CHAPTER 7
Polynomials
Curriculum Outcomes
Key Words
algebraic model
numerical coefficient
literal coefficient
like terms
distributive property
Get Ready Words
term
numerical coefficient
literal coefficient
polynomial
monomial
binomial
trinomial
factors
greatest common factor
Major Outcomes
B8
B9
B10
B11
B12
B13
B14
add and subtract polynomial expressions symbolically to solve problems
factor algebraic expressions with common monomial factors, concretely,
pictorially, and symbolically
recognize that the dimensions of a rectangular area model of a polynomial
are its factors
find products of two monomials, a monomial and a polynomial, and two
binomials, concretely, pictorially, and symbolically
find quotients of polynomials with monomial divisors
evaluate polynomial expressions
demonstrate an understanding of the applicability of commutative, associative, distributive, identity, and inverse properties to operations involving
algebraic expressions
Contributing Outcomes
B4
demonstrate an understanding of, and apply the exponent laws for, integral
exponents
Chapter Problem
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 summative 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 • M athematics 9: Fo cus on Understanding Te acher ’s Resource
Planning Chart
Section
Suggested Timing
Chapter Opener
• 10 min (optional)
Get Ready
• 60 min
7.1 Add and Subtract
Polynomials
• 60 min
7.2 Common Factors
• 60 min
Teacher’s Resource
Blackline Masters
Chapter Problem WrapUp
• 60 min
Adaptations
Materials and
Technology Tools
• BLM 7GR Letter to
• BLM 7GR Divisibility
Parents
Rules
• BLM 7GR Extra Practice
• BLM 7.1 Extra Practice Formative Assessment:
question #26
• algebra tiles
• BLM 7.2 Extra Practice Formative Assessment:
question #12
• algebra tiles
7.3 Multiply a Monomial • BLM 7.3 Extra Practice
by a Polynomial
• 60 min
7.4 Multiply Two
• BLM 7.4 Extra Practice
Binomials
• 60 min
7.5 Polynomial Division • BLM 7.5 Extra Practice
• 60 min
7.6 Apply Algebraic
Modelling
• 60 min
Chapter 7 Review
• 60 min
Chapter 7 Practice Test
• 60 min
Assessment Tools
• algebra tiles
Formative Assessment:
• BLM 7.3 Assessment
Question, #20
Formative Assessment:
question #13
• algebra tiles
Formative Assessment:
• BLM 7.5 Assessment
Question, #13
• BLM 7.6 Extra Practice Formative Assessment:
• BLM 7.6 Assessment
Question, #13
• BLM 7R Extra Practice
• algebra tiles
Summative Assessment:
• BLM 7PT Chapter 7
Practice Test
• BLM 7CP Chapter
Problem Wrap-Up
Rubric
• algebra tiles
• algebra tiles
• algebra tiles
Chapter 7 • MHR
245
Get Ready
WA R M - U P
Evaluate. Use the distributive property to help you.
Materials
• algebra tiles
Related Resources
• BLM 7GR Letter to Parents
• BLM 7GR Extra Practice
• BLM 7GR Divisibility Rules
Suggested Timing
60 min
1. 7 45 3 45
<450>
2. 13 22 3 22
<220>
3. 17 (9.5) 16 (9.5)
1
1
4. 4 9 2 9
2
2
<ⴚ9.5>
<63>
Evaluate. Use the commutative property to help you.
5. 7.3 6.5 8.3 1.5
<9>
7
3
3
2
<2>
5
10
5
10
7. 5 14 2
<140>
1
1
8.
86
<4>
3
4
6.
Evaluate.
1
18
<81>
2
10. 1.5 (16)
<24>
9. 4
Find the two numbers that have …
11.
12.
13.
14.
15.
a sum of 9 and a product of 14.
<2, 7>
a sum of 6 and a product of 5.
<1, 5>
a sum of 11 and a product of 24.
<3, 8>
a sum of 1 and a product of 6. <ⴚ2, 3>
a sum of 2 and a product of 24.
<4, ⴚ6>
DIAGNOSTIC ASSESSMENT
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 independently when working with a manipulative for the first time. Ask students to identify 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 teaching modality at this level.
246 MHR • M athematics 9: Fo cus on Understanding Te acher ’s Resource
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 principle with the instructions. Keep close track that all students understand this principle 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.
Reinforce the Concepts
Have those students who need more reinforcement of these prerequisite skills complete BLM 7GR Extra Practice.
TEACHING SUGGESTIONS
Suggested Le sson Plan
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 polynomials. 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
1
2x
m2
2
and
. (The numerical coefficients are
and
; the lit
3
7
3
7
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”
are in the terms
Chapter 7 • MHR
247
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
x
(one term as it is given; two terms if you express it as 3).
2
Ongoing Assessment
•
•
•
•
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 polynomials? (See the definitions at the top of page 322.)
Can students concretely represent an expression using tiles?
Common Errors
•
Rx
•
Rx
Students add exponents when collecting like terms. For example,
2x 3x 5x2.
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.
Use question 9 as an example and assign more questions like this to discuss
in small groups and consolidate understanding.
Inter vention
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.
Literac y Connec tions
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.
248 MHR • M athematics 9: Fo cus on Understanding Te acher ’s Resource
7.1
Add and Subtract Polynomials
WA R M - U P
Combine like terms.
