polynomials - cape breton (e)/files/teacher's... · 2. 13 22 3 22 3. 17 ( 9.5) 16 ( 9.5) <...

Key Wordsalgebraic modelnumerical coefficientliteral coefficientlike termsdistributive property

Get Ready Wordstermnumerical coefficientliteral coefficientpolynomialmonomialbinomialtrinomialfactorsgreatest common factor

CHAPTER 7 Polynomials

Cu rriculum Outcomes

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

B8 add and subtract polynomial expressions symbolically to solve problems

B9 factor algebraic expressions with common monomial factors, concretely,

pictorially, and symbolically

B10 recognize that the dimensions of a rectangular area model of a polynomial

are its factors

B11 find products of two monomials, a monomial and a polynomial, and two

binomials, concretely, pictorially, and symbolically

B12 find quotients of polynomials with monomial divisors

B13 evaluate polynomial expressions

B14 demonstrate an understanding of the applicability of commutative, associa-

tive, distributive, identity, and inverse properties to operations involving

algebraic expressions

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

B4 demonstrate an understanding of, and apply the exponent laws for, integral

exponents

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

A chapter problem is introduced in the chapter opener. Having students discuss their

understanding of how to answer the chapter problem will provide you with an idea

of what students currently know about this topic. You may wish to have students

complete the chapter problem revisits that occur throughout the chapter. These mini

chapter problems are particularly useful for some students because the revisits will

assist them in doing the Chapter Problem Wrap-Up on page 365.

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 sum-

mative assessment.

When working on this chapter problem, students should definitely have access

to concrete models (e.g., algebra tiles). This situation provides an opportunity to link

the visual nature of the tiles to the real situation that they are modelling. You may

also consider having students use virtual tiles that are freely available. (See The

Geometer’s Sketchpad®, particularly when solving the parts in section 7.4 and the

Chapter Problem Wrap-Up.)

244 MHR • Mathematics 9 : Focus on Understanding Teacher ’s Resource

Planning Chart

SectionSuggested Timing

Teacher’s ResourceBlackline Masters Assessment Tools Adaptations

Materials andTechnology Tools

Chapter Opener• 10 min (optional)

Get Ready• 60 min

• BLM 7GR Letter toParents • BLM 7GR Extra Practice

• BLM 7GR DivisibilityRules

• algebra tiles

7.1 Add and SubtractPolynomials• 60 min

• BLM 7.1 Extra Practice Formative Assessment:question #26

• algebra tiles

7.2 Common Factors• 60 min


• algebra tiles

7.3 Multiply a Monomialby a Polynomial• 60 min

• BLM 7.3 Extra Practice Formative Assessment:• BLM 7.3 AssessmentQuestion, #20

• algebra tiles

7.4 Multiply TwoBinomials• 60 min


• algebra tiles

7.5 Polynomial Division• 60 min


• algebra tiles

7.6 Apply AlgebraicModelling• 60 min


Chapter 7 Review• 60 min

• BLM 7R Extra Practice • algebra tiles

Chapter 7 Practice Test• 60 min

Summative Assessment:• BLM 7PT Chapter 7Practice Test

• algebra tiles

Chapter Problem Wrap-Up• 60 min

• BLM 7CP ChapterProblem Wrap-UpRubric

Chapter 7 • MHR 245

Get Ready

W A R M - U P

Evaluate. Use the distributive property to help you.

1. 7 � 45 � 3 � 45 <450>2. 13 � 22 � 3 � 22 <220>3. 17 � (�9.5) � 16 � (�9.5) <�9.5>

4. 4 � 9 � 2 � 9 <63>

Evaluate. Use the commutative property to help you.

5. �7.3 � 6.5 � 8.3 � 1.5 <9>

6. <2>

7. 5 � 14 � 2 <140>

8. � 8 � 6 � <4>

Evaluate.

9. 4 � 18 <81>

10. �1.5 � (�16) <24>

Find the two numbers that have …

11. a sum of 9 and a product of 14. <2, 7>12. a sum of 6 and a product of 5. <1, 5>13. a sum of 11 and a product of 24. <3, 8>14. a sum of 1 and a product of �6. <�2, 3>15. a sum of �2 and a product of �24. <4, �6>

D I A G N O S T I C A S S E S S M E N T

Algebra is typically a challenging topic for many students and sometimes a source of

math anxiety. Take sufficient time to review the fundamental concepts in the Get

Ready section.

Students should have prior experience working with algebra tiles, but a

refresher is a good idea. Always provide an opportunity for students to explore inde-

pendently when working with a manipulative for the first time. Ask students to iden-

tify common and distinguishing features of the various tiles before discussing their

formal properties. When working with tiles, always try to strengthen the conceptual

link between the physical and visual model, and the symbolic representation. Facility

with symbolic manipulation is the eventual goal, but should not be the only teach-

ing modality at this level.

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Materials• algebra tiles

Related Resources• BLM 7GR Letter to Parents• BLM 7GR Extra Practice• BLM 7GR Divisibility Rules

Suggested Timing60 min

There is a fair bit of terminology to be reviewed to begin this chapter. Consider

using a game (e.g., Jeopardy) or a literacy strategy (e.g., Word Wall or Frayer Model)

in order to make the review more interesting and effective. (See Mathematics 9: A

Teaching Resource, pages 71-84, for other examples, templates, and implementation

strategies.)

After reviewing the nature of algebra tiles, have students work through the Get

Ready questions. Depending on the needs of your students, you may wish to take

extra time to provide additional reinforcement on key topics. (See Reinforce the

Concepts.)

If students are not familiar with algebra tiles at all, spend time allowing them

to get use to the x-tile, x2-tile, and unit or 1-tile for the first day. Blend the zero prin-ciple with the instructions. Keep close track that all students understand this princi-

ple and can connect the zero principle to their integer work in grade 7. This allows

students to build on previous knowledge. Give students the opportunity to rename

the shapes using different letters. On the second day, introduce the other shapes (y2

and xy) and continue to represent zero in a variety of ways.

Although finding common factors is studied in grade 7, it is important that

students can easily find the GCF to be successful in this chapter. It may need a daily

warm-up of five minutes over the first week or longer if difficulties persist.

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

Have those students who need more reinforcement of these prerequisite skills com-

plete BLM 7GR Extra Practice.

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

S u g g e s t e d Le s s o n P l a n

1. Warm-up. Ask the class questions such as: What is algebra? What are algebra

tiles? How do you use them?

2. Hand out algebra tiles. After initial exploration, discuss their formal properties.

Assign the first page of the Get Ready section.

3. Review algebraic terminology (see the Diagnostic Assessment for suggestions).

Assign the second page of the Get Ready section.

4. Assess students’ readiness to proceed with section 7.1. Provide additional reme-

diation as necessary.

It is important that students have the language under control and that they recognize

what a term is and how terms are combined (added and subtracted) to form poly-

nomials. It is important to spend some time discussing responses to the questions

and helping students focus on applying the definitions to distinguish the number of

terms. As further examples, you might ask what the numerical and literal coefficients

are in the terms and . (The numerical coefficients are and ; the lit

eral coefficients are x and m2). Note that numerical coefficient and literal coefficient

may not be prior knowledge to some students.

If students look at the expression 4(2x � 1) from question 8, part d) as a

whole, there is one term. The bracketted expression is considered to be a “package”

�1

7

2

3

�m2

7

2x

3


and since it is multiplied by 4, it is still one term. If students look inside the bracket,

however, there are two terms. Which is why, when you multiply 4 by each of the

inside terms, you get the expression 8x � 4, which has two terms. Part f) is similar

(one term as it is given; two terms if you express it as � 3).

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

• Do students understand the zero principle?

• Do students understand that the name of the tile is related to the area of the

tile?

• Can students easily use the proper terminology connected with polyno-

mials? (See the definitions at the top of page 322.)

• Can students concretely represent an expression using tiles?

Co m m o n E r ro r s

• Students add exponents when collecting like terms. For example,

2x � 3x � 5x2.

Rx Ask students to use the tiles to reinforce that collecting like terms is similar

to combining groups of the same object and then counting the total. The

nature of the object does not change.

• Students confuse numerical parts of a variable (i.e., exponents) with the

numerical coefficient of a term.

Rx Use question 9 as an example and assign more questions like this to discuss

in small groups and consolidate understanding.

I nt e r ve nt i o n

A quick review of divisibility rules may also help students become more efficient in

finding the Greatest Common Factor. A reference sheet with the rules clearly stated

and examples given could be helpful for some students. See BLM 7GR Divisibility

Rules in the Extra Practice folder.

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

Interesting contexts that relate to functions may be found in the article, Children’s

Literature: a Motivating Context to Explore Functions, in Mathematics Teaching in the

Middle School, May 2005. See section 7.1 for more information.

x

2


7.1 Add and Subtract Polynomials

W A R M - U P

Combine like terms.

a) 13a � 87a <100a> b) 4 a � 1 a <6a>

c) 6a2 � 4a2 <10a2> d) 2.8ab � 2.2ab <5ab>

e) �8.5a � 9a <0.5a> f) �6 a � 3 a <�10a>

g) 17a � 19a � 13a <49a> h) 7.7a � 3.8a � 2.3a <13.8a>i) �23ab � 94ab � 24ab <95ab> j) 87a2 � 69a2 �11a2 <7a2>k) 3.1a � 5.4b � 1.9a � 2.6b <5a � 8b>

l) 8 a2 � 5 a � 3 a2 � 1 a <5a2 � 7a>

m) �73a � 95 � 74a � 96 <a � 1>

n) a � 2b � a � b < a � 3b>

o) 43a � 16b � 57a � 14b <100a � 30b>

Us i n g t h e D i s t r i b u t i ve Pro p e r t y

With the class, discuss how to use the distributive property when collecting like

terms.

