the coordinate grid below? y 4 2 x -4

Item 5 To find the point that the graph of y = 10 + x does not pass through, students may need to graph each point on a coordinate grid to see that choice C is the correct answer.

Item 4 Students often will get the independent and dependent quantities backward in problems, thereby choosing A for the answer. Remind students that the dependent quantity depends on the independent quantity. Therefore, the number of points earned depends on the number of prizes captured.

O

2

4

-2

G

-4

Avoid Common Errors Item 1 Students may forget what the first and second numbers in an ordered pair mean. Remind students that the first number is the x-coordinate and the second is the y-coordinate.

-2

A (4, 3)

C

B (4, -3)

D (-4, -3)

(-4, 3)

2. A point is located in quadrant II of a coordinate plane. Which of the following could be the coordinates of that point? A (-5, -7)

C

B (5, 7)

D (5, -7)

(-5, 7)

3. Matt had 5 library books. He checked 1 additional book out every week without returning any books. Which equation describes the number of books he has, y, after x weeks? A y = 5x

C

y = 1 + 5x

B y= 5-x

D y= 5+x

4. Stewart is playing a video game. He earns the same number of points for each prize he captures. He earned 1,200 points for 6 prizes, 2,000 points for 10 prizes, and 2,600 points for 13 prizes. Which is the dependent variable in the situation? A the number of prizes captured B the number of points earned C

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Online Assessment and Intervention

5. Dwayne graphed the equation y = 10 + x. Which point does the graph not pass through? A (0, 10)

C

B (3, 13)

D (5, 15)

(8, 2)

6. Amy gets paid by the hour. Her little sister helps. As shown below, Amy gives her sister part of her earnings. Which equation represents Amy’s pay when her sister’s pay is \$13? Amy’s pay in dollars

10

20

30

40

Sister’s pay in dollars

2

4

6

8

13 A y = __ 5 x B 13 = __ 5

C

5y = 13

D 13 = 5x

Gridded Response 7. Betty earns \$7.50 per hour at a part-time job. Let x be the number of hours and y be the amount she earns. Betty makes a graph to show how x and y are related. If she earns \$60, how many hours did she work?

8

.

0

0

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0

0

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1

1

1

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© Houghton Mifflin Harcourt Publishing Company

Texas Test Prep

the number of hours

D the number of prizes available

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Unit 4

Texas Essential Knowledge and Skills Items

Mathematical Process TEKS

1

6.11

6.1.E

2

6.11

6.1.F

3

6.6.C

6.1.A

4

6.6.A

6.1.A

5

6.11

6.1.F

6

6.6.B

6.1.A, 6.1.E

7*

6.2.E, 6.6.C

6.1.A, 6.1.F

* Item integrates mixed review concepts from previous modules or a previous course.

Relationships in Two Variables

406

UNIT 4

Expressions, Equations, and Relationships

Additional Resources Personal Math Trainer my.hrw.com

Online Assessment and Intervention

Assessment Resources • Leveled Unit Tests: A, B, C, D • Performance Assessment

Study Guide Review Vocabulary Development Integrating the ELPS Encourage English learners to refer to their notes and the illustrated, bilingual glossary as they review the unit content. c.4.E Read linguistically accommodated content area material with a decreasing need for linguistic accommodations as more English is learned.

MODULE 10 Generating Equivalent Numerical Expressions 6.7.A

Key Concepts • A power is a number that is formed by repeated multiplication by the same factor. An exponent and a base can be used to write a power. (Lesson 10.1) • Factors are whole numbers that are multiplied to find a product. (Lesson 10.2) • To simplify an expression with more than one operation, there is a specific order in which to apply the operations. (Lesson 10.3)

MODULE 11 Generating Equivalent Algebraic Expressions 6.7.C, 6.7.D

Key Concepts • An algebraic expression is an expression that contains one or more variables and may also contain operation symbols, such as + or -. A variable is a letter or symbol used to represent an unknown number. (Lesson 11.1) • To evaluate an expression, substitute a number for the variables and find the value of the expression. (Lesson 11.2) • To generate equivalent expressions, use the properties of operations to combine like terms. (Lesson 11.3)

407

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DO NOT EDIT--Changes must be made through "File info" CorrectionKey=A

Study Guide MODULE MODULE

?

10

Review

Find the value of each power. (Lesson 10.1)

Generating Equivalent Numerical Expressions

base (base (en numeración)) exponent (exponente)

7. 75

power (potencia)

EXAMPLE 1 Find the value of each power.

