CHAPTER 7 Algebraic Expressions and Solving Equations Specific Curriculum Outcomes Major Outcomes 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 Contributing Outcomes 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 Chapter Problem 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 Words variable expression equation polynomial numerical coefficient term Get Ready Words variables constants polynomial terms zero principle simplify 258 MHR • Mathematics 8: Focus on Understanding Teacher’s Resource
26
Embed
CHAPTER 7 Algebraic Expressions and Solving Equations 8_TR/Teachers Resource... · CHAPTER 7 Algebraic Expressions and Solving Equations Specific Curriculum Outcomes Major Outcomes
This document is posted to help you gain knowledge. Please leave a comment to let me know what you think about it! Share it to your friends and learn new things together.
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?
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.
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
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.
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.
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