CHAPTER 7 Algebraic Expressions and Solving Equations 8_TR/Teachers Resource... · CHAPTER 7 Algebraic Expressions and Solving Equations Specific Curriculum Outcomes Major Outcomes

CHAPTER 7 Algebraic Expressions andSolving Equations

S p e c i f i c Cu r r i c u l u m O u t co m e s

M a j o r O u t c o m e s

B14 add and subtract algebraic terms concretely, pictorially, and symbolically to

solve simple algebraic problems

B15 explore addition and subtraction of polynomial expressions, concretely and

pictorially

B16 demonstrate an understanding of multiplication of a polynomial by a scalar,

concretely, pictorially, and symbolically

C6 solve and verify simple linear equations algebraically

C o n t r i b u t i n g O u t c o m e s

C1 represent patterns and relationships in a variety of formats and use these

representations to predict unknown values

C7 create and solve problems, using linear equations

C h a p t e r Pro b l e m

A chapter problem is introduced in the chapter opener. This chapter problem invites

students to use algebra to plan a class trip and solve various problems about the trip.

The chapter problem is revisited in section 7.1, questions 17 and 18, section 7.2,

question 14, and section 7.3, question 18. You may wish to have students complete

the chapter problem revisits that occur throughout the chapter. These simpler

versions provide scaffolding for the chapter problem and offer struggling students

some support. The revisits will assist students in preparing their response for the

Chapter Problem Wrap-Up on page 325.

Alternatively, you may wish to assign only the Chapter Problem Wrap-Up

when students have completed Chapter 7. The Chapter Problem Wrap-Up is a

summative assessment.

Key Wordsvariableexpressionequationpolynomialnumerical coefficientterm

Get Ready Wordsvariablesconstantspolynomialtermszero principlesimplify

258 MHR • Mathematics 8 : Focus on Understanding Teacher ’s Resource

Planning Chart

SectionSuggested Timing

Teacher’s ResourceBlackline Masters Assessment Tools Adaptations

Materials andTechnology Tools

Chapter Opener• 15 min (optional)

Get Ready• 60 min

• BLM 7GR Parent Letter• BLM 7GR Extra Practice

• algebra tiles

7.1 Add and SubtractAlgebraic Expressions• 180 min

• BLM 7.1 Extra Practice Formative Assessment:• BLM 7.1 AssessmentQuestion, #21

• algebra tiles

7.2 Multiply PolynomialExpressions• 90 min


• algebra tilesOptional:• trays

7.3 Solve LinearEquations• 240 min


• algebra tiles

Chapter 7 Review• 90 min

• BLM 7R Extra Practice • algebra tiles

Chapter 7 Practice Test• 90 min

Summative Assessment:• BLM 7PT Chapter 7Test

• algebra tiles

Chapter Problem Wrap-Up• 60 min

• BLM 7CP ChapterProblem Wrap-UpRubric

Chapter 7 • MHR 259

Get Ready

W A R M - U P

Use the properties of operations to evaluate each expression.

1. 7(–13) + 3(–13) <–130> 2. –19 + 42 + (–21) <2>3. 0.5(14 � 9) <63> 4. 43 + (–41 + 77) <79>5. 9(–23) – 7(–23) <–46> 6. 247 + 139 + 253 <639>7. 34 + (–22) + 16 + (– 28) <0>8. 33(–19) + 40(–19) + 27(–19) <–1900>

Follow the order of operations to evaluate each expression.

9. 2 � 52 <50> 10. (2 � 5)2 <100>11. 72 + 42 <65> 12. 5 + 6(9.6 – 9.1) <8>13. 7(8) + 3(4) <68> 14. 24 + 12 ÷ 3 � 4 <40>15. (24 ÷ 3)2 + (72 ÷ 12)2 <100>

A S S E S S M E N T F O R L E A R N I N G

Before starting Chapter 7, explain that the topic is algebra and solving equations. The

chapter involves the study of addition, subtraction and multiplication of polynomials

and solving simple linear equations.

Discuss with students when they have combined like terms, added or subtracted

polynomial expressions and solved equations before, and what they know about

these concepts. You may wish to brainstorm and develop a mind map for each topic

or start the development of a graphic organizer to be used throughout the chapter.

Students might find it helpful to keep a journal of new vocabulary learned in this

chapter.

After students have discussed what they already know about algebra, have them

complete the assessment suggestions below in pairs or individually. This assessment

is designed to provide you and your students with information about their readiness

for the chapter. After strengths and weaknesses have been identified, students can

work on appropriate sections of the Get Ready.

Method 1: Have students develop a journal entry to explain what they know about

the topics and how they use expressions or equations in their everyday language or

in their everyday lives.

Method 2: Challenge students to show how much they know about algebra and

solving equations. Encourage them to use words, numbers, and diagrams to show

what they know.

R e i n fo rce t h e Co n ce p t s

Have those students who need more reinforcement of the prerequisite skills

complete BLM 7GR Extra Practice.

Materials• algebra tiles

Related Resources• BLM 7GR Parent Letter• BLM 7GR Extra Practice

Suggested Timing60 min


T E A C H I N G S U G G E S T I O N S

The Get Ready provides students with the skills they require to fully understand the

topics developed in Chapter 7. Start the class with a brainstorming session or by

drawing a concept map covering the topics in the Get Ready section to find out stu-

dents’ prior knowledge. You may wish to have students complete all of the Get Ready

questions before starting the chapter or complete portions of the Get Ready ques-

tions as they work on the various sections of the chapter.

When working through Order of Operations, check that students understand

operations must be done in a specific order to get the correct answer. Have students

work with a starting number and then perform an addition and then a multiplica-

tion. Then have the students perform the same operations in reverse. They will get

different answers and should see the importance of agreeing on the order in which

the operations are done.

When working through Represent Expressions Using Algebra Tiles, check

that students understand how algebra tiles can be used to model expressions and

which tiles are used to represent the variables. Have students define the vocabulary

in their math journals or record the terms on the word wall.

When working through Solve Equations by Inspection, check that students

understand that the variable represents a number in the same way that an open space

represented a number in earlier grades. When working through Solve EquationsUsing a Model, check that students understand the concept of equality and that they

are forming equivalent statements while solving the equation.