0.9 = 0.9 × 0.9 = 0.81

MODULE MODULE

4 C. (_41 )

( ) ( )( )( )( )

Any number raised to the power of 0 is 1.

2

1 4 1 _ 1 _ 1 _ 1 1 _ _ ___ 4 =  4 4 4 4 = 256

?

180 = 1

EXAMPLE 2 Find the prime factorization of 60.

© Houghton Mifflin Harcourt Publishing Company

B. 27 ÷ 32 × 6

= 4 × 13 = 52

= 27 ÷ 9 × 6

32 = 9

=3×6

Divide.

Multiply.

= 18

Multiply.

3.62

2. 9 × 9 × 9 × 9

8 ___ 343

(_27 )

3

11

23 × 3 × 7

9. 168

22 - (3 + 4) 12. __________ 12 ÷ 4 2

Generating Equivalent Algebraic Expressions

How can you generate equivalent algebraic expressions and use them to solve real-world problems?

21; 6, 14; 7, 12 3 Key Vocabulary algebraic expression (expresión algebraica) coefficients (coeficiente) constant (constante) equivalent expressions (expresiónes equivalente) evaluating (evaluar) term (término (en una expresión))

B. w - y 2 + 3w; w = 2, y = 6

2(52 - 9)

52 = 25

2 - 62 + 3(2)

62 = 36

= 2(16)

Subtract.

= 2 - 36 + 6

Multiply.

= 32

Multiply.

= -28

Add and subtract from left to right.

When w = 2 and y = 6, w - y 2 + 3w = -28.

EXAMPLE 2 Determine whether the algebraic expressions are equivalent: 5(x + 2) and 10 + 5x. 5(x + 2) = 5x + 10

= 10 + 5x

Distributive Property Commutative Property

5(x + 2) is equal to 10 + 5x. They are equivalent expressions.

EXERCISES Use exponents to write each expression. (Lesson 10.1) 1. 3.6 × 3.6

45

When x = 5, 2(x2 – 9) = 32.

EXAMPLE 3 Simplify each expression.

23 = 8

29

ESSENTIAL QUESTION

The prime factorization of 60 is 22 × 3 × 5.

= 4 × (8 + 5)

8. 29

A. 2(x2 - 9); x = 5

60 = 22 × 3 × 5

A. 4 × (23 + 5)

6.

EXAMPLE 1 Evaluate each expression for the given value of the variable.

60 = 2 × 2 × 3 × 5

2 60 2 30 3 15 5 5 1

52 × 3

11. 2 × 52 – (4 + 1)

B. 180

169

10. Eduardo is building a sandbox that has an area of 84 square feet. What are the possible whole number measurements for the length and width of the sandbox? (Lesson 10.2) 1, 84; 2, 42; 3, 28; 4,

order of operations (orden de las operaciones)

How can you generate equivalent numerical expressions and use them to solve real-world problems?

5. 132

Write the prime factorization of each number. (Lesson 10.2)

Key Vocabulary

ESSENTIAL QUESTION

A. 0.92

1

4. 120

© Houghton Mifflin Harcourt Publishing Company

UNIT 4

DO NOT EDIT--Changes must be made through “File info” CorrectionKey=B

(_45 )

3

94

3. _45 × _45 × _54

EXERCISES Write each phrase as an algebraic expression. (Lesson 11.1) 1. x subtracted from 15

Unit 4

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InCopy Notes

InDesign Notes

1. This is a list

1. This is a list

408

15 - x

2. 12 divided by t

12 __ t

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InCopy Notes 1. This is a list Bold, Italic, Strickthrough.

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InDesign Notes 1. This is a list

Expressions, Equations, and Relationships

408

MODULE 12 Equations and Relationships 6.7.B, 6.9.A, 6.9.B, 6.9.C, 6.10.A, 6.10.B

Key Concepts • An equation is a mathematical statement that two expressions are equal, which, if it includes a variable, has a solution. (Lesson 12.1) • Both sides of an equation remain equal after adding, or subtracting, the same number from both sides. (Lesson 12.2) • Both sides of an equation remain equal after multiplying, or dividing, both sides by the same number. (Lesson 12.3)

409

Unit 4

DO NOT EDIT--Changes must be made through “File info” CorrectionKey=B

DO NOT EDIT--Changes must be made through "File info" CorrectionKey=A

EXAMPLE 2

Write a phrase for each algebraic expression. (Lesson 11.1)

4. s + 7

67

7. s - 5t + s2; s = 4, t = -1

Add 12 to both sides. ? Check: 22 - 12 = 10 Substitute.

