SAMPLE: Math in Focus: Course 1: Grade 6: Chapter 17: Algebra: TE

Teacher’s EditionSampler

CourSE 1

Teacher’s EditionSampler

Grade 6

CourSE

1

B

Title

advanced proofs not final pages

This Math in Focus Course 1 sampler provides one full chapter from the Grade 6 program that will be available for Fall 2011.

Courses 2 and 3 for Grades 7 and 8 will be coming soon!

ArrivingFall 2011

Course 2

A

Arriving2012

Arriving2012

CourSE 1 Teacher’s Edition Sampler

* PaGes in This samPler are advanced ProoFs and noT Final PaGes

Grades 1–5

1

Table of Contents

The Story Behind Math in Focus™ ...................2

Singapore Math Framework ...........................6

Instructional Pathway .....................................8

Extensive Teacher Support ........................... 12

Transition Support ........................................ 16

CourSE 1 Table of Contents .......................... 18

CourSE 1 Chapter 7 Algebraic Expressions ...................................................27

Scope and Sequence: Grades 4–6 ...............65

Program Components ..................................78

CourSE 1 Teacher’s Edition Sampler

2 advanced proofs not final pages

Background: History of the Development of Singapore Math TextbooksFrom 1965 to 1979, many of the primary and secondary textbooks (and all mathematics textbooks) used in singapore had been imported from other nations. But starting in 1980, singapore began to take a new approach. The curriculum development institute of singapore (cdis) was set up to develop primary and secondary textbooks for the new education system. This initiative produced new developments in learning that propelled singapore’s math students to the top of the international community. most schools in singapore today use programs published by marshall cavendish education.

1. Gonzales, Patrick, Juan carlos Guzmán, lisette Partelow, erin Pahlke, leslie Jocelyn, david Kastberg, and Trevor Williams. Highlights From the Trends in International Mathematics and Science Study: Timss 2003. U.s. department of education, national center for education statistics, 2004.

2. national council of Teachers of mathematics. Curriculum Focal Points for Prekindergarten through Grade 8 Mathematics, 2006.

3. national mathematics advisory Panel. Foundations for Success. U.s. department of education, 2008.

4. american institutes for research® What the United States Can Learn from Singapore’s World-Class Mathematics System. U.s. department of education Policy and Program studies services, 2005.

5. national research council. adding It Up: Helping Children Learn Mathematics. Washington, dc, national academy Press, 2001.

6. ministry of education, singapore. Mathematics Syllabus: Primary, 2007.

Footnotes

Math in Focus: Singapore Math Course 1 is a mathematics program created specifically to address the recommendations for instructional materials agreed upon by national and international panels of mathematics education specialists. Aligned to the Common Core State Standards, Math in Focus provides world-class mathematics instruction that meets the specific needs of U.S. students.

Meeting the Needs of u.S. Classrooms...Top performing countries have gained ground on, and now surpass the U.s. in mathematics education, as shown by the Trends in international math and science study (Timss)1. efforts to reverse this trend have led to a large body of solid research. analysis of the research base has led the national council of Teachers of mathematics (ncTm)2, the national math advisory Panel3, the american institutes for research®4, and the national research council5 to make several undisputed key recommendations:

1 a focused, coherent curriculum, without significant repetition year after year

2 an equal emphasis on conceptual understanding and fluency with skills

3 Use of concrete and pictorial representations

4 multi-step and non-routine problem solving

...by Drawing on Success in SingaporeThe research base used to guide the common core state standards noted that singapore students have been top performers on the Timss assessment since 1995. Their success can be largely attributed to the mathematics curriculum revision implemented by the singapore ministry of education6 in the 1980s.

Key requirements of their instructional materials that parallel the recommendations of u.s. specialists are:

• Precise framework of concepts and skills (specifics of what to exclude, as well as what to include, provides hierarchy and linkage)

• Skills and concepts taught in depth to allow for mastery (consolidation of concepts and skills)

• use of a concrete-to-visual-to-symbolic development of concepts using model drawings to connect visual representation to problem solving

• Emphasis on problem solving considered central to all mathematics study

Grades 1–5

3advanced proofs not final pages

Foreshadow Specific Future Concepts

Draw on Solid Prior Knowledge

Skill Building

Concept Building

the WHY

Math in Focus is the SolutionMath in Focus: Singapore Math by Marshall Cavendish embodies the world-class Singaporean pedagogy, methodology, and instructional materials while adapting the Singaporean mathematics standards to correspond to the Common Core State Standards for Mathematics outlined by the National Governors Association for Best Practices and the Council of Chief State School Officers in the United States. The Common Core State Standards conclusions parallel the assumptions behind Math in Focus.

market research for Math in Focus included multiple rounds of research including focus group testing and discussions with experienced educators. regional and national studies ensured that the student books and teacher support meet the current needs of students and teachers across the U.s.

For more details, visit greatsource.com/mathinfocus

• Teach to Mastery Math in Focus helps students build solid conceptual understanding through the use of visual models. Math in Focus authors created a strategic, articulated sequence of topics developed in depth to mastery. rather than repeating topics, students master them in a grade level and subsequent grades develop them to more advanced levels. students learn the “how” and the “why” through instruction, hands-on activities, and practice.

• Focused and Coherent CurriculumMath in Focus is organized to teach fewer concepts at each level, but to teach them thoroughly. When a concept appears in a subsequent grade level, it is always at a higher level.

• Clear Visuals and Use of ModelsMath in Focus consistently employs a concrete-visual-symbolic progression. clear and engaging visuals that present concepts and model solutions allow all students to focus and better understand abstract concepts.

• Organize Topics by Big Ideasin course 1, the common core state standards have identified four big ideas in ratio and proportion, number, algebra, and statistics. Math in Focus is divided into two books, roughly a semester each. about 75% of Book a is devoted to the big ideas in ratio and proportion, number, and algebra. These key topics are in the beginning of the school year so students have a whole year to master and review them. common core state standards correlations are provided for every chapter.

how can i simplify this equation?


