mif teacher's edition sample pages - course 1

CHAPTER

Chapter at a Glance7

CHAPTER OPENERAlgebraic Expressions

Recall Prior Knowledge

LESSON 7.1Writing Algebraic Expressions

Pages 428

LESSON 7.2Evaluating Algebraic Expressions

Pages 9211

LESS

ON

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Pacing 1 day 2 days 1 day

Objectives

Algebraic

expressions can

be used to describe

situations and solve

real-world problems.

• Usevariablestowritealgebraic

expressions.

• Evaluatealgebraicexpressions

for given values of the variable.

Vocabulary

variable

algebraic expression

terms

evaluate

substitute

RE

SOU

RC

ES

Materials

Lesson Resources

Student Book A, pp. 123 Assessments, Chapter 7

Pre-Test

Transition Guide, Course 1, Skills 24226

Student Book A, pp. 428Extra Practice A, Lesson 7.1Reteach A, Lesson 7.1


Common Core State Standards

6.EE.2a, b Write, read, and evaluate expressions in which letters stand for numbers. Identify parts of an expression using mathematical terms...; 6.EE.6Usevariables...when solving a real-world or mathematical problem...

6.EE.2 Write, read, and evaluate expressions in which letters stand for numbers. 6.EE.2cEvaluateexpressions at specific values of their variables...

GRADE 5 COURSE 1 COURSE 2

• Evaluateexpressionsusing

grouping symbols. (5.OA.1)

• Writeandinterpretnumerical

expressions to record calculations.

(5.OA.2)

• Identifynumericalpatternsand

rules. (5.OA.3)

• Write,read,andevaluatevariable

expressions.(6.EE.2,6.EE.2a,

6.EE.2b,6.EE.2c)

• Usepropertiestogenerateand

identify equivalent expressions.

(6.EE.3,6.EE.4)

• Usevariablestowriteexpressions

when solving real-world or

mathematicalproblems.(6.EE.6)

• Applypropertiesofoperationsto

add, subtract, factor, expand, and

rewritelinearexpressions.(7.EE.1,

7.EE.2)

• Usealgebraicexpressionstosolve

multi-step problems with rational

numbers.(7.EE.3)

• Usevariablestorepresent

quantities in real-world or

mathematicalproblems.(7.EE.4)

Concepts and Skills Across the Courses

1A Chapter 7 Chapter at a Glance

1BChapter 7 Chapter at a Glance

LESSON 7.3Simplifying Algebraic Expressions

Pages 12221

LESSON 7.4Expanding and Factoring

Algebraic ExpressionsPages 22228

LESSON 7.5Real-Word Problems: Algebraic Expressions

Pages 29235

3 days 2 days 2 days

• Simplifyalgebraicexpressionsinone variable.

• Recognizethattheexpressionobtained after simplifying is equivalent to the original expression.

• Expandsimplealgebraicexpressions.

• Factorsimplealgebraicexpressions.

• Solvereal-worldproblemsinvolvingalgebraic expressions.

simplifycoefficientlike termsequivalent expressions

expandfactor

paper,ruler,scissors,TR14* paper, ruler, scissors, yardsticks




6.EE.3 Apply the properties of operations to generate equivalent expressions. 6.EE.4 Identify when two expressions are equivalent. 6.EE.6 Usevariables...whensolvingareal-world or mathematical problem...

6.EE.2 Write, read, and evaluate expressions in which letters stand for numbers. 6.EE.3 Apply the properties of operations to generate equivalent expressions. 6.EE.4 Identify when two expressions are equivalent...

6.EE.6Usevariablestorepresentnumbers and write expressions when solving a real-world or mathematical problem; understand that a variable can represent an unknown number, or, depending on the purpose at hand, any number in a specified set.

TECHNOLOGYEvery Day CountsTM

ALGEBRA READINESS

• OnlineStudenteBook

• Virtual Manipulatives

• TeacherOneStopCD-ROM

• ExamViewAssessmentSuiteCD-ROMCourse 1

• OnlineProfessionalDevelopmentVideos

The January activities in the Pacing Chart provide:

• Review of factors, square numbers, number sequences,

and divisibility (Ch1: 6.NS.4)

• Review of integers and other rational numbers including

decimals, fractions, and percents (Ch224, 6: 6.NS.5,

6.NS.6)

• Preview of patterns and attributes in quadrilaterals

(Ch10: 6.G.3)

• Preview of data collection and analysis and statistical

terms (Ch13214: 6.SP.1, 6.SP.2)

Additional Chapter Resources

PR

OFESSIONAL

LEARNING

*Teacher resources (TR) are available on the Teacher One Stop CD-ROM.

CHAPTER

Chapter at a Glance7

Chapter 7 Chapter at a Glance1C

CHAPTERWRAP UP/REVIEW/TEST

Brain@Work Pages 35–38

LESS

ON

AT

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CE

Pacing 1 day

Objectives

Reinforce,consolidate,andextendchapter skills and concepts.

Vocabulary

RE

SOU

RC

ES

Materials

Lesson Resources

Student Book A, pp. 35–38 Activity Book, Chapter 7 Project Enrichment, Chapter 7Assessments, Chapter 7 Test

ExamViewAssessmentSuite CD-ROMCourse1

Common Core State Standards

TEACHER NOTES

Algebraic Twists to Familiar Ideas • Inthischapter,studentswilllearnhowtowrite

algebraic expressions to represent situations in

the world around them. Algebraic expressions are

sometimes called variable expressions because they

contain one or more variables.

New algebraic notation• Studentslearntousevariablestorepresentunknown

quantities. They learn to write “6 2 n” to represent

“6 minus a number.”

• Studentslearntocorrectlyidentifytermsinalgebraic

expressions.Forexample,thetermsin3x 1 5 are 3x

and 5.

Working with algebraic expressions• Studentslearnhowtoevaluatealgebraicexpressions

for given values. Asked to evaluate 2x 1 5 for x 5 4,

they substitute 4 for x in the expression, then simplify

to find the value.

2(4) 1 5 5 8 1 5 5 13

• Studentssimplifyalgebraicexpressionsandexpand

and factor them, such as 4(p 2 3) 5 4p 2 12.

• Studentsrecognizeequivalentalgebraicexpressions,

such as 3(x 1 2) 5 3x 1 6.

• Studentssolvereal-worldproblemsusingalgebraic

expressions.

Sarahas3packsofbatteries.Eachpackhasn

batteries. She also has 2 single batteries. How many

batteries does she have in all? Write an expression

to represent the number of batteries Sara has.

3n 1 2

If the packs have 4 batteries each, how many

batteries does she have?

3(4) 1 2 5 12 1 2 5 14

Chapter 7 Algebraic Expressions

PR

OFESSIONAL

LEARNING Math Background

Bar Models • Inthischapter,studentswillrelatewhattheyknow

about bar models to algebraic expressions. The part-

part-whole model was used in earlier grades to solve

problems such as the following:

Peter bought 3 identical packs of baseball cards.

He gave 5 cards away. This left Peter with 31 cards.

How many cards were in each pack?

Solution Draw1longbaranddivideitinto3equalparts.This

represents the identical, full packs of baseball cards.

Sub-divide one of the parts and add the labels 31

and 5 in the model.

31 5

?

FromthemodelyoucanseethatPeterstartedwith

31 1 5 5 36 cards, so each pack had 36 3 5

12 cards.

• Nowstudentswillsolvesimilarproblems,suchasthis:

Peterbought3packsofbaseballcards.Eachpack

had c cards. Then Peter gave 5 cards away. Write

an expression to represent the number of cards

Peter has left.

Solution The bar model looks the same; only the labels have

changed.Fromthemodelyoucanseethathehad

c 1 c 1 (c 2 5) cards left.

c c

31 5

? c

1DChapter 7 Math Background

Additional Teaching Support

Transition Guide, Course 1Online Professional DevelopmentVideos

Continued on next page

Chapter 7 Math Background1E

• Similarly,inearliergrades,studentusedbarmodels

torepresentdivisionproblems.Forinstance,students

can use either of the models below to represent the

division probem 36 3.

36

?

? groups

36

3 3 3 3

• Nowstudentswilllearntousebarmodelstosolve

algebraic problems such as the following.

Ericaearnedx dollars in 3 days. If she earned the

same amount each day, how many dollars did she

earn each day?

Solution

x

1 day 1 day 1 day

?

Again, the bar model looks the same; only the labels

havechanged.Fromthemodel,youcan

seethatEricaearnedx3 dollars each day.

Recognizing equivalent expressions• Studentsusebarmodelstorecognizeequivalent

expressions.

Simplify x 1 3x.

3xx

?

x 1 3x 5 4x

PR

OFESSIONAL

LEARNING Math Background

Expanding algebraic expressions• Studentsusebarmodelsandnumberpropertiesto

expand algebraic expressions.

Expand3(p 1 2).

p 2 p 2 p 2

1 group

Rearrangetocollectliketerms.

p p p 2 2 2

3 3 p 3 3 2

3(p 1 2) 5 3 3 ( p 1 2)

5 3 3 p 1 3 3 2

5 3p 1 6

Solving real-world problems• Finally,studentsusebarmodelsandalgebraic

expressions to represent and solve real-world problems.

Jeff is x years old. His sister Sara is 3 years older and

hisbrotherRafaelis2yearsyounger.Writealgebraic

expressionsforSaraandRafael’sages.

x

x

3 2

?

?

Sara: x 13 Rafael:x 2 2

• Usingbarmodelscanhelpstudentssetupexpressions,

so that they can then evaluate them for a given value of

thevariable.Forexample:

IfJeffis10yearsold,howoldareSaraandRafael?

Solution Evaluatetheexpressionsabove.

Sara: x 1 3 5 10 1 3

5 13

Rafael: x 2 2 5 10 2 2

5 8

1FChapter 7 DifferentiatedInstruction

ASSESSMENT1

2

3

Response to InterventionSTRUGGLING LEARNERS

DIAGNOSTIC• QuickCheckinRecallPriorKnowledge

in Student Book A, pp. 223

• Chapter7Pre-Testin Assessments

• Skills24–26inTransition Guide, Course 1

ON-GOING• GuidedPractice

• LessonCheck

• TicketOuttheDoor

• Reteachworksheets

• ExtraPracticeworksheets

• ActivitiesinActivity Book, Chapter 7

END-OF-CHAPTER

• ChapterReview/Test

• Chapter7TestinAssessments • ExamViewAssessmentSuite

CD-ROMCourse1

• Reteachworksheets

Assessment and Intervention

ADVANCED LEARNERS

• Studentscanbuildvisualpatternsfromanysetof

identical building blocks, toothpicks, grid paper, or

dotpaper.Forexample,studentscouldusebuilding

blocks to build perfect cubes or dot paper to form a

sequence of triangular numbers.

• Havethemlisttermsintheirpatternsandwrite

expressions for the nth term in their patterns. They

may need help in writing expressions for complex

patterns.

• Patternsintwocolorscanbeusedtowrite

expressions for each color, and then for the two colors

combined as an application of combining like terms.

To provide additional challenges use:

• Enrichment, Chapter 7

• StudentBookA,Brain@Workproblem

ELL ENGLISH LANGUAGE LEARNERS

Reviewthetermsvariable, algebraic expression,

and bar model.

Say You can use the letter n to stand for a number you

do not know. The letter n is called a variable. (Write

n 1 2 on the board.) This expression contains a variable

and a number. It is called an algebraic expression.

ModelDrawabarmodeltoshowthealgebraic

expression.

2

?

n

Fordefinitions,seeGlossaryattheendofStudentBook.

Online Multi-Lingual Glossary.

Differentiated Instruction

1Chapter 7 Algebraic Expressions

Algebraic Expressions How safe is it?Imagine this: You are standing on a bridge, about to experience the thrill

of bungee jumping. A fast-flowing river rushes by beneath you, and a

bungee cord is strapped around your ankles. How safe is it for you to

make the jump? Is the bungee cord the right length?