33

(Lesson 12.1)

40 in2

1. 7x = 14; x = 3

13. 7x + 4(2x - 6)

© Houghton Mifflin Harcourt Publishing Company

MODULE MODULE

?

12

yes

2. y + 13 = -4; y = -17

3. Don has three times as much money as his brother,

equivalent

d __ = 25 3

who has \$25.

4. There are s students enrolled in Mr. Rodriguez’s class. There are 6 students absent and 18 students present

m2 - 2m - 5

today.

15x - 24

s - 6 = 18

Equations and Relationships

Key Vocabulary equation (ecuación) solution (solución)

ESSENTIAL QUESTION

12 is not a solution of r - 5 = 17.

-42 is a solution of _6x = -7.

InDesign Notes

1. This is a list

1. This is a list

q = 2.1

8. 3.5 + x = 7

x = 49

10. _27 = 2x

x = 3.5

x = _17

x – 12.50 = 34.25; \$46.75

equation to solve the problem. (Lesson 12.3)

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InCopy Notes

7. 9q = 18.9

t = -48

12. Tom read 132 pages in 4 days. He read the same number of pages each day. How many pages did he read each day? Write and solve an

Unit 4

6_MTXESE051676_U4EM.indd 409

6. _4t = -12

the problem. (Lesson 12.2)

B. _6x = -7; x = -42 -42 ? ____ = -7 Substitute. 6 -7 = -7

p = 23

11. Sonia used \$12.50 to buy a new journal. She has \$34.25 left in her savings account. How much money did Sonia have before she bought the journal? Write and solve an equation to solve

EXAMPLE 1 Determine if the given value is a solution of the equation.

7 ≠ 17

5. p - 5 = 18

9. 18 = x - 31

How can you use equations and relationships to solve real-world problems?

A. r - 5 = 17; r = 12 ? 12 - 5 = 17 Substitute.

no

Write an equation to represent the situation. (Lesson 12.1)

not equivalent

Combine like terms. (Lesson 11.3) 12. 3m - 6 + m2 - 5m + 1

-30 = -30

EXERCISES Determine whether the given value is a solution of the equation.

Determine if the expressions are equivalent. (Lesson 11.3)

11. 2.5(3 + x); 2.5x + 7.5

Divide both sides by 5. ? Check: 5(-6) = -30 Substitute.

10 = 10

-34

8. x - y3; x = -7, y = 3

9. The expression _12 (h)(b1 + b2) gives the area of a trapezoid, with b1 and b2 representing the two base lengths of a trapezoid and h representing the height. Find the area of a trapezoid with base lengths 4 in. and 6 in. and a height of 8 in. (Lesson 11.2)

10. 7 + 7x; 7(x + _17 )

p = -6

y = 22

6. 3(7 + x2); x = 2

25

5p -30 __ = ____ 5 5

+12 = +12

Evaluate each expression for the given value of the variable. (Lesson 11.2) 5. 8z + 3; z = 8

B. 5p = -30

A. y - 12 = 10

the sum of s and 7

410

© Houghton Mifflin Harcourt Publishing Company

3. 8p

the product of 8 and p

4p = 132; 33 pages

Unit 4

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InCopy Notes 1. This is a list Bold, Italic, Strickthrough.

29/11/12 12:11 PM

InDesign Notes 1. This is a list

Expressions, Equations, and Relationships

410

MODULE 13 Inequalities and Relationships 6.9.A, 6.9.B, 6.9.C, 6.10.A, 6.10.B

Key Concepts • An inequality is a mathematical statement that uses one of the following inequality symbols: greater than, >, less than, <, greater than or equal to, ≥, or less than or equal to, ≤. (Lesson 13.1) • All inequalities have many solutions. (Lesson 13.2) • Reverse the inequality symbol when multiplying or dividing both sides of an inequality by a negative number. (Lesson 13.4)

MODULE 14 Relationships in Two Variables 6.6.A, 6.6.B, 6.6.C, 6.11

Key Concepts • An ordered pair is a pair of numbers in the form (x, y) that gives the location of a point on a coordinate plane. (Lesson 14.1) • The quantity that depends on the other quantity is called the dependent variable, and the quantity it depends on is called the independent variable. (Lesson 14.2) • Tables and graphs can be used to represent the relationship between an independent and dependent variable. (Lesson 14.4)

411

Unit 4

?