Co m m o n E r ro r s

• Students may misinterpret BEDMAS and do all of the multiplication before

doing any division, regardless of order. The same applies to addition and

subtraction.

Rx Remind students that they must perform the operations of multiplication

and division, and then addition and subtract, in the order they occur from

left to right. To illustrate, you might have students evaluate an expression

such as 6 ÷ 2 � 3 both ways: multiplication first, and multiplication and

division in the order in which they appear. This should illustrate the need

for an agreed upon order.

L i t e ra c y Co n n e c t i o n s

Concept Cards: Concept cards are a tool all students may find useful, but they are

especially helpful for ESL students and students who require adaptations and modi-

fications. To make a concept card, students take important vocabulary or procedures

and create a “cue” card that will help them remember the topic. It is particularly

important for students to put the content into their own words; this will help them

consolidate and demonstrate their understanding of the concept. Including exam-

ples for each concept helps students build connections for the concept and shows

that they can apply it. ESL students and students on adaptations may find the cards

useful for testing situations or for general class use. All students will find these an

effective way to create study notes for assessments.


Students could create a table of contents to be placed at the beginning of the

concept card collection. Or they could use colour coding to highlight cards with

related topics, either by using coloured index cards or by highlighting the edge of the

cards with coloured markers. All cards can be held together by placing a ring or piece

of ribbon or string through a hole in one corner of each card. Placing the topic title

at the bottom of each card can help students find a particular card more easily when

flipping through the collection. A low priced photograph album could be an alter-

native way of organizing the collection.

Placemat Activities: Placemat activities allow for group discussion around a partic-

ular topic. Place students in groups and give each group a sheet of paper or chart

paper. Write a different polynomial expression in the centre of each group’s page.

Divide the paper into sections, at least one for each student in the group. Students

should think for a few minutes and then each find a different way to represent their

group’s polynomial expression in their section of the page. Once students have com-

pleted their sections, they can discuss their entries and choice of representation. Each

group can share with the rest of the class their expression and how they represented

it. Since students have added, subtracted, and multiplied using area models and

repeated addition, students should represent their expression with at least one of

each of the methods used in the chapter.

By listening to students’ explanations and their understanding of the concepts,

the teacher can make decisions about whether learning is complete or if more

instruction is necessary. This activity might also be an interesting assessment activ-

ity if students each create their own mini-poster.

G e t R e a d y An s we r s

1. a) –230 b) 63

2. a) 7 b) 8

3. a) 1 x-tile and 1 –x-tile b) 4 pairs of opposite unit tiles

c) No. After the opposite tiles are grouped to form zero, there would always be

one tile left over. Examples may vary.

4. a) 3n and n; 2 and –7 b) –y and 3y; 4x and –7x

5. a) 2 x-tiles, 3 y-tiles, 1 –x-tile, and 1 y-tile; x + 4y

b) 1 unit tile, 1 negative unit tile, 2 y-tiles, 2 –x-tiles, 3 –y-tiles, and 2 unit tiles;

–3x – y + 3

6. a) 6x + 4y – 3 b) 2x2 + 5x + 2y c) 10x + 3y d) 5m – n

7. a) 8. What number added to 10 is equal to 18? b) 12. What number added to 4 is

equal to 16? c) –25. What number added to 15 is equal to –10? d) 14. What

number subtracted from 12 is equal to –2? e) –6. What number multiplied by 4

is equal to –24? f) 12. What number divided by 3 is equal to 4?

8. Step 1: 4x + 3 = 11. Step 2: 4x + 3 – 3 = 11 – 3. Step 3: 4x = 8. Step 4: x = 2

9. a) 3x b) x + 10 c) x ÷ 2 d) x – 4

10. a) Step 1: 4x + 2 = 10. Step 2: 4x + 2 – 2 = 10 – 2. Step 3: 4x = 8.

Step 4: 4x ÷ 4 = 8 ÷ 4. Step 5: x = 2

b) Step 1: 4x = 8. Step 2: 4x ÷ 4 = 8 ÷ 4. Step 3: x = 2 c) Step 1: 3y + 3 = 9.

Step 2: 3y + 3 – 3 = 9 – 3. Step 3: 3y = 6. Step 4: 3y ÷ 3 = 6 ÷ 3. Step 5: y = 2.


7.1 Add and Subtract Algebraic Expressions

W A R M - U P

Use the Compensation Strategy for Addition to evaluate each expression.

1. 45 + 29 <74> 2. 77 + 17 <94>3. 166 + 398 <564> 4. 519 + 296 <815>5. 3764 + 1999 <5763> 6. 1863 + 7998 <9861>7. 5.5 + 7.8 <13.3> 8. 2.9 + 2.9 <5.8>9. 8.4 + 1.9 <10.3> 10. 12.7 + 7.6 <20.3>11. 0.77 + 0.18 <0.95> 12. 2.65 + 1.97 <4.62>

13. 6 + 2 <9 > 14. 4 + 1 <6 >

15. 7 + 3 <11 >

Co m p e n s at i o n St rat e g y fo r Ad d i t i o n

This strategy involves changing one number in a sum to a “nice” number, doing the

addition, and then adjusting the answer to compensate for the change. The number

is changed to make it easier to add. But you have to remember how much you

changed it by so you can subtract it later.

Examples:

54 + 27 Think: 54 + 30 = 84. But I added 3 too many. To compensate I must

subtract 3 from my answer to get 81.

568 + 396 Think: 568 + 400 = 968. But I added 4 too many. To compensate I

must subtract 4 from my answer to get 964.

3.6 + 5.8 Think: 3.6 + 6 = 9.6. But I added 0.2 too much. To compensate I must

subtract 0.2 from my answer to get 9.4.

0.37 + 0.49 Think: 0.37 + 0.5 = 0.87. But I added 0.01 too much. To compensate I

must subtract 0.01 from my answer to get 0.86.

3 + 1 Think: 3 + 2 = 5 . But I added too much. To compensate I must

subtract from my answer to get 5 which is equal to 5 .


In this section, students continue to learn about the addition and subtraction of poly-

nomial expressions and algebraic terms. Review how algebra tiles can be used to repre-

sent simple terms and expressions. Ask students how algebraic terms could be used to

represent the numbers of buses and kayaks needed in the chapter problem on page 291.