To answer this question, you can use an algebraic expression to calculate

how much the bungee cord will stretch when you jump. For example,

the amount the cord stretches is 80.9 feet for a 100-pound person, and

111.5 feet for a 150-pound person.

In this chapter, you will learn how variables and algebraic expressions

can be used in daily life. For example, the manufacturer of the bungee

cords uses an algebraic expression to find the weights that are safe

for jumping.

BIG IDEA

CHAPTER

7

7.1 Writing Algebraic Expressions

7.2 Evaluating Algebraic Expressions

7.3 Simplifying Algebraic Expressions

7.4 Expanding and Factoring Algebraic Expressions

Algebraic expressions can be used to describe situations and solve real-world problems.

7.5 Real-World Problems: Algebraic Expressions

2 Chapter 7 AlgebraicExpressions


Algebraic Expressions How safe is it?Imagine this: You are standing on a bridge, about to experience the thrill

of bungee jumping. A fast-flowing river rushes by beneath you, and a

bungee cord is strapped around your ankles. How safe is it for you to

make the jump? Is the bungee cord the right length?

To answer this question, you can use an algebraic expression to calculate

how much the bungee cord will stretch when you jump. For example,

the amount the cord stretches is 80.9 feet for a 100-pound person, and

111.5 feet for a 150-pound person.

In this chapter, you will learn how variables and algebraic expressions

can be used in daily life. For example, the manufacturer of the bungee

cords uses an algebraic expression to find the weights that are safe

for jumping.

BIG IDEA

CHAPTER

7





Algebraic expressions can be used to describe situations and solve real-world problems.


1Chapter 7 AlgebraicExpressions

CHAPTER OPENER

Usethechapteropenertotalkabouttheuseofalgebrain

a real life situation.

Ask Has anyone ever watched bungee jumping? What

makes the cord stretch when a person jumps? Possible

answer: The weight of the person pulls down on the cord,

causing it to stretch.

Explain The extension in the bungee cord can be

calculated using algebra. By substituting different values of

aperson’sweightintothealgebraicformulaforcalculating

the extension, the heaviest weight that the cord can safely

takecanbeevaluated.Forsafetyreasons,ajumper’s

weight must be less than this weight.

In this chapter, you will learn how to write and evaluate

algebraic expressions for many different situations, as

summarizedintheBig Idea.

Chapter Vocabulary

Vocabulary terms are used in context inthestudenttext.Fordefinitions,see the Glossary at the end of the Student Book and the online Multi-Lingual Glossary.

algebraic expression An expression that contains at least one variable.

3y 2 2 and y4

are algebraic

expressions.

coefficient The numerical factor in a term of an algebraic expression. In the term 8z, the coefficient of z is 8.

equivalent expressionsExpressionsthat are equal for all values of the variables. 2x 1 x and 3x are equivalent expressions because 2x 1 x 5 3x for all values of x.

evaluate To find the value.

expand To write an expression that uses parentheses as an equivalent expression without parentheses. Forexample,4(y 1 1) 5 4y 1 4.

CHAPTER

77CHAPTER


Finding common factors and greatest common factor of two whole numbers

List the common factors of 6 and 14. Then find their greatest common factor.

6 5 1 3 6 14 5 1 3 14

5 2 3 3 5 2 3 7

Factors of 6: 1, 2, 3, 6

Factors of 14: 1, 2, 7, 14

The common factors of 6 and 14 are 1 and 2.

The greatest common factor of 6 and 14 is 2.

Quick CheckFind the common factors and greatest common factor of each pair of numbers.

5 6 and 9 6 4 and 12

7 5 and 15 8 8 and 28

Meaning of mathematical terms

The sum of 3 and 4 is 3 1 4.

The difference of 4 and 3 is 4 2 3.

The product of 3 and 4 is 3 3 4.

The quotient of 3 and 4 is 3 4 4 or 34

. 3 is the dividend and 4 is the divisor.

Quick CheckComplete with quotient, sum, difference, product, dividend, or divisor.

9 The ? of 7 and 5 is 7 2 5.

10 The ? of 5 and 7 is 57

. 7 is the ? and 5 is the ? .

11 The ? of 5 and 7 is 7 3 5.

12 The ? of 5 and 7 is 5 1 7.

1 and 3; 3

1 and 5; 5 1, 2, and 4; 4

1, 2, and 4; 4

difference

quotient; divisor; dividend

product

sum?

5 ?

15 4

17

?


Quick Check

1 2

3 4


RECALL PRIOR KNOWLEDGE

Usethe Quick Checkexercisesasadiagnostictooltoassessstudents’levelofprerequisiteknowledgebeforetheyprogresstothechapter.Forintervention

suggestions see the chart on the next page.

Chapter Vocabulary

factor To write an expression that does not use parentheses as an equivalent expression with parentheses.Forexample, 4x 1 4 5 4(x 1 1).

like terms Terms that have the same variables with the same corresponding exponents. In the expression 2x 1 4 1 x 1 1, the terms 2x and x are like terms, as are 4 and 1.

simplify To write an equivalent expression by combining like terms.

substitute To replace the variable by a number.

term A number, variable, product or quotient found in an expression. In the expression 5x 1 3, the terms are 5x and 3.

variable A quantity represented by a letter that can take different values. In the expression 2x 1 1, x is the variable.


Finding common factors and greatest common factor of two whole numbers

List the common factors of 6 and 14. Then find their greatest common factor.

6 5 1 3 6 14 5 1 3 14

5 2 3 3 5 2 3 7

Factors of 6: 1, 2, 3, 6

Factors of 14: 1, 2, 7, 14

The common factors of 6 and 14 are 1 and 2.

The greatest common factor of 6 and 14 is 2.

Quick CheckFind the common factors and greatest common factor of each pair of numbers.

5 6 and 9 6 4 and 12

7 5 and 15 8 8 and 28

Meaning of mathematical terms

The sum of 3 and 4 is 3 1 4.

The difference of 4 and 3 is 4 2 3.

The product of 3 and 4 is 3 3 4.

The quotient of 3 and 4 is 3 4 4 or 34

. 3 is the dividend and 4 is the divisor.

Quick CheckComplete with quotient, sum, difference, product, dividend, or divisor.

9 The ? of 7 and 5 is 7 2 5.

10 The ? of 5 and 7 is 57

. 7 is the ? and 5 is the ? .

11 The ? of 5 and 7 is 7 3 5.

12 The ? of 5 and 7 is 5 1 7.

1 and 3; 3

1 and 5; 5 1, 2, and 4; 4

1, 2, and 4; 4

difference

quotient; divisor; dividend

product

sum?

5 ?

15 4

17

?


Quick Check

1 2

3 4


1

2

3

Response to Intervention ASSESSING PRIOR KNOWLEDGE

Foradditionalassessmentof

students’priorknowledgeand

chapter readiness, use the

Chapter 7 Pre-Test in

Assessments, Course 1.

1

2

3

Response to Intervention Assessing Prior Knowledge

Exercises Skill or Concept Intervene with Transition Guide

1 to 4 Usebarmodelstoshowthefouroperations. Skill 24

5 to 8Findcommonfactorsandthegreatestcommonfactoroftwo

whole numbers.Skill 25

9 to 12 Understandthemeaningofmathematicalterms. Skill 26

Le

arn


Vocabulary

b)

5Lesson 7.1 Writing Algebraic Expressions

Le

arn Use variables to write subtraction expressions.

a) A straw of length 10 centimeters is cut from a straw of length 24 centimeters. What is

the length of the remaining straw?

24 cm

10 cm?

24 2 10 5 14

The length of the remaining straw is 14 centimeters.

b) A straw of length 6 centimeters is cut from a straw of length y centimeters. What is


? 6 cm

y cm

The length of the remaining straw is ( y 2 6) centimeters.

y 2 6 is an algebraic expression in terms of y.

The total length of the two ribbons is (x 1 9) inches.

x 1 9 is an algebraic expression in terms of x.

x and 9 are the terms of this expression.

You can say that

x 1 9 is the sum

of x and 9.

y and 6 are the terms of the

expression y 2 6.

You can say that y 2 6 is the

difference of y and 6.


a) Ask How can you find the total length of the two

ribbons? 5 1 8 5 13; 13 inches

Model Remindstudentsthatabarmodelcanbe

used to represent the addition sentence.

b) Ask How do you represent the unknown length of

the ribbon? x

ExplainExplainthatwhentheactualnumberis

not known, you can use a letter to represent this

unknown number. The letter x is called a variable,

because it can take different values depending on

what the actual length of the first ribbon is.

Explain If the length of the first ribbon is 12 in.,

then x 5 12. If the length is 24 in., then x 5 24.

Emphasizethatx represents a number.

KEY CONCEPTS

• Youcanuseavariabletorepresent

an unknown number or numbers.

• Youcanusevariablestowrite

algebraic expressions.

DAY 1

PACING

DAY 1 Pages 425

DAY 2 Pages 628

Materials: none

Learn Use variables to represent unknown numbers and write addition expressions.

Writing Algebraic Expressions7.1

5 5-minute Warm Up

1. John is 12 years old. His sister

is 6 years older.

Ask: How do you find his

sister’sage?Add 6 to 12.

2. Joyce is 15 years old. Her

brother is 4 years younger.

Ask: How do you find her

brother’sage?Subtract 4

from 15.

Also available on

TeacherOneStopCD-ROM.

Le

arn


Vocabulary

b)


Le

arn Use variables to write subtraction expressions.

a) A straw of length 10 centimeters is cut from a straw of length 24 centimeters. What is


24 cm

10 cm?

24 2 10 5 14

The length of the remaining straw is 14 centimeters.

b) A straw of length 6 centimeters is cut from a straw of length y centimeters. What is


? 6 cm

y cm

The length of the remaining straw is ( y 2 6) centimeters.

y 2 6 is an algebraic expression in terms of y.

The total length of the two ribbons is (x 1 9) inches.

x 1 9 is an algebraic expression in terms of x.

x and 9 are the terms of this expression.

You can say that

x 1 9 is the sum

of x and 9.

y and 6 are the terms of the

expression y 2 6.

You can say that y 2 6 is the

difference of y and 6.

5Lesson 7.1 WritingAlgebraicExpressions

ELL Vocabulary Highlight

Remindstudentsthat51 2 and

9 26areexpressions.Reinforce

that the term algebraic means

that one of the terms in the

expression must be a variable

term.

CautionExplaintostudentsthatterms

can be written in any order in an

expression using addition without

affecting the sum. However, the

order of the terms in algebraic

expressions using subtraction is

important. A straw of length

6 centimeters cut from a straw of

length y centimeters can be only

be written as y 2 6, not 6 2 y.

Usethenumericalexampleinpart a to introduce the algebraic

example in part b.

b) Explain In part a, you used a model to show the subtraction

24–10.Nowsupposeyouhadastrawoflengthy centimeters.

Ask How can you show the subtraction of 6 centimeters from

y centimeters? Drawabary cm long and mark off 6 cm. How do

you write the length remaining after 6 centimeters are subtracted?

( y 2 6) cm

Explain Point out that parentheses are used in ( y 26) to indicate

that “centimeters” describes the entire expression, not just the

“6.”

Learn continued

Ask What is the total length of

the two ribbons? (x 1 9) inches

Explain Tell students that x 1 9

is called an algebraic expression

in terms of x. In the expression

x 1 9, the variable x and the

number 9 are called the terms of

the expression. The expression

x 1 9 has two terms, x and 9.

Ask What are the terms in the

expression x 1 4? x and 4

Learn

Use variables to write subtraction expressions.

6 Chapter 7 Algebraic Expressions

Le

arn

4z is the only term of the

expression 4z.

You can say that 4z is the

product of z and 4.

Use variables to write multiplication expressions.

a) There are 12 crackers in each box. How many crackers are there in 2 boxes?

?

12

2 3 12 5 24

There are 24 crackers in 2 boxes.

b) There are z crackers in each box. How many crackers are there in 4 boxes?