13

Inequalities and Relationships

9.

Key Vocabulary solution of an inequality (solución de una desigualdad)

ESSENTIAL QUESTION

Sample answer: Juan’s dog lost 3 pounds and still weighs at least 11 pounds.

How can you use inequalities and relationships to solve real-world problems?

10. Omar wants a rectangular vegetable garden. He only has enough space to make the garden 5 feet wide, and he wants the area of the garden to be more than 80 square feet. Write and solve an inequality to find the possible lengths of the garden. (Lesson 13.3)

EXAMPLE 1 Write and graph an inequality to represent each situation. A. There are at least 5 gallons of water in an aquarium.

5ℓ > 80, ℓ > 16

B. The temperature today will be less than 35 °F.

g≥5

MODULE

t < 35

?

30 31 32 33 34 35 36 37 38 39 40

0 1 2 3 4 5 6 7 8 9 10

Solve each inequality. Graph and check your solutions. A. x - 7 ≤ 2

B. -5y < -15 y>3

Divide by -5. Reverse the symbol.

© Houghton Mifflin Harcourt Publishing Company

0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5

s < 2.5

2. Tina got a haircut, and her hair is still at least 15 inches long.

h ≥ 15

5. 9q > 10.8 7. -_45 x < 8

q > 15

4. _4t ≤ -1

t ≤ -4

q > 1.2

6. 87 ≤ 25 + x

x ≥ 62

x > -10

8. -4 ≥ -0.5x

x≥8

(4, -2) is in quadrant IV.

EXAMPLE 2 Tim is paid \$8 more than the number of bags of peanuts he sells at the baseball stadium. The table shows the relationship between the money Tim earns and the number of bags of peanuts Tim sells. Identify the independent and dependent variables, and write an equation that represents the relationship.

10 11 12 13 14 15 16 17 18 19 20

Solve each inequality. Graph and check your solutions. (Lessons 13.2, 13.3, 13.4) 3. q - 12 > 3

̵5 ̵ 4 ̵3 ̵2 ̵1 O 1 2 3 4 5 ̵1 ̵2 (4, ̵2) ̵3 Quadrant III Quadrant IV ̵4 ̵5

(Lesson 13.1)

\$2.50 per share.

axes (ejes)

(4, -2) is located 4 units to the right of the origin and 2 units down from the origin.

y 5 4 Quadrant II 3 2 1

Key Vocabulary coordinate plane (plano cartesiano)

EXAMPLE 1 Graph the point (4, -2) and identify the quadrant where it is located.

EXERCISES Write and graph an inequality to represent each situation.

1. Orange Tech’s stock is worth less than

Relationships in Two Variables

How can you use relationships in two variables to solve real-world problems?

0 1 2 3 4 5 6 7 8 9 10

5 6 7 8 9 10 11 12 13 14 15

14

ESSENTIAL QUESTION

EXAMPLE 2

x≤9

Write a real-world comparison that can be described by x - 3 ≥ 11. (Lesson 13.2)

# of bags of peanuts, x

0

1

2

Money earned, y

8

9

10 11

3

© Houghton Mifflin Harcourt Publishing Company

MODULE

The number of bags is the independent variable, and the money Tim earns is the dependent variable. The equation y = x + 8 expresses the relationship between the number of bags Tim sells and the amount he earns.

Unit 4

411

412

Unit 4

Expressions, Equations, and Relationships

412

Unit 4 Performance Tasks The Performance Tasks provide students with the opportunity to apply concepts from this unit in real-world problem situations.

CAREERS IN MATH For more information about careers in mathematics as well as various mathematics appreciation topics, visit the American Mathematical Society at www.ams.org

CAREERS IN MATH Botanist In Performance Task Item 1, students can see how a botanist uses mathematics on the job.

Possible Points (Total: 6)

a

1 point for correctly writing the expression 205 + 2d.

b

1 point for correctly setting the expression 205 + 2d = 235. 1 point for correct answer: 15 days

c

1 point for correct expression for Suntracker: 195 +2.5d. 1 point for determining the heights of each sunflower variety after 22 days: Suntracker: h = 195 + 2.5(22) = 250 Sunny Yellow: h = 205 + 2(22) = 249 1 point for stating that Suntracker is taller.