D i s cove r t h e M at h

Read through the Discover the Math activity before presenting it to the class to become

familiar with the manipulation of the tiles. It is strongly recommended that the class

2

3

4

6

1

6

1

6

5

6

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6

5

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5

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8

7

8

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Related Resources• BLM 7.1 Assessment

Question• BLM 7.1 Extra Practice

Specific CurriculumOutcomesB14add and subtract

algebraic termsconcretely, pictoriallyand symbolically tosolve simple algebraicproblems

B15explore addition andsubtraction ofpolynomial expressions,concretely andpictorially


Link to Get ReadyStudents should havedemonstratedunderstanding of RepresentExpressions Using AlgebraTiles in the Get Ready priorto beginning this section.


do the activity prior to working through the lesson. The activities will help students

develop their understanding and fluency with the meaning of algebra tiles and will

improve their ability to manipulate the tiles. Do Part A one day and Part B the next day

to allow students time to internalize what they have learned. Consider presenting these

activities to the whole class while giving carefully guided instructions.

D i s cove r t h e M at h An s we r s

P a r t A

1. 2x + 2

2.

3. a) x + 2x + 2 + x + 2x + 2 b) 2(2x + 2) + 2x

4. 6x + 4

5. x + 1

6. x + x + 1 + x + x + 1 + x + x + 1 + x + x + 1; 8x + 4

7. longer; (8x + 4) – (6x + 4); 2x

8. ; + 2x + 2 + + 2x + 2; 5x + 4

9. (5x + 4) – (4x + 2); x + 2

10. 10x + 8

11. Saturday; 2x + 4 more

P a r t B

1. x2

2. a) x b) x2

3. a) 1 x2-tile, 4 x-tiles, and 4 unit tiles b) x2 + 4x + 4 c) no; no like terms

4. a) x-tile and y-tile b) i) x ii) y iii) xy c) –xy

5. Answers may vary. x2-tile: area of a square; xy-tile: area of a rectangle.

Example 1 shows how to add like terms by using algebra tiles. It might help students

to build the model of the pathway to see the tile pattern. Examples 2 and 3 show how

to subtract polynomial expressions using the take-away method and adding the

opposite (Example 2) and using the comparison method and by finding the missing

addend (Example 3). The two methods in Example 3 are similar. Ensure that

students know all four methods, as they will be asked to subtract expressions using

these methods throughout this section. Example 4 shows how to collect like terms.

Ensure that students realize that by combining an x2-tile and a – x2-tile or a y2-tile

and a –y2-tile, they apply the zero principle. They are not cancelling the opposite

pairs; they are adding the pairs and the sum is zero.

2x + 2

2x + 2

x–2

x–2

x

2

x

2

x

2

x + 1

x + 1

xx

2x + 2

2x + 2

xx


Co m m u n i c ate t h e Key I d e a s

Have students work in groups to answer and discuss all of the Communicate the Key

Ideas questions. For question 2, they could model each method for a classmate and

have the classmate check their simplification. Use this opportunity to assess student

readiness for the Check Your Understanding questions.

Co m m u n i c ate t h e Key I d e a s An s we r s

1. Use the commutative property to rearrange the terms and group like terms.

Simplify by adding like terms. Examples may vary.

2. a) Take away method: Model 5x + 2 using algebra tiles. Add two pairs of opposite

unit tiles. Take away 3 x-tiles and 4 unit tiles. 2 x-tiles and 2 negative unit tiles

remain.

Adding the opposite: (5x + 2) – (3x + 4) becomes (5x + 2) + (–3x – 4);

5x + 2 –3x – 4, 2x – 2.

Finding the missing addend: What must be added to 3x + 4 to get 5x + 2? Add 2x

to 3x to get 5x, and add –2 to 4 to get 2. The missing addend is 2x – 2.

b), c) Answers may vary.

3. Two parallel sides are the length of an x-tile and two parallel sides are the length

of a y-tile, so the area of the tile is x � y, or xy.

4. The lengths or areas that the tiles represent are unknown until the variables x

and y are given a value. You cannot add or subtract different unknown values so

unlike terms cannot be added or subtracted. Examples may vary.

5. Once the pairs of opposite variables are removed, both models simplify to the

expression 2x2.

O n g o i n g A s s e s s m e nt

• Can students use algebra tiles to represent polynomial expressions?

• Can students use algebra tiles to represent addition and subtraction of

positive and negative terms in algebraic expressions?

C h e c k Yo u r U n d e r s t a n d i n g

Q u e s t i o n P l a n n i n g C h a r t

For question 8, students could model with tiles to check their answers. Questions 12to 14 should be assigned as a group. Students can use their answers to question 14 to

check their answers to questions 12 and 13. For question 20, students will need to

have a classmate check their work.


• Students sometimes only change the sign on the first term when subtracting

polynomial expressions. (This is the error shown in question 16, part b).)

Level 1 Knowledge andUnderstanding

Level 2 Comprehension of

Concepts and Procedures

Level 3 Application and Problem Solving

1, 3–5 2, 6, 7, 9–14, 20 8, 15–19, 21–23


Rx Have students write out the expression showing the subtraction of each

term before they proceed. For example, 9 – (7x + 2) = 9 – 7x – 2. Review all

methods in Example 2 and Example 3 to ensure that students understand

how to correctly subtract terms.

I nt e r ve nt i o n

• For some students, you may need to review the nature of integers. Remind

students that subtracting an integer has the same effect as adding its oppo-

site. Once students understand this property of integers, they should be

ready to apply it to variables.

A S S E S S M E N T

Q u e s t i o n 2 1 , p a g e 3 0 4 , An s we r s

a)

b) w + 3w + 9 + w + 3w + 9

c) 98 cm

d) 8w + 18; 98 cm

e) Yes. The expression in part d) since it is shorter.

f) 6w + 4

g) (8w + 18) – (6w + 4); 2w + 14; 34 cm

A D A P T A T I O N S

BLM 7.1 Assessment Question provides scaffolding for question 21.

BLM 7.1 Extra Practice provides additional reinforcement for those who need it.

V i s u a l / Pe rce p t u a l / S p at i a l / M o t o r

• Algebra tiles are an ideal tool to help students visualize variable expressions

and equations. Encourage students to use algebra tiles to model each prob-

lem before solving it.

