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
• BLM 7.2 Extra Practice Formative Assessment:• BLM 7.2 AssessmentQuestion, #13
• algebra tilesOptional:• trays
7.3 Solve LinearEquations• 240 min
• BLM 7.3 Extra Practice Formative Assessment:• BLM 7.3 AssessmentQuestion, #19
• 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
260 MHR • Mathematics 8 : Focus on Understanding Teacher ’s Resource
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.
Chapter 7 • MHR 261
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.
262 MHR • Mathematics 8 : Focus on Understanding Teacher ’s Resource
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 .
T E A C H I N G S U G G E S T I O N S
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
5
6
5
6
5
6
3
8
7
8
1
2
2
5
4
5
3
5
1
9
8
9
2
9
Materials• algebra tiles
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
Suggested Timing180 min
Link to Get ReadyStudents should havedemonstratedunderstanding of RepresentExpressions Using AlgebraTiles in the Get Ready priorto beginning this section.
Chapter 7 • MHR 263
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
264 MHR • Mathematics 8 : Focus on Understanding Teacher ’s Resource
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.
Co m m o n E r ro r s
• 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
Chapter 7 • MHR 265
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
266 MHR • Mathematics 8 : Focus on Understanding Teacher ’s Resource
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
term is not squared.
Chapter 7 • MHR 267
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>
T E A C H I N G S U G G E S T I O N S
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.
D i s cove r t h e M at h
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
Related Resources• BLM 7.2 Assessment
Question• BLM 7.2 Extra Practice
Specific CurriculumOutcomesB16demonstrate an
understanding ofmultiplication of apolynomial by a scalar,concretely, pictorially,and symbolically
Suggested Timing90 min
Link to Get ReadyStudents should havedemonstratedunderstanding of RepresentExpressions Using AlgebraTiles in the Get Ready priorto beginning this section.
268 MHR • Mathematics 8 : Focus on Understanding Teacher ’s Resource
D i s cove r t h e M at h An s we r s
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.
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 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.
O n g o i n g A s s e s s m e nt
• 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?
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
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
Level 1 Knowledge andUnderstanding
Level 2 Comprehension of
Concepts and Procedures
Level 3 Application and Problem Solving
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
Chapter 7 • MHR 269
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.
Co m m o n E r ro r s
• 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
Q u e s t i o n 1 3 , p a g e 3 0 9 , An s we r s
a) 12x + 6; 30x + 20
b) 30 m2; 80 m2
A D A P T A T I O N S
BLM 7.2 Assessment Question provides scaffolding for question 13.
BLM 7.2 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
• 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.
270 MHR • Mathematics 8 : Focus on Understanding Teacher ’s Resource
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 …
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.
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.
Chapter 7 • MHR 271
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 .
T E A C H I N G S U G G E S T I O N S
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.
D i s cove r t h e M at h
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.
1
2
2
4
1
4
1
4
1
4
1
4
3
4
1
4
5
6
5
6
2
3
1
2
4
8
7
8
3
8
2
3
2
3
1
3
Materials• algebra tiles
Related Resources• BLM 7.3 Assessment
Question• BLM 7.3 Extra Practice
Specific CurriculumOutcomesC6 solve and verify simple
linear equationsalgebraically
C7 create and solveproblems, using linearequations
Suggested Timing240 min
Link to Get ReadyStudents should havedemonstratedunderstanding of Order ofOperations, Solve Equationsby Inspection, and SolveEquations Using a Model inthe Get Ready prior tobeginning this section.
272 MHR • Mathematics 8 : Focus on Understanding Teacher ’s Resource
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.
D i s cove r t h e M at h An s we r s
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.
1
2
Chapter 7 • MHR 273
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. 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.
O n g o i n g A s s e s s m e nt
• 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?
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
Have algebra tiles available. Students may benefit from drawing a diagram for each
question to help them conceptualize the problem before writing an equation.
Co m m o n E r ro r s
• 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
Level 1 Knowledge andUnderstanding
Level 2 Comprehension of
Concepts and Procedures
Level 3 Application and Problem Solving
1, 9 2–5, 7, 8 6, 10–22
274 MHR • Mathematics 8 : Focus on Understanding Teacher ’s Resource
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
Q u e s t i o n 1 9 , p a g e 3 2 0 , An s we r s
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
A D A P T A T I O N S
BLM 7.3 Assessment Question provides scaffolding for question 19.
BLM 7.3 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• 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.
1
4˛x
1
4˛x
x
4
x
4
x
4
x
4
Chapter 7 • MHR 275
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. 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)
1
3
2
3
1
2l
b
5
276 MHR • Mathematics 8 : Focus on Understanding Teacher ’s Resource
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>
T E A C H I N G S U G G E S T I O N S
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.
1
3
1
2
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2
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2
1
2
1
4
1
2
3
4
Materials• algebra tiles
Related Resources• BLM 7R Extra Practice
Suggested Timing90 min
Chapter 7 • MHR 277
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?
A D A P T A T I O N S
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
2
3
1
6
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
term is not squared.
–2x2 + 3x 2 –x2-tiles and 3 x-tiles unlike termsVariable in second
term is not squared.
278 MHR • Mathematics 8 : Focus on Understanding Teacher ’s Resource
Chapter 7 Practice Test
T E A C H I N G S U G G E S T I O N S
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.
A D A P T A T I O N S
V i s u a l / Pe rce p t u a l / S p at i a l / M o t o r
• 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
Materials• algebra tiles
Related Resources• BLM 7PT Chapter 7 Test
Suggested Timing90 min
Chapter 7 • MHR 279
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
280 MHR • Mathematics 8 : Focus on Understanding Teacher ’s Resource
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.
Chapter 7 • MHR 281
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
282 MHR • Mathematics 8 : Focus on Understanding Teacher ’s Resource