4 � 19 � 6 � 19: you have four 19s added to six 19s to make ten 19s.

2 � (�7) � 1 � (�7): you have two and one-half (�7)s added to one and one-

half (�7)s to make four (�7)s.

7n � 3n � 7 � n � 3 � n: you have seven n’s added to three n’s to make ten n’s.

Us i n g t h e Co m m u t at i ve Pro p e r t y a n d Co m p at i b l eN u m b e r s

Use the commutative property or look for compatible numbers when collecting like

terms.

3a � 6b � 2a � 5b � 3a � 2a � 6b � 5b

� 5a � 11b

7.3n � 1.7n � 1.1n � 9n � 1.1n

� 10.1n


Open the section with a discussion about rectangular swimming pools of different

sizes, each with a length that is twice its width. There are connections to geometry

here, and you may wish to lightly review the concept of similar figures (i.e., figures

that have the same shape but different sizes). Because of this relationship, a single

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Related Resources• BLM 7.1 Extra Practice

Specific CurriculumOutcomesB8 add and subtract

polynomial expressionssymbolically to solveproblems

B13 evaluate polynomialexpressions


Link to Get ReadyStudents should havedemonstratedunderstanding of allconcepts, except Factors, inthe Get Ready prior tobeginning this section.

variable can be used in both expressions for the length and width: x and 2x. Students’

review of like terms should prepare them to realize that the simplified expression for

the perimeter of such a pool would have one term.

Connect the algebraic expressions for the pool with finding the perimeter of

the pool with a given width. For example, if the expression for the perimeter is 6x for

question 1, part b), then in question 2, finding the perimeter if the width is 6 m

should look like this: 6x � 6(6 m) � 36 m. This is a connection that should be

emphasized in the lesson; otherwise, students will likely not get past the arithmetic

approach (6 m � 12 m � 6 m � 12 m � 36 m). Students need to see the advantage

of setting up a formula, while recognizing that algebra generalizes the arithmetic.

D i s cove r t h e M at h

The purpose of the investigation is to introduce and solve a real problem that

requires collection of like terms. Algebra tiles are used to model the problem, to con-

solidate the process of adding and subtracting terms involving variables, and to help

students make sense of the result in symbolic terms.

Spend a little time talking about the margin notes; in particular, why 2x and 7x2

are not considered to be like terms. You might also consider, as an example, 5ab and

�4ba. These are like terms because ab and ba are equivalent expressions (the order

in which you multiply does not change the result). Ask students for examples and

non-examples related to like terms.


• Are students collecting like terms properly?

• Are students using the simplified form of the perimeter and substitution to

find a given perimeter?

• Are students connecting algebraic expressions with a correct diagram of the

pool?

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

1. a) For the width, the numerical coefficient is 1 and the literal coefficient is x. For

the length, the numerical coefficient is 2 and the literal coefficient is x.

b) x � 2x � x � 2x � 6x c) The perimeter of the rectangle shown is also 6x.

2. a) The perimeter of the pool is 6(6 m) � 36 m.

b) The perimeter of the pool is 6(16 m) � 96 m.

3. a)

b) x � (2x � 4) � x � (2x � 4) � 6x � 8

c) The perimeter is 6(9 m) � 8 m � 62 m.

4. Answers may vary.

a) b)

c) x � (x � 3) � x � (x � 3) � 4x � 6

x + 3

x

xxxx2

xxxxx2x2


5. a) When using objects and pictures, the similar objects or pictures can be

grouped together and then recounted. When using symbols, the expression for

each side can be summed and the like terms can be collected. All these represen-

tations use one form of grouping or another.

b) It is useful to simplify the expression for perimeter because, once the expres-

sion is simplified, less work is required to calculate the perimeter when given a

side length.

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

Have students work in pairs to answer the Communicate the Key Ideas questions.

Then ask for volunteers to demonstrate their answers using the overhead tiles.

In question 2, post a few of the best responses on the wall in the classroom for

reference and as a reminder.

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

1. a) Answers may vary. x � 4 � 3x � 5; x � 3y � 2y � x

b) Answers may vary.

�2x � 9; 5y

Unlike terms cannot be combined because they do not have the same dimensions.

2. a) The sum of all of the original numerical coefficients becomes the new numer-

ical coefficient. For example, 5x � 2x � 3x.

b) The literal coefficients do not change when like terms are collected.

For example, x � x � 2x.

E x a m p l e s

Work through the three Examples with the class, using overhead algebra tiles to

illustrate each method of solution.

Although students will probably tend toward the adding the opposite strategy

when subtracting, emphasize that while this algorithm produces the same result as

using the zero principle, it is not the same situation.

Subtraction is difficult for some students and time will be needed to develop

the strategies described in Example 2 and 3. The most common mistake is for stu-

dents to change the sign of the first term in the bracket and leave the other terms as

they are. For example:

3x � (4 � 5x) � 3x � 4 � 5x.

Notice the �4 was changed to �4 but the �5x was not changed.

In Example 1, part a), only positive terms are used to illustrate how two differ-

ent polynomial expressions can be modelled and grouped when adding. In this type

of question, the tiles are used simply as counters. In part b), negative terms are intro-

duced, and represented using a different-coloured tile. As mentioned in the Get

Ready, students should be familiar with the zero principle based on prior work with

;-x -x

+++++

y y y y y

++++


integers and/or algebra in previous grades. This is a critical concept that is useful

when terms to be added have opposite signs.

Example 2, part a) introduces the concept of polynomial subtraction. Two

methods are illustrated: take away, in which tiles are removed to leave the result, and

comparison of the two polynomials to identify the difference. In part b), the zero

principle is applied to solve a seemingly impossible take away situation. Adding zero

pairs does not change the expression; it simply introduces additional tiles that can be

used to perform the subtraction.

Example 3 illustrates two alternative methods to polynomial subtraction. The

first method involves adding the opposite polynomial. The second method involves

identifying the missing addend (i.e., what polynomial must you add to the sub-

tracted polynomial to give the first polynomial?).

It is important to have students explore several different tactile approaches to

polynomial addition and subtraction prior to a pure symbolic treatment. This will

help them develop a strong comprehension of procedure, rather than simply relying

on memorization of seemingly meaningless rules. Not all students will appreciate the

value of all the different methods; however, different pedagogical approaches may

resonate with different learners and their styles.

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

Questions 5 to 7 provide an opportunity to address most of the errors students

make. This is a good opportunity to have students work independently first, then in

pairs or groups to compare and explain their responses. Then a full class discussion

is helpful to check for understanding.

Assign question 19, and consider taking it up before assigning question 20, so that

students understand what a “magic number problem” is before trying to create one.

In question 21, the parking rows can be represented by x-tiles and the wildlife

park by an x2-tile.

Different methods can be used to solve question 26, part c), such as systematic

trial, set up and solve an equation, The Geometer’s Sketchpad®, etc. Encourage stu-

dents to share different methods when taking up the question.


• Are students adding or subtracting the numerical coefficients of the terms

when combining like terms and using the correct literal coefficient in the

result?

• Are the expressions simplified as far as possible in all cases?

• When subtracting a polynomial inside brackets, are students adding the

opposite of all terms inside the brackets?

• Are students using substitution for applications, such as question 24, part

c), after simplifying the expression for perimeter?

Level 1 Knowledge andUnderstanding

Level 2 Comprehension of

Concepts and Procedures

Level 3 Application and Problem Solving

1–4, 11, 15, 17, 21b) 5–10, 12–14, 16, 21a) 18–20, 22–29



• Students add exponents when collecting like terms. (e.g., 2x � 3x � 5x2).

Rx Ask students to use the tiles to reinforce that collecting like terms is similar

to combining groups of the same object and then counting the total. The

nature of the object does not change.

• Students confuse numerical parts of a variable (i.e., exponents) with the

numerical coefficient of a term.

Rx Use question 9 from the Get Ready as an example and assign more ques-

tions like this to discuss in small groups to consolidate understanding.

A S S E S S M E N T

Q u e s t i o n 2 6 , p a g e 3 3 1 , An s we r s

a) 8w b) 2400 m c) 200 m by 600 m

A D A P T A T I O N S

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

Have students work in pairs. Heterogeneous pairings will improve the accessibility of

the activity for students who struggle with reading, mathematics, vocabulary, and

symbolic reasoning. Some vocabulary review may be useful prior to the activity.

Some students may require more practice modelling expressions using algebra

tiles. Allow the use of algebra tiles throughout the chapter. Demonstrating their work

with the tiles on an overhead projector allows visual learners one more representa-

tion of the question.

E x t e n s i o n

Consider having students explore some of the later questions using The Geometer’s

Sketchpad®, particularly questions 28 and 29. Draw connections between algebra

and geometry wherever possible. Try to expose all students to various learning

modalities: manipulatives (algebra tiles), technology (e.g., The Geometer’s

Sketchpad®, computer algebra systems), and symbolism. All students will benefit

from exposure to multiple, diverse representations.

Te c h n o l o g y Ad a p t at i o n

Algebra tile demonstration videos are useful for students who have difficulty with

modelling. Go to www.mcgrawhill.ca/books/math9NS for a link to a Grade 9 site.

These demonstrations could be used for any math topic, not just algebra tiles.

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

Adapt the information in The Story of The Young Map Colorer. Go to www.mcgrawhill.ca/

books/math9NS for a link to the site that is suitable for students. Use coloured tokens


to represent the colours assigned to the regions instead of using crayons. This will pro-

vide students with a physical example of the concept of a variable. The area on the map

is the variable, and the colour given to the area is the value. Students might not be inter-

ested in this concept yet, but will recognize that using non-permanent colour tokens is

a very good problem-solving strategy

J o u r n a l Ac t i v i t y

Have students write a simple plot for a children’s story where a function directly

arises from the story line. An example is the story of Goldilocks and the Three Bears,

where sequencing of events occurs. You might use this journal prompt:

• Once upon a time….