The Story Behind Math in Focus

Singapore Math Foundations are Supported by research

singapore ministry of education

“mathematical problem solving is central tomathematics learning. it involves the acquisition and application of mathematics concepts and skills in a wide range of situations, including non-routine,open-ended, and real-world problems.”

—Mathematics Syllabus: Primary, 2006

Math in Focusaddresses fewer topics in greater depth

at each level.

• Knowledgeisbuiltcarefullyandthoroughlywithboth multi-page lessons and multi-day lessons.

• Timeisbuiltintotheprogramtodevelopunderstanding with activities, as well as ample scaffolded, guided practice with every learn and extensive skills practice.

Math in Focusdevelops concepts to mastery

at each level.• Studentsaregivenscaffoldedsupportwith

instruction, activities, and ample guided practice.

• Skillsareconnectedtoconceptsthroughvisualrepresentations for understanding the “how” and the “why.”

• Extensiveproblemsolvingmergesconceptualunderstanding with computational skills for mastery.

national council of Teachers of mathematics

“a curriculum is more than a collection of activities: it

must be coherent, focused on important mathematics,

and well articulated across the grades.”

—Curriculum Focal Points for Prekindergarten through Grade 8 Mathematics, 2006

Focused and Coherent Curriculum Teach to Mastery

Skill Building

Concept Building

the WHY

the HoW

7.1 Writing Algebraic expressions

7.2 evaluating Algebraic expressions

7.3 simplifying Algebraic expressions

7.4 expanding and Factoring Algebraic expressions

7.5 real-World Problems Algebraic expressions

Chapter 7- Algebraic expressions


The Story Behind Math in Focus

national research council

“opportunities should involve connecting symbolic representations and operations with physical or pictorial representations, as well as translating between various symbolic representations.”

—Adding it Up: Helping Children Learn Mathematics, 2001

Bar Model: a visual representation of a word problem (Grade 2)

Math in Focususes visual models for presenting concepts,

focusing on a meaningful transition to the symbolic.

• Thevisualrepresentationofwordproblemswithbar models leads to symbolic solutions of rich and complex problems.

• Minimaltextanddirectvisualsallowstudentstofocus on key concepts.

• Consistentuseofconcrete-visual-symbolicpedagogy leads to conceptual understanding.

Clear Visuals and Use of Models

common core state standards for mathematics

in Grade 6, instructional time should focus on four critical areas: (1) connecting ratio and rate to whole number multiplication and division and using concepts of ratio and rate to solve problems; (2) completing understanding of division of fractions and extending the notion of number to the system of rational numbers, which includes negative numbers; (3) writing, interpreting, and using expressions and equations; and (4) developing understanding of statistical thinking.

—Common Core State Standards for Mathematics: corestandards.org

Math in Focusfollows an articulated scope and sequence

focused on key middle school math concepts.• Lessoncontentbuildsstrongnumberconcepts

and algebraic thinking.

• Contentisfocusedonbuildingfoundationsfornumbers, operations, ratios, proportions, algebraic expressions, and equations.

Organize Topics by Big Ideas

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.

© 2010. National Governors Association Center for Best Practices and Council of Chief State School Officers. All rights reserved.


Chapter Wrap UpConcept Map

are combined

and simplified

to form

can be

translated to

can be

Algebraic Expressions

Numbers

(coefficients)

Real-lifesituations

(in word problems)

Operational signs

1, 2, 3, 4

Variables

Evaluated FactoredExpanded

Key Concepts

A letter or variable in an algebraic expression represents an unknown specific

number or any number.

The expression x 1 x 1 x 1 x… (20 terms) is equivalent to 20x.

Expanding and factoring are inverse operations.


Singapore Math Framework

Math in Focus student Books help students build problem solving ability through an instructional pathway of:

• learning concepts and skills through visual lessons and teacher instruction

• consolidating concepts and skills through practice, discussion, and activities

• applying concepts and skills with extensive problem solving practice and challenges

Singapore’s Approachto Becoming Strategic Mathematical Problem SolversMath in Focus is built around the Singapore Ministry of Education’s pentagon that emphasizes conceptual understanding, skill development, strategies for solving problems, attitudes towards math, and metacognition.

Singapore‘s Mathematics Framework

From the Singapore Ministry of Education

The center of the pentagon is problem solving. The primary goal of Math in Focus is to enable students to become strategic mathematical problem solvers.

Grades 1–5


Building a Solid Foundation for Problem Solving

diagrams are used to present examples with solutions.

Visual

only numerals, mathematical notation, and symbols are used once students are familiar with the abstract representation.

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.


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.


Le

arn

Le

arn

Use order of operations to simplify algebraic expressions.

a) Simplify x 1 6x 1 2x.

x 1 6x 1 2x 5 7x 1 2x Work from left to right.

5 9x Add.

b) Simplify 7x 2 5x 2 x.

7x 2 5x 2 x 5 2x 2 x Work from left to right.

5 x Subtract.

c) Simplify 9x 2 3x 1 2x.

9x 2 3x 1 2x 5 6x 1 2x Work from left to right.

5 8x Add.

Guided PracticeSimplify each expression.

21 j 1 3j 1 2j 5 ? 1 2j 22 4j 1 5j 1 2j

5 ?

23 9t 2 3t 2 4t 24 5t 2 t 2 4t

25 8w 2 6w 1 3w 26 7w 1 2w 2 6w

CautionWhen adding and subtracting

algebraic terms with no parentheses,

always work from left to right.

For example:

7x 2 5x 2 x 7x 2 4x9x 2 3x 1 2x 9x 2 5x

Collect like terms to simplify algebraic expressions.

a) The figure shows a parallelogram. Find the perimeter of the parallelogram.