?

z

4 3 z 5 4z There are 4z crackers in 4 boxes.

4z is an algebraic expression in terms of z.

Guided PracticeWrite an algebraic expression for each of the following.

1 The sum of x and 10.

2 The difference of y and 7.

3 Jim is now z years old.

a) His brother is 4 years older than Jim. Find his brother’s age in terms of z.

b) His sister is 3 years younger than Jim. Find his sister’s age in terms of z.

x 1 10

z 1 4

z 2 3

y 2 7


Guided Practice 1 and 2 Explaintostudentsthat

the terms sum and difference are

used in the same way with variables

as with numbers.

3 Watch out for students who

may not understand that “4 years

older than” implies addition, and

that “3 years younger than” implies

subtraction.

Learn

a) Ask How many crackers are there in 2 boxes?

2 3 12 5 24

Model Remindstudentsthatabarmodelcanbe

used to represent the multiplication sentence.

b) Ask If there are z crackers in each box, how many

crackers will there be in 4 boxes? 4z How many

terms are in the expression 4z? 1

Ask If there are 5 boxes, what is the total number of

crackers? 5z What will the algebraic expression for the

total number of crackers be if there are 12 boxes? 12z

Explain Tell students that 4z, 5z, and 12z are algebraic

expressions in terms of z.

Ask What are the terms of the expressions 4z, 5z, and

12z? 4z, 5z, and 12z

DAY 2


Point out to students that the

number in a multiplication

expression such as 4x is usually

written in front of the variable.

So, the “product of z and 4” is

written as 4z.

Best Practices


Le

arn

4z is the only term of the

expression 4z.

You can say that 4z is the

product of z and 4.


a) There are 12 crackers in each box. How many crackers are there in 2 boxes?

?

12

2 3 12 5 24

There are 24 crackers in 2 boxes.

b) There are z crackers in each box. How many crackers are there in 4 boxes?

?

z

4 3 z 5 4z There are 4z crackers in 4 boxes.

4z is an algebraic expression in terms of z.


1 The sum of x and 10.

2 The difference of y and 7.

3 Jim is now z years old.

a) His brother is 4 years older than Jim. Find his brother’s age in terms of z.

b) His sister is 3 years younger than Jim. Find his sister’s age in terms of z.

x 1 10

z 1 4

z 2 3

y 2 7


Le

arn

?

12 in.

Use variables to write division expressions.

a) A 12-inch rod is divided into 3 parts of equal length.

What is the length of each part?

12 4 3 5 4

The length of each part is 4 inches.

b) A rod of length w inches is divided into 7 parts of equal length. What is the

length of each part?

?

w in.

The length of each part is (w 4 7) inches or w7

inches.

w7

is an algebraic expression in terms of w.


4 The product of z and 6.

5 The quotient of w and 8.

6 Mia bought a pair of shoes for p dollars. She also bought a dress that cost 5 times as

much as the shoes, and a belt that cost 14

of the price of the shoes.

a) Find the cost of the dress in terms of p.

b) Find the cost of the belt in terms of p.

Math Note

w7

can also be written as 17

w.

w7

is the only term of the expression w7

.

You can say that w7

is the quotient of w and 7.

w is the dividend and 7 is the divisor.

6z

5p dollars

, or p dollarsp4

14

w8

7Lesson 7.1 WritingAlgebraicExpressions

Guided Practice4 and 5 Explaintostudentsthat

the terms product and quotient are

used in the same way with variables

aswithnumbers.Remindstudents

that 6z means 6 3 z and not six z’s.

Ask students to describe

situations in which they might

have to use division, such as

finding the cost for each friend if

5 friends pay a total of x dollars

for movie tickets. Have them

describe the situation using a

variable and model it with an

algebraic expression.

Best PracticesLe

arn

a) Ask What is the length of each part of the rod?

12 3 5 4; 4 inches

Model Remindstudentsthatabarmodelcanbeused

to represent the division sentence.

b) Ask If a rod of w inches is divided into 7 parts of equal

length, how can you find the length of each part?

w 7 5 w7

; w7

inches Point out that w7

is

the only term of the expression.

Ask If the rod is divided into 9 parts of equal length,

what will the length of each part be? w9

Use variables to write division expressions.

Explain Tell students that w7

and w9

are algebraic

expressions in terms of w.

Ask What are the terms of the expressions w7

and w9

?w7

and w9

Explain Show students thatw7

5 w 7 5 w 3 17

5 17

3 w 5 17

w.

Point out that the order of the

terms in an algebraic expression

involving multiplication does

not affect the product, but such

expressions are usually written

withthevariablelast.Explain

that the order of the terms in a

division expression does matter. If

a rod of w inches is divided into 7

parts of equal length, the correct

expression is w 7, not 7 w.

Caution

9Lesson 7.2 Evaluating Algebraic Expressions

Le

arn


Vocabularyevaluate

substitute

Lesson Objective• Evaluatealgebraicexpressionsforgivenvaluesofthevariable.

Algebraic expressions can be evaluated for given values of the variable.

a) Simon has x marbles and Cynthia has 3 marbles. How many more marbles does

Simon have than Cynthia?

x

?3

Simon

Cynthia

From the model, Simon has (x 2 3) more marbles than Cynthia.

To know exactly how many more marbles Simon has than Cynthia, you need to

know the value of x.

When x 5 5, x 2 3 5 5 2 3

5 2

When x 5 5, Simon has 2 more marbles than Cynthia.

When x 5 9, x 2 3 5 9 2 3

5 6


When x 5 17, x 2 3 5 17 2 3

5 14


Continue on next page


Write an algebraic expression for each of the following.

1 The sum of 4 and p. 2 The difference of q and 8.

3 The product of 3 and r. 4 The quotient of s and 5.

5 Cheryl is now x years old.

a) Her father is 24 years older than Cheryl. Find her father’s age in terms of x.

b) Her brother is 2 years younger than Cheryl. Find her brother’s age in terms of x.

c) Her sister is twice as old as Cheryl. Find her sister’s age in terms of x.

d) Her cousin is 13

Cheryl’s age. Find her cousin’s age in terms of x.

6 Multiply k by 5, and then add 3 to the product.

7 Divide m by 7, and then subtract 4 from the quotient.

8 Divide j by 9, and then multiply the quotient by 2.

9 The sum of 13

of z and 15

of z.

Solve.

10 Jeremy bought 5 pencils for w dollars. Each pen costs 35¢ more than a pencil.

Write an algebraic expression for each of the following in terms of w.

a) The cost, in dollars, of a pen.

b) The number of pencils that Jeremy can buy with $20.

11 The figure shown is formed by a rectangle and a square. Express the area of the

figure in terms of x.

x cm

7 cm

3 cm

Practice 7.1

dollarsw5

0 351 .

Basic 1 – 5

Intermediate 6 – 9

Advanced 10 – 11

4 1 p

3r

(7x 1 9) cm2

5k 1 3

x 1 24

2x

x 2 2

q 2 8

s5

x3

2 4m7

3 2j9

z1 z13

15

pencils100w


1

2

3

Response to Intervention Lesson Check

Before assigning homework, use the following … to make sure students … Intervene with …

Exercises 1 and 2• can identify and write simple algebraic

expressionsReteach7.1

• can write an algebraic expression to

represent a situation

Practice 7.1

Assignment Guide

DAY 1 All students should

complete 1 – 2 .


complete 3 – 9 .

10 – 11 provide additional

challenge.

Optional: Extra Practice 7.1

Benis3timesasoldasKyra.Dilip

is4yearsyoungerthanKyra.If

Kyraisx years old, how old is

Ben?HowoldisDilip?Writean

algebraic expression for each

boy’sage.

Ben’sage:3x;Dilip’sage:x 2 3

Also available on



Le

arn


Vocabularyevaluate

substitute

Lesson Objective• Evaluatealgebraicexpressionsforgivenvaluesofthevariable.


a) Simon has x marbles and Cynthia has 3 marbles. How many more marbles does

Simon have than Cynthia?

x

?3

Simon

Cynthia

From the model, Simon has (x 2 3) more marbles than Cynthia.

To know exactly how many more marbles Simon has than Cynthia, you need to

know the value of x.

When x 5 5, x 2 3 5 5 2 3

5 2


When x 5 9, x 2 3 5 9 2 3

5 6


When x 5 17, x 2 3 5 17 2 3

5 14




Write an algebraic expression for each of the following.

1 The sum of 4 and p. 2 The difference of q and 8.

3 The product of 3 and r. 4 The quotient of s and 5.

5 Cheryl is now x years old.

a) Her father is 24 years older than Cheryl. Find her father’s age in terms of x.

b) Her brother is 2 years younger than Cheryl. Find her brother’s age in terms of x.

c) Her sister is twice as old as Cheryl. Find her sister’s age in terms of x.

d) Her cousin is 13

Cheryl’s age. Find her cousin’s age in terms of x.

6 Multiply k by 5, and then add 3 to the product.

7 Divide m by 7, and then subtract 4 from the quotient.

8 Divide j by 9, and then multiply the quotient by 2.

9 The sum of 13

of z and 15

of z.

Solve.

10 Jeremy bought 5 pencils for w dollars. Each pen costs 35¢ more than a pencil.

Write an algebraic expression for each of the following in terms of w.

a) The cost, in dollars, of a pen.

b) The number of pencils that Jeremy can buy with $20.

11 The figure shown is formed by a rectangle and a square. Express the area of the

figure in terms of x.

x cm

7 cm

3 cm

Practice 7.1

dollarsw5

0 351 .

Basic 1 – 5


Advanced 10 – 11

4 1 p

3r

(7x 1 9) cm2

5k 1 3

x 1 24

2x

x 2 2

q 2 8

s5

x3

2 4m7

3 2j9

z1 z13

15

pencils100w

9Lesson 7.2 EvaluatingAlgebraicExpressions

5

KEY CONCEPT

• You can evaluate algebraic

expressions for given values of

the variables.

Evaluating Algebraic Expressions

7.2

5-minute Warm Up

Reviewhowtoevaluatethese

numerical expressions:

1. (3 3 5) 1 8 23

2. 18 2 (4 3 4) 2

3. 15 2 ( )2 6

33

11

4. ( )8 3

43 1

( )22 872

2 186

5

Also available on


PACING

DAY 1 Pages 9–11

Materials: none

Learn

a) ModelUseabarmodeltoshowthatSimonhas

(x 23)moremarblesthanCynthia.Explainthat,in

order to know exactly how many more marbles, they

need the exact value of x.

Explain Show students that when different values of

x are substituted into the expression x 2 3, they get

different values for the number of marbles Simon has

more than Cynthia.

Ask If x = 5, how can you find out how many more

marbles Simon has than Cynthia? Substitute 5 for x in

the expression x − 3 to get 5 − 3 = 2. How can you

find out how many more marbles Simon has if x = 9?

Substitute 9 in the expression to get 9 − 3 = 6.

DAY 1


9

4

w4

20

Guided Practice

1

43x

x5

225– x

7

24

5

5

12

18

25


In e), you may want to contrast

the given expression,w4

2 4, with the expression

w 2 44

. Have students evaluate

each expression for w 5 20 to

see that the expressions are not

equivalent.

Best Practices

DIFFERENTIATED INSTRUCTION

Through Multiple Representations

You may want to point out the

efficient use of a table in Guided

Practice exercise. Some students

maybenefitfromorganizingtheir

work in Practice exercises 1 – 16

in a table.

Guided Practice 1 Remindstudentsthatan

expression such as 2x means 2 times

x. Be on the lookout for students who

forget to use the order of operations

when evaluating expressions.

b) Ask How do you evaluate x 1 12 when x 5 5?

Substitute 5 for x in the expression x 1 12. What

answer do you get? 5 1 12 5 17 What is the value

of x 1 12 when x 5 10? 22 when x 5 100? 112

c) – e) Explain Work through these examples with

students. As in b), you may want to have them evaluate

each expression for other values of the variable. Ask

students to evaluate 16 2 y, 3z 1 6, andw4

2 4 for other values of the variables.