Related Resource: Billings, Esther M. H. and Charlene E. Beckmann. “Children’s

Literature: A Motivating Context to Explore Functions.” Mathematics Teaching in the

Middle School 10.9. Reston: NCTM, 2005: 470.

Journal

Use this prompt for the journal entry.

• Like terms mean … . Two examples are …

• Unlike terms mean … . Two examples are …

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

C h a p t e r P r o b l e m

21. a) Explanations may vary.

Since you cannot be sure of the width of the park, let the side length be x. Also,

the park is a square, so all the sides are length x. Then let the width of each

parking row be 1, which means that each parking row is x by 1. Assuming that

the size of the picnic area will be determined by the number of parking rows, let

there be 8 parking rows along the bottom and 7 parking rows along the side.

The dimensions of the park would be x � 8 by x � 7.

b) P � 4x � 30

P u z z l e rx x x x

x x x xx

xx

xx

x

xx

xxx

x xx

xx

x x

xxx

x

x x x xx

xx

xxx

111

1 1 1

x

y

z

x

x...

x. . .


7.2 Common Factors

W A R M - U P

Evaluate.

1. 1 � 15 <25> 2. �3.5 � 16 <�56>

3. � � 14 <7 > 4. a � b � 14 <14>

5. 11 � 18 <198> 6. � (�47) � � (�47) <�47>

7. 64 � a�2 b � 62 � a�2 b <�5>

8. 13.7 � (�43) � 2.3 � 44 <17>9. 5.5 � 11 � 2 <121>10. 3 � 52 <75>

Simplify.

11. �27a � 33a <�60a>

12. 4 a2 � 6 a2 <11a2>

13. 1.4a � 9.1a � 3.6a � 2.9a <17a>14. �27a � 52b � 28a � 28b <a � 80b>

15. 3 a2 � 1 a � 3 a2 � a <7a2 � 2a>

Us i n g t h e Co m m u t at i ve Pro p e r t y a n d Co m p at i b l eN u m b e r s

Use the commutative property or look for compatible numbers when collecting like

terms.

1.3a � 6.8b � 2.7a � 1.8b � 1.3a � 2.7a � 6.8b � 1.8b

� 4a � 5b

18 � a b � 12 � a b � 6 � a b� 4


Introduce common factors by building on students’ prior experience in elementary school.

Students identified numeric factors of a composite number as being the length and width of a

rectangle whose area has the value of the composite number.The dimensions (length and width)

of the parking lot mentioned in the section opener are numeric factors of 48.

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Specific CurriculumOutcomesB9 factor algebraic

expressions withcommon monomialfactors, concretely,pictorially, andsymbolically

B10 recognize that thedimensions of arectangular area modelof a polynomial are itsfactors


Link to Get ReadyStudents should havedemonstratedunderstanding of allconcepts, including Factors,in the Get Ready prior tobeginning this section.


The purpose of the activity is to discover two concrete or visual methods to illustrate

the relationship between a polynomial expression and its algebraic factors.

• Sharing method: tiles are split evenly into a number of groups. The number of

groups represents the numeric common factor and the expression that

describes each group is the polynomial factor.

• Area model: tiles are arranged into a rectangular array whose area is equal to the

polynomial expression. Expressions that describe the length and width of the

resulting rectangle represent the factors of the polynomial expression.

Parallels between the factoring of numbers and expressions should be drawn out

through this activity. For example, some numbers and expressions can be factored in

more than one way and some cannot be factored at all.

Refer to the Making Connections feature on page 332. After working through

questions 1 and 2, talk about the definition of factors that is presented in the mar-

gin notes again. How are the factors of a number the same as the factors of an alge-

braic expression? (In both cases, the factors multiply to give the original number or

expression.) How are they different? (The factors of a number are just numbers; the

factors of an algebraic expression will include terms and polynomials.)

Refer to the Communicating Mathematically feature on page 333. Ask students

why is it important to use a centred dot (for example, 3 • 2 compared to 3.2).

As you work through the section, be sure that the meaning of the two factors

in relation to the sharing model and the area model is not lost. In the expression

3(x � 5), it is important that students understand the 3 describes the number of

groups of x � 5. This may seem obvious but students who have challenges in math-

ematics will often misinterpret this expression. Frequently assess students’ under-

standing by asking them to represent concretely, or in words, factored expressions

from Part A and Part B. Question 5, part c) is a good example: x2 � 4x � x(x � 4)


1. a)

b) There are 3 groups. c) x � 2 d) 3x � 6 � 3 � (x � 2)

2. a)

4x � 10 � 2 � (2x � 5)

x

+++++

x

x

+++++

x

x

+++

x

+++

x

+++


b) 2y � 6 � 2 � (y � 3)

c) It is not possible to factor 3x � 5 because it cannot be split into multiple

identical groups.

d) You cannot use the sharing model to find factors of this expression because

you may not necessarily be able to represent x identical groups.

3. a) b) The length is x � 3 and the width is 5.

c) 5x � 15 � 5 • (x � 3)

4. a)

b) The length of the first rectangle is x � 4 and the width is 6. There are two

other rectangles with dimensions 2x � 8 by 3 and 3x � 12 by 2.

c) 6x � 24 � 6 • (x � 4) 6x � 24 � 3 • (2x � 8) 6x � 24 � 2 • (3x � 12)

5. a) The width is 2 and the length is x � 4.

2x � 8 � 2 • (x � 4)x

+++

x

+++

+ +

x

+++

x

+++

x

+++

x

+++

x

+++

+ + + + +

x

++++

x

++++

x

+++

x

+++

x

+++

x

+++

x

+++

+ + +

+ +

x

++++

x

++++

x

+++

x

+++

x

+++

x

+++

+ +

+ +

x

x xx

++++

+++

+++

+ +

++++

x

+++

x

+++

x

+++

x

+++

x

+++

+++

y

+++

y


b)

The width is 2 and the length is 2x � 3.

4x � 6 � 2 • (2x � 3)

c)

The width is x and the length is x � 4.

x2 � 4x � x • (x � 4)

d)

The length is 2x and the width is x � 3.

2x2 � 6x � 2x • (x � 3)

e) 5x � 6 cannot be expressed as more than one rectangle because the numerical

coefficient on the x-term does not share any common factors with 6.

f)

The length of the first rectangle is 2x � 6 and the width is 2x.

The other rectangle has dimensions x � 3 by 4x.

4x2 � 12x � 2x • (2x � 6)

4x2 � 12x � 4x • (x � 3)

6. Algebraic expressions can be expressed as a product of factors if the terms in

the expression have a common factor. This means that if the numerical coeffi-

xxxx2

xxxx2

xxxx2

xxxx2

xxxx2

xxx

xxx

xxxx2

x2

x2

xxxx2

xxxx2

xxxxx2

x

+++

x

x x

+++


cients have a common factor greater than one, the expression can be expressed

as a product of that factor and a simpler algebraic expression. Also, if there are

terms that have a common literal coefficient, that literal coefficient can be fac-

tored to further simplify the original algebraic expression.

E x a m p l e s

Example 1 illustrates the sharing method and area model to factor a simple binomial

that contains a numeric common factor. Students should see both methods but recog-

nize that the sharing model works best when the common factor is purely numerical.

Note that there is a natural connection between factoring and the distributive

property which students will explore further in the next section: the product of the two

resulting factors will give the original polynomial. Thus, students can check their work by

multiplying the common factor by the polynomial factor. Refer to the Communicating

Mathematically feature on page 334 for an illustration of this relationship.

The difference between an area model and the sharing model can be subtle, as

they sometimes look very similar. If you take the sharing model of Method 1, and push

the equal groups up into a rectangle, the resulting model will look like the area model.

It may be useful to give an example where the sharing model cannot be rearranged into

an area model. If you take the algebra tiles for the expression 2x2 � 4 and try to form

a rectangle, it will not work; but you can put the tiles into equal groups.

Example 2 presents a situation in which the common factor is purely literal.

The area model works best to illustrate this concept. The sharing method can be

thought of, but can be a little confusing since it requires an unknown group of tiles

to be split up into x groups. Abstract thinkers, however, may benefit from consider-

ing the concept in this way.

Example 3 presents polynomials whose common factors are a combination of

numerical and literal coefficients. The challenge and objective is to identify and

extract the greatest common factor. This process can be verified by considering

whether or not the resulting polynomial factor is factorable. If it is not, then the poly-

nomial has been completely factored. If it is, then the greatest common factor has not

been found. The tile arrays shown on page 335 illustrate this method visually. Part b)

provides a systematic means for finding the greatest common factor by writing and

comparing the factors of each term.

To help students understand visually and concretely what it means to fully fac-

tor an expression, take one of the rows in the area model and try to break it up into

smaller equal groups. For example, the solution to part b) at the top of page 335

shows the rearrangement of tiles into three different rectangles. The first solution is

2(3y � 6). The top row, which is 3y � 6, can be broken down into three smaller

groups of y � 2. Similarly, the top row of the solution 3(2y � 4) is 2y � 4 which

breaks into two smaller groups of y � 2. In the third solution, 6(y � 2), the top row

of y � 2 cannot be broken down into smaller groups.

Remind students that the algebra tile model that most closely resembles a

square will represent the fully factored form of the algebraic expression.

Explore the language in the Communicating Mathematically feature on page

334. Help students make connections by asking:

• How many terms are there in the expression 3(x � 4)? (one)

• How many terms are there in the expression 3x � 12? (two)

If you simply count the number of terms, it makes sense that the second


expression is greater (has more terms) and is, therefore, expanded. The first expres-

sion is factored because it is expressed as a product.


