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A sample of the Math in Focus Course 1 (grade 6) Teacher's Edition, available for Fall 2011. These pages are advanced proofs and are not final.
Citation preview
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
AT
A G
LAN
CE
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
Student Book A, pp. 9211Extra Practice A, Lesson 7.2Reteach A, Lesson 7.2
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
Student Book A, pp. 12221Extra Practice A, Lesson 7.3Reteach A, Lesson 7.3
Student Book A, pp. 22228Extra Practice A, Lesson 7.4Reteach A, Lesson 7.4
Student Book A, pp. 29235Extra Practice A, Lesson 7.5Reteach A, Lesson 7.5
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
A G
LAN
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
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
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
3Chapter 7 Algebraic Expressions
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
?
Recall Prior Knowledge
Quick Check
1 2
3 4
2 Chapter 7 AlgebraicExpressions
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.
3Chapter 7 Algebraic Expressions
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
?
Recall Prior Knowledge
Quick Check
1 2
3 4
3Chapter 7 AlgebraicExpressions
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
7.1 Writing Algebraic Expressions
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
the length of the remaining straw?
? 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.
4 Chapter 7 AlgebraicExpressions
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
7.1 Writing Algebraic Expressions
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
the length of the remaining straw?
? 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
6 Chapter 7 AlgebraicExpressions
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
Use variables to write multiplication expressions.
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
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
7Lesson 7.1 Writing Algebraic Expressions
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.
Guided PracticeWrite an algebraic expression for each of the following.
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
7.2 Evaluating Algebraic Expressions
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 9, Simon has 6 more marbles than Cynthia.
When x 5 17, x 2 3 5 17 2 3
5 14
When x 5 17, Simon has 14 more marbles than Cynthia.
Continue on next page
8 Chapter 7 Algebraic Expressions
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
8 Chapter 7 AlgebraicExpressions
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 .
DAY 2 All students should
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
TeacherOneStopCD-ROM.
9Lesson 7.2 Evaluating Algebraic Expressions
Le
arn
7.2 Evaluating Algebraic Expressions
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 9, Simon has 6 more marbles than Cynthia.
When x 5 17, x 2 3 5 17 2 3
5 14
When x 5 17, Simon has 14 more marbles than Cynthia.
Continue on next page
8 Chapter 7 Algebraic Expressions
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
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
TeacherOneStopCD-ROM.
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
Algebraic expressions can be evaluated for given values of the variable.
9
4
w4
20
Guided Practice
1
43x
x5
225– x
7
24
5
5
12
18
25
10 Chapter 7 AlgebraicExpressions
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
11Lesson 7.2 Evaluating Algebraic Expressions
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
Intermediate 13 – 20
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 .
21 – 28 provide additional
challenge.
Optional: Extra Practice 7.2
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
TeacherOneStopCD-ROM.
1
2
3
Response to Intervention Lesson Check
Before assigning homework, use the following … to make sure students … Intervene with …
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
7.3 Simplifying Algebraic Expressions
Vocabulary
12 Chapter 7 AlgebraicExpressions
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
TeacherOneStopCD-ROM.
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.
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
7.3 Simplifying Algebraic Expressions
Vocabulary
13Lesson 7.3 SimplifyingAlgebraicExpressions
DIFFERENTIATED INSTRUCTION
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
15Lesson 7.3 Simplifying Algebraic Expressions
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
14 Chapter 7 AlgebraicExpressions
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.
15Lesson 7.3 Simplifying Algebraic Expressions
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
15Lesson 7.3 SimplifyingAlgebraicExpressions
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.
17Lesson 7.3 Simplifying Algebraic Expressions
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
16 Chapter 7 AlgebraicExpressions
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.
17Lesson 7.3 Simplifying Algebraic Expressions
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
17Lesson 7.3 SimplifyingAlgebraicExpressions
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.
19Lesson 7.3 Simplifying Algebraic Expressions
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.
Guided PracticeComplete.
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.
Simplify each expression.
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.
Work from left to right.
Add.
Guided Practice
21 22
24
26
Caution
8 8 Identify like terms.
Change the order of terms
to collect like terms.
Simplify.
Math Note
4j 11j
3w
0
6j
2t
5w
18 Chapter 7 AlgebraicExpressions
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
19Lesson 7.3 Simplifying Algebraic Expressions
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.
Guided PracticeComplete.
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.
Simplify each expression.
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.
Work from left to right.
Add.
Guided Practice
21 22
24
26
Caution
8 8 Identify like terms.
Change the order of terms
to collect like terms.
Simplify.
Math Note
4j 11j
3w
0
6j
2t
5w
19Lesson 7.3 SimplifyingAlgebraicExpressions
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
20 Chapter 7 AlgebraicExpressions
Assignment Guide
DAY 1 All students should
complete 1 – 3 .
DAY 2 All students should
complete 4 – 6 .
DAY 3 All students should
complete 7 – 23 , 26 .
24 – 25 provide additional
challenge.
Optional: Extra Practice 7.3
1
2
3
Response to Intervention Lesson Check
Before assigning homework, use the following … to make sure students … Intervene with …
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
21Lesson 7.3 Simplifying Algebraic Expressions
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
21Lesson 7.3 SimplifyingAlgebraicExpressions
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
TeacherOneStopCD-ROM.
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)
State whether each pair of expressions are equivalent.