Summarize To evaluate an expression, substitute the

given value of the variable into the expression.

Learn continued


Evaluate each expression for the given value of the variable.

1 x 1 x 1 5 when x 5 7 2 3x 1 5 when x 5 5

3 5y 2 8 when y 5 3 4 40 2 9y when y 5 2

5 33 2 7w 1 6 when w 5 4 6 76w when w 5 18

7 4 1 56z

when z 5 12 8 4 56

1 z when z 5 12

9 20 2 45r when r 5 10 10 8

9r 2 15 when r 5 27

11 16 2 2 43

z 2 when z 5 18 12 16 2 23z 2 4 when z 5 18

Evaluate each expression when x 5 3.

13 x 1 12

1 5 310

x 2 14 1121 x 2 9 3

4x 2

15 7 63

x 2 1 4(8 1 2x) 16 13(11 2 3x) 2 5 16 42

( )2 x

17 5(x 1 2) 1 2(6 2 x) 1 2 33

x 1 18 5 3

4x 2 1 5 5

8( )x 1 1 3(13 2 2x)

19 2 4

5x 1 2

x 1 14

1 x6

20 7x 2 x5

1 7

92 x

Evaluate each of the following when y 5 7.

21 The difference of (5y 1 2) and (2y 1 5).

22 The sum of y3

and 49y .

23 The product of ( y 1 1) and ( y 2 1).

24 The difference of 8(2y 2 1) and 14 375

y 1 .

25 The quotient of 9(7y 2 15) and 110 642 y .

26 The sum of 56y and 4

37

2y y1

.

27 The quotient of y y2

23

1

and

56 3y y

2

.

Practice 7.2

12

Basic 1 – 12


Advanced 21 – 27

19 20

7 22

11 21

14

12

1

9

0

34 29

61 16

18

48

77

13845

20

49

5

13

2

56

73

23

10

18

135

153

11Lesson 7.2 EvaluatingAlgebraicExpressions

Practice 7.2

Assignment GuideAll students should complete

1 – 20 .


challenge.


Write two algebraic expressions

using x that when evaluated for

x 5 3 give the same value. Show

your work. Possible answer:

3x 1 2 and 2x 1 5. When

evaluated for x 5 3,

3x 1 2 5 3(3) 1 2 5 9 1 2 5 11

2x 1 5 5 2(3) 1 5 5 6 1 5 5 11

Also available on


1

2

3



Exercises 1 , 3 and 11• can evaluate simple algebraic expressions in

one variableReteach7.2

• can write and evaluate simple algebraic

expressions

In 23 , students can write the

product of ( y + 1) and ( y − 1)

before evaluating. Some students

may run into difficulties if they

try to find the product before

evaluating inside parentheses.

Best Practices

13Lesson 7.3 Simplifying Algebraic Expressions

p cm p cm p cm

?

p 1 p 1 p 5 3 3 p 5 3p

The perimeter of the triangle is 3p centimeters.

In the term 3p, the coefficient of p is 3.

c) The figure shows six rods and their lengths. Find the total length of the six rods in

terms of z. Then state the coefficient of the variable in the expression.

z cm z cm z cm

z cm

5 cm2 cm

z cm z cm z cm z cm 5 cm2 cm

?

z 1 z 1 z 1 z 1 2 1 5 5 (4 3 z) 1 2 1 5

5 4z 1 7

The total length of the six rods is (4z 1 7) centimeters.

In the term 4z, the coefficient of z is 4.

7 7 7

7 1 7 1 7 5 3 3 7

p p p

p 1 p 1 p 5 3 3 p

3 3 p is the same as 3p.

Add the variables

together. Then add

the numbers.

Le

arn


Vocabulary


KEY CONCEPTS

• Algebraic expressions in one

variable can be simplified by

combining like terms.

• Theexpressionobtainedafter

simplifying is equivalent to the

original expression.

Simplifying Algebraic Expressions

7.3

5 5-minute Warm Up

Reviewtheconceptof

multiplication as repeated

addition. Ask students how they

can write these sums as products:

2 1 2 2 3 2 2 1 2 1 2 3 3 2

3 1 3 2 3 3 3 1 3 1 3 3 3 3

12 1 12 2 3 12

12 1 12 1 12 3 3 12

Also available on


PACING

DAY 1 Pages 12 –15

DAY 2 Pages 16 –17

DAY 3 Pages 18 – 21

Materials: paper,ruler,scissors,TR14

Learn

a) ModelUsetwobarsofthesamelengthto

represent the addition of two straws of length

yinches.Remindstudentsthatavariablerepresents

a number. Here, the variable y represents a number

that is the unknown length of a straw.

Ask What expression represents the total length of

the two straws with a length of y inches each? y 1 y

ExplainRemindstudentswhattheyhavejust

reviewed: 3 1 3 5 2 3 3, 4 1 4 5 2 3 4.

Ask Since 3 1 3 5 2 3 3 and 4 1 4 5 2 3 4, what is

y 1 y equal to? 2y If y represents any number, what

does y 1 y 5 2y mean? It means that any number

added to itself is 2 times the number.

Explain Tell students that in the term 2y, 2 is called

the coefficient of y. The term 2y means 2 times the

value of y.

Ask Given that you can represent the addition of

the same number using multiplication, how do you

show 5 1 5 1 5 using multiplication? 3 3 5

b) Ask What does perimeter mean? Sum of all sides

DAY 1

Algebraic expressions can be simplified.


p cm p cm p cm

?

p 1 p 1 p 5 3 3 p 5 3p

The perimeter of the triangle is 3p centimeters.

In the term 3p, the coefficient of p is 3.

c) The figure shows six rods and their lengths. Find the total length of the six rods in

terms of z. Then state the coefficient of the variable in the expression.

z cm z cm z cm

z cm

5 cm2 cm

z cm z cm z cm z cm 5 cm2 cm

?

z 1 z 1 z 1 z 1 2 1 5 5 (4 3 z) 1 2 1 5

5 4z 1 7

The total length of the six rods is (4z 1 7) centimeters.

In the term 4z, the coefficient of z is 4.

7 7 7

7 1 7 1 7 5 3 3 7

p p p

p 1 p 1 p 5 3 3 p

3 3 p is the same as 3p.

Add the variables

together. Then add

the numbers.

Le

arn


Vocabulary

13Lesson 7.3 SimplifyingAlgebraicExpressions


Through Visual Cues

You may want to point out the

different lengths of the rods.

Students should note that the first

four rods have the same length,

z centimeters, while the two

remaining rods have lengths of

2 centimeters and 5 centimeters

respectively.

Ask What sum represents the perimeter of the

triangle, with each side of length p centimeters?

p 1 p 1 p

ModelUsethemodeltohelpstudentstoseethat

p 1 p 1 p 5 3 3 p 5 3p.

Ask What is the coefficient of p in the term 3p? 3

c) ModelUsefourbarsofthesamelengthto

represent the addition of four rods of length

zcentimeters.Usetwootherbarsofdifferent

lengths to represent rods of length 2 centimeters

and 5 centimeters respectively.

Ask What is the sum of the variable terms? z 1 z 1 z 1

z 5 4 3 z 5 4z What is the sum of the numerical terms?

7 What is the total length of the six rods?

(4z 1 7) centimeters What is the coefficient of z in the

term 4z? 4

Summarize To simplify algebraic expressions involving

addition, first group all the variable terms together and

find their sum. Then group all the numerical terms and

find their sum.

Learn continued


Work in pairs.

STEP

1 Make the following set of paper strips.

Let the length of the shortest strip be m units. Make and label 5 such strips.

RECOGNIZE THAT SIMPLIFIED EXPRESSIONS ARE EQUIVALENT

Materials:

• paper

• ruler

• scissors

m mm m m

Make and label 4 more strips of lengths 2m units, 3m units, 4m units, and 5m units.

2m 3m

4m 5m

STEP

2 Take one of the longer strips and place it horizontally.

Example

3m

STEP

3 Ask your partner to use the pieces of the shortest strips to match the length of the

chosen strip in STEP

2 .

Example

STEP

4 Write an algebraic expression to describe the number of short strips used, and

simplify it. For example in STEP

3 , write m 1 m 1 m 5 3m.

STEP

5 Repeattheactivitywithotherlengthsofstrips.

How do the lengths of the strips show that the expressions

are equivalent? In each case, the combined lengths of the short strips is equal to the length of the long strip.

5

6

4x

(3w 1 10)

x; x; x

x; x; x; 4x

3w 1 10

w; w; 10; 3w 1 10

w; w; 10

4x

Guided Practice

1 2

3 4

5x; 5 2y 1 6; 2

6n 1 4; 63m 1 9; 3


Guided Practice2 and 3 Forstudentswhohave

difficulty simplifying the expressions,

have them first draw bar models to

helpthemvisualizetheterms.

5 and 6 Askstudentstoverbalize

their work to ensure that they have

thecorrectconcept.Forexample,

w plus w plus w is equal to 3 times

w. Check in 6 that students are only

combining the like terms.


Work in pairs.

STEP

1 Make the following set of paper strips.

Let the length of the shortest strip be m units. Make and label 5 such strips.

RECOGNIZE THAT SIMPLIFIED EXPRESSIONS ARE EQUIVALENT

Materials:

• paper

• ruler

• scissors

m mm m m

Make and label 4 more strips of lengths 2m units, 3m units, 4m units, and 5m units.

2m 3m

4m 5m

STEP

2 Take one of the longer strips and place it horizontally.

Example

3m

STEP

3 Ask your partner to use the pieces of the shortest strips to match the length of the

chosen strip in STEP

2 .

Example

STEP

4 Write an algebraic expression to describe the number of short strips used, and

simplify it. For example in STEP

3 , write m 1 m 1 m 5 3m.

STEP

5 Repeattheactivitywithotherlengthsofstrips.

How do the lengths of the strips show that the expressions

are equivalent? In each case, the combined lengths of the short strips is equal to the length of the long strip.

5

6

4x

(3w 1 10)

x; x; x

x; x; x; 4x

3w 1 10

w; w; 10; 3w 1 10

w; w; 10

4x

Guided Practice

1 2

3 4

5x; 5 2y 1 6; 2

6n 1 4; 63m 1 9; 3


Hands-On ActivityThis activity reinforces the concept

of simplifying algebraic expressions

through a concrete approach. Have

students work in pairs.

Optional Materials:TR14,Paper

Strips

1 Make sure students cut the

same length for all the 5 strips

of length m units. When making

the strips for lengths of 2m units,

3m units, 4m units, and 5m units,

tell students that the length of

2m must be 2 times the length

of m, the length of 3m must be

3 times the length of m, and

so on.

4 Make sure students understand

that writing the algebraic

expression represents the action

carried out in 3 .

Guide

students to see that by

matching the number of

individual strips against a single

strip of the same length, they are

demonstrating that an addition

expression and the simplified form

of the expression are equivalent.

Forexample,thesumm 1 m 1 m is

equal to the product of 3 and m.


Le

arn Like terms can be subtracted.

a) Simplify 2v 2 v.

2v

v v

2v 2 v 5 v

b) Simplify 5w 2 3w.

5w

3w

w ww w w

5w 2 3w 5 2w

c) Simplify y 2 y.

y 2 y 5 0

2v 2 v and v are equivalent

expressions because they are

equal for all values of v.

If v 5 2, 2v 2 v 5 2 and v 5 2.

If v 5 3, 2v 2 v 5 3 and v 5 3.

Math Note

Any term that is subtracted from

itself is equal to zero.

Guided PracticeComplete.

16 Simplify 4s 2 s.

s ss s

?

?

4s 2 s 5 ?

Simplify each expression.

17 12z 2 7z 18 3p 2 3p

State whether each pair of expressions are equivalent.