These questions prompt students to reflect upon and demonstrate understanding of

the factoring concepts explored in the Discover the Math activity and Examples.

Asking students to provide their own examples serves to formatively assess their

degree of understanding. Consider having some students share their own examples

using the chalkboard or overhead tiles before assigning the Check Your

Understanding questions.


1. a) A polynomial is the product of its factors.

For example, 4x � 6 � 2 • (2x � 3) and 4x2 � 8x � 4x • (x � 2).

b) The product can be interpreted as the area of the rectangle using the area

model. The factors are the width and the length of the rectangle.

2. a) The sharing method of factoring will work because both terms have a com-

mon numeric factor of 2.

b) The sharing method of factoring will not work because without the numerical

coefficient, it is difficult to “share” the common elements of the terms. Instead,

the area model will be a better factoring method.

c) Since this expression cannot be factored, the sharing method of factoring will

not work.

3. a) Answers may vary.

3x2 � 9x � 3(x2 � 3x) � 3x(x � 3); 6x � 12 � 6(x � 2)� 2(3x � 6) � 3(2x � 4)

b) A polynomial has been fully factored when the remaining terms have no com-

mon factors other than 1. The fully factored forms of the above examples are

3x(x � 3) and 6(x � 2).



Question 10 provides an opportunity for students to develop their skills in prepar-

ing study notes. By trading with a classmate, they have the opportunity to learn orga-

nizational methods from each other.

Question 11 provides an opportunity for students to apply a unique and pow-

erful method of mathematical reasoning: the use of a counter-example to prove or

disprove a statement. Use the Communicating Mathematically on page 337 as a dis-

cussion vehicle for illustrating this technique, which can be transferred to other sub-

ject areas beyond mathematics.

Question 12 is useful in providing a link between algebra and problem solving.

Some students find the concept of a variable inherently confusing and abstract. It

should be noted that algebraic expressions reduce to numerical values when a given

value is substituted for the variable (the resultant numerical value will depend on the





1–4, 6, 7, 12e) 5, 8–10, 12a)-c) 11, 12d), 13–16


number chosen for the variable). It should be pointed out that algebraic expressions

are powerful in their symbolic form because they summarize many different situa-

tions simultaneously. Forms of technology (e.g., graphing calculators, The

Geometer’s Sketchpad®) can be useful in illustrating simultaneous situations, or fam-

ilies of solutions, such as those described in this question.


• A common factor is identified, but not divided out. For example,

5x � 15 � 5(5x � 15).

Rx Ask students to use multiplication to verify that the polynomial has been

factored correctly. For example, 5(5x � 15) � 5x � 15.

• A polynomial is incompletely factored. For example,

8y2 � 4y � 2(4y2 � 2y).

Rx Ask students to examine the resultant polynomial factor to see if it can be

factored further. Note that in this example, both an additional numerical

factor and a literal factor can still be divided out.


• Are students using the algebra tiles properly in factoring the polynomials?

• Are students identifying the Greatest Common Factor of each polynomial

easily or do they need prompts?

• Are some students factoring without the algebra tiles?

• Can students tell the difference between the sharing model and the area

model?

A S S E S S M E N T


a)

b) x by 12x � 4; 2x by 6x � 2; 4x by 3x � 1

c) 4x(3x � 1)

d) Factor out common factors until the binomial that was left is not factorable.

There are no other common factors that can be taken out.

x2 x2 x2 x2 x2 x2 x2 x2 x2 x2 x2 x2 x x x x

x2 x2 x2 x2 x2 x2 x x

x2 x2 x2 x2 x2 x2 x x

x2 x2 x2 x

x2 x2 x2 x

x2 x2 x2 x

x2 x2 x2 x


e) 85 000 000 m2




Students lacking a sound fundamental understanding of multiplication may benefit

first from reviewing the relationship between a composite number and its numeric

factors. Students can explore this concept in various ways, by using algebra/integer

tiles, dot array diagrams, scale diagrams, or computer software such as The

Geometer’s Sketchpad®.

Students often struggle with factoring, particularly when working in strictly

symbolic terms. Encourage the use of tiles to assist understanding. Provide addi-

tional practice with worksheets as needed. There are natural links to factoring and

the distributive property that should provide opportunities to revisit and consolidate

understanding. Revisit the concept throughout the chapter wherever possible.

Some students may need to use algebra tiles for some, or all, of the factoring

questions in the Check Your Understanding section. If this is the case, questions such

as question 8, part d), should not be included in students’ assignments.

A class discussion should be used to summarize student work and come to a

conclusion for Check Your Understanding question 11. Some samples of student

work could also be posted in the classroom for reference.

E x t e n s i o n

Question 15 extends the concept of common factoring to binomial common factors,

which students will apply in future courses when they learn advanced techniques

(e.g., grouping, decomposition). Although not necessary at this level, students who

demonstrate a firm understanding of factoring in symbolic terms may benefit from

this exposure.


Write an incorrect statement about the factors of 24. Show that it is incorrect using

a counter-example.

You might use these journal prompts:

• The factors of 24 are …

• By using tiles to model the different groups of 24, I discovered that …

Journal


• The way to identify the greatest common factor of a polynomial is …

For example …

• The way to factor the polynomial using the greatest common factor is to …

For example …


7.3 Multiply a Monomial by a Polynomial

W A R M - U P

Simplify.

1. (39)(34) <313> 2. a ab(24a) <8a2>

3. (�8a4)a a3b <�4a7> 4. (25)(2�3)(2) <23>

5. (10a3)(15a7) <150a10> 6. (�4a2)(�2.5a4) <10a6>7. (a3b)(8a2b) <8a5b2> 8. (4a7)(�5a�4) <�20a3>

9. a a6b(12a2) <10a8> 10. (18a�5)(1.5a6) <27a>

11. (2a)(12a2)(5a2) <120a5> 12. (3.5a)(7a)(2a) <49a3>

13. a2 abb(10ab3) <25a2b4> 14. a a4b(9a)a a2b <9a7>

15. (�5)(7.5a2)(�2a) <75a3>

M u l t i p l i c at i o n

When multiplying terms, use these tips to help you.

Power Rule: When multiplying powers with the same base, add the exponents.

a5 � a2 � a7

Coefficients: When multiplying powers with coefficients, you must multiply the

coefficients.

3a6 � 5a3 � 15a9

Use methods such as the following to help with the multiplication.

Distributive Property: 3 � 12 � 3 � 12 � � 12

� 36 � 6

� 42

Double and Halve: 3.5 � 14 � 7 � 7

� 49

Compatible Factors: Look for numbers that make an easy pair to multiply.

4.5 � 8 � 2 � 9 � 8

� 72


Students should be familiar with perimeter of a rectangle from work in previous

grades. Students can write a simplified expression for the perimeter by collecting like

terms. Recognizing that there are two different ways or expressions for finding

1

2

1

2

3

5

5

3

1

2

5

6

1

2

1

3



Related Resources• BLM 7.3 Assessment

Question• BLM 7.3 Extra Practice

Specific CurriculumOutcomesB10 recognize that the

dimensions of arectangular area modelof a polynomial are itsfactors

B11 find products of twomonomials, a monomialand a polynomial, andtwo binomials,concretely, pictorially,and symbolically


B14 demonstrate anunderstanding of theapplicability ofcommutative, associative,distributive, identity, andinverse properties tooperations involvingalgebraic expressions


Link to Get ReadyStudents should havedemonstrated understandingof all concepts in the GetReady prior to beginning thissection.

perimeter lays a contextual framework for students to realize that some algebraic

expressions can appear in different forms. The distributive property provides a

means for converting from one form to another in a number of useful situations.


The purpose of the activity is to use a familiar situation (perimeter of a rectangle) to

illustrate that an algebraic expression can be simplified by applying the distributive

property. As students work through the investigation, they should become convinced

that both forms of the algebraic expression 2l � 2w and 2(l � w) are equivalent. This

understanding can be consolidated by using the numerical example shown in the

distributive property definition on page 338.

Alternative approaches are to use Geoboards or The Geometer’s Sketchpad® or

both. These approaches may serve to enhance student interest and address alterna-

tive learning styles. Continue to identify for students how algebra, geometry, and

measurement concepts link together.

When investigating the similarities and differences between the expressions

l � w � l � w and 2(l � w), it may be helpful to also include the expression 2l � 2w.

Students are familiar with this latter formula and will more than likely mention it as

a third alternative to the others given in the textbook. If it is brought up in class dis-

cussion, add it to the investigation of questions 2, 3, and 4. This investigation allows

for students to discover that all the formulas are equivalent but in different forms.

In question 2, allow students time to come up with their own responses and

perhaps share with a partner. Use full class discussion to collect and discuss a com-

prehensive list of similarities and differences. Be sure to look for references to the

language that has been developed: number of terms, expanded form, factored form,

like terms, combining, product, sum, order of operations, etc.

In Parts A and B, students derive methods for multiplying a monomial by a

monomial, and multiplying a monomial by a polynomial using the distributive

property. Algebra tiles should be used to provide a concrete and visual means of find-

ing the products. Draw out conceptual links to work done earlier with exponent

laws, for example in Chapter 1. (Refer to the Making Connections on page 341.)

In Part B, question 5, examine the language: why is distributive a good name

for this property? Connect this to students’ understanding of the word distribute.


• Do students understand the connection between the three versions of the

formula for perimeter of a rectangle?

• Can students move easily from one formula to another?


1. 2l � 2w

2. Both expressions for the perimeter are equivalent, but one is fully factored while

the other is fully expanded.


3. a) 2l � 2w � 2(150 m) � 2(50 m)

� 300 m � 100 m

� 400 m

2(l � w) � 2(150 m � 50 m)

� 2(200 m)

� 400 m

b) Answers may vary.