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
Le
arn
7.4 Expanding and Factoring Algebraic Expressions
2
Vocabulary
22 Chapter 7 AlgebraicExpressions
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
TeacherOneStopCD-ROM.
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.
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)
State whether each pair of expressions are equivalent.
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
Le
arn
7.4 Expanding and Factoring Algebraic Expressions
2
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
25Lesson 7.4 Expanding and Factoring Algebraic Expressions
Le
arn
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.
Continue on next page
24 Chapter 7 AlgebraicExpressions
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
25Lesson 7.4 Expanding and Factoring Algebraic Expressions
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.
Continue on next page
25Lesson 7.4 ExpandingandFactoringAlgebraicExpressions
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 Not equivalent
Not equivalent
26 Chapter 7 AlgebraicExpressions
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
27Lesson 7.4 Expanding and Factoring Algebraic Expressions
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
State whether each pair of expressions are equivalent.
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
Expand each expression.
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
Intermediate 17 – 22
Advanced 23 – 27
27Lesson 7.4 ExpandingandFactoringAlgebraicExpressions
Practice 7.4
Assignment Guide
DAY 1 All students should
complete 1 – 6
and 17 – 19 .
DAY 2 All students should
complete 7 – 16 .
and 20 – 22 .
23 – 27 provide additional
challenge.
Optional: Extra Practice 7.4
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
Response to Intervention Lesson Check
Before assigning homework, use the following … to make sure students … Intervene with …
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
Le
arn
7.5 Real-World Problems: Algebraic Expressions
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
28 Chapter 7 AlgebraicExpressions
DIFFERENTIATED INSTRUCTION
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
TeacherOneStopCD-ROM.
29Lesson 7.5 Real-WorldProblems:AlgebraicExpressions
Le
arn
7.5 Real-World Problems: Algebraic Expressions
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
29Lesson 7.5 Real-WorldProblems:AlgebraicExpressions
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
TeacherOneStopCD-ROM.
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
Write an addition or subtraction algebraic expression for a real-world problem and evaluate it.
Le
arn
Guided Practice
1
a)
y 1 6
y 2 4
y ; 4; 12; 4
12; 8; 20
20
8
31Lesson 7.5 Real-WorldProblems:AlgebraicExpressions
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
Guided PracticeComplete.
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.
Continue on next page
14
3p; 14; 42p
42p
30 Chapter 7 AlgebraicExpressions
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.
Le
arn
Guided Practice
1
a)
y 1 6
y 2 4
y ; 4; 12; 4
12; 8; 20
20
8
31Lesson 7.5 Real-WorldProblems:AlgebraicExpressions
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
Guided PracticeComplete.
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.
Continue on next page
14
3p; 14; 42p
42p
31Lesson 7.5 Real-WorldProblems:AlgebraicExpressions
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
33Lesson 7.5 Real-WorldProblems:AlgebraicExpressions
Guided PracticeComplete.
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
32 Chapter 7 AlgebraicExpressions
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
33Lesson 7.5 Real-WorldProblems:AlgebraicExpressions
Guided PracticeComplete.
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
33Lesson 7.5 Real-WorldProblems:AlgebraicExpressions
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).
DIFFERENTIATED INSTRUCTION
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
35Lesson 7.5 Real-WorldProblems:AlgebraicExpressions
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
34 Chapter 7 AlgebraicExpressions
Practice 7.5
Assignment Guide
DAY 1 All students should
complete 1 – 2 .
DAY 2 All students should
complete 3 – 4 .
5 – 6 provide additional
challenge.
Optional: Extra Practice 7.5
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 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
35Lesson 7.5 Real-WorldProblems:AlgebraicExpressions
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
35Lesson 7.5 Real-WorldProblems:AlgebraicExpressions
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
TeacherOneStopCD-ROM.
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.
DIFFERENTIATED INSTRUCTION
Through Enrichment
Becauseallstudentsshouldbechallenged,haveallstudentstrytheBrain@Work
exercise on this page.
Foradditionalchallengingpracticeandproblemsolving,seeEnrichment, Course 1,
Chapter 7.
Chapter Wrap UpConcept Map
Algebraic Expressions
Key Concepts
37Chapter 7 Algebraic Expressions
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.
Evaluate each expression for the given value of the variable.
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
Simplify each expression.
7 24k 1 11 2 5k 2 4 8 10 1 13h 2 6 2 4h 1 9 1 12h
Expand each expression.
9 5(m 1 3) 1 2(m 1 8) 10 9(x 1 2) 1 4(5 1 x)
Factor each expression.
11 5a 2 25 12 28 2 7x 13 12z 1 28 2 7z 2 3
State whether each pair of expressions are equivalent.
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
36 Chapter 7 AlgebraicExpressions
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
37Chapter 7 Algebraic Expressions
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.
Evaluate each expression for the given value of the variable.
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
Simplify each expression.
7 24k 1 11 2 5k 2 4 8 10 1 13h 2 6 2 4h 1 9 1 12h
Expand each expression.
9 5(m 1 3) 1 2(m 1 8) 10 9(x 1 2) 1 4(5 1 x)
Factor each expression.
11 5a 2 25 12 28 2 7x 13 12z 1 28 2 7z 2 3
State whether each pair of expressions are equivalent.
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
37Chapter 7 AlgebraicExpressions
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
38 Chapter 7 AlgebraicExpressions