19 f 2 6 and 6 2 f 20 5c 2 5c and a 2 a EquivalentNot equivalent

4s

3s

5z 0

s

Le

arn

b)

Guided Practice

7

8 9

11

13

15

9x

5r

Equivalent

Not equivalent Not equivalent

EquivalentNot equivalent

Equivalent

11y

x 8x


Students may not understand that

the variable x has a coefficient of

1.Emphasizethat,whenusing

a bar model for the expression,

you use one bar for the x term, to

show one group of x.

Best Practices

Guided PracticeUse 7 to reinforce the idea that

the coefficient of x is 1, so that

x 1 8x 5 9x.

8 and 9 Remindstudentsthatthe

coefficient of a term tells you how

manygroupsarebeingadded.For

example, 3r 1 2r means 3 groups

of r plus 2 groups of r.

Learn

a) ModelUsefourbarsofthesamelengthtomodel

3x 1 x.

ExplainExplainthatsince3x 5 x 1 x 1 x, the

expression 3x 1 x means x 1 x 1 x 1 x, or 4x.

Students should interpret 3x 1 x as 3 groups of x

plus 1 group of x, giving a total of 4 groups of x,

or 4x.

Explain Tell students that 3x and x are called like

terms.Emphasizethatonlyliketermscanbeadded.

Ask What are some other examples of like terms?

Possible answers: 5m and 7m, 4 and 6

Explain Point out that when two expressions

are equal for all values of the variables, they are

equivalent expressions. Since 3x 1 x 5 4x for all

values of x, 3x 1 x and 4x are equivalent expressions.

b) Model Model 4z 1 2z as in a).

Ask When you add 4 groups of z to 2 groups of

z, how many groups of z do you have? 6 What

expression represents 6 groups of z? 6z

Ask What are the equivalent expressions in the

example 4z 1 2z 5 6z? 4z 1 2z and 6z

DAY 2

Like terms can be added.


Le

arn Like terms can be subtracted.

a) Simplify 2v 2 v.

2v

v v

2v 2 v 5 v

b) Simplify 5w 2 3w.

5w

3w

w ww w w

5w 2 3w 5 2w

c) Simplify y 2 y.

y 2 y 5 0

2v 2 v and v are equivalent

expressions because they are

equal for all values of v.

If v 5 2, 2v 2 v 5 2 and v 5 2.

If v 5 3, 2v 2 v 5 3 and v 5 3.

Math Note

Any term that is subtracted from

itself is equal to zero.


16 Simplify 4s 2 s.

s ss s

?

?

4s 2 s 5 ?


17 12z 2 7z 18 3p 2 3p


19 f 2 6 and 6 2 f 20 5c 2 5c and a 2 a EquivalentNot equivalent

4s

3s

5z 0

s

Le

arn

b)

Guided Practice

7

8 9

11

13

15

9x

5r

Equivalent


EquivalentNot equivalent

Equivalent

11y

x 8x


Guided PracticeLook out for students who claim

that the expressions in 20 are not

equivalent. They may be thinking that

the word “equivalent” applies only to

an expression and the simplified form

ofthatexpression.Remindthemthat

“equivalent” means having the same value.

Learn

Like terms can be subtracted.

a) Ask What does 2v 2 v mean? 2 groups of v minus

1 group of v

Model Usetwobarsofthesamelengthto

represent 2v. Show that one bar is to be removed

from the diagram. Lead students to see that

2v 2 v 5 v.

b) Model Have students interpret the model 5w 2 3w.

Ask How many groups of w do you start with? 5

How many groups do you subtract? 3From

the model, how many groups of w are left after

subtraction? 2

c) Ask If a ribbon of length y centimeters is used up to

tie a gift, what is the length of the ribbon left?

0 centimeters

Explain Reinforcebyshowingexamplessuchas

2 2 2 5 0, and 15 2 15 5 0. Therefore, y 2 y 5 0.

Ask When you subtract a number from itself, what

answer do you get? 0

Summarize Any term that is subtracted from itself is

equaltozero.


b) Simplify 5x 2 2 1 3x.

5x 2 2 1 3x Identify like terms.

5 5x 1 3x 2 2 Change the order of terms

to collect like terms.

5 8x 2 2 Simplify.

Caution8x 2 2 6x because 8x and 2 are

not like terms. 8x 2 2 cannot be

simplified further.

5x 2 2 1 3x and 8x 2 2 are equivalent

expressions because they are equal for all

values of x.

If x 5 2, 5x 2 2 1 3x 5 14 and 8x 2 2 5 14.

If x 5 3, 5x 2 2 1 3x 5 22 and 8x 2 2 5 22.


27 The figure shows a quadrilateral. Find the perimeter of the quadrilateral.

6x 1 6 1 2x 1 2 5 6x 1 2x 1 6 1 2

5 ? 1 ?

The perimeter of the quadrilateral is ? units.


28 4x 2 3 1 3x 29 5y 1 4 2 2y

30 8y 2 7 2 4y 31 7z 1 9 2 2z 2 2

32 5 1 11z 2 4 1 6z 33 8g 1 10 2 3g 1 7

34 12 1 6g 2 5 2 4g 35 27 1 3r 2 9 1 15r

2x units

6x units

2 units6 units

8x; 8

7x 2 3

4y 2 7

17z 11

2g 17

8x 18

5g 117

3y 14

5z 17

18r 118

Le

arn

Le

arn

Add.

Work from left to right.

Subtract.


Add.

Guided Practice

21 22

24

26

Caution

8 8 Identify like terms.

Change the order of terms


Simplify.

Math Note

4j 11j

3w

0

6j

2t

5w


Guided PracticeWatch out for students who forget

to work from left to right in 23 to 26 .

Remindthemtofollowtheorderof

operations.

DAY 3

Learn

a) Explain Compare simplifying a numerical expression to simplifying an

algebraic one.

Ask How do you evaluate this expression 1 + 6 + 2? Work from left to

right: 1 + 6 + 2 = 7 + 2 = 9. Given the expression x + 6x + 2x, what is

your first step in simplifying it? Simplify x + 6x to get 7x. What is the next

step? Simplify 7x + 2x to get 9x.

Explain Make sure students see how the expressions in parts b and c

are different from the one in part a. In each case, the process for

simplifying is to work from left to right.

Summarize When adding and subtracting algebraic terms without

parentheses, always work from left to right.

Use order of operations to simplify algebraic expressions.

Learn

a) Ask Since the perimeter of

the parallelogram is r 1 8 1

r 1 8, how can you simplify

the expression? Reorderand

combine the like terms:

r 1 r 1 8 1 8 5 2r 1 16.

Can 2r 1 16 be simplified

further? No Why? 2r and 16

are not like terms.

Collect like terms to simplify algebraic expressions.

To reinforce understanding,

you may want to have students

analyzemathematicalstatements

in the Caution. Make sure they

understand that in each of the

“wrong” examples, the addition

and subtraction have not been

performed from left to right,

which leads to an incorrect result.

Best Practices


b) Simplify 5x 2 2 1 3x.

5x 2 2 1 3x Identify like terms.

5 5x 1 3x 2 2 Change the order of terms


5 8x 2 2 Simplify.

Caution8x 2 2 6x because 8x and 2 are

not like terms. 8x 2 2 cannot be

simplified further.

5x 2 2 1 3x and 8x 2 2 are equivalent

expressions because they are equal for all

values of x.

If x 5 2, 5x 2 2 1 3x 5 14 and 8x 2 2 5 14.

If x 5 3, 5x 2 2 1 3x 5 22 and 8x 2 2 5 22.


27 The figure shows a quadrilateral. Find the perimeter of the quadrilateral.

6x 1 6 1 2x 1 2 5 6x 1 2x 1 6 1 2

5 ? 1 ?

The perimeter of the quadrilateral is ? units.


28 4x 2 3 1 3x 29 5y 1 4 2 2y

30 8y 2 7 2 4y 31 7z 1 9 2 2z 2 2

32 5 1 11z 2 4 1 6z 33 8g 1 10 2 3g 1 7

34 12 1 6g 2 5 2 4g 35 27 1 3r 2 9 1 15r

2x units

6x units

2 units6 units

8x; 8

7x 2 3

4y 2 7

17z 11

2g 17

8x 18

5g 117

3y 14

5z 17

18r 118

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Add.


Subtract.


Add.

Guided Practice

21 22

24

26

Caution

8 8 Identify like terms.

Change the order of terms


Simplify.

Math Note

4j 11j

3w

0

6j

2t

5w


Guided Practice27 to 34 and 36 Check that students

have two terms in each of their final

expressions. Students who end up

with one term do not understand that

unlike terms cannot be combined.

b) Explain Make sure students understand that they

must change the order of the terms so that they can

add 5x and 3x to get 8x, and that 2 is subtracted

from the result.

Ask What are the like terms in 5x 2 2 1 3x? 5x and

3x If you want to add the like terms, how can you

rewrite the expression? 5x 1 3x 2 2 5 8x 2 2

Explain Emphasizethat8x 2 2 cannot be simplified

further because 8x and 2 are not like terms.

Learn continued

As students develop ideas

about combining like terms,

make sure they understand that

the operation symbol in front

of a term determines how it is

combined with other like terms.

Forexample,inb), students

should see that 2 is being

subtracted. When the terms

are reordered, 2 must still be

subtracted from 5.

Best Practices

1

2

3

4 5 6

7 8

9

11 12

13 123v

15

17 18

19

20

Practice 7.3 Basic 1 – 20

Intermediate 21 – 23 , 26

Advanced 24 – 25

Not equivalent

Not equivalent

Not equivalent

Equivalent

4u; 4

4p 9p 5p

p6p

2v 1 3; 2

6w 1 8; 6

7x 1 11 8x 1 2

3w 1 5 3u 1 6

(4b 1 4) inches

(3z 1 15) cm

Equivalent

Equivalent

Practice 7.3


Assignment Guide


complete 1 – 3 .


complete 4 – 6 .


complete 7 – 23 , 26 .


challenge.


1

2

3



Exercises 1 , 3 and 8

• can simplify simple algebraic expressions

involving addition and subtraction of like

terms

Reteach7.3Exercises 9 and 14

• check whether two algebraic expressions are

equivalent

• write an algebraic expression to represent a

situation and simplify that expression

Practice 7.3

1

2

3

4 5 6

7 8

9

11 12

13 123v

15

17 18

19

20

Practice 7.3 Basic 1 – 20

Intermediate 21 – 23 , 26

Advanced 24 – 25

Not equivalent

Not equivalent

Not equivalent

Equivalent

4u; 4

4p 9p 5p

p6p

2v 1 3; 2

6w 1 8; 6

7x 1 11 8x 1 2

3w 1 5 3u 1 6

(4b 1 4) inches

(3z 1 15) cm

Equivalent

Equivalent


21 Anne is currently h years old. Bill is currently 2h years old and Charles is

currently 8 years old. Find an expression for each person’s age after h years.

Then find an expression for the sum of their ages after h years.

22 There are 18 boys in a class. There are w fewer boys than girls. How many

students are there in the class?

23 A rectangular garden has a length of ( y 1 2) yards and a width of (4y 2 1) yards.

Find the perimeter of the garden in terms of y.

24 Kayla had 64b dollars. She gave 18

of it to Luke and spent $45. How much

money did Kayla have left? Express your answer in terms of b.

25 A rectangle has a length of (2m 1 1) units and a width of (10 2 m ) units.

A square has sides of length 2 1

2m 1

units.

a) Find the perimeter of the rectangle.

b) Find the perimeter of the square.

c) Find the sum of the perimeters of the two figures if m 5 6.

d) Find the difference between the perimeter of the rectangle and the

perimeter of the square if m 5 6.

26 Ritasimplifiedtheexpression10w 2 5w 1 2w in this way:

10w – 5w + 2w = 10w – 7w = 3w

IsRita’sanswercorrect?Ifnot,explainwhyitisincorrect.

(w 1 36) students

(10y 1 2) yards

(2m 1 22) units

(4m 1 2) units

60 units

8 units

No,itisnotcorrect.Ritadidnot work from left to right when simplifying the expression.