The first rectangle has dimensions of 10 m by 15 m.

2(l � w) � 2(10 � 15)

� 2(25)

� 50

2l � 2w � 2 � 10 � 2 � 15

� 20 � 30

� 50

The second rectangle has dimensions of 123 m by 456 m.

2(l � w) � 2(123 � 456)

� 2(579)

� 1158

2l � 2w � 2 � 123 � 2 � 456

� 246 � 912

� 1158

4. Both expressions are equivalent. When you evaluate 2l � 2w, the result is always

the same as evaluating 2(l � w).

P a r t A

1. a) The length is x and the width is 4. The total area is 4x.

b) The length is 3y and the width is 2. The total area is 6y.

c) The length is 4x and the width is x. The total area is 4x2.

c) The length is 3y and the width is y. The total area is 3y2.

2. Answers may vary.

a)

b) The length is 2x and the width is x. The total area is 2x2.

P a r t B

3. a)

The length is x � 1 and the width is 4. The total area is 4x � 4.

b)

The length is y � 2 and the width is y. The total area is y2 � 2y.

y2 y y

x

+

x

+

x

+

x

+

x2x2


c)

The length is 2x and the width is x � 3. The total area is 2x2 � 6x.

d)

The length is x � 3 and the width is 2. The total area is 2x � 6.

e)

The length is x � 1 and the width is x. The total area is x2 � x.

f)

The length is 2y and the width is y � 4. The total area is 2y2 � 8y.

4. 4(x � 5)

� 4(x) � 4(5)

� 4x � 20

5. Each term of the polynomial is multiplied by the monomial expression. The

monomial term is distributed to each of the terms of the polynomial.

2(x � y � z) � 2x � 2y � 2z


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

Ideas questions. Use this opportunity to assess student readiness for the Check Your

Understanding questions. These questions focus on the need for the distributive

property (i.e., when you want to simplify an expression but cannot add unlike

terms). Preemptively pose some common errors for students to discuss.


1. You cannot add the terms in the brackets because they are unlike terms.

2. a) He did not multiply 3x and x together properly. When two like literal coeffi-

y2 y y y y

y2 y y y y

xx2

x

+++

x

+++

xxxx2

xxxx2


cients are multiplied together, the exponent will change. 3x(x) � 3x2

He did not multiply 3x and 2 together properly. The numeric coefficients

should be multiplied together. 3x(2) � 6x

He subtracted when he should have added.

The correct solution is 3x2 � 6x.

b) Let x � 1. Then his answer is 1, but 3(1)(1 � 2) � 9.

E x a m p l e s

Example 1 illustrates how to apply the distributive property using algebra tiles and

symbolic reasoning. Note that it does not matter whether the monomial precedes or

follows the polynomial as in part b), because multiplication is commutative

(sequence of the operands does not matter). Note also that the exponent laws are

useful when multiplying expressions involving powers of variables as in part c).

When demonstrating the method of algebra tiles, it is useful to reinforce the concept

that the length and width of the rectangle represent the factors of the polynomial,

which was the key concept from the previous section.

Example 2 illustrates how the distributive property can be useful in simplify-

ing more complicated algebraic expressions involving a number of terms. When sub-

tracting a polynomial you can think of adding the opposite polynomial, as shown in

an earlier section, or you can think of distributing �1 to the polynomial.


• Are students able to analyse the expression in the following manner? “For

3(x) or 3(x � 2), the expression in the brackets tells me the tile pieces I need

in a group, and the number outside the bracket tells me how many groups I

need. Three groups of x � 2.”

• Can students extend the above thinking to reach a conclusion? “Therefore I

have three x-tiles and three 2s.”

• Can students extend the distributive property to a variable as the scalar?

• Are students applying the scalar to all terms inside the brackets?



In question 10, students should recognize that it is simpler to substitute the given value

after applying the distributive property in order to evaluate the expression for area.

Throughout latter parts of the exercises, opportunities are provided for stu-

dents to apply operations involving integers and fractions. These are areas in which

many students struggle and would benefit from remediation and practice. Assign

more or fewer of these questions depending on the needs of your students.

Lead students to recognize that they must add area expressions in question 13and subtract them in question 14.

A type of skill like the distributive property outlined in question 18 is useful in





1–6, 9 7, 8, 10–12, 15–17, 19 13, 14, 18, 20–23


everyday life for estimating costs, etc. It is worth learning.


• The monomial is only partially distributed to the polynomial. For example,

2(w � 3) � 2w � 3.

Rx Ask students to use tiles to illustrate that they need two groups of three unit

tiles. Also, you can verify a correct solution by substituting a numeric value

into both expressions and checking for equality.

• Integer distribution is handled improperly. For example,

�2(x � 3) � �2x � 6.

Rx Review and reinforce integer operations, as needed.

• Variables are improperly multiplied. For example, 3x(2x2) � 6x2.

Rx Review and reinforce exponent laws, as needed. Ask what exponent x has.


A common mistake when students use the distributive property to multiply is to

apply the scalar only to the first term inside the brackets. For example:

x(3x � 4) � 3x2 � 4. If this error is occurring, ask students to represent what is hap-

pening concretely.


• Are students expanding the expressions properly: multiplying scalar by each

term inside and accurately applying the integer signs?

• Is students’ work organized? Are they working down the page rather than

across? This manner of simplifying expressions mimics the style used in

solving equations and should be practised here.

• Are students correctly combining like terms?

• Do students recognize when the expression is simplified?

A S S E S S M E N T


a) 10x � 20 b) 6x2 � 20x c) 20x � 40, 24x2 � 80x d) Yes. Each term is twice the orig-

inal. e) No. Each term is 4 times the original.


BLM 7.3 Assessment Question provides scaffolding for question 20.



Partner students who can build the models in dialogue with a student who needs this

kind of help. Students who learn well with technology might benefit from working


with virtual algebra tiles.

Some students may need to continue solving these algebraic expressions using

algebra tiles. For this reason, questions 3 and 5, and others like them should be omit-

ted from students assignments, since using the negative factor is difficult with the

algebra tiles.

BLM 7.3 Assessment Question might be helpful for some students who strug-

gle with question 20. Additional scaffolding can be provided Ask for an expression

for the perimeter in part a) before asking for a simplified expression, and do the same

for the area in part b). Break up part c) to ask for expressions for double the length

and double the width first.

E x t e n s i o n

Question 21 is useful in developing students’ skills in formulating and defending

mathematical arguments.

Question 22 leads into the next section. Assign with discretion, but do not

spend a lot of time taking it up, since it is the focus of the next lesson.

Question 23 is similar, but more challenging than questions 10, 13, and 14.

Consider assigning fewer of the latter questions in lieu of this more challenging ques-

tion for your stronger students.


P u z z l e r

4 13 9

6

1510


7.4 Multiply Two Binomials

W A R M - U P

Evaluate.

1. 66 � 79 � 34 � 79 <7900>2. 7 � 92 <644>

3. � (�48) � � (�48) <�48>

4. �8.7 � 9.3 � (�10.3) � 9.7 <0>5. (�1.2 � 3.8)2 <25>

6. �12 � a � 1 b <�6>

7. 4.5 � (�80) <�360>8. �24 � (�3.5) <84>

Simplify.

9. �69n � 31n <�100n>

10. n � n � n <n>

11. 1.5n2 � 6.2n � 0.5n2 � 5.2n <2n2 � n>

12. n � n < n>

13. (8�2)(8�5)(811) <84>14. (5.5n2)(7n3)(2n) <77n6>

15. a3 n4b(8n3) <26n7>


Pose the scenario in the section opener to introduce the idea of multiplication of two

numerical expressions that appear similar to binomials. Then, with the class, simplify

the expression (2 � 3)(2 � 1) in two ways.

First, by following the order of operations:

(2 � 3)(2 � 1)

� (5)(3)

� 15

Second, by applying the distributive property twice:

(2 � 3)(2 � 1)

� 2(2 � 1) � 3(2 � 1)

� 2(2) � 2(1) � 3(2) � 3(1)

� 4 � 2 � 6 � 3

� 15

This numerical argument should pave the way for the algebraic treatment that follows.

1

4

1

9

1

3

4

9

1

2

1

8

3

8

2

3

1

3

5

9

4

9




Specific CurriculumOutcomesB10 recognize that the

dimensions of arectangular area modelof a polynomial are itsfactors






The purpose of the activity is to learn that you can express some trinomials as a

product of two binomial factors. Algebra tiles are used to illustrate a geometric rep-

resentation in which the expressions for the length and width of the rectangle form

the factors and, when multiplied together, give the polynomial that represents the

area. This is another opportunity to emphasize the inverse relationships between the

processes of factoring, expanding, and simplifying. Avoid applications involving neg-

ative coefficients at this stage.

Use the Math Tip in the margin as a guide to help students build their area

models in a systematic way. Take the time to do many examples of giving the area and

then finding the dimensions as in question 2. One method to help develop the con-

nection between area and dimensions is in recording the partial areas once the rec-

tangle is formed. In the first example x2 � 7x � 12, the partial areas can be recorded

as x2 � 4x � 3x � 12. However, it may be more helpful for students to see those same

terms recorded as:

x2 � 3x

� 4x � 12

x2 � 7x � 12

This focuses attention on the coefficients of the x-terms and their connection to the

constant term.


1. a)

b) The length is x � 4 and the width is x � 3.

c) x2 � 7x � 12 � (x � 4)(x � 3)

d) The small rectangle is 3 by 4. The numerical coefficient on the x-term is 3 � 4

while the constant is (3�4).