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6.1.A, 6.1.F

6.1.A

Possible Points (Total: 6)

a

1 point for correctly defining a variable: Let w = the number of hours Vernon practiced soccer over the weekend. 1 point for the correct equation: 4__13 + w = 5__34 .

b

1 point for correctly finding the LCM, 12, and showing how to find it. Students can use a number line, a list of multiples, or prime factorization to find the LCM. 4 = 2 × 2; 3 = 3; LCM = 2 × 2 × 3 = 12

c

5 1 point for correctly solving the equation: w = 1__ . 12 1 point for showing stepped-out solution to equation. 1 point for correctly interpreting the equation in terms of the problem: Vernon 5 practiced 1__ hours over the weekend. 12

DO NOT EDIT--Changes must be made through "File info" CorrectionKey=A

DO NOT EDIT--Changes must be made through “File info” CorrectionKey=B

EXERCISES Graph and label each point on the coordinate plane. (Lesson 14.1)

1.

y

1. (4, 4) 2. (-3, -1)

(-1, 4)

4

(4, 4)

2

3. (-1, 4)

x ̵4

̵2

(-3, -1)

O

2

CAREERS IN MATH Botanist Dr. Adama is a botanist. She measures the daily height of a particular variety of sunflower, Sunny Yellow, beginning when the sunflower is 60 days old. At 60 days, the height of the sunflower is 205 centimeters. Dr. Adama finds that the growth rate of this sunflower is 2 centimeters per day after the first 60 days.

a. Write an expression to represent the sunflower’s height d days after the 60th day.

4

̵2

205 + 2d

̵4

b. How many days after the 60th day does it take for the sunflower to reach 235 centimeters? Show your work.

235 = 205 + 2d; 30 = 2d; 15 = d

Use the graph to answer the questions. (Lesson 14.2)

It takes 15 days for the sunflower to reach 235 centimeters. c. Dr. Adama is studying a different variety of sunflower, Suntracker, which grows at a rate of 2.5 centimeters per day after the first 60 days. If this sunflower is 195 centimeters tall when it is 60 days old, write an expression to represent Suntracker’s height d days after the 60th day. Which sunflower will be taller 22 days after the 60th day? Explain how you found your answer.

8 6 4

Suntracker: 195 + 2.5d; after 22 days: Suntracker: h = 195 + 2.5(22)

2 O

2

4

6

8

Time (h)

= 250; Sunny Yellow: h = 205 + 2(22) = 249. Suntracker is taller.

10

2. Vernon practiced soccer 5_43 hours this week. He practiced 4_13 hours on weekdays and the rest over the weekend.

time

4. What is the independent variable?

a. Write an equation that represents the situation. Define your variable.

distance

5. What is the dependent variable?

Let w = the number of hours Vernon practiced soccer over the weekend; 4_13 + w = 5_34

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6. Describe the relationship between the independent variable and the dependent variable.

b. What is the least common multiple of the denominators of 5_34 and 4_13 ? Show your work.

The dependent variable is 3 times the independent variable.

Using prime factorization: 4 = 2 × 2; 3 = 3; LCM = (2)(2)(3) = 12

7. Use the data on the table to write an equation to express y in terms of x. Then graph the equation. (Lessons 14.3, 14.4) x y

0

1

-2 -1

2

3

0

1

y= x-2

c. Solve the equation and interpret the solution. Show your work.

5 9 9 4 4 4_13 + w = 5_34; 4__ + w = 5__ ; w = 5__ - 4__ = 1__ 12 12 12 12 12

y 4 2

(2, 0)

̵4

̵2

O ̵2 ̵4

2

5 hours over the weekend. Vernon practiced 1__ 12

(3, 1)

© Houghton Mifflin Harcourt Publishing Company

Distance (km)

10

x

4

(1, -1) (0, -2) Unit 4

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Expressions, Equations, and Relationships

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Expressions, Equations, and Relationships

Additional Resources Personal Math Trainer my.hrw.com

Online Assessment and Intervention

Assessment Resources • Leveled Unit Tests: A, B, C, D • Performance Assessment

MIXED REVIEW

Texas Test Prep Texas Testing Tip Students should always read each question carefully to identify key words or phrases, such as no more than, to help identify what the question is really asking. Item 5 Students should underline the phrase no more than; it will help them to realize that n cannot be larger than 7, but it can be equal to 7. This will help them choose the correct inequality. Item 7 Students who do not read the problem carefully may choose B because they see the word times. But reading carefully they will see the phrase “3 more times than,” which will reveal choice D as the correct answer.