(56b 2 45) dollars

Anne: 2h; Bill: 3h; Charles: 8 1 h; Sum: 6h 1 8


Write an algebraic expression for

the perimeter of a rectangular

garden with a width of 3 feet and

a length of 2y feet. Simplify your

expression and find the perimeter

of the garden if y 5 6. Show your

work. 3 1 2y 1 3 1 2y; 6 1 4y;

30 ft

Also available on


You may want to suggest students

use a bar model for 22 .Usinga

model may help them see that the

total number of girls is 18 + w.

Best Practices

23Lesson 7.4 Expanding and Factoring Algebraic Expressions

3(k 1 6) and 3k 1 18 are equivalent

expressions because they are equal for

all values of k.

3 3 (k 1 6)

5 (k 1 6) 1 (k 1 6) 1 (k 1 6)

5 k 1 k 1 k 1 6 1 6 1 6

5 3k 1 18

Guided PracticeExpand each expression.

1 3(x 1 4) 2 6(2x 1 3) 3 2(7 1 6x)

4 5( y 2 3) 5 4(4y 2 1) 6 9(5x 2 2)


7 6(x 1 5) and 6x 1 30 8 7(x 1 3) and 21 1 7x

9 4( y 2 4) and 4y 2 4 10 3( y 2 6) and 18 2 3y

b) Expand 3(k 1 6).

3(k 1 6) means 3 groups of k 1 6:

1 group

k k k6 6 6

Rearrangethetermstocollecttheliketerms:

3 3 k 3 3 6

k k k 6 6 6

From the models,

3(k 1 6) 5 3 3 (k 1 6)

5 3 3 k 1 3 3 6

5 3k 1 18

3k 1 18 is the expanded form of 3(k 1 6).

Equivalent

Not equivalentNot equivalent

Equivalent

3x 1 12 12x 1 18 14 1 12x

5y 2 15 16y 2 4 45x 2 18

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2

Vocabulary


5

KEY CONCEPT

• Expandingistheoppositeprocess

of factoring.

Expanding and Factoring Algebraic Expressions

7.4

5-minute Warm Up

Reviewwithstudentsthe

distributive property of

multiplication over addition and

subtraction.

6 3 (5 1 2) 5 6 3 5 1 6 3 2

4 3 (7 2 3) 5 4 3 7 2 4 3 3

Also available on


PACING

DAY 1 Pages 22–24

DAY 2 Pages 25–28

Materials: paper, ruler, scissors,

yardsticks

Learn

a) Ask What does 2(r 1 8) mean? 2 groups of (r 1 8)

ModelUseabarmodeltoshow2groupsof

(r 1 8). Then rearrange the model such that the two

bars of “r” are put together and the two bars

of “8” are put together.

Explain Tell students that the two bars of “r”

represent 2 3 r and the two bars of “8” represent

2 3 8.

Model Show students how 2(r 1 8) is expanded

using the distributive property. Mention that 2r 1 16

is the expanded form of 2(r 1 8).

Explain Get students to deduce from the

model that 2(r 1 8) 5 2 3 r 1 2 3 8 5 2r 1 16.

Alternatively, lead students to see that

2 3 (r 1 8) 5 (r 1 8) 1 (r 1 8) 5 r 1 r 1 8 1 8

5 2r 1 16.

Ask Since 2(r 1 8) 5 2r 1 16, what do you call

the expressions 2(r 1 8) and 2r 116? Equivalent

expressions How do you check if the expressions

are equivalent? Substitute any value of r into the

expressions and check if they are equal.

DAY 1

Use the distributive property to expand algebraic expressions.


3(k 1 6) and 3k 1 18 are equivalent

expressions because they are equal for

all values of k.

3 3 (k 1 6)

5 (k 1 6) 1 (k 1 6) 1 (k 1 6)

5 k 1 k 1 k 1 6 1 6 1 6

5 3k 1 18

Guided PracticeExpand each expression.

1 3(x 1 4) 2 6(2x 1 3) 3 2(7 1 6x)

4 5( y 2 3) 5 4(4y 2 1) 6 9(5x 2 2)


7 6(x 1 5) and 6x 1 30 8 7(x 1 3) and 21 1 7x

9 4( y 2 4) and 4y 2 4 10 3( y 2 6) and 18 2 3y

b) Expand 3(k 1 6).

3(k 1 6) means 3 groups of k 1 6:

1 group

k k k6 6 6

Rearrangethetermstocollecttheliketerms:

3 3 k 3 3 6

k k k 6 6 6

From the models,

3(k 1 6) 5 3 3 (k 1 6)

5 3 3 k 1 3 3 6

5 3k 1 18

3k 1 18 is the expanded form of 3(k 1 6).

Equivalent

Not equivalentNot equivalent

Equivalent

3x 1 12 12x 1 18 14 1 12x

5y 2 15 16y 2 4 45x 2 18

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Vocabulary

23Lesson 7.4 ExpandingandFactoringAlgebraicExpressions

ELL Vocabulary Highlight

Make sure that students

understand that equivalent

expressions evaluated for the

same value of the variable are

equal.Forexample,forx 5 3,

2x 2 4 5 2

2(x 2 2) 5 2

Guided PracticeTell students it is best to use the

distributive property to expand

algebraic expressions.

4 to 6 Remindstudentstowatch

the signs in the parentheses when

expanding expressions.

7 to 10 Remindstudentstoonly

combine like terms after expanding.

Be sure that students understand

that variable terms with a

coefficient such as 6x are

expanded the same way as

variables without a coefficient.

They are multiplied by the factor

outsidetheparentheses.For

example, in 3 , the 6x becomes

2 3 6x or 12x.

Best Practices

b) Model Have students interpret the model 3(k 1 6).

Then rearrange the bars such that the three bars of

“k” are put together and the three bars of “6” are

put together.

Explain Tell students that the three bars of “k”

represent 3 3 k and the three bars of “6” represent

3 3 6.

Ask How do you expand 3(k 1 6)? 3(k 1 6) 5

3 3 k 1 3 3 6 5 3k 1 18 What do you call the

expressions 3(k 1 6) and 3k 1 18? Equivalent

expressions How do you check if the expressions

are equivalent? Substitute any value of k into the

expressions and check if they are equal.

Learn continued

STEP

1

STEP

2

STEP

3

STEP

4

STEP

5STEP

2STEP

4

STEP

6

RECOGNIZE THAT EXPANDED EXPRESSIONS ARE EQUIVALENT

Materials:

(8p 1 24) cm2

8p cm2, 24 cm2

Area of big rectangle 5 Area of rectangle A 1 Area of rectangle B


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To factor an

expression, look

for common factors

in the terms of the

expression.

Since they are equal for all values of y, 2y 1 10

and 2( y 1 5) are equivalent expressions.

Factoring is the inverse

of expanding. You can

use expanding to check

if you have factored an

expression correctly.

Algebraic expressions can be factored by taking out a common factor.

You can expand the expression 3(4z 1 1) by writing it as 12z 1 3.

You can also start with the expression 12z 1 3 and write it as 3(4z 1 1).

When you write 12z 1 3 as 3(4z 1 1), you have factored 12z 1 3.

a) Factor 2y 1 10.

List the factors of each term in the expression.

10 5 1 3 10 2y 5 1 3 2y 5 2 3 5 5 2 3 y

The factors of 10 are 1, 2, 5, and 10.

The factors of 2y are 1, 2, y, and 2y.

Excluding 1, the common factor of 10 and 2y is 2.

2y 1 10 5 2 3 y 1 2 3 5 5 2 3 ( y 1 5) Take out the common factor 2.

5 2( y 1 5)

2( y 1 5) is the factored form of 2y 1 10.

Check: Expand the expression 2( y 1 5) to

check the factoring.

2( y 1 5) 5 2 3 y 1 2 3 5

5 2y 1 10

2y 1 10 is factored correctly.



Hands-On ActivityThis activity shows a concrete

representation of the distributive

property used for expanding

algebraic expressions such as

8(p 1 3). By equating the area of

the large rectangle to the sum

of the areas of the two smaller

rectangles, students can see the

concrete representation of

8(p 1 3) 5 8p 1 24.

2 Guide students to find the area

of the rectangle by reminding

them about the formula for area

of a rectangle, length 3 width.

Make sure students remember

how to expand the expression.

5 Make sure students see that the

sum of the area of rectangle A

and the area of rectangle B is

equal to the area of the original

rectangle.

STEP

1

STEP

2

STEP

3

STEP

4

STEP

5STEP

2STEP

4

STEP

6

RECOGNIZE THAT EXPANDED EXPRESSIONS ARE EQUIVALENT

Materials:

(8p 1 24) cm2

8p cm2, 24 cm2

Area of big rectangle 5 Area of rectangle A 1 Area of rectangle B


Learn

To factor an

expression, look

for common factors

in the terms of the

expression.

Since they are equal for all values of y, 2y 1 10

and 2( y 1 5) are equivalent expressions.

Factoring is the inverse

of expanding. You can

use expanding to check

if you have factored an

expression correctly.

Algebraic expressions can be factored by taking out a common factor.

You can expand the expression 3(4z 1 1) by writing it as 12z 1 3.

You can also start with the expression 12z 1 3 and write it as 3(4z 1 1).

When you write 12z 1 3 as 3(4z 1 1), you have factored 12z 1 3.

a) Factor 2y 1 10.

List the factors of each term in the expression.

10 5 1 3 10 2y 5 1 3 2y 5 2 3 5 5 2 3 y

The factors of 10 are 1, 2, 5, and 10.

The factors of 2y are 1, 2, y, and 2y.

Excluding 1, the common factor of 10 and 2y is 2.

2y 1 10 5 2 3 y 1 2 3 5 5 2 3 ( y 1 5) Take out the common factor 2.

5 2( y 1 5)

2( y 1 5) is the factored form of 2y 1 10.

Check: Expand the expression 2( y 1 5) to

check the factoring.

2( y 1 5) 5 2 3 y 1 2 3 5

5 2y 1 10

2y 1 10 is factored correctly.



CautionStudents may not understand

that the phrase “Take out” means

moving the common factor and

placing the y 1 5 in parentheses.

In a), the common factor is 2.

Doingthisistheoppositeof

applying the distributive property.

Learn

AskExpand3(4z 1 1). What do you get? 12z 1 3

Explain 12z 1 3 is the expanded form of 3(4z 1 1).

Since 12z 1 3 and 3(4z 1 1) are equivalent expressions,

12z 1 3 5 3(4z 1 1). Tell students that they have

factored 12z 1 3. Make sure students see how factoring

and expanding are related.

a) Explain To factor 2y 1 10, find the common

factor(s) of 2y and 10. In order to find the common

factor(s) of the two terms, you list the factors of

each term. Point out to students that 1 is excluded

because 1 is a factor of every term. Write 2y 1 10

5 2 3 y 1 2 35ontheboard.Next,takeoutthe

common factor and write 2( 1 ).

Ask What terms should you write in the blank

spaces? y and 5 How can you check your answer? By

expanding 2( y 1 5) Since 2y 1 10 5 2( y 1 5), what

do you call the expressions 2y 1 10 and 2( y 1 5)?

Equivalentexpressions

Explain Tell students that factoring is the opposite

process of expanding. You can use expanding to

check if you have factored an expression correctly.

Algebraic expressions can be factored by taking a common factor.

DAY 2

Take out the common factor 3.

Guided Practice

11 12

14

16

18

20

21 22

24

26Equivalent

Equivalent

3(x 1 1)

5( y 2 2)

2(2 2 5z) 4(3 2 2x)

5(3 1 q)

4(3t 2 2)2(4f 1 3)

2(2x 1 3)

8(4m 2 5)

2(4 1 3y)

Equivalent


Not equivalent


Guided Practice16 , 18 and 20 Remindstudentsthat

in factoring, they should look for the

greatest common factor of the terms

in the expression.

b) Ask Whatarethetermsintheexpression6z2 9?

6zand9

Explain To factor 6z 2 9, find the common factor(s)

of 6zand9.Remindstudentsthat1isexcluded

because 1 is a factor of every term.