E x t e n s i o n

Assign question 23. You may wish to reduce the number of Check Your Understanding

questions to provide students with extra time to work on the Extend question. Students

will need to infer the length of three sides before they solve this problem. You might ask

them to think about how the perimeter would change if the garden were a complete

rectangle with width 3g and length 6.

Journal

Students could use these prompts for question 16.

• The error in the solution was made when…

• To fix this you have to …

• The way to show this in pictures is …

3w + 9

3w + 9

ww


C h e c k Yo u r U n d e r s t a n d i n g An s we r s

1. a) 2 terms b) 1 term c) 3 terms d) 1 term

2. a) 8x b) Not possible; variable in second term is not squared. c) 2x2 d) –2xy

3.

4. Niall; 5 and 7 are like terms since they are both units.

5. Answers may vary.

6. a) –3x2 + x – 7 b) –4y2 + 3y + c) –3n2 – 2n – 4

d) – m3 + m2 – 3m e) –y5 + 6n2 f) –8w4 + 0.8w2 – 2

7. a) 14x + 7 b) b – 1 c) 3r + 2 d) 1.2m – 1.5

e) 2p5 + p4 – 3q2 f) x3y + 3x2y + y2 + 4xy g) 1 a + b

8. Answers may vary.

9. a) 4 x2-tiles, 5 y-tiles, and 3 negative unit tiles b) Let x represent n and

y represent m: 3 –x2-tiles, 1 –xy-tile, 2 y2-tiles, and 4 unit tiles.

10. a) 5y + 1 b) 2x2 + x – 1

11. a) 8x – 10y + 7 b) –2n2 + 7n – 7

12. a) 6y + 1 b) 2x – 1 c) 4m2 + 3 d) p3 – 6p2

13. a) 7s b) 4m + 3 c) –w2 d) –h2 – 4xy

14. See answers to questions 12 and 13.

15. a) –2y2 + 5y b) 10y2 – y Similar: both have a y2-term and a y-term.

Different: the like terms have opposite signs.

16. a) 4x2 should be subtracted from –6x2 not added; –10x2 + 8xy

b) 5y should be added to –4y not subtracted; 4x2 + y

17. a) t + 2f + 3w + 5u b) (2) + 2(3) + 3(5) + 5(1); 28

18. a) 6x + 10 b) 4x + 12 c) (6x + 10) – (4x + 12); 2x – 2

19. a) x + 2y + 3z b) 20 points

20. a)

b) Answers may vary.

22. 3g + 6s + 17b

23. a) 6g + 12

b) 15 m c) $273.60

6 – 2g g

6

2g

2g

3g

4n2 + 0.7pq

–3n2 + 0.4pq

–8n2 + 5pq

n2 + 0.3pq

–11n2 + 4.6pq–10n2 + 4.9pq

–31n2 + 7.5pq–21n2 + 2.6pq

–10n2 + 2pq–2n2 + 7pq

1

2

1

4

3

2

4

5

1

2

Expression Model TermsLike/Unlike

Terms Justification

4x2 – 3x2 4 x2-tiles, 3 –x2-tiles 4x2, –3x2 like termsVariable in both

terms is x2.

2y2 + 4y 2 y2-tiles, 4 y-tiles 2y2, 4y unlike termsVariable in second

term is not squared.

–3x2 + 3x 3 –x2-tiles, 3 x-tiles –3x2, 3x unlike termsVariable in second



7.2 Multiply Polynomial Expressions

W A R M - U P

Simplify.

1. 17x + 45x + 13x <75x>2. –5.3x – 2.7x <–8x>3. –63x + 57x + 62x <56x>

4. 3 x + 5 x <9x>

5. 5x – 9x + 7x – 3x <0>6. –4.3x + (–1.7x) + 3.5x + 0.5x <–2x>7. 6x + 7y + 9x + 4y <15x + 11y>8. –8x + 6y + 11y + 8x <17y>9. 1.5x – 3.1+ 4.5x – 1.9 <6x – 5>10. 1.7x2 + 0.3x + 5.3x2 + 0.7x <7x2 + 1x>11. –4xy – 6x – 8xy + 10x <–12xy + 4x>12. (5x – 12) + (–x – 5) <4x – 17>13. (14x + 6y) – (9x + 4y) <5x + 2y>

Use the Compensation Strategy for Addition to evaluate each expression.

14. 274 + 598 <872>15. 23.7 + 9.9 <33.6>


In this section, students continue to learn about multiplication of a polynomial by a

scalar concretely, pictorially, and symbolically. Review the usefulness of representing

calculations in different ways. Using algebra tiles or pictures can help students to

keep track of terms as they multiply, then simplify by collecting like terms.


The activity is relatively short but provides a useful base for students to build their

understanding of multiplying a polynomial by a scalar. Multiplying polynomials

requires a large number of algebra tiles. Ensure there are enough tiles available before

doing this lesson. You may need to use paper tiles or other manipulatives to repre-

sent the algebra tiles.

2

3

1

3

Materials• algebra tilesOptional:• trays



Specific CurriculumOutcomesB16demonstrate an

understanding ofmultiplication of apolynomial by a scalar,concretely, pictorially,and symbolically


Link to Get ReadyStudents should havedemonstratedunderstanding of RepresentExpressions Using AlgebraTiles in the Get Ready priorto beginning this section.



1. 2 x-tiles and 7 unit tiles; 2x + 7

2. a) 3(2x + 7) b) 2x + 7 + 2x + 7 + 2x + 7 c) 6x + 21

3. Similar: they are all equivalent expressions that have two or more positive terms

being added. Different: they have a different number of terms. Preferences may

vary. The simplified algebraic expression will be easier to evaluate since it

requires fewer calculations.

Example 1 shows how to multiply polynomials by a scalar. Method 1 uses repeated

addition and Method 2 uses an area model. Modelling these methods on the over-

head would be helpful for students to see as they will need to understand both meth-

ods when completing this section.

Example 2 shows how to multiply polynomials by a scalar to find the volume

of solids. Reinforce the fact that volume is found by multiplying the area of the base

by the height. Students may be unsure about x as a side length. Review the use of the

variable with them. x represents a length that is not known.