8 cm

8 cm

r cm r cm

r 1 8 1 r 1 8 Identify like terms.

5 r 1 r 1 8 1 8 Change the order of terms

to collect like terms.

5 2r 1 16 Simplify.

The perimeter of the parallelogram is

(2r 1 16) centimeters.

Math Note

Commutative Property of Addition:

Two numbers can be added in

any order.

So, 4 1 a 5 a 1 4.


Le

arn

7.1 Writing Algebraic Expressions

Vocabularyvariable

algebraic expression

terms

Lesson Objective• Usevariablestowritealgebraicexpressions.

A variable can be

replaced by different

values. x is a variable,

so it can represent

different values.

If the length of the first

ribbon is 12 in., then

x is 12.

You can use a letter, called

a variable, to represent

the unknown length.

Use variables to represent unknown numbers and write addition expressions.

a) A 5-inch ribbon is taped to an 8-inch ribbon. What is the total length of the

two ribbons?

?

8 in.5 in.

5 1 8 5 13

The total length of the two ribbons is 13 inches.

b) A ribbon of unknown length is taped to a 9-inch ribbon. What is the total length

of the two ribbons?

Let the length of the first ribbon be x inches.

?

9 in.x in.

Math Notes contain helpful reminders to students.

Cautions contain common mistakes and misconceptions.

Models are used to help students visualize mathematical concepts.

side notes are used to reinforce student understanding.

Symbolic

2x + x = 3x

each lesson in the student Book is introduced with a Learn element. mathematical concepts are presented in a straightforward visual format, with specific and structured learning tasks and worked-out examples.


consolidates learning.


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


Le

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

A car uses 1 gallon of gas for every 25 miles traveled.

a) How far can the car travel on w gallons of gas?

1 gallon

w gallons

25 miles 25 miles 25 miles25 miles

1 gallon 25 miles

w gallons w 3 25 5 25w miles

The car can travel 25w miles on w gallons of gas.

Guided PracticeComplete.

1 Raoulisyyearsold.Kaylais6yearsolderthanRaoulandIsaacis4yearsyounger thanRaoul.

a) Find Kayla’s age.

Kayla is ? years old.

y 6

?

b) Find Isaac’s age.

Isaac is ? years old.

y

4?

c) If y 512,findthesumofRaoul’sageandIsaac’sage.

When y 5 12,

Isaac’s age:

? 2 ? 5 ? 2 ?

5 ?

SumofRaoul’sageandIsaac’sage:

? 1 ? 5 ?

ThesumofRaoul’sageandIsaac’sageis ? .


Solve.

5 A square has sides of length x centimeters. Find the perimeter of the square

in terms of x.

x cm

?

? ? ?

x 1 ? 1 ? 1 ? 5 ?

The perimeter of the square is ? centimeters.

6 The figure shows a trapezoid. The length of each side is given as shown. Find the

perimeter of the trapezoid in terms of w.

w cm

?

? ? ?

w 1 ? 1 ? 1 ? 5 ?

The perimeter of the trapezoid is ? centimeters.

Guided PracticeSimplify each expression. Then state the coefficient of the variable in the expression.

1 x 1 x 1 x 1 x 1 x 2 y 1 y 1 6

3 m 1 m 1 m 1 5 1 4 4 n 1 n 1 n 1 n 1 n 1 n 1 12 2 8

x cm

10 cm

w cm w cm

w cm

Consolidating Concepts and Skillsfor Deep Math Understanding

Extensive Practiceeach Learn portion of the lesson is followed by opportunities to develop deeper understanding through these features:

• carefullycraftedskillspracticeinthelesson using Guided Practice and Practice

• real-worldproblems

• additionalpracticeproblemsintheExtra Practice Book and advanced practice in the enrichment Book

Guided Practice allows students to check their understanding while working with some guidance.


Practice 7.5

1 Jenny is x years old. Thomas is 3 times as old as she is. Jenny is 5 years older

than Alexis.

a) Find Alexis’s age in terms of x.

b) Find Thomas’s age in terms of x.

c) If x 5 12, how much older is Thomas than Jenny?

2 A van travels from Town A to Town B. It uses 1 gallon of gas for every 24 miles

traveled.

a) How many gallons of gas does the van use if it travels 3x miles?

b) The van uses 2y gallons of gas for its journey from Town A to Town B.

Find the distance between Town A and Town B.

3 Brian bought x apples and some oranges. Brian bought 3 more oranges

than apples.

a) Find the total number of fruits Brian bought in terms of x.

b) Find the total amount of money, in cents, that Brian spent on the fruits.

Give your answer in terms of x.

c) If Brian could have bought exactly 12 pears with the amount of money

that was spent on the apples and oranges, find the cost of each pear,

in cents, in terms of x.

4 A rectangle has a width of x centimeters and a perimeter of 8x centimeters.

A square has sides of length 14

that of the length of the rectangle.

a) Find the length of the rectangle.

b) Find the perimeter of the square.

c) Find the difference between the perimeter of the square and the

perimeter of the rectangle if x 5 4.

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

the square if x 5 4.

Instructional Pathway

Practice


Hands-on Work in Pairs and Small Groupsstudents develop concepts and explore connections as they practice skills and communicate their reasoning processes.

Hands-on and Technology Activities reinforce skills, concepts, and problem solving strategies in small-group or partner settings.

Chapter Wrap up serves as a recap of the key concepts in the chapter. The concept map is a graphic organizer that shows the relationships between key concepts.


STEP

1 Draw a rectangle that is greater than 3 centimeters long and 8 centimeters

wide on a piece of paper. Then cut out the rectangle.

(p 1 3) cm

8 cm

STEP

2 Find the area of the rectangle in terms of p.

STEP

3 Then cut the rectangle into two rectangles A and B as shown.

(p 1 3) cm p cm

A B8 cm 8 cm

3 cm

8 cm

STEP

4 Find the areas of rectangle A and rectangle B.

STEP

5 UsingyouranswersfoundinSTEP

2 and STEP

4 , state how the three areas are related.