2. a)

The length is x � 4 and the width is x � 2.

x2 � 6x � 8 � (x � 4)(x � 2)

xxxxx2

++

++

++

++x

x

xxxxx2

+++

+++

+++

+++x

xx


b)


x2 � 9x � 20 � (x � 5)(x � 4)

c)


x2 � 7x � 6 � (x � 6)(x � 1)

3. a)

b) Answers may vary. x2 � 3x � 2

c)

The area is (x � 2) by (x � 1).

d) x2 � 3x � 2 � (x � 2)(x � 1)

4. a)

b) Answers may vary. x2 � 7x � 10

c)

The area is (x � 2) by (x � 5).

d) x2 � 7x � 10 � (x � 2)(x � 5)

5. Answers may vary. When multiplying two binomials, add the products of the

two first terms, the two outer terms, the two inner terms, and the two last

terms. Collect like terms. This diagram shows the multiplication of (2 � 3) and

(2 � 1), and demonstrates why each of the four products mentioned above

needs to be calculated and then added.

x

xx

xx2

+ +

x x x

+ + ++ + + + +

x + 5

x + 2

x

x

xx2

+ +

x + 2

x + 1

xxxxx2

+ + + +

xx

+ +x

xxxxx2

+++

+++

+++

+++x

xx

++

++

++

++x

x


A shortcut you can use when multiplying two binomials in the form (x � first

number)(x � second number) is to use the sum and product of the two num-

bers in the following way, x2 � (sum of numbers)x � (product of numbers).


Consider using a think-pair-share approach, particularly with question 2. Get stu-

dents to share models with their classmates in small groups first, and then have a

couple of models presented using overhead tiles. This will let students see several

examples and share in a small group before presenting to the larger class as a whole.

This should help build confidence with this abstract topic, prior to assigning the

Check Your Understanding questions.


1. a) The length is x � 3, the width is x � 2, and the area is x2 � 5x � 6.

b) x2 � 5x � 6 � (x � 3)(x � 2)

c) The factors are x � 3 and x � 2.

2. a) Answers may vary.

b) (x � 2)(x � 1) � x2 � 3x � 2

3. Answers may vary. The expression can be found by using either algebra or tile

diagrams. Using the algebraic shortcut:

(x � 3)(x � 4) � x2 � (3 � 4)x � (3)(4)

� x2 � 7x � 12

When using tile diagrams, count the total number of tiles in a rectangle with

the given dimensions.

So, (x � 3)(x � 4) � x2 � 7x � 12.

x

xx

xx2

+ +

x x

+ ++ + + +

x + + + +

x + 2

x + 1

(2 + 1)

(2 + 3)3

2

2 1


E x a m p l e s

The Example illustrates how to expand two binomials using tiles and using symbolic

reasoning. It is recommended that the tile model be restricted to situations in which

all numerical coefficients are positive. Eventually, students should become comfort-

able with the algebraic treatment (i.e., application of the distributive property). Only

once students have developed a comfort level with symbolic reasoning should

expressions with negative coefficients be introduced, at which point the tile model

should be shelved. The concept of negative length (e.g., x � 3) is counter-intuitive

when building tile models, and you can also run into problems when finding bino-

mial factors of polynomials having negative coefficients.

Before reviewing the solution to the Example, students should have time to

practice modelling the product of a binomial and a binomial using algebra tiles.

Students could work in pairs to model the multiplication expressions. Each group

should compare their models with other groups.

Refer to the margin notes for Method 2. Ask students to make the connections

between the diagrams and each step of the written solutions.



Question 8 involves the expansion of perfect square trinomials. Consider introducing

the term and the alternative way of writing the multiplication in power form, (x � 2)2.

Question 10 provides an opportunity to apply the concept of binomial prod-

ucts to the context of the chapter problem. Strongly encourage the use of tiles; how-

ever, some students may benefit from using technology such as The Geometer’s

Sketchpad® or drawing the tile representation, especially if the quantity of tiles avail-

able is an issue.

Question 12 illustrates a very important point: not all trinomials can be repre-

sented as products of binomial factors involving integers. It is worth noting that

there are far more polynomials that cannot be factored than those that can.


• Students have trouble building their area rectangle.

Rx Follow the strategy illustrated in the Math Tip on page 346 to systematically

build the rectangle.

• Negative terms are improperly distributed. For example,

(x � 2)(x � 3) � x2 � 3x � 2x � 6.

Rx Review integer operations as needed. Emphasize that �2 is being distrib-

uted, not 2.





1, 2, 4–6, 10a), b) 3, 7–9, 10c), d), 11, 13, 15 10e). 12, 14, 16, 17


• Expressions are expanded, but not simplified. For example,

(x � 3)(x � 5) � x2 � 5x � 3x � 15.

Rx Remind students to always express final answers in simplified form. In this

type of situation, ask if there are like terms that can be collected.


• Are students connecting factors of the constant term with the coefficient of

the middle term? For example, in the expression x2 � 7x � 12, 12 has many

factor pairs (2 and 6, 1 and 12, 3 and 4) but only 3 � 4 � 7, so the dimen-

sions should be x � 3 and x � 4.

• Are students able to use algebra tiles to find the dimensions given the area

and to find the area given the dimensions?

• Are students able to work with finding dimensions or finding the area sym-

bolically as well as concretely?

A S S E S S M E N T

Q u e s t i o n 1 3 , p a g e 3 5 1 , An s we r

The girl on the right is correct.




Some students may have trouble envisioning what is being asked for in the chapter

problem revisit. It may be helpful to include a couple of examples on a worksheet.

Students can use these examples to build alternative models to find the best design.

A pre-made sketch in The Geometer’s Sketchpad® may also provide greater

accessibility for students who enjoy working with technology.

E x t e n s i o n

Questions 15 to 17 are good questions to assign to students who are strong in alge-

bra. Students will face questions of this type in future math courses, and it is within

their abilities to figure out how to simplify them. Assign these questions instead of

some of the routine skill-reinforcement questions that appear earlier in the exercise

set, depending on students’ needs.

x xxxxx2

xxxxx

+ + + +

++

++

++

++

++

++

++

++

++

++

+


Note you could ask students what conditions must be met for a, b, and c in

question 17 for the diagram to be true with respect to size. That is, a > b > c.


Have students name the wildlife park after a mathematician and explain their choice.

You might use this journal prompt:

• I chose to name the wildlife part after _______________ because he/she …

Journal


• To apply the distributive property to (x � 6)(x � 4), first you …


C h a p t e r P r o b l e m

10. a), b), c) Answers may vary.

x2 � 30x � 200

x2 � 30x � 225

x x x x x x x x x x x x x x xx2

xxxxxxxxxxxxxxx

+ + + + + + + + + ++ + + + + + + + + ++ + + + + + + + + ++ + + + + + + + + ++ + + + + + + + + ++ + + + + + + + + ++ + + + + + + + + ++ + + + + + + + + ++ + + + + + + + + ++ + + + + + + + + ++ + + + + + + + + ++ + + + + + + + + ++ + + + + + + + + ++ + + + + + + + + ++ + + + + + + + + +

+ + + + ++ + + + ++ + + + ++ + + + ++ + + + ++ + + + ++ + + + ++ + + + ++ + + + ++ + + + ++ + + + ++ + + + ++ + + + ++ + + + ++ + + + +

x x x x x x x x x xx2

xxxxxxxxxxx

+ + + + + + + + + ++ + + + + + + + + ++ + + + + + + + + ++ + + + + + + + + ++ + + + + + + + + ++ + + + + + + + + ++ + + + + + + + + ++ + + + + + + + + ++ + + + + + + + + ++ + + + + + + + + ++ + + + + + + + + +


x2 � 30x � 209

d) Answers may vary. (x � 10)(x � 20); (x � 15)(x � 15); (x � 11)(x � 19);

Each factor represents a side length.

e) Answers may vary. The park design layout of (x � 15)(x � 15) maximizes the

picnic area because multiplying two numbers that are in the middle (close to

each other) gives a greater product than multiplying a larger and a smaller

number.

P u z z l e r

E � 2


xxxxxxxxxxx

+ + + + + + + + + ++ + + + + + + + + ++ + + + + + + + + ++ + + + + + + + + ++ + + + + + + + + ++ + + + + + + + + ++ + + + + + + + + ++ + + + + + + + + ++ + + + + + + + + ++ + + + + + + + + ++ + + + + + + + + +

x x x x x x x x x x

+ + + + + + + + + ++ + + + + + + + + ++ + + + + + + + + ++ + + + + + + + + ++ + + + + + + + + ++ + + + + + + + + ++ + + + + + + + + ++ + + + + + + + + ++ + + + + + + + + ++ + + + + + + + + ++ + + + + + + + + +


7.5 Polynomial Division

W A R M - U P

Evaluate.

1. 1 � <5> 2. 2.1 � 0.5 <4.2>

3. � < > 4. 6 � 1 <4>

Simplify.

5. 68 � 63 <65> 6. 25 � 2�3 <28>

7. 3a4 � a3 <6a> 8. 1.6a7 � 0.5a2 <3.2a5>

9. �8.5a6 � (�0.5a3) <17a3> 10. a9 � a2 < a7>

11. 4a2 � 0.2a�1 <20a3> 12. a4 � a�3 <6a7>

13. 1.1a10 � 0.1a6 <11a4> 14. 10a7 � 2.5a5 <4a2>15. 9a7 � 0.5a <18a6>

D i v i s i o n

When dividing terms, here are a few tips you should keep in mind.

Power Rule: When dividing powers with the same base, keep the base the same and

subtract the exponents.

a7 � a3 � a4

Coefficients: When dividing powers with coefficients, you still must divide the coef-

ficients. It is a division problem!

12a5 � 3a3 � 4a2

Try strategies such as the following to help with division.

Think “How many?”: 3 � � 10 How many thirds are there in 3 ?