Have students work in groups to answer and discuss the Communicate the Key Ideas

question. This question demonstrates the usefulness of algebra tiles to model and

calculate with polynomials. Use this opportunity to assess student readiness for the

Check Your Understanding questions.

Co m m u n i c ate t h e Key I d e a s An s we r

1. Julien did not multiply 5 by –2. Five groups of (x – 2) equals 5x - 10.


• Can students use algebra tiles to model repeated addition and multiplica-

tion of a polynomial by a scalar?

• Can students model and calculate multiplication of a polynomial by a scalar

using symbols?



Have algebra tiles available. Multiplication requires a large number of tiles; consider

using paper tiles or other manipulatives. For questions 1 and 2, students will need





1–5 6, 7, 10, 11, 15 8, 9, 12–14, 16

+x

–1–1

+x

–1–1

+x

–1–1

+x

–1–1

+x

–1–1


some way of marking positive and negative tiles on their drawings, either by using

colour or by using symbols. For question 9, students will need to divide the figures

into rectangles before finding the total areas. For part b), students will probably find

it easier to calculate the area of the large rectangle and then subtract the area of the

small one.


• Students might try to add the terms in brackets before multiplying. For

example, attempt adding x + 7 in 4(x + 7).

Rx Remind students that only like terms can be combined. These are not like

terms and can therefore not be combined.

I nt e r ve nt i o n

• For some students, you might need to review how to collect like terms from

section 7.1.

A S S E S S M E N T


a) 12x + 6; 30x + 20

b) 30 m2; 80 m2





• Although not required for question 9, students could use algebra tiles to

build each backyard then find each area.

• Students could work in pairs to develop all possible rectangles for question 12.

E x t e n s i o n

Assign questions 15 and 16. You may wish to reduce the number of Check Your

Understanding questions to provide students with extra time to work on the Extend

questions. If necessary, review how to find surface area with students or suggest that

they draw a net of the box in question 16 to help with their calculations. Trial and

error is an effective strategy to find the dimensions of the box in question 16, part d).

Technology

Use Internet resources to explore demonstrations of operations involving polynomials.

Go to www.mcgrawhill.ca/books/math8NS for some interesting Web sites.


Journal

Students could use these prompts for question 12.

• My area model of 12x + 6 has dimensions …

• I could also model the area like this �because …


1.

2.

3. a) (x + 1) + (x + 1) + (x + 1) b) (3x – 2) + (3x – 2)

c) (x2 – x + 1) + (x2 – x + 1) + (x2 – x + 1)

4. a) 3(x + 1) b) 2(3x – 2) c) 3(x2 – x + 1)

5. a) A b) D c) F d) B e) E

6. a) 2(5x); 10x b) 4(x + 1); 4x + 4 c) 2(3x – 2); 6x – 4

7. a) 2(x + 3), or 2x + 6 b) 3(2y + 3), or 6y + 9 c) x(x), or x2

8. a) 2(4 – n), or 8 – 2n b) 4(6 – b), or 24 – 4b c) 3(7 – 2m), or 21 – 6m

9. a) 23x + 16 b) 30x + 30

10. C

11. D

12. Models may vary. Yes. There are four possible models because 6 and 12 have

four common factors: 6(2x + 1), 3(4x + 2), 2(6x + 3), 1(12x + 6).

14. a) 12x + 4 b) 6x + 9 c) 6x – 5 d) 7 m2

15. a) 17x + 16 b) 10x – 1.6

16. a) 26x + 132 b) 40x + 80 c) 512 cm3 d) 8 cm � 8 cm � 8 cm

Model Repeated Addition Multiplication Result

b) two sets of 1 x2-tile and1 x-tile

(x2 + x) + (x2 + x) 2(x2 + x) 2x2 + 2x

c) two sets of 3 x-tiles and3 negative unit tiles

(3x – 3) + (3x – 3) 2(3x – 3) 6x – 6

d) Models may vary.

Model Width Length Area

b) 6 x-tiles and 6 unit tiles 3 2x + 2 3(2x + 2) = 6x + 6

c) 6 x- tiles and 6 unit tiles 2 3x + 3 2(3x + 3) = 6x + 6

d) Models may vary.


7.3 Solve Linear Equations

W A R M - U P

Use the Compensation Strategy for Subtraction to evaluate each expression.

1. 94 – 38 <56> 2. 41 – 27 <14>3. 688 – 299 <389> 4. 743 – 294 <449>5. 1774 – 897 <877> 6. 5089 – 2995 <2094>7. 8.6 – 1.8 <6.8> 8. 5.1 – 3.8 <1.3>9. 24.5 – 19.9 <4.6> 10. 0.88 – 0.59 <0.29>11. 0.62 – 0.27 <0.35> 12. 9.16 – 4.98 <4.18>

13. 5 – 1 <3 > 14. 7 – 1 <5 or 5 >

15. 8 – 4 <3 >

Co m p e n s at i o n St rat e g y fo r S u b t ra c t i o n

The compensation strategy also works for subtraction. As with addition, it involves

changing one number to a “nice” number. This time, however, you do the subtrac-

tion and then adjust the answer to compensate for the change. The second number

(the subtrahend) is changed to make it easier to subtract. You have to remember how

much you changed it by so you can add the amount later.

Examples:

64 – 19 Think: 69 – 20 = 44. But I subtracted 1 too many. To compensate I

must add 1 to my answer to get 45.

373 – 295 Think: 373 – 300 = 73. But I subtracted 5 too many. To compensate I

must add 5 to my answer to get 78.

0.84 – 0.58 Think: 0.84 – 0.6 = 0.24. But I subtracted 0.02 too much. To compen-

sate I must add 0.02 to my answer to get 0.26.

5 – 2 Think: 5 – 3 = 2 . But I subtracted too much. To compensate I

must add to my answer to get 2 which is equal to 2 .


In this section, students continue to learn about solving and verifying simple linear

equations algebraically. Review order of operations, solving equations by inspection

and solving equations using a model.


The situation at the beginning of the activity is easily solved mentally, which lends

itself to teaching the method of working backward to solve for x. Have students work

through each subsequent step, working to build a solid understanding of each

method. The Examples that follow build on this understanding to demonstrate solv-

ing equations by multiplying and dividing, and solving multi-step equations.