Ask What are the factors of 6z and 9? 6: 3 and 2;

9: 3 What is the common factor of 6z and 9? 3

Explain Write 6z 2 9 5 3 3 2z 2 3 3 3 on the

board.Next,takeoutthecommonfactorandwrite

3( 2 ).

Ask What terms should you write in the blank spaces?

2z and 3 How can you check your answer? By expanding

3(2z 2 3) Since 6z 2 9 5 3(2z 2 3), what do you call

the expressions 6z 2 9 and 3(2z 2 3)? Equivalent

expressions

Explain Tell students that 3(2z 2 3) is the factored form

of 6z 2 9.

Learn continued


Expand each expression.

1 5(x 1 2) 2 7(2x 2 3)

3 4( y 2 3) 4 8(3y 2 4)

5 3(x 1 11) 6 9(4x 2 7)

Factor each expression.

7 6p 1 6 8 3p 1 18

9 12 1 3q 10 4w 2 16

11 14r 2 8 12 12r 2 12


13 4x 1 12 and 4(x 1 3) 14 5(x 2 1) and 5x 2 1

15 7(5 1 y) and 7y 1 35 16 9( y 2 2) and 18 2 9y


17 3(m 1 2) 1 4(6 1 m)

18 5(2p 1 5) 1 4(2p 2 3)

19 4(6k 1 7) 1 9 2 14k

Simplify each expression. Then factor the expression.

20 14x 1 13 2 8x 2 1

21 8( y 1 3) 1 6 2 3y

22 4( 3z 1 7) 1 5(8 1 6z)

Solve.

23 Expand and simplify the expression 3(x 2 2) 1 9(x 1 1) 1 5(1 1 2x) 1 2(3x 2 4).

Practice 7.4

Not equivalentEquivalent

Not equivalentEquivalent

2(21z 1 34)

5( y 1 6)

6( x 1 2)

10k 1 37

18p 1 13

7m 1 30

12(r 2 1)

4(w 2 4)

3(p 1 6)6(p 1 1)

5x 1 10 14x 2 21

4y 2 12 24y 2 32

36x 2 633x 1 33

3(4 1 q)

2(7r 2 4)

28x

Basic 1 – 16


Advanced 23 – 27


Practice 7.4

Assignment Guide


complete 1 – 6

and 17 – 19 .


complete 7 – 16 .

and 20 – 22 .


challenge.


You may want to highlight the

grouping symbols: parentheses.

Have students work with the

parentheses first. So for 18 ,

students would first expand

5(2p 1 5) to 10p 1 25 and

4(2p 2 3) to 8p 2 12. Then,

they would collect like terms,

10p 1 8p 1 25 2 12 5 18p 1 13.

Best Practices

1

2

3



Exercises 1 , 3 and 6 • can expand simple algebraic expressions

Reteach7.4Exercises 9 and 12 • can factor simple algebraic expressions

• can write, expand, and evaluate simple

algebraic expressions

24

25

26

27

a)

Equivalent

(3w 1 80) cents

(10m 1 9) pounds

(x 1 2) cm; (3x 1 6) cm2

Unshadedrectangle5 3x cm2, shaded rectangle 5 6 cm2

Sum of area of smaller rectangles 5 Area of rectangle ABCD 3x 1 6 5 3(x 1 2)Thus, the expressions 3x 1 6 and 3(x 1 2) are equivalent.

29Lesson 7.5 Real-WorldProblems:AlgebraicExpressions

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Lesson Objective• Solvereal-worldproblemsinvolvingalgebraicexpressions.

Write an addition or subtraction algebraic expression for a real-world problem and evaluate it.

The figure shows a triangle ABC.

a) What is the perimeter of the triangle ABC in terms of s?

?

s cm s cm 10 cm

s 1 s 1 10 5 2s 1 10

The perimeter of the triangle ABC is (2s 1 10) centimeters.

b) The perimeter of a trapezoid is 7 cm shorter than the perimeter of triangle ABC.

Find the perimeter of the trapezoid.

(2s 1 10) cm

? 7 cm

2s 1 10 2 7 5 2s 1 3

The perimeter of the trapezoid is (2s 1 3) centimeters.

c) If s 5 7, find the perimeter of the triangle ABC.

When s 5 7,

2s 1 10 5 (2 3 7) 1 10

5 14 1 10

5 24

The perimeter of the triangle ABC is 24 centimeters.

AC 5 s cm. Since

AC 5 7 cm, s 5 7.

s cm s cm

10 cm

A

B C



Through Visual Cues

For25 , use three yardsticks, one

to represent a yard of lace, and

the other two to represent

2 yards of fabric. Ask students to

write an expression for the cost of

the lace, w, and 2 yards of fabric

2(w 1 40). Have students find the

total cost of the lace and fabric,

w 1 2w 1 80 5 (3w 1 80) cents.

Write an algebraic expression

and expand it by multiplying by a

factor.Evaluatebothexpressions

for the same value of the variable.

Check that the answers to the

evaluations are the same. Possible

answer:

4z 2 5, 3(4z 2 5) 5 12z 2 15;

Let z 5 2.

3(4z 2 5) 5 3(4 3 2 2 5)

5 3(8 2 5)

5 3(3) 5 9 3

12z 2 15 5 12 3 2 2 15

5 24 2 15 5 9 3

Also available on



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Lesson Objective• Solvereal-worldproblemsinvolvingalgebraicexpressions.


The figure shows a triangle ABC.

a) What is the perimeter of the triangle ABC in terms of s?

?

s cm s cm 10 cm

s 1 s 1 10 5 2s 1 10

The perimeter of the triangle ABC is (2s 1 10) centimeters.

b) The perimeter of a trapezoid is 7 cm shorter than the perimeter of triangle ABC.

Find the perimeter of the trapezoid.

(2s 1 10) cm

? 7 cm

2s 1 10 2 7 5 2s 1 3

The perimeter of the trapezoid is (2s 1 3) centimeters.

c) If s 5 7, find the perimeter of the triangle ABC.

When s 5 7,

2s 1 10 5 (2 3 7) 1 10

5 14 1 10

5 24

The perimeter of the triangle ABC is 24 centimeters.

AC 5 s cm. Since

AC 5 7 cm, s 5 7.

s cm s cm

10 cm

A

B C


5

KEY CONCEPT

• The process of problem solving

involves the application of

concepts, skills and strategies.

Real-World Problems: Algebraic Expressions

7.5

5-minute Warm Up

Demonstratehowtosolvethis

problem:

Jim has x stamps. His friend gives

him 20 stamps and he gives 15 in

return. How many stamps does

Jim have now?

x 1 20 215 5 x 1 5

Also available on


PACING

DAY 1 Pages 29–32

DAY 2 Pages 32–35

Materials: none

Learn

a) Model Work through a) with students to

demonstrate the problem solving process.

Step 1Understandtheproblem.

Ask What is given in the problem? The lengths of

the sides of the triangle What are you asked to find?

The perimeter of the triangle

Step 2 Decideonastrategytouse.

Ask How can you find the perimeter of the triangle?

Add the side lengths. Are the lengths of the triangle

given? Yes What are they? s cm, s cm, and 10 cm

Step 3 Solve the problem.

Ask What expression do you get? s 1 s 1 10 Can the

expression be simplified? Yes, s 1 s 1 10 5 2s 1 10

b) Ask Usingabarmodel,whatexpressioncanyou

writefortheperimeterofthetrapezoid?2s 1

10 2 7 Can the expression be simplified? Yes, 2s 1

10 2 7 5 2s 1 3

c) Ask How do you find the perimeter of triangle

ABC? Substitute 7 for s in the expression 2s 1 10.

2s 1 10 5 2 3 7 1 10 5 24.

DAY 1


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Guided Practice

1

a)

y 1 6

y 2 4

y ; 4; 12; 4

12; 8; 20

20

8


b) How much gas is used if the car travels 5w miles? Evaluate this expression when

w 5 72.

5w miles

? groups

25 miles 25 miles 25 miles25 miles

25 miles 1 gallon

5w miles 5w 4 25 5 525w

gallons

525w

gallons of gas is used.

When w 5 72,

525w

5 5 72253

5 36025

5 14.4


2 A pick up truck uses 1 gallon of gas for every 14 miles traveled.

a) How far can it travel on 3p gallons of gas?

1 gallon

3p gallons

14 miles 14 miles 14 miles 14 miles

1 gallon ? miles

3p gallons ? 3 ? 5 ? miles

It can travel ___?___ miles on 3p gallons of gas.


14

3p; 14; 42p

42p


Guided Practice1 Students who have difficulty

writing the expressions may not have

internalizedtheconceptthatletters

are used to represent numbers.

Assist these students by replacing

the letters with numbers and check if

they can then solve the problem.

Forc), encourage students to check

their answers by comparing the ages

theyfoundforKaylaandIsaacagainst

the facts in the original problem

statement. The ages they found

should make the statement true.

Since y 512,Raoulis12yearsold.

Kaylashouldbe18yearsold,and

Isaac should be 8 years old.

Learn

Model Useabarmodeltorepresentthescenarioina).

Ask What are you required to find in the problem? How

far the car can travel on w gallons of gas.

Ask Suppose the car can travel 25 miles on 1 gallon

of gas. What expression can you write that shows how

far the car can travel on 3 gallons of gas? 3 3 25 5 75

What expression can you write to show how far the car

can travel on w gallons of gas? w 3 25 5 25w

Explain Help students to see the relationship between

numbers and algebra. If students have difficulty

understanding the relationship in a), you may want to

set up a table showing gallons in one column and miles

traveled in a second column. Work with students to

fill out the table for 1 gallon, 2 gallons, 3 gallons, and

so on, so that they see the pattern of multiplying the

number of gallons (w) by 25.

Write a multiplication or division algebraic expression for a real-world problem and evaluate it.

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Guided Practice

1

a)

y 1 6

y 2 4

y ; 4; 12; 4

12; 8; 20

20

8


b) How much gas is used if the car travels 5w miles? Evaluate this expression when

w 5 72.

5w miles

? groups

25 miles 25 miles 25 miles25 miles

25 miles 1 gallon

5w miles 5w 4 25 5 525w

gallons

525w

gallons of gas is used.

When w 5 72,

525w

5 5 72253

5 36025

5 14.4


2 A pick up truck uses 1 gallon of gas for every 14 miles traveled.

a) How far can it travel on 3p gallons of gas?

1 gallon

3p gallons

14 miles 14 miles 14 miles 14 miles

1 gallon ? miles

3p gallons ? 3 ? 5 ? miles

It can travel ___?___ miles on 3p gallons of gas.


14

3p; 14; 42p

42p


Guided Practice2 In this problem, students can use

unit rates to understand whether they

write a multiplication expression or

division expression. In a), the answer

is in miles so multiply: miles per

gallon 3 gallons 5 miles.

Learn continued

b) Ask What are you required to find in the problem?

Amount of gas used if the car traveled 5w miles

What information in the problem can help you solve

the problem? To travel 25 miles, the car will need

1 gallon of gas.

Ask How much gas will be used after traveling

50 miles? 2 gallons How did you get the answer?

5025

3 1 What expression can you write for the

amount of gas used after traveling 5w miles?

525w

3 1 5

525w

gallons

Ask What do you need to do next? To evaluate the

expression when w = 72.

ExplainRemindstudentsofthemeaningof

“evaluate”.

Explain To evaluate 525w when w 5 72, students

need to substitute 72 for w in the expression 525w

.

Point out to students that they can evaluate 5w

before dividing by 25, the denominator.

Ask What answer do you get? 5 72253 5 14.4.

Le

arn

1

4

;v14

5614

v14

v; 14;v

14



3 There were three questions in a mathematics test. Salma earned m points for the

first question and twice the number of points for the second question.

a) How many points did she earn for the first two questions?

First question:

points?

Second question:

points?

? 1 ? 5 ?

She earned ? points for the first two questions.

b) If she received a total of 25 points on the test, how many points did she

earn for the third question?