STEP

6 You may repeat the activity using rectangles of other sizes.

RECOGNIZE THAT EXPANDED EXPRESSIONS ARE EQUIVALENT

Materials:

• paper

• ruler

• scissors

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?

Chapter review/Test can be used as either review exercises or formal assessment. additional chapter Tests are also provided in the assessments component.

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.



are combined

and simplified

to form

can be

translated to

can be


Numbers

(coefficients)

Real-lifesituations

(in word problems)

Operational signs

1, 2, 3, 4

Variables

Evaluated FactoredExpanded

Key Concepts

A letter or variable in an algebraic expression represents an unknown specific

number or any number.

The expression x 1 x 1 x 1 x… (20 terms) is equivalent to 20x.

Expanding and factoring are inverse operations.


Text

Applying Concepts and SkillsBuilds Real-World Problem Solvers

Frequent ExposureMath in Focus embeds problem solving throughout the lessons.

Learn elements use models to explain computation concepts. students become accustomed to seeing and using visual models to form mental images of mathematical ideas.

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.


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

Model-Drawing Strategiesstudents become familiar with this systematic way to translate complex word problems into mathematical equations, and avoid the common issue of not knowing where to start.

Model Drawing:• helpsstudentssolvesimple

and complex word problems

• developsalgebraicthinking

• followstheintroductionof operational skills

• helpsvisualizethealgebraic structure of the problem

• developsoperationalsense

• fostersproportionalreasoning


Le

arn Like terms can be added.

a) Simplify 3x 1 x.

3x x

x xx x

3x 1 x 5 x 1 x 1 x 1 x 5 4x

3x and x are the terms of the expression 3x 1 x.

3x and x are called like terms.

In the expression 2y 1 4y 1 6 1 3, 2y and 4y are

like terms. So are 6 and 3.

b) Simplify 4z 1 2z.

4z 2z

z zz z z z

4z 1 2z 5 z 1 z 1 z 1 z 1 z 1 z 5 6z

3x 1 x and 4x are equivalent


equal for all values of x.

If x 5 1, 3x 1 x 5 4 and 4x 5 4.

If x 5 2, 3x 1 x 5 8 and 4x 5 8.


7 Simplify x 1 8x.

x 1 8x 5 ?


8 3r 1 2r 9 5y 1 6y


10 3a and a 1 a 1 a 11 2h 1 2h and 4h

12 2k 1 5 and (k 1 k) 3 5 13 6z 1 4z and 10 1 2z

14 1p 1 3p and 13p 15 3n 1 2 1 4n and 2 1 7n

x x x xx xx x x

? ?


Le

arn


Vocabularyvariable


terms

Lesson Objective• Usevariablestowritealgebraicexpressions.

A variable can be

replaced by different

values. x is a variable,

so it can represent

different values.

If the length of the first

ribbon is 12 in., then

x is 12.

You can use a letter, called

a variable, to represent

the unknown length.

Use variables to represent unknown numbers and write addition expressions.

a) A 5-inch ribbon is taped to an 8-inch ribbon. What is the total length of the

two ribbons?

?

8 in.5 in.

5 1 8 5 13

The total length of the two ribbons is 13 inches.

b) A ribbon of unknown length is taped to a 9-inch ribbon. What is the total length

of the two ribbons?

Let the length of the first ribbon be x inches.

?

9 in.x in.


Le

arn

7.3 Simplifying Algebraic Expressions

Lesson Objectives• Simplifyalgebraicexpressionsinonevariable.

• Recognizethattheexpressionobtainedafter

simplifying is equivalent to the original expression.

4 4

4 1 4 5 2 3 4

5 5

5 1 5 5 2 3 5

y y

y 1 y 5 2 3 y

2 3 y is the same as 2y.

p cm p cm

p cm

Algebraic expressions can be simplified.

a) A straw of length y inches is joined to another straw of the same length. What is

the total length of the two straws?

?

y in. y in.

The total length of the two straws is (y 1 y) inches.

y 1 y 5 2 3 y 5 2y y 1 y has been simplified to 2y.

In the term 2y, 2 is called the coefficient of y.

b) Each side of the triangle below has length p centimeters.

Find the perimeter of the triangle in terms of p. Then state the coefficient of the

variable.

Vocabularysimplify

like terms

coefficient

equivalent expressions

Instructional Pathway (CoNTINuED)



Practice 7.5

1 Jenny is x years old. Thomas is 3 times as old as she is. Jenny is 5 years older

than Alexis.

a) Find Alexis’s age in terms of x.

b) Find Thomas’s age in terms of x.

c) If x 5 12, how much older is Thomas than Jenny?

2 A van travels from Town A to Town B. It uses 1 gallon of gas for every 24 miles

traveled.

a) How many gallons of gas does the van use if it travels 3x miles?

b) The van uses 2y gallons of gas for its journey from Town A to Town B.

Find the distance between Town A and Town B.

3 Brian bought x apples and some oranges. Brian bought 3 more oranges

than apples.

a) Find the total number of fruits Brian bought in terms of x.

b) Find the total amount of money, in cents, that Brian spent on the fruits.

Give your answer in terms of x.

c) If Brian could have bought exactly 12 pears with the amount of money

that was spent on the apples and oranges, find the cost of each pear,

in cents, in terms of x.

4 A rectangle has a width of x centimeters and a perimeter of 8x centimeters.

A square has sides of length 14

that of the length of the rectangle.

a) Find the length of the rectangle.


c) Find the difference between the perimeter of the square and the

perimeter of the rectangle if x 5 4.

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

the square if x 5 4.

Challenging Problemseach Math in Focus chapter concludes with Brain at Work which challenges students to solve non-routine questions.