Divide by Balancing: When you multiply the numbers in a division expression by

the same number, the answer will remain the same. Multiplying both parts by the

same number balances the expression.

6 � 3 � (6 � 2) � (3 � 2)

� 12 � 6

� 2

7 � 0.2 � (7 � 5) � (0.2 � 5)

� 35 � 1

� 35

1

3

1

3

1

3

1

6

5

2

1

5

1

2

1

2

1

2

1

2

1

2

1

4

1

5





Specific CurriculumOutcomesB9 factor algebraic

expressions withcommon monomialfactors, concretely,pictorially, andsymbolically

B12 find quotients ofpolynomials withmonomial divisors




4.2 � 0.5 � (4.2 � 2) � (0.5 � 2)

� 8.4 � 1

� 8.4

� � a � 3b � a � 3b� � 1

�


Pose the problem presented in the section opener. Students should have some

understanding of the area of a rectangle and its relationship to the length and width,

from previous work in this chapter and in previous grades. Some will recognize that

you can divide area by one dimension to find the other dimension, while others will

recognize this if you provide some numbers.


From their work in section 7.2, students should understand that a polynomial can be

expressed as a product of factors. Students should also understand that the distribu-

tive property involves multiplying two such factors together to produce a polynomial.

Students should arrive at an understanding, through this activity, that polynomial

division is simply an inverse process to applying the distributive property.

The sharing model in Part A is used to illustrate how a polynomial can be

divided into n groups, where n is a natural number. Algebra tiles are useful as coun-

ters that can be reorganized to form the groups. The sharing model is most useful

when the divisor is purely numerical, and positive.

The area model in Part B is useful for reinforcing the relationship between the

polynomial expression that represents the area (dividend) and the factors that repre-

sent the length and width (divisor and quotient). The area model is useful when the

divisor consists of either positive numerical or literal coefficients, or both.


1. a)

b) 2x c)

d) When 6x is divided by 3, the result is 2x. This means that 3 � 2x � 6x.

2. a)

x x x x x x x x x x x xxx x

6x

3� 2x

x x x x x x

3

2

3

2

1

3

1

2

1

3

1

2


There are three x-tiles in each group.

When 15x is divided by 5, the result is 3x. This means that 5 � 3x � 15x.

b)

There are two x2 tiles in each group.

When 4x2 is divided by 2, the result is 2x2. This means that 2 � 2x2 � 4x2.

c)

There are one y and two unit tiles in each group.

When 3y � 6 is divided by 3, the result is y � 2.

This means that 3 � (y � 2) � 3y � 6.

3. a)

b) The other dimension is 2x. c)

d) When 8x is divided by 4, the result is 2x. This means that 4 � 2x � 8x.

4. a)

The other dimension is 2x.

When 10x is divided by 5, the result is 2x. This means that 5 � 2x � 10x.

b)

+ ++ ++ +

yy

yy

10x

5� 2x

x x x x

x x x x

x

x

8x

4� 2x

x x x x

x x x x

3y � 6

3� y � 2

++

y ++

y ++

y

4x2

2� 2x2

x2

x2

x2

x2

15x

5� 3x


The other dimension is 2y � 3.

When 4y � 6 is divided by 2, the result is 2y � 3.

This means that 2 � (2y � 3) � 4y � 6.

c)

The other dimension is x � 2.

When 2x2 � 4x is divided by 2x, the result is x � 2.

This means that 2x � (x � 2) � 2x2 � 4x.

5. Answers may vary. When using the sharing model, you can divide your tiles into

a number of equal groups, and then count the tiles in one grouping.

To divide 4y � 2 by 2, divide the tiles into two groups.

So, � 2y � 1.

When using the area model, try to arrange the tiles into a rectangle. It is not

always possible, but when successful, the length of each side of the rectangle

represents each of the two factors.

The length of this rectangle is 2y � 1 and the width is 2.

++

yy

yy

4y � 2

2

+ +yy yy

2x2 � 4x

2x� x � 2

x xx2

x xx2

4y � 6

2� 2y � 3



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

Ideas questions. Use this opportunity to assess student readiness for the Check Your

Understanding questions.

Relate the statement at the bottom of page 353 to the statement made earlier:

Multiplication is the inverse of division. Ask: How can you use multiplication to

check your answer? a(x � y) � ax � ay


1. When you find the quotient, which is 2x � 3, you can check your answer by

multiplying this quotient and the original divisor. 5 � (2x � 3) � 10x � 15

2. The length of this rectangle is 3x � 1 and the width is 3. The total area is

9x � 3. Since the area of a rectangle is the product of its length and width, then

3 � (3x � 1) � 9x � 3. So, if a polynomial has two factors, and it is divided by

one of them, the quotient will be the other factor. As an example, if the area is

divided by the width, the quotient is the length, (9x � 3) � 3 � 3x � 1.

3. a) The sharing model would not be effective if the divisor of an expression either

includes a literal coefficient or is not a whole number. An example in which the

sharing model would not be effective is 10x2 � 2x.

b) The area model would not be effective if the divisor is not a whole number.

An example in which it would not be effective is 7x � 2.

E x a m p l e s

Part a) in the Example shows how polynomial division can be done, using three

approaches: the sharing model (with tiles), the area model (with tiles), and symbolic

reasoning. Note that the latter method serves to reinforce the inverse nature of the

mathematical processes of polynomial division and the distributive property. The

divisor consists of a positive numerical coefficient to provide strong conceptual

development.

Part b) introduces a slightly more complicated divisor, consisting of both a

positive numerical and a literal coefficient. The area model can still be applied, but

the sharing model becomes awkward (i.e., how can you split a group of items into

“2x” groups?). Students should begin to be more comfortable with symbolic reason-

ing, which is the ultimate goal.

In Part c), a trinomial is divided by a negative monomial. Neither the sharing

nor area models are useful in this situation, as they are counter-intuitive. By now,

most students should be able to follow a purely symbolic treatment.

When one of the terms is the same or the opposite of the divisor, students

sometimes get confused about how to divide. Use the Math Tip on page 355 to pre-

vent this common error. Also see Common Errors.




Questions 8 and 9 illustrate why it is useful to simplify an expression. There are fewer

computational steps required when evaluating simplified expressions than unsimpli-

fied ones.

In question 12, encourage students to simplify expressions using the algebraic

techniques learned in this chapter before substituting.


• Division is incomplete. For example, (6x � 3) � 3 � 2x � 3.

Rx Use substitution into both expressions to demonstrate inequality. Then

remind students of the distributive property in reverse: divide all terms by

the divisor, not just the first term.

• Variables are improperly divided. For example,

(x2 � 2x) � x � x2 � 2.

Rx Review exponent laws, as needed. Also remind students that x is the same as x1.

• Signs of terms are mixed up. For example, � y � 2.

Rx Review operations with integers, as needed. Ask what the equations would

be if the variable was 1.

• Terms resulting in 1 or �1 are omitted. For example, � 3x.

Rx Use substitution into both expressions to demonstrate inequality. Refer back

to the Math Tip on page 355 or use a similar example to illustrate why there

must be a �1 to represent the second term.


In this section, a reminder of the law of exponents pertaining to division may be of

benefit.

, x � 0


• Are students able to divide both concretely, using algebra tiles, and symboli-

cally?

• Are students dividing all terms by the divisor?

• Are students keeping in mind both the laws of exponents and the integer

rules of division?

x5

x3� x5�3 � x2

6x2 � 2x

2x

3y � 6

�3





1–6 7–12 13–15


A S S E S S M E N T


a) For the garden, the missing dimension is x � 3. For the entire area, the missing

dimension is x � 2. b) 3x2 � 5x c) The garden length triples (just like the area) and

the missing dimension of the entire area also become 5 times larger.





Pair or group students heterogeneously across learning styles (i.e., pair a strong sym-

bolic reasoner with a kinesthetic learner), if possible.

Some students who use algebra tiles to solve all the division problems may not

be expected to complete question 7, part f) or similar ones where a negative divisor

is used or an exponent greater than 2 is used.

In question 13, you may wish to break down the problem by splitting the dia-

gram into two separate rectangles. This may help students to visualize the problem

more clearly. For part c), encourage students who are stuck to draw diagrams and

label them with the known information.

E x t e n s i o n

Question 14 is a good challenge for students who are doing well with algebra in sym-

bolic terms and who have demonstrated some facility in mechanical manipulation

involving fractions within contexts. Prompt students to think of the formula for the

area of a triangle and how that can help in this situation.

Question 15 should be assigned to students who have a strong three-dimen-

sional visual sense. There is an opportunity to draw links between volume, area, and

length, and to make connections between algebra, exponents, measurement, and

three-dimensional geometry.

Journal


• The advantages of using polynomial division to simplify an algebraic

expression are … For example …


7.6 Apply Algebraic Modelling

W A R M - U P

Evaluate.

1. �4 � (�63) <252>

2. 2 � (�60) <�150>

3. 4 � <14>

4. 4.4 � 0.5 <8.8>

Simplify.

5. (53)(55) <58>

6. a3 a2b(10a) <35a3>

7. (1.5a5)(22a5) <33a10>

8. a a6b(�8a4) <�7a10>

9. (3.5a)(9a2)(2a) <63a4>

10. a a�4b(6a5) <3a>

11. 45 � 4�1 <46>

12. a3 � a <4a2>

13. 3.2a9 � 0.5a4 <6.4a5>

14. 5 a5 � a�1 <11a6>

15. a7 � a3 < a4>5

4

1

5

1

4

1

2

1

2

1

4

1

2

7

8

1

2

1

3

2

3

1

2




Specific CurriculumOutcomesB8 add and subtract

polynomial expressionssymbolically to solveproblems

B9 factor algebraicexpressions withcommon monomialfactors, concretely,pictorially, andsymbolically

B10 recognize that thedimensions of arectangular area modelof a polynomial are itsfactors


B12 find quotients ofpolynomials withmonomial divisors





If you have a flamboyant flair, you may wish to dress up as a magician and lead the

whole class through the Discover the Math activity. Alternatively, you could select

and train a student to become the magician. The mystery number is found by sim-

plifying an algebraic expression (although students will have no way to suspect this

until the trick is actually performed).