25 points

??

She earned ? points for the third question.

c) If m 5 5, find the points she earned for each question.

First question: m 5 5

Second question: 2m 5 2 3 ?

5 ?

Third question: 25 2 3m 5 25 2 (3 3 ? )

5 25 2 ?

5 ?

She earned ? points for the first question, ? points for the second

question and ? points for the third question.

m

2m

m; 2m; 3m

3m

3m

5

5

15

5; 10; 10

10

10

25 2 3m


Guided PracticeIn b), students divide:

miles miles per gallon 5 gallons.

Explain Ask students to think of a random number, say

5. Ask them to multiply it by 3 and then subtract 9 from

the product. Ask them for the answer. 6

Ask How did you find the answer? First,multiply5

by3.Next,subtract9fromtheproductof5and3.

Explain Look back at the question. Instead of the

number 5, you can replace it with the term y.Usingthe bar models, model for students the process of

multiplying y by 3 and then subtracting 9 from the

product.

Ask What expression do you get when you multiply the

term y by 3? 3y What expression do you get when you

subtract 9 from the product? 3y 2 9 What do you need

to do in the last part of the question? To evaluate the

expression when y 5 12

Explain Explaintheprocessofevaluatingexpressions.

Remindstudentstomultiplybeforesubtracting.

Learn

DAY 2

Write an algebraic expression using several operations and evaluate it.

Le

arn

1

4

;v14

5614

v14

v; 14;v

14



3 There were three questions in a mathematics test. Salma earned m points for the

first question and twice the number of points for the second question.

a) How many points did she earn for the first two questions?

First question:

points?

Second question:

points?

? 1 ? 5 ?

She earned ? points for the first two questions.

b) If she received a total of 25 points on the test, how many points did she

earn for the third question?

25 points

??

She earned ? points for the third question.

c) If m 5 5, find the points she earned for each question.

First question: m 5 5

Second question: 2m 5 2 3 ?

5 ?

Third question: 25 2 3m 5 25 2 (3 3 ? )

5 25 2 ?

5 ?

She earned ? points for the first question, ? points for the second

question and ? points for the third question.

m

2m

m; 2m; 3m

3m

3m

5

5

15

5; 10; 10

10

10

25 2 3m


Guided Practice3 In a), students build the algebraic

expression 3m by adding the

number of points. In b), they solve

the problem by using a part-whole

bar model. Point out that to find

a missing part, subtract. Guide

students to connect the pieces of

information given and found in a) and b) to solve c).


Through Modeling

You may want to highlight the

relationship between the two

bars in a) and the bar model in b) by making the intermediate step

explicit. Have students use three

bars of equal length to model the

equation they completed in a): m 1 2m 5 3m. Point out that

their model for 3m matches one

of the bars of the bar model in b).

Practice 7.5

1

c)

2

3

c)

4 14

c)

Basic 1 – 3

Intermediate 4

Advanced 5 – 6

x 2 5

3x

24 years older

48y miles

(2x + 3) fruits

(90x + 150) cents

3x cm

3x cm

20 cm

39 cm2

x8

gallons

15x 1 252

cents


5 José bought 4 comic books and 2 nonfiction books. The 4 comic books cost

him 8y dollars. If the cost of one nonfiction book is (3 1 7y) dollars more

expensive than the cost of one comic book, find

a) the cost of the 2 nonfiction books in terms of y.

b) the total amount that José spent on the books if y 5 4.

6 Wyatt has (2x 2 1) one-dollar bills and (4x 1 2) five-dollar bills. Susan has

3x dollars more than Wyatt.

a) Find the total amount of money that Wyatt has in terms of x.

b) Find the number of pens that Wyatt can buy if each pen costs 50¢.

c) If x 5 21, find how much money Susan will have now if Wyatt gives her

half the number of five-dollar bills that he has.

Find the perimeter of the figure in terms of x, given that all the angles in the

figure are right angles. If x 5 5.5, evaluate this expression.

16 cm

x cm

x cm

x cm

(22x 1 9) dollars

(18y 1 6) dollars

$110

(44x 1 18) pens

$749

(6x 1 32) cm; 65 cm


Practice 7.5

Assignment Guide


complete 1 – 2 .


complete 3 – 4 .


challenge.


1

2

3



Exercises 1 and 2 • can solve a real-world problem using algebra

Reteach7.5• can write and evaluate expressions

Practice 7.5

1

c)

2

3

c)

4 14

c)

Basic 1 – 3

Intermediate 4

Advanced 5 – 6

x 2 5

3x

24 years older

48y miles

(2x + 3) fruits

(90x + 150) cents

3x cm

3x cm

20 cm

39 cm2

x8

gallons

15x 1 252

cents


5 José bought 4 comic books and 2 nonfiction books. The 4 comic books cost

him 8y dollars. If the cost of one nonfiction book is (3 1 7y) dollars more

expensive than the cost of one comic book, find

a) the cost of the 2 nonfiction books in terms of y.

b) the total amount that José spent on the books if y 5 4.

6 Wyatt has (2x 2 1) one-dollar bills and (4x 1 2) five-dollar bills. Susan has

3x dollars more than Wyatt.

a) Find the total amount of money that Wyatt has in terms of x.

b) Find the number of pens that Wyatt can buy if each pen costs 50¢.

c) If x 5 21, find how much money Susan will have now if Wyatt gives her

half the number of five-dollar bills that he has.

Find the perimeter of the figure in terms of x, given that all the angles in the

figure are right angles. If x 5 5.5, evaluate this expression.

16 cm

x cm

x cm

x cm

(22x 1 9) dollars

(18y 1 6) dollars

$110

(44x 1 18) pens

$749

(6x 1 32) cm; 65 cm


Briefly describe a situation in

your classroom that can be

described algebraically. Then,

write an algebraic expression for

it.Finally,evaluatetheexpression

by substituting a realistic number

for the variable. Possible answer:

Tonight I have 3 times as many

math problems to do as I had last

night. Luckily, I have already done

4 problems. Algebraic expression:

3x 2 4. I had 6 problems to do

last night, so I substitute 6 for x

and solve: 3 3 6 2 4 5 14.

I have 14 problems left.

Also available on


Focusstudents’attentiononthe

vertical height of the figure. Guide

them to think of how they can get the

measurement of this vertical height.

Then, have students look at the

threehorizontalportionsofunknown

lengths. Ask students to figure out

what their sum is equal to. Then have

students solve the problem.


Through Enrichment

Becauseallstudentsshouldbechallenged,haveallstudentstrytheBrain@Work

exercise on this page.

Foradditionalchallengingpracticeandproblemsolving,seeEnrichment, Course 1,

Chapter 7.

Chapter Wrap UpConcept Map

Algebraic Expressions

Key Concepts


Chapter Review/TestConcepts and SkillsWrite an algebraic expression for each of the following.

1 A number that is 5 more than twice x.

2 The total cost, in dollars, of 4 pencils and 5 pens if each pencil costs w cents

and each pen costs 2w cents.

3 The length of a side of a square whose perimeter is r units.

4 The perimeter of a rectangle whose sides are of lengths (3z 1 2) units and

(2z 1 3) units.


5 3(x 1 4) 2 x2

when x 5 2 6 5 9

2p 1

1 2 5

3p 1

when p 5 5


7 24k 1 11 2 5k 2 4 8 10 1 13h 2 6 2 4h 1 9 1 12h


9 5(m 1 3) 1 2(m 1 8) 10 9(x 1 2) 1 4(5 1 x)


11 5a 2 25 12 28 2 7x 13 12z 1 28 2 7z 2 3


14 3(x 1 5) and 5(x 1 3) 15 6y 2 26 and 2(3y 2 13)

16 18 2 12p and 3(5 1 6p) 1 3(2p 1 1) 17 15 2 5q and 5(q 2 3 )

Problem SolvingSolve. Show your work.

18 Juan is g years old and Eva is 2 years younger than Juan.

a) Find the sum of their ages in terms of g.

b) Find the sum of their ages in g years’ time, in terms of g.

2x 1 5

(10z 1 10) units

7m 1 31

19k 1 7

5(a 2 5) 7(4 2 x) 5(z 1 5)

Equivalent

Not equivalent

Not equivalent

Not equivalent

2g 2 2

4g 2 2

13x 1 38

13 1 21h

17 22

7w50

dollarsr4

units


CHAPTER WRAP UP

Usethenotesandtheexamplesin

the concept map to review writing,

simplifying, evaluating, expanding,

and factoring algebraic expressions.

CHAPTER PROJECT

Towidenstudent’smathematical

horizonsandtoencouragethemto

think beyond the concepts taught in

this chapter, you may want to assign

the Chapter 7 project, available in

Activity Book, Course 1.

Vocabulary Review

Usethesequestionstoreviewchaptervocabularywith

students.

1. A letter used to represent a number is called a ? . variable

2. In the algebraic expression 2x 1 7, 2x and 7 are the ? of the expression. terms

3. When two expressions are equal for all values of

the variables, they are called ? ? . equivalent

expressions

4. In the expression 3y 1 y 1 6, the terms 3y and y are ? ? . like terms

5. In the expression 4x 2 5, 4 is the ? of x.

coefficient

AlsoavailableonTeacherOneStopCD-ROM.

Chapter Wrap UpConcept Map

Algebraic Expressions

Key Concepts


Chapter Review/TestConcepts and SkillsWrite an algebraic expression for each of the following.

1 A number that is 5 more than twice x.

2 The total cost, in dollars, of 4 pencils and 5 pens if each pencil costs w cents

and each pen costs 2w cents.

3 The length of a side of a square whose perimeter is r units.

4 The perimeter of a rectangle whose sides are of lengths (3z 1 2) units and

(2z 1 3) units.


5 3(x 1 4) 2 x2

when x 5 2 6 5 9

2p 1

1 2 5

3p 1

when p 5 5


7 24k 1 11 2 5k 2 4 8 10 1 13h 2 6 2 4h 1 9 1 12h


9 5(m 1 3) 1 2(m 1 8) 10 9(x 1 2) 1 4(5 1 x)


11 5a 2 25 12 28 2 7x 13 12z 1 28 2 7z 2 3


14 3(x 1 5) and 5(x 1 3) 15 6y 2 26 and 2(3y 2 13)

16 18 2 12p and 3(5 1 6p) 1 3(2p 1 1) 17 15 2 5q and 5(q 2 3 )

Problem SolvingSolve. Show your work.

18 Juan is g years old and Eva is 2 years younger than Juan.

a) Find the sum of their ages in terms of g.

b) Find the sum of their ages in g years’ time, in terms of g.

2x 1 5

(10z 1 10) units

7m 1 31

19k 1 7

5(a 2 5) 7(4 2 x) 5(z 1 5)

Equivalent

Not equivalent

Not equivalent

Not equivalent

2g 2 2

4g 2 2

13x 1 38

13 1 21h

17 22

7w50

dollarsr4

units


TEST PREPARATION

Foradditionaltestprep

Examview Assessment Suite

CD-ROM Course 1

CHAPTER REVIEW/TEST

Chapter Assessment

UsetheChapter7Testin

Assessments, Course 1 to assess

how well students have learned

the material in this chapter. This

assessment is appropriate for

reporting results to adults at

home and administrators.

1

2

3

Response to Intervention Use the table for reteaching recommendations.

Exercises Intervene with Reteach worksheet…

1 to 4 7.1 Writing algebraic expressions

5 to 6 7.2 Evaluatingalgebraicexpressions

7 to 8 7.3 Simplifying algebraic expressions

9 to 17 7.4 Expandingandfactoringalgebraicexpressions

18 to 24 7.5 Solving real-world problems involving algebraic expressions

19

20

21

22

23

24

c)

60t

chairs7t20

chairs

(4p 2 5) marbles

(h 2 4) muffins

(8y 1 8) m

(7m 1 10) yards

31 yards

(11p 2 4) quarts

84 quarts

5 liters


mif teacher's edition sample pages - course 1

Documents