These problems ask children to draw on deep prior knowledge, as well as recently acquired concepts, combining problem solving strategies with critical thinking skills.

critical thinking skills students develop with Math in Focus include:

• classifying

• comparing

• sequencing

• analyzing

• identifyingpatternsand relationships

• induction(fromspecifictogeneral)

• deduction(fromgeneraltospecific)

• spatialvisualization

Frequent PracticePractice opportunities include both computation and problem solving sections.

• Eachsetofproblemsencompasses previous skills and concepts.

• Wordproblemsgrowincomplexity from 1-step to 2-step to multi-step.

• TheTeacher’sEditionlevelsandlabelsthe problems as Basic, intermediate, and advanced for differentiated instruction.

27Lesson 7.4 Expanding and Factoring Algebraic Expressions


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)


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


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

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


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


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


Concepts and Skills Across the Courses

1A Chapter 7 Chapter at a Glance1BChapter 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.

Extensive Teacher Support

Step-by-Step Supportand Embedded Professional DevelopmentMath in Focus Teacher’s Editions provide comprehensive lesson plans with pacing suggestions, step-by-step instructional support, and embedded professional development, including math background discussions and classroom management tips.

skills Trace shows concepts and skills learned in the previous level on which the chapter is based, as well as concepts and skills in the following level to which the chapter will lead. This shows the vertical skills alignment across the three-course series.

every Day Counts® Algebra readiness is an interactive companion piece that encourages classroom discussion through all the strands, with a particular focus on algebraic thinking and reasoning.

Grades 1–5


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

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


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

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


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

Differentiated Instruction opportunities are called out for struggling learners, english language learners, and advanced learners.

Math Background clearly outlines the mathematical significance of key concepts. This is embedded professional development for teachers.

Built-in Assessment resources



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

9?


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.

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




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




5 6 and 9 6 4 and 12

7 5 and 15 8 8 and 28








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



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


product

sum

?

5 ?

?

15 4

17

9?


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.

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 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.

definitions of key chapter Vocabulary are available for easy reference.

each chapter begins with a day of guiding students to recall Prior Knowledge and a quick check to assess students’ readiness to proceed.

Extensive Teacher Support (CoNTINuED)

Modeled dialogues are included for communication and discussion during instruction.

Grades 1–5




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.

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

Best Practices are highlighted throughout the Teacher’s edition.

Ticket out the Door activities check for understanding at the end of the class period.

Key Concepts










(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 )





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.

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

each chapter ends with a chapter review and a comprehensive assessment. additional test preparation opportunities are referenced in the Teacher’s edition for students who need extra support.


Transition Support

Transition Guide and online Transition resource Mapfor Program Implementation Grades 6–8The Math in Focus Transition Guide provides a map for teachers and math supervisors to help transition students into the Math in Focus program.

Math BackgroundWithin the Transition Guide and Teacher’s edition, educators will find background information on important concepts covered in the program. The mathematics Background topics were chosen because of their uniqueness to Math in Focus and singapore math. Teachers can use the mathematics Backgrounds within the Transition Guide to ensure they understand the strategies used to teach singapore math and can teach and demonstrate them to students who may not have used them in their previous math programs.

Algebratic Twists to Familiar Ideas

Bar Models

Math Background information in the Transition Guide and Teacher’s edition cover concepts that are unique to singapore math, such as bar modeling.

Grades 1–5


Transition WorksheetsFor each skill objective, the Transition Guide includes Transition Worksheet Blackline masters. These pages provide step-by-step instruction, practice, and review. a student can follow the steps independently or with someone’s help. corresponding Teacher Guides provide strategies for teaching, Quick checks, and intervention suggestions.

an online Transition resource Map is also available, which makes it easy to locate and print transition materials. reteach worksheets and extra Practice worksheets from previous grade levels will be available online for further reinforcement and assessment.

recognizing equivalent expressions solving real-World Problems

Math Background information in the Transition Guide ensures that teachers are prepared to teach and demonstrate singapore math strategies to students who may not have used them in their previous math programs.


Chapter Table of Contents CourSE 1

Chapter 1 Positive numbers and the number line

Chapter 2 negative numbers and the number line

Chapter 3 multiplying and dividing Fractions and decimals

Chapter 4 ratios

Chapter 5 rates and Unit rates

Chapter 6 Percents

Chapter 7* Algebraic Expressions

chapter at a Glance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1a

math Background. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1d

differentiated instruction. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .1F

recall Prior Knowledge. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2

•Usingbarmodelstoshowthefouroperations •Findingcommonfactorsandgreatestcommonfactor

of two whole numbers •Meaningofmathematicalterms

Lesson 7.1* Writing Algebraic Expressions. . . . . . . . . . . . . . . . . . . . . . . . . . . . 4

Learn •Usevariablestorepresentunknownnumbersandwrite addition expressions •Usevariablestowritesubtractionexpressions •Usevariablestowritemultiplicationexpressions •Usevariablestowritedivisionexpressions

Lesson 7.2* Evaluating Algebraic Expressions. . . . . . . . . . . . . . . . . . . . . . . . . 9 Learn algebraic expressions can be evaluated for given values of the variable

Lesson 7.3* Simplifying Algebraic Expressions . . . . . . . . . . . . . . . . . . . . . . . 12 Learn •Algebraicexpressionscanbesimplified •Liketermscanbeadded •Liketermscanbesubtracted •Useorderofoperationstosimplifyalgebraicexpressions •Collectliketermstosimplifyalgebraicexpressions Hands-On Activity recognize that simplified expressions are equivalent



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Lesson 7.4* expanding and Factoring Algebraic expressions. . . . . . . . . . . . . 22 Learn •Usethedistributivepropertytoexpandalgebraic expressions •Algebraicexpressionscanbefactoredbytakingout a common factor Hands-On Activity recognize that expanded expressions are equivalent