The purpose of the activity is to show how algebra can be used to model and sim-

plify a real situation. When the time comes to reveal the secret, and students have the

opportunity to reverse-engineer the magic trick, encourage them to write down the

steps in algebraic terms. (They should start by representing the number with a vari-

able, such as n.) When students have the entire expression written out, they will dis-

cover how it can be simplified by applying the distributive property, collecting like

terms, etc.

Tricks like the one in question 2 are common and students may have some of

their own to share. As a class, spend time analysing these new tricks. Other sources

of this type of trick can be found on the Internet or in puzzle books.


6. a) No, the magician used algebra to predict how the final result was related to

the original number.

b) This trick will work for any number because the series of algebraic operations

that was performed can be applied to any number, regardless of sign or size.

c) You can select several operations and apply them to any initial number n.

After simplifying the expression, you will be able to see how to work backwards

from this final number and discover the original number.

E x a m p l e s

The Example illustrates another context in which algebraic modelling can be applied

to solve a problem. There are connections to be made with partial variation.

Application of algebra tiles is extended: unit tiles represent fixed costs in increments

of $100 and x-tiles represent variable costs, at $50/h. Note that the area of the alge-

bra tiles does not represent a dollar value.

It is always critical to state clearly what your choice of variable represents. Only

then can a reader put meaning to any expressions that are used. Encourage students

to examine expressions and talk about what they mean by the variable that is used as

they work through examples and questions.


Question 1 asks students to translate word statements into algebraic expressions.

This is a very important skill when using algebraic modelling to solve problems.

Provide extra practice, as needed.

It might be helpful in question 1 to suggest word expressions that correspond

to the other algebraic expressions, since the differences can be very subtle. In partic-


ular, the model 2(x � 4) corresponds to “four less than a number, doubled.”

Alternatives include “a number minus four, times two” or “double the result of a

number decreased by four.”

Question 2 illustrates partial commission, a common method of payment for

certain types of employees, where part of the income depends on performance. Have

the class brainstorm the type of workers who would be paid in this way (e.g., sales-

persons, professional athletes, etc.).


1. a) 2x b) x � 5 c) 3x � 2 d) x2 � 2

2. a) Let t be the total number of hours Sanjay works. Sanjay’s pay will be

100 � 23t. b) Let s be the total sales. Wendy’s pay will be 200 � 0.15s.

3. a) Add six to a number. b) Double a number and then subtract 1.

c) Multiply a number by 15 and then add 50.



The previous sections in this chapter provided many level 3 questions. This section

may be optional. For additional practice in writing algebraic models for problems,

assign questions 5 to 13. These questions will reinforce the skills students learned in

chapters 2 and 3.

In question 11, suggest that students split the diagram into two parts to help

visualize the problem.

Have some students explore question 12, using The Geometer’s Sketchpad®.

For enrichment activities, assign the Extend questions.


• Are students able to translate between contextual and symbolic representa-

tions? Can students take the applied example and translate the relationship

into an algebraic expression?

• Can students use a variable correctly for the context given and identify its

meaning in the problem?

• Are students able to correctly substitute values into their expression and

solve for fixed values?

A S S E S S M E N T


a) Let a be the total number of adults and s be the total number of students. The total

cost, in dollars, for the adults is 17a. The total cost, in dollars, for the students is 14s.





1, 2 3, 11 4–10, 12–15


b) 17a � 14s c) $331





If students are struggling with how the magic trick works, you can provide them with

all, or part, of the algebraic expression . Have them discuss and

explain where the various parts come from. Then, have them simplify the expression

and consider how the result can be used in conjunction with the last secret steps

from the magician in order to determine the mystery number.

For students who struggle, provide the expressions for question 13, part a) and

ask students to explain what each part represents.

E x t e n s i o n

Assign question 14 to students who would benefit from a challenging problem.

Other students can attempt this question, with some scaffolding (e.g., they may need

help interpreting what means and how to use it).

Question 15 leads to optimization, a concept that is explored more fully when

students learn calculus. The treatment at this level should be informal. Students may

approach this question using systematic trial, for example. The Geometer’s

Sketchpad® could be a very useful tool in solving this question. Encourage some stu-

dents to explore this approach.

Te c h n o l o g y Ad a p t at i o n s

Modelling of specific problems may be accomplished with the use of The Geometer’s

Sketchpad®. Prepare several questions similar to question 15 to maximize the use of

the computer during a class period.


Choose a question in this chapter that you found more difficult than the others. Tell

what you did to overcome that difficulty.

You might use this prompt:

• I had difficulty with question … I found that I did not … I was able to

understand and solve the question by …

w

3

3(2n � 50) � 100

2


Chapter 7 Review

W A R M - U P

Simplify.

1. 2.4a � 11.6a <14a> 2. 5 a2 � 3 a2 <9a2>

3. 4.7ab � 18.6ab � 3.7ab <19.6ab>4. 14a � 28a � 15a <a>

5. �3 a � 17b � 4 a � 18b <a � b>

6. (72)(7)(75) <78> 7. (1.5a)(10a) <15a2>

8. a a4b(12a5) <4a9> 9. (�2ab2)(�11a4b�1) <22a5b>

10. (4.5a2)(11a3)(2a2) <99a7> 11. (35) � (3�2) <37>

12. (2a4) � a a2b <8a2> 13. (1.4a9) � (0.5a) <2.8a8>

14.a3 a6b � a a2b <10a4> 15. (2a2) � (0.5a�2) <4a4>


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

Students might work independently to complete the Chapter Review, and then com-

pare solutions in pairs. Alternatively, the Chapter Review could be assigned for rein-

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

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

ask about strategies and problems they found difficult.

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

questions that caused them difficulty, and include the related sections. They can use

this list to focus their studying for a final test on the chapter’s content.

A S S E S S M E N T

This is an opportunity for 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.

1

3

1

3

1

4

1

3

1

2

1

2

5

7

2

7



Related Resources• BLM 7R Extra Practice


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 identify and collect like terms, using concrete materials, dia-

grams, and symbols?

• Can students recognize that the dimensions of a rectangular area model of a

polynomial are its factors?

• Can students fully factor a polynomial?

• Can students use concrete materials, diagrams, and symbols to multiply a

monomial by a monomial? a monomial by a polynomial? a binomial by a

binomial?

• Can students apply the distributive property to expand and simplify alge-

braic expressions?

• Can students divide a polynomial by a monomial, using concrete materials,

diagrams, and symbols?

• Can students construct an algebraic model to describe a real situation?

• Can students apply algebraic modeling to solve problems?

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 7 Test as a summative assessment.

Question Refer to Section

1, 9, 15 7.1

5, 12 7.2

2, 10, 11 7.3

3, 6, 7, 8, 11 7.4

4, 9, 14 7.5

13 7.6



Related Resources• BLM 7PT Chapter 7 Test




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

recording.

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

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

card files, or notes.


Chapter 7 Chapter Problem Wrap-Up

1. Introduce the problem.

2. Remind students that they have worked on the chapter problem during chapter

problem revisits throughout the chapter and that these will help them. Students

can also be directed to section 7.1 question 21 and section 7.4 question 10.

3. Clarify the assessment criteria by reviewing BLM 7CP Chapter Problem Wrap-

Up Rubric with students.

4. Brainstorm with students on how to approach the problem.

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

Students should prepare separate reports.

6. Consider sharing with all students the example from this Teacher’s Resource

after they have completed their work on the problem. Keep copies of your own

students’ work to show in future years.

O v e r v i e w o f t h e P r o b l e m

Students have designed a wildlife park in the mini chapter problems. Now they need to

modify their design and find the new values for the area and dimensions of the park.

Provide access to The Geometer’s Sketchpad® for students to use for this prob-

lem, if possible.

A S S E S S M E N T

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

H i g h S co r i n g S a m p l e R e s p o n s e

Refer to the Chapter Problem Wrap Up Answer for a level 4 response.

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 clearly summarizes information from chapter problem revisits.

• Student draws an accurate tile model that represents the new wildlife park,

and labels each part of the park. A scale may or may not be included.

• Student explains how the expressions representing the value of each dimen-

sion are derived.

• Student successfully determines the algebraic expression for each dimension

and the new algebraic expression for the total area.

• Student accurately evaluates the new area when x � 100 and checks their

work.

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

At this level, look for the following:

• Student may not successfully represent the area of the park algebraically.

• Student may not make an accurate drawing of the remodeled park.


• Student may successfully evaluate the area of the park when x � 100 m but

the algebraic model is incorrect.

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

tions using some representation; just cannot finish.

C h a p te r Pro b l e m Wra p - Up, p a g e 3 6 5 , An s we r s

1. Total area � Product of new dimensions

� (x � 14)(x � 12)

� x2 � 26x � 168

2. Total area � Product of new dimensions Check:

� (x � 14)(x � 12) Total area � x2 � 26x � 168

� (100 � 14)(100 � 12) � (100)2 � 26(100) � 168

� 114(112) � 10 000 � 2600 � 168

� 12 768 � 12 768

The area of the park is 12 768 m2.

3. Total area � x2 � 26x � 168

2208 � x2 � 26x � 168


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x x

+ ++ ++ ++ ++ ++ ++ ++ ++ ++ ++ +

x + + + + + + + + + + + +

x x

+ ++ ++ ++ ++ ++ ++ ++ ++ ++ ++ ++ +

x + 12

x + 14

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