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Specific CurriculumOutcomesC6 solve and verify simple

linear equationsalgebraically

C7 create and solveproblems, using linearequations


Link to Get ReadyStudents should havedemonstratedunderstanding of Order ofOperations, Solve Equationsby Inspection, and SolveEquations Using a Model inthe Get Ready prior tobeginning this section.


Journal

Students can use these prompts for question 5.• Miriam’s method is like the pan-balance method because the bags in the

pan-balance method are the same as the �tiles in the algebraic

method. The red circles are the same as the �tiles.

• When you solve an algebraic equation you have to�just like in the

pan-balance method.


1. a), b) Step 1: 2 bags and 3 erasers on the left and box with 17 items on the right,

2x + 3 = 17. Step 2: remove 3 erasers and 3 items from the box,

2x + 3 – 3 = 17 – 3. Step 3: simplify, 2x = 14. Step 4: divide the bags and items

into two groups, 2x ÷ 2 = 14 ÷ 2. Step 5: simplify, x = 7.

c) There is 1 bag on the left side and 7 items on the right side, so 1 bag must

contain 7 items.

2. a) Jake used x-tiles to represent the quantity of items in a grab bag and a unit tile

to represent each additional item. He removed 3 unit tiles from the left and the

right, simplified, divided the remaining x-tiles and unit tiles into two groups, and

removed one group to find the value of 1 x-tile: 1 x-tile = 7 unit tiles.

b) Miriam wrote an equation, using x to represent the quantity of items in a

grab bag. She subtracted 3 from both sides of the equation, simplified, divided

both sides by 2, then simplified to find the value of x: x = 7.

c) The solutions all undo operations in steps and the result is an unknown

amount on one side and a known value on the other.

d) Substitute 7 for x in the equation and check that the left side equals the right

side: 2(7) + 3 = 14 + 3 = 17.

3. a) x = 2 b) n = 3 c) x = 2 d) x = –3 e) y = 0 f) x = 1

4. To keep both sides balanced or equal. If an operation is done on only one side

of an equation, the expressions will no longer be equal.

5. The same operations are performed on both sides to isolate the unknown

quantity on one side.

Example 1 shows how to solve one-step equations by multiplying using a pan-balance

in Method 1 and using the cover-up method in Method 2. Make sure that students

understand the link between the pictorial and symbolic models in Method 1. Have

students verify by substitution. It is an important skill for them to learn.

Examples 2 to 4 show how to solve multi-step equations first by using tiles and

then by using symbols. The cover-up method is also shown for all three Examples.

Make sure that students understand the meaning of the equal sign and remember

that they must maintain equality while solving the equation. Students maintain bal-

ance by using either additive or multiplicative reasoning.

Examples 5 and 6 show how to use equations to solve problems. Encourage

students to draw a picture to help solve Example 5. Have students look back to see if

they answered the question fully. Many students may stop at x = 35° thinking that

they have completed the question when they have not.

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Have students work in groups to answer and discuss all of the Communicate the Key

Ideas questions. Questions 1 and 2 are especially well suited to group discussion.

Students should write their answers to questions 3 and 4 in their math journals. Use this

opportunity to assess student readiness for the Check Your Understanding questions.

Co m m u n i c ate t h e Key I d e a s An s we r s

1. 4x + 1 = 9. Pan-balance: model the diagram using 4 bags and 1 counter on the

left and 9 counters on the right. Remove 1 counter from each side, divide the

remaining bags and counters into four groups, remove three groups to find the

value of 1 bag; 1 bag = 2 counters.

Algebra tiles: same as pan-balance, except use x-tiles for bags and unit tiles for

counters: x = 2.

Algebraic symbols: 4x + 1 = 9, 4x + 1 – 1 = 9 – 1, 4x = 8, 4x ÷ 4 = 8 ÷ 4, x = 2.

Cover up method: 4x + 1 = 9;

+ 1 = 9 (8) + 1 = 9 so 4x = 8.

4 = 8 4(2) = 8 so x = 2.

2. a) In third line, 2y should be negative; y = –4. b) In fourth line, only right side is

divided by 4. Both sides should be multiplied by 4; n = –32. c) In third line, 15

should be negative; x = –15.

3. y = –12. Subtract 1 from both sides, multiply both sides 2.

4. Brandon used x to represent the amount of money he needs to save each week.

He has 12 weeks in which to save money (12x) plus $60 in savings already to

buy a $240 cell phone. He needs to save $15 each week.


• Can students solve algebraic equations by always performing the same

operation on both sides of the equal sign?

• Can students locate and correct an error in a solution to an algebraic equation?

• Can students write an algebraic equation that will help them solve a problem?



Have algebra tiles available. Students may benefit from drawing a diagram for each

question to help them conceptualize the problem before writing an equation.


• When multiplying both sides of an equation by a scalar to eliminate a

denominator, students may inadvertently multiply one side of the equation

twice, eliminating the denominator and multiplying the numerator by the





1, 9 2–5, 7, 8 6, 10–22


scalar. For example, they may write:

= 21

� 4 = 21 � 4

4x = 84

Rx Review with students that � 4 = x. Have them write as , then

multiply by 4. This way of writing the term may help them see that

� 4 = 1x, and help them to remember this when solving equations in

the future.

I nt e r ve nt i o n• For some students, you may need to review the Pythagorean relationship for

question 4.

A S S E S S M E N T


a) Let x represent the unknown length of side CD; 16.6 + 2x

b) 7.4 cm

c) Sides BC and CD are twice the length of sides AB and AD;

16.6 + 2(16.6) = 49.8 or 3x = 49.8




V i s u a l / Pe rce p t u a l / S p at i a l / M o t o r• Pair students who have difficulty reading or understanding the word prob-

lems with stronger students. Have them discuss each problem before writing

an equation and solving it.

E x t e n s i o nAssign questions 21 and 22. You may wish to reduce the number of Check Your

Understanding questions to provide students with extra time to work on the Extend

questions. Question 21 emphasizes the connections between patterns and algebra.

Students could make up their own questions involving patterns for others to solve.

Question 22 is a multi-step problem where students use the Pythagorean relation-

ship to write equations for area and perimeter.