Lesson 7.5* Real-World Problems: Algebraic Expressions. . . . . . . . . . . . . . . 29 Learn •Writeanadditionorsubtractionalgebraicexpression for a real-world problem and evaluate it •Writeamultiplicationordivisionalgebraicexpression for a real-world problem and evaluate it •Writeanalgebraicexpressionusingseveraloperations and evaluate it chapter Wrap Up. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36 chapter review/Test . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37

Chapter 8 equations and inequalities

Chapter 9 The coordinate Plane

Chapter 10 area of Polygons

Chapter 11 circumference and area of circles

Chapter 12 volume and surface area of solids

Chapter 13 introduction to statistics

Chapter 14 measures of central Tendency

*Chapter 7 included in sampler

20

Chapter overview COUrse 1 • ChapTer 7


CHAPTER


CHAPTER OPENERAlgebraic Expressions


LESSON 7.1Writing Algebraic Expressions

Pages 428

LESSON 7.2Evaluating Algebraic Expressions

Pages 9211

LESS

ON

AT

A G

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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


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




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


• 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


Concepts and Skills Across the Courses

1A Chapter 7 Chapter at a Glance


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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.

22



CHAPTER


Chapter 7 Chapter at a Glance1C

CHAPTERWRAP UP/REVIEW/TEST

Brain@Work Pages 35–38

LESS

ON

AT

A G

LAN

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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


TEACHER NOTES


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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


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


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


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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


• ExtraPracticeworksheets

• ActivitiesinActivity Book, Chapter 7

END-OF-CHAPTER

• ChapterReview/Test

• Chapter7TestinAssessments • ExamViewAssessmentSuite

CD-ROMCourse1


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

26

Chapter opener COUrse 1 • ChapTer 7

The following is Chapter 7 Algebraic Expressions from the Math in Focus CourSE 1 Teacher’s Edition


Teacher’s editio

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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


7.2 Evaluating 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


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


recall Prior Knowledge COUrse 1 • ChapTer 7




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




5 6 and 9 6 4 and 12

7 5 and 15 8 8 and 28








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



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


product

sum

?

5 ?

?

15 4

17

9?


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.


Teacher’s editio

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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




5 6 and 9 6 4 and 12

7 5 and 15 8 8 and 28








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



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


product

sum

?

5 ?

?

15 4

17

9?


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


Lesson 1 COUrse 1 • ChapTer 7

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Vocabulary

b)


Le




24 cm

10 cm?

24 2 10 5 14




? 6 cm

y cm






You can say that

x 1 9 is the sum

of x and 9.


expression y 2 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.

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Teacher’s editio

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Vocabulary

b)


Le




24 cm

10 cm?

24 2 10 5 14




? 6 cm

y cm






You can say that

x 1 9 is the sum

of x and 9.


expression y 2 6.





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.




Le

arn

?

12 in.




12 4 3 5 4




?

w in.


inches.

w7










Math Note

w7


w.

w7


.

You can say that w7



6z

5p dollars

, or p dollarsp4

14

w8

7Lesson 7.2 Evaluating Algebraic Expressions

Le

arn


expression 4z.


product of z and 4.



?

12

2 3 12 5 24



?

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


Teacher’s editio

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Le

arn

?

12 in.




12 4 3 5 4




?

w in.


inches.

w7










Math Note

w7


w.

w7


.

You can say that w7



6z

5p dollars

, or p dollarsp4

14

w8

7Lesson 7.2 Evaluating Algebraic Expressions

Le

arn


expression 4z.


product of z and 4.



?

12

2 3 12 5 24



?

z









x 1 10

z 1 4

z 2 3

y 2 7


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 Practices

Learn

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


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


Lessons 1 and 2 COUrse 1 • ChapTer 7


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

1 2

3 4

5

a)

13

6

7

8

913

15

a)

11

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


1

2

3



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 .


challenge.


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

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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




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




c)

9

w4

w 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


Teacher’s editio

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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


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




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.

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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.


Teacher’s editio

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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.

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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




Materials:

• paper

• ruler

• scissors

m mm m m


2m 3m

4m 5m

STEP


Example

3m

STEP



2 .

Example

STEP



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

STEP



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.


Teacher’s editio

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Work in pairs.

STEP




Materials:

• paper

• ruler

• scissors

m mm m m


2m 3m

4m 5m

STEP


Example

3m

STEP



2 .

Example

STEP



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

STEP



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


a) Simplify 2v 2 v.

2v

v v

2v 2 v 5 v


5w

3w

w ww w w

5w 2 3w 5 2w

c) Simplify y 2 y.

y 2 y 5 0




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




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

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Guided Practice

7

8 9

10 11

12 13

14 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.


Teacher’s editio

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Le


a) Simplify 2v 2 v.

2v

v v

2v 2 v 5 v


5w

3w

w ww w w

5w 2 3w 5 2w

c) Simplify y 2 y.

y 2 y 5 0




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




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

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Guided Practice

7

8 9

10 11

12 13

14 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


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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Work from left to right.

Add.


Subtract.

c)


Add.

Guided Practice

21 22

23 24

25 26

Caution

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


Teacher’s editio

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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.

c)


Add.

Guided Practice

21 22

23 24

25 26

Caution

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 10

11 12

13 123v

15 16

17 18

19

20

Practice 7.3 Basic 1 – Intermediate 21 – 23 , 26

Advanced 24 –

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.


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

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


Teacher’s editio

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1

2

3

4 5 6

7 8

9 10

11 12

13 123v

15 16

17 18

19

20

Practice 7.3 Basic 1 – Intermediate 21 – 23 , 26

Advanced 24 –

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.


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




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

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.


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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


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


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.



Teacher’s editio

nSTEP

1

STEP

2

STEP

3

STEP

4

STEP

5STEP

2STEP

4

STEP

6


Materials:

(8p 1 24) cm2

8p cm2, 24 cm2

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


Le

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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.