Avoid Common Errors Item 8 Some students may fail to pay attention to the direction the graph is pointing. As a result, they may choose answer choice A or D, because they both include the number 4. 4 in the solution. Remind the students that the direction of the inequality is as important as its endpoint. Item 13 Some students may give 2 as their answer because they transposed the x- and y-coordinates. Remind students that the y-coordinate is the second number in an ordered pair.

Texas Essential Knowledge and Skills Items

Mathematical Process TEKS

1

6.7.A

6.1.D

2

6.7.A

6.1.F

3

6.7.C

6.1.A

4

6.9.B, 6.10.A

6.1.D

5

6.9.A

6.1.A, 6.1.F

6

6.10.B

6.1.F

7

6.9.A

6.1.A, 6.1.F

8

6.9.B, 6.10.A

6.1.D

9

6.6.A

6.1.A

10*

6.4.C, 6.4.E

6.1.F

11*

6.3.E

6.1.F

12

6.10.A

6.1.A

13

6.11

6.1.E

14

6.7.A, 6.7.C

6.1.A

* Item integrates mixed review concepts from previous modules or a previous course.

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DO NOT EDIT--Changes must be made through “File info” CorrectionKey=B

DO NOT EDIT--Changes must be made through “File info” CorrectionKey=B

Personal Math Trainer

Texas Test Prep Selected Response 1. Which expression is equivalent to 2.3 × 2.3 × 2.3 × 2.3 × 2.3? A 2.3 × 5

2. Which operation should you perform first when you simplify 63 – (2 + 54 × 6) ÷ 5? A addition

3. Sheena was organizing items in a scrapbook. She took 25 photos and divided them evenly among p pages. Which algebraic expression represents the number of photos on each page? A p – 25

© Houghton Mifflin Harcourt Publishing Company

B 25 – p p __ C 25  25 D __ p

D 4 – 6 = –2

6. For which of the inequalities below is v = 4 a solution? A v+5≥9

̵4

5

6 C __ 15 18 __ D 45

A 1.8 centimeters B 11.4 centimeters C 13.7 centimeters D 114 centimeters

A j–9=3

Gridded Response

C j–3=9

12. The area of a rectangular mural is 84 square feet. The mural’s width is 7 feet. What is its length in feet?

B 3j = 9

5

10

m A __  > 1.1 4 m B __  < 1.2 3 C 2m < 8.8 D 5m > 22

Hot ! Tip

When possible, use logic to eliminate at least two answer choices.

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O ̵2

11. One inch is 2.54 centimeters. About how many centimeters is 4.5 inches?

7. Sarah has read aloud in class 3 more times than Joel. Sarah has read 9 times. Which equation represents this situation?

0

̵2

D the number of stars available

2 _ A 5 __ B 12 25

D j+3=9

0

x ̵4

C the number of hours played

8. The number line below represents the solution to which inequality?

4. The number line below represents which equation?

G 2

10. Which ratio is not equivalent to the other three?

D v+5<8

D subtraction

C 4 + 6 = –2

A n<7

C v+5≤8

C multiplication

y 4

B the number of points earned

B v+5>9

B division

13. What is the y-coordinate of point G on the coordinate grid below?

A the number of stars picked up

D n≥7

D 2.35

B –2 – 6 = 4

5. No more than 7 copies of a newspaper are left in the newspaper rack. Which inequality represents this situation?

C n>7

C 25 × 35

A –2 + 6 = 4

9. Brian is playing a video game. He earns the same number of points for each star he picks up. He earned 2,400 points for 6 stars, 4,000 points for 10 stars, and 5,200 points for 13 stars. Which is the independent variable in the situation?

Online Assessment and Intervention

B n≤7

B 235

-5

my.hrw.com

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Gridded responses cannot be negative numbers. If you get a negative value, you likely made an error. Check your work!

Hot ! Tip

14. When traveling in Canada, Patricia converts the temperature given in degrees Celsius to a Fahrenheit temperature by using the expression 9x ÷ 5 + 32, where x is the Celsius temperature. Find the temperature in degrees Fahrenheit when it is 25 °C.

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Unit 4

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InCopy Notes 1. This is a list Bold, Italic, Strickthrough.

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Expressions, Equations, and Relationships

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Expressions, Equations, and Relationships

UNIT 4 Expressions, Equations, and Relationships Contents 6.7.A 6.7.A 6.7.A 6.7.C 6.7.A 6.7.D 6.9.A 6.10.A 6.10.A 6.9.A 6.10.A 6.10.A 6.9.B MODUL...

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