Te c h n o l o g yUse Internet resources to access interactive algebra activities, such as pan-balance

applications for solving linear equations. There are also online lessons which explore

linear relations using spreadsheet software such as Excel®. Go to www.mcgrawhill.ca/

books/math8NS for some interesting Web sites.

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1. 2x + 2 = 8; 2x + 2 – 2 = 8 – 2; 2x = 6; 2x ÷ 2 = 6 ÷ 2; x = 3

2. a) y = 6 b) x = –2

3. a) 2 b) 2 c) 4 d) –7

4. TM = 8.4 m

5. a) x = 2 b) n = –14 c) m = 16 d) x = 12

6. Situations may vary.

7. a) –2n – 4 = –8 b) 0.8y – 7 = 9.8 c) – 7 = –2

8. a) n = 2 b) y = 21 c) b = 25

9. Both separate the two expressions or two quantities and show the two

expressions or quantities are equal. Examples may vary.

10. a) 3x + 1 = 10 b) 3 questions

11. a) Ravi’s savings, s, tripled plus $35 will equal $500, the total cost of the trip.

b) $155 c) Strategies may vary.

12. Let w represent the width, so length (l) = (2w + 6); 6w + 12 = 36; w = 4 m,

l = 14 m

13. 3t + 4 = 17.5; t = 4.5 cm

14. Let l represent the length, so width (w) = – 1; 3l – 2 = 36; l =12 m, w = 5 m

15. Let p represent the number of packages of seeds; 3.75p + 5.5 = 58; 14 packages

16. Let m represent the number of additional minutes. 0.45m + 29.95 = 144.70;

255 additional minutes

17. Let t represent the number of tickets; 10t = 1100 + 600; 170 tickets

18. a) Let s represent the number of students and C represent the total cost for one

day; C = 100 + 25s. b) $850 c) 8 students

20. x + 2x + 3x = 180; 6x = 180; �A = 30°, �B = 60°, �C = 90°

21. a)

b) wheel 15

22. a) P = 32x b) A = 48x2 c) P = 80 cm; A = 300 cm2

Ad d i t i o n a l St u d e nt Tex t b o o k An s we r s

P u z z l e r

Triangled) The sums are the same.

e) Let x, y, z, represent the numbers at the vertices and w represent the number in

the centre. Side xy = x + (z + w) + y, side xz = x + (y + w) + z,

side yz = y + (x + w) + z. All three sums are equal to x + y + z + w.

f) yes

Geoboard: 30 squares

Wheel Circumference, C (cm) Pattern

1 32� 2�(15 + 1)

2 34� 2�(15 + 2)

3 36� 2�(15 + 3)

4 38� 2�(15 + 4)

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Chapter 7 Review

W A R M - U P

Simplify.

1. –34x + 97x + (–66x) <–3x>

2. 6 x + 3 x + 1 x <11 x>

3. 5x2 – 9y2 – 6x2 + 8y2 <–x2 – y2>4. –4xy + 11xz – 7xy – 6xz <–11xy + 5xz>5. (5x – 3y) + (8x + 7y) <13x + 4y>

Multiply.

6. 3(4x + 7) <12x + 21> 7. 8a x + 7b <4x + 56>

8. (9x – 3) <6x – 2> 9. 2.5(4x + 12) <10x + 30>

10. 18(1.5x + 5) <27x + 90>

Solve.

11. x + 9 = 2 <x = –7> 12. 6x = –42 <x = –7>

13. x = –6 <x = –18> 14. 5x + 1 = –24 <x = –5>

15. x + 13 = 16 <x = 6>


Us i n g t h e C h a p t e r R ev i ew

The students might work independently to complete the Chapter Review, and then

compare solutions in pairs. Alternatively, the Chapter Review could be assigned for

reinforcing skills and concepts in preparation for the Practice Test. Provide an

opportunity for the students to discuss any questions, consider alternative strategies,

and ask about questions they find difficult.

Provide algebra tiles. Question 10 may be the first time students have seen a

fractional numerical coefficient multiplying two terms in brackets, i.e.,

3(2x + 5 – 4x) + (6x + 3). Point out to students that when they multiply both sides

of the equation by 3 to eliminate the denominator in an equation like this one, they

should only multiply the terms outside of the brackets. In turn, this will result in the

whole expression being multiplied by 3.

After students complete the Chapter Review, encourage them to make a list of

questions they found difficult, and to include the related sections. They can use this

list as a guide on what to concentrate their efforts on when preparing for the final

chapter test.

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Related Resources• BLM 7R Extra Practice



A S S E S S M E N T

Chapter Review

This is an opportunity for the students to assess themselves by completing selected

questions and checking the answers. They can then revisit any questions that they

found difficult.

Upon completing the Chapter Review, students can also answer questions such

as the following:

• Did you work by yourself or with others?

• What questions did you find easy? difficult? Why?

• How often did you have to ask a classmate to help you with a question? For

which questions?


Have students use BLM 7R Extra Practice for more practice.

R ev i ew An s we r s

1. a) D b) A c) C d) G e) E f) B g) F

2. D 3. C

4. Models may vary. The x-tile has an area x(1), the y-tile has an area y(1), and the

xy-tile has an area x(y). So, the area 2x(1) + 4y(1) will only equal the area

6xy = 2x(y) + 4x(y) when x and y equal 1. Since x and y are unknowns, the

terms 2x and 4y cannot be combined.

5.

6. a) 3r + 3t + 4 b) 4c2 + d2 + 2 c) 4x – 3y + 2.5

d) –3q4 + 3p3 + pq + 7 e) b4 – 2b3 – 7b2 + 1 f) 2y5 – 5xy + 3

7. a) 2(x + 2); 2x + 4 b) 2(2y + 3); 4y + 6 c) (x + 1)(2x); 2x2 + 2x d) (y – 3)(2y); 2y2 – 6y8. a) i) (x + 2) + (x + 2) + (x + 2) + (x + 2) ii) 4(x + 2)b) i) (y + 1) + (y + 1) ii) 2(y + 1)

c) i) (y2 + 2y) + (y2 + 2y) + (y2 + 2y) ii) 3(y2 + 2y)9. Let p represent the number of people; 11.25p + 125 = 1812.50; 150 people10. x = –911. a) x + y b) –x2 + 2y2 – 2xy12. a) (–2x + 3y + 5) – (x – 2y + 3); –3x + 5y + 2. (x – 2y + 3) – (–2x + 3y + 5);

3x – 5y – 2.b) (2x2 + x + 2y + 2xy – 2) – (–3x2 + 2y2 + x + y – xy – 1);5x2 – 2y2 + y + 3xy – 1. (–3x2 + 2y2 + x + y – xy – 1) – (2x2 + x + 2y + 2xy – 2);–5x2 + 2y2 – y – 3xy + 1.