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


3

3 Take out the common factor 3.

Guided Practice

11 12

13 14

15 16

17 18

19 20

21 22

23 24

25 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



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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)


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

BC

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.


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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


Lessons 4 and 5 COUrse 1 • ChapTer 7


Teacher’s editio

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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

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 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.



Teacher’s editio

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Guided Practice

1

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 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.


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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.



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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

2

3

4 14

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















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



Teacher’s editio

n

Practice 7.5

1

2

3

4 14

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















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 up and review/Test CourSE 1 • CHAPTEr 7

Key Concepts




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.


Teacher’s editio

nChapter Wrap up and review/Test CourSE 1 • CHAPTEr 7

Key Concepts










(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 )





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


Chapter review/Test COUrse 1 • ChapTer 7

19

20

21

22

23

24

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


advanced proofs not final pages64

Grades 1–5

65

scope and sequence


MIF Scope and Sequence GrADES 4–6

Grade 4 Grade 5 Grade 6 (COuRSE 1)

Number and operations

sets and Numbers

explore negative numbers in context

Understand the systems of integers and rational numbers

Number representa-tion

represent numbers to 100,000

represent numbers to 10,000,000

represent negative numbers on a number line

represent whole numbers using roots and exponents (squares and cubes only)

represent the absolute value of a number

Compare and order

compare and order whole numbers to 100,000

compare and order decimal numbers

compare and order whole numbers to 10,000,000

compare and order integers and rational numbers using < and >

Place Value express numbers to 100,000 in three forms

express numbers to 10,000,000 in various forms

apply place value concepts in standard algorithms

Fraction Concepts

recognize, write, and name mixed numbers and improper fractions

Find a fraction of a set, equivalent fractions, and equivalent mixed numbers/ improper fractions

convert fractions to decimals

relate fraction and division expressions

recognize, name, and write positive and negative integers and rational numbers


Scope and Sequence GrADES 4–6


Number and operations (continued)

Decimal Concepts

model decimals using tenths and hundredths

Understand decimal notation (hundredths) as an extension of the base-ten system

read and write decimals that are greater than 1 and less than 1

compare and order decimal numbers

identify equivalent decimals and fractions

model decimal numbers using thousandths

Understand place value concepts through thousandths

convert decimals to fractions

apply place value concepts in standard algorithms

convert decimals to fractions and percents.

ratio and Proportion

Understand concept of ratio

Use ratios to solve problems

Find equivalent ratios

solve problems involving unit rates and speed

construct tables of equivalent ratios and plot them on coordinate plane

recognize linear relationship of plotted pairs of equivalent ratios

Use proportional reasoning to draw conclusions and predict relative frequencies of outcomes for situations involving randomness

Grades 1–5

67

scope and sequence




Percents solve problems with percents

convert fractions to percents

Find a percent of a number

Write fractions and decimals as percents

Find the whole, given a percentage and a percent

solve multi-step problems involving simple interest, tax, percent increase or decrease, percent error

Whole Number Computation: Multiplication and Division

apply an understanding of models of multiplication and division

recall multiplication facts and related division facts

develop fluency in multiplying multi-digit numbers

divide by a 1-digit number with a remainder

solve multi-digit multiplication and division problems

multiply multi-digit numbers

Find quotients involving multi-digit dividends

solve whole-number multiplication and division problems

select the most useful form of the quotient and interpret the remainder

Find whole-number quotients involving multi-digit divisors

Find factors of whole numbers, square roots of perfect squares, and cube roots of perfect cubes

Find the greatest common factor or least common multiple of two whole numbers





Fraction Computation

add and subtract unlike fractions that have related denominators

add and subtract unlike fractions and mixed numbers

multiply proper fractions, improper fractions, mixed numbers, and whole numbers

divide fractions by whole numbers

solve word problems by adding, subtracting, multiplying, and dividing fractions

divide whole numbers, fractions, and mixed numbers by fractions

solve word problems by adding, subtracting, multiplying, and dividing fractions

Decimal Computation

add and subtract decimal numbers

solve problems by adding and subtracting decimals

add and subtract decimals to the thousandths place

multiply and divide decimals by whole numbers

solve problems by multiplying and dividing decimals

multiply and divide decimals by decimals

solve problems by multiplying and dividing decimals

estimation and Mental Math

Use mental math and estimation strategies to find sums, differences, products, and quotients

decide whether an estimate or an exact answer is needed

estimate relative sizes of amounts or distances

round and estimate with decimals

Use mental math and estimation strategies to find sums, differences, products, and quotients

estimate sums and differences of fraction and decimal operations

estimate products and quotients with decimals

estimate to check reasonableness of fraction and decimal computations

Grades 1–5

69

scope and sequence



Algebra

Patterns identify, describe, and extend numeric and nonnumeric patterns

Use a rule to describe a sequence of numbers or objects

identify, describe, and extend numeric patterns involving all operations

Find a rule to complete a number pattern

construct and analyze tables of quantities in equivalent ratios and use equations to describe the relationship

Properties represent division as the inverse of multiplication

Write equivalent algebraic expressions by applying number properties

Number Theory

Find the greatest common factor and least common multiple of two numbers

identify prime and composite numbers

Write prime factorization of whole numbers

apply prime factorizations to finding square roots, cube roots, GcF, and lcm

Functional relationships

Understand the relationships between the numbers and symbols in formulas for area and perimeter

describe number relationships in context

Understand the relationships between the numbers and symbols in formulas for surface area and volume

describe number relationships in context

Use variables to represent two varying quantities in a given context

identify independent and dependent variables

express the relationship between two quantities as an equation

Graphing Functions

Graph equations from function tables

Graph data given in a table, such as conversion of °F to °c or equivalent ratios




Algebra (continued)

expressions/ Models

Use a variety of concrete, pictorial, and symbolic models for multiplication and division; and addition and subtraction with fractions and decimals