13. a) x = –6 b) t = 3 c) x = –50 d) x = 15 e) x = –16 f) x = 0.9 g) y = 18 h) w = –314. a) Let w represent a win, t represent an overtime win, and l represent an

overtime loss; 3w + 2t + l, b) 20 points

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Expression Model Like/Unlike Terms Justification

3x2 – 4x2 3x2-tiles and 4 –x2-tiles like termsVariable in both

terms is x2.

4y2 + 4y 4 y2-tiles and 4 y-tiles unlike termsVariable in second


–2x2 + 3x 2 –x2-tiles and 3 x-tiles unlike termsVariable in second



Chapter 7 Practice Test


Us i n g t h e Pra c t i ce Te s t

This Practice Test can be assigned as an in-class or take-home assignment. If it is used

as an assessment, use the following guidelines to help you evaluate the students.

• Can students add and subtract algebraic terms concretely, pictorially, and

symbolically?

• Can students multiply a polynomial by a scalar concretely, pictorially, and

symbolically?

• Can students solve and verify simple linear equations algebraically?

St u d y G u i d e

Use the following study guide to direct students who have difficulty with specific

questions to appropriate areas to review.

A S S E S S M E N T

After students complete the Practice Test, you may wish to use BLM 7PT Chapter 7Test as a summative assessment.



• Allow the use of calculators.

• Let students give their answers verbally, either in an interview setting or

recorded.

L a n g u a g e / M e m o r y

• Allow students to refer to personal math dictionaries, journals, index card

files, or notes.

Question Refer to Section

5, 6, 8, 9 7.1

2, 3 7.2

1, 4, 7, 10–14 7.3


Related Resources• BLM 7PT Chapter 7 Test



Pra c t i ce Te s t An s we r s

1. C

2. D

3. A

4. B

5. a) like terms: 2p, p; unlike terms: 3q, –2, 3q2

b) like terms: 5x2, –5x2, 3x2, and 5x, x; unlike term: –5

c) like terms: 4y4, y4, and 5xy, 2xy; unlike term: 6y3

6. 14c – 10

7. a) 2x – 4 + 4 = 10 + 4 b) simplify: 2x = 14

8. a) –3x + 5y – 3 b) x2 – 2y2 + 2xy

9. a) (–2x – y – 3) – (–x + 2y – 2); –x – 3y – 1. (–x + 2y – 2) – (–2x – y – 3);

x + 3y + 1.

b) (–x2 + y2 + 2x + y – 1) – (2x2 + 2y2 – x + y – 2); –3x2 – y2 + 3x + 1.

(2x2 + 2y2 – x + y – 2) – (–x2 + y2 + 2x + y – 1); 3x2 + y2 – 3x – 1.

10. Let p represent the regular price; p � 0.25 = 15; $60

11. Evan made the equation easier to work with by changing the decimal to a whole

number, while Jerod simplified the equation by undoing the operation 0.4 � n.

12. Let p represent the number of photocopies; 0.03p + 2 = 101; 3300 photocopies

13. a) x = 100 b) q = 7 c) r = 2 d) n = 3.25 e) x = 9 f) x = 8 g) y = 36 h) w = –2

14. Let p represent the price of a case of pop; 6p + 8p = 56; $4 per case


Chapter 7 Chapter Problem Wrap-Up

1. Introduce the problem.

2. Clarify the assessment criteria by reviewing BLM 7CP Chapter Problem Wrap-Up Rubric with students.

3. Remind individual students that they have worked on the chapter problem

throughout the chapter and that these will help them. Students can also be

directed to section 7.1, question 17, section 7.2, question 14, and section 7.3,

question 18 at this point.

4. Brainstorm with students what they might include in an equation to solve each

problem.

5. Allow students time to work on the problem, either individually or in a group.

Students should prepare separate reports.

O ve r v i ew o f t h e Pro b l e m

Students have worked on calculating different costs for a trip as well as combining

like terms in a fossil hunt. The chapter problem wrap-up describes a different class

trip and has students determine the cost. They also examine patterns and develop an

equation to model the number of lifejackets hanging on a number of pegs.

A S S E S S M E N T

Use BLM 7CP Chapter Problem Wrap-Up Rubric to assess student achievement.

C r i t e r i a fo r a H i g h S co r i n g R e s p o n s e

• Student successfully identifies the unknown quantity or quantities in each

situation.

• Student represents each problem situation clearly in a well-organized equation.

• Student explains the relationship between lifejackets and pegs clearly and

accurately.

Wh at D i s t i n g u i s h e s Lowe r S co r i n g R e s p o n s e s

• Student may not identify the unknown quantity or quantities in each situation.

• Student may not successfully generalize and represent the problem situation

algebraically.

• Student may not use the patterns to generalize and describe the relationship

between life jackets and pegs.

• Student basically understands the problem and can make some generaliza-

tions using some representations—just cannot finish.


C h a p te r Pro b l e m Wra p - Up An s we r

1. a) C = 0.25k + 100, where C is the cost of renting the bus and k is the number of

kilometres driven.

b) T = 0.25k + 80p + 100, where T is the total cost of the trip, k is the number of

kilometres driven by the bus, and p is the number of people going on the trip.

2. a)

b) The total number of lifejackets is one less than three times the number of pegs.

c) L = 3p – 1

d) Substitute 35 for L and then solve for p: 35 = 3p – 1, 36 = 3p, 12 = p; 12 pegs

Number of Pegs Total Number of Lifejackets

1 2

2 5

3 8

4 11

5 14


Documents

CHAPTER 7 Algebraic Expressions and Solving Equations 8_TR/Teachers Resource... · CHAPTER 7 Algebraic Expressions and Solving Equations Specific Curriculum Outcomes Major Outcomes