Use letters as variables

simplify algebraic expressions

Use the order of operations in numeric expressions with two or more operations

Use, write, and interpret conventions of algebraic notation, including order of operations (including exponents)

evaluate expressions given specific values for variables

Write equivalent expressions using number properties

Number sentences and equations

Write and solve number sentences for one-, two-, and three-step real-world problems

Use bar models and number sentences for one-, two-, and three-step problems

determine the missing parts (quantities or symbols) in number sentences

Write and solve number sentences and equations for one- and two-step real-world problems

Write and solve one- and two-step equations

solve linear equations of the form x + p = q, p < q, using properties of equality informally

Find unknown quantities in problems involving equal ratios

Write equations and inequalities to solve problems in context

equality and Inequality

Understand equality and inequality

Understand equality and inequality

Understand and apply the properties of equality informally

represent solutions to inequalities of the form x > c or x ≤ c on a number line

Grades 1–5

71

scope and sequence



Geometry

Lines and Angles

draw perpendicular and parallel lines

construct and measure angles

solve problems involving angles on a straight line

solve problems involving angles about a point

Polygons apply the properties of squares and rectangles

Find unknown angle measures and side lengths of squares and rectangles

identify figures that form tessellations

Understand the relationships between the numbers and symbols in formulas for area and perimeter

apply the properties of right, isosceles, and equilateral triangles

apply the sum of the angle measures of a triangle formula

apply the properties of a parallelogram, rhombus, and trapezoid

show that the sum of any two side lengths of a triangle is greater than the length of the third

Find the area of a triangle

identify polygons in the coordinate plane

Find area of polygons by decomposition into triangles

derive area formula of special quadrilaterals by decomposition and reassembly

solve word problems involving area of plane figures




Geometry (continued)

Circles identify the parts of a circle

relate the radius of a circle to its diameter, circumference, and area

calculate the circumferences and areas of circles, semicircles, and quarter circles

solid Figures identify and classify prisms and pyramids

identify the solid that can be formed from a given net

identify cylinders, spheres, and cones

describe cylinders, spheres, and cones by the number and types of faces, edges, and vertices

Find volume of a prism with a uniform cross-section base

Find the surface areas of cubes, prisms, and pyramids

explore rectangular prisms with same surface area and different volumes

explore rectangular prisms with same volume and different surface areas

Grades 1–5

73

scope and sequence



Geometry (continued)

Congruence and symmetry

identify line and rotational symmetry

relate rotational symmetry to turns and congruency

Transforma-tions

Use transformations to form tessellations

Coordinate Geometry

develop coordinate readiness with tables and line graphs

Plot points on a coordinate grid (first quadrant only)

Plot points on a coordinate grid (all four quadrants)

draw polygons in the coordinate plane, given coordinates of the vertices

Find distance between two points with the same x-coordinate or the same y-coordinate




Measurement

Angles estimate and measure angles with a protractor

classify angles by angle measures

relate ¼-, ½-, ¾-, and full turns to the number of right angles

apply the sum of the angles in a straight line to solve problems

apply the vertical angles property of intersecting lines

apply the sum of the angles about a point to solve problems

Perimeter and Circumference

Find the perimeter of composite figures

solve problems involving the perimeter of squares, rectangles, and composite figures

solve problems involving the perimeter of squares, rectangles, and composite figures

Give an informal derivation of the formula relating the radius and circumference of a circle

calculate circumferences of circles, semicircles, and quarter circles

Grades 1–5

75

scope and sequence



Measurement (continued)

Area identify area as an attribute of two-dimensional figures

connect area formula for a rectangle to the area model for multiplication

estimate area in square units

compare the areas and perimeters of two-plane figures

Find the area of rectangles and composite figures

Find the area of triangles Find area of polygons by decomposing them into triangles

derive area formulas for special quadrilaterals by decomposition and reassembly

derive formula for area of a circle

solve word problems involving area of plane figures

surface Area and Volume

Use the net of a prism to find its surface area

estimate and measure volume in cubic units

Find surface area of cubes, prisms, and pyramids

Find volume of prisms with uniform cross-section base

exhibit prisms with different surface areas but same volumes and vice versa

solve word problems involving solids




Data Analysis

Classifying and sorting

construct line plots, stem-and-leaf plots, tables, and line graphs

represent data in a double bar graph

collect data and summarize the results using tables

represent appropriate data using dot plots (scatter plots), histograms, and box plots

Interpret/ Analyze Data

interpret tally charts, bar graphs, picture graphs, tables, and line graphs

Find the mean (average), median, mode, and range of a data set

analyze data in a double bar graph

describe and summarize distributions of data

measure the center of a data set as a single number that describes all the values of the set

Find the mean, median, range, and interquartile range of a given set of data

measure variation as a single number that describes how the values of a numerical data set vary

Grades 1–5scope and sequence

advanced proofs not final pages 77


Probability

outcomes decide whether an outcome is certain, more likely, equally likely, less likely, or impossible

Find all possible outcomes of a compound event by listing, making a tree diagram, or multiplying

expressing Probability

express the probability of an event as a fraction

determine the experimental probability of a simple event

compare the results of an experiment with the theoretical probability

Math in Focus CourSE 1

CoMPoNeNTsstudent Books A and Bwith full examples, guided practice, and hands-on activities

Teacher’s editions A and Breteach, enrichment, and extra Practice

Transition Guide and online resource Map for comprehensive transition support and guidance with access to previous grade material

Assessments

solutions Key

online student Book and Teacher’s edition

Interactive Whiteboard Lessons

Activity Book

Test Generator powered by examview®

Manipulative Kit

Virtual Manipulatives

online resources for Teachers, students, and Parents

Program Components

78

Grades 1–5

7979

80

Notes


Grades 1–5


ArrivingFall 2011

Course 3

A

ArrivingFall 2012

Course 2

A

ArrivingFall 2012

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