Materials
• algebra tiles
1
2
a 1 a <6a>
3
3
6a2 4a2 <10a2>
d) 2.8ab 2.2ab
<5ab>
1
1
8.5a 9a <0.5a>
f) 6 a 3 a
<ⴚ10a>
2
2
17a 19a 13a <49a>
h) 7.7a 3.8a 2.3a
<13.8a>
2
2
2
23ab 94ab 24ab
<95ab> j) 87a 69a 11a
<7a2>
3.1a 5.4b 1.9a 2.6b <5a ⴙ 8b>
1
1
1
1
8 a2 5 a 3 a2 1 a <5a2 ⴙ 7a>
3
2
3
2
73a 95 74a 96 <a ⴚ 1>
1
1
3
a 2b a b
< a ⴙ 3b>
2
4
4
43a 16b 57a 14b
<100a 30b>
a) 13a 87a
Related Resources
• BLM 7.1 Extra Practice
c)
e)
Specific Curriculum
Outcomes
B8 add and subtract
polynomial expressions
symbolically to solve
problems
B13 evaluate polynomial
expressions
Suggested Timing
60 min
g)
i)
k)
l)
m)
n)
Link to Get Ready
Students should have
demonstrated
understanding of all
concepts, except Factors, in
the Get Ready prior to
beginning this section.
o)
<100a>
b) 4
Usi 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.
1
1
2 (7) 1 (7): you have two and one-half (7)s added to one and one2
2
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.
Usi ng t h e Com m u t at i ve Pro p e r t y an d Com p atib le
Numb ers
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
TEACHING SUGGESTIONS
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
Chapter 7 • MHR
249
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 consolidate 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.
Ongoing Assessment
•
•
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 cover 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)
x2
x2
x x x x
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)
x2
b)
x
x x x
x+3
c) x (x 3) x (x 3) 4x 6
250 MHR • M athematics 9: Fo cus on Understanding Te acher ’s Resource
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 representations use one form of grouping or another.
b) It is useful to simplify the expression for perimeter because, once the expression is simplified, less work is required to calculate the perimeter when given a
side length.
Com 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.
Com 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.
+
+
-x -x +
+
+
+
+
+ ; y y y y y
+
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.
Examples
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 students 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 different 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 introduced, 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
Chapter 7 • MHR
251
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 subtracted 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.
Check Your Understanding
Question Planning Chart
Level 1
Knowledge and
Understanding
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
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 students to share different methods when taking up the question.
Ongoing Assessment
•
•
•
•
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?
252 MHR • M athematics 9: Fo cus on Understanding Te acher ’s Resource
Common Errors
• 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.
•
Rx
Students confuse numerical parts of a variable (i.e., exponents) with the
numerical coefficient of a term.
Use question 9 from the Get Ready as an example and assign more questions like this to discuss in small groups to consolidate understanding.
ASSESSMENT
Qu es t i o n 2 6 , p ag e 3 3 1 , An s we r s
a) 8w b) 2400 m c) 200 m by 600 m
A D A P TAT I O N S
BLM 7.1 Extra Practice provides additional reinforcement for those who need it.
Visual/Perce ptual/Spatial/Motor
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 representation of the question.
Ex tension
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 chnology Adaptation
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.
Literac y Connec tion
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
Chapter 7 • MHR
253
to represent the colours assigned to the regions instead of using crayons. This will provide 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 interested in this concept yet, but will recognize that using non-permanent colour tokens is
a very good problem-solving strategy
Journal Ac tivity
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 …
Addi ti ona l St u d e nt Tex tb o o k An s we r s
Chapter Problem
21. a) Explanations may vary.
z
x.
..
y
1
1
1
x
...
x
1 1 1
x
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
Puzzler
x x x
x
x
x
x
x
x
x
x
x
x
x x
x x x x
x
x
x
x x
x
x
x x x x
x x x x
x
x
x x
x
x
254 MHR • M athematics 9: Fo cus on Understanding Te acher ’s Resource
7.2
Common Factors
WA R M - U P
Evaluate.
Materials
• algebra tiles
2
⫻ 15 <25>
3
1
1
1
3.
⫹ ⫻ 14 <7 >
2
2
2
1. 1
Related Resources
• BLM 7.2 Extra Practice
Specific Curriculum
Outcomes
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
Suggested Timing
60 min
Link to Get Ready
Students should have
demonstrated
understanding of all
concepts, including Factors,
in the Get Ready prior to
beginning this section.
5. 11 ⫻ 18
<198>
2. ⫺3.5 ⫻ 16
<ⴚ56>
4. a
1
1
⫹ b ⫻ 14 <14>
2
2
3
5
6.
⫻ (⫺47) ⫹ ⫻ (⫺47)
8
8
<ⴚ47>
7. 64 ⫻ a⫺2
1
1
b ⫺ 62 ⫻ a⫺2 b <ⴚ5>
2
2
8. 13.7 ⫹ (⫺43) ⫹ 2.3 ⫹ 44
<17>
9. 5.5 ⫻ 11 ⫻ 2
<121>
10. 3 ⫻ 52
<75>
Simplify.
11. ⫺27a ⫺ 33a
<ⴚ60a>
5 2
2 2
12. 4 a ⫹ 6 a
<11a2>
7
7
13. 1.4a ⫹ 9.1a ⫹ 3.6a ⫹ 2.9a
<17a>
14. ⫺27a ⫹ 52b ⫹ 28a ⫹ 28b
<a ⴙ 80b>
1
1
3
1
15. 3 a2 ⫺ 1 a ⫹ 3 a2 ⫺ a
<7a2 ⴚ 2a>
4
2
4
2
Usi ng t h e Com m u t at i ve Pro p e r t y an d Com p atib le
Numb ers
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
2
2
2
18 ⫻ a b ⫺ 12 ⫻ a b ⫽ 6 ⫻ a b
3
3
3
⫽4
TEACHING SUGGESTIONS
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.
Chapter 7 • MHR
255
D i s cove r t h e M at h
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 margin notes again. How are the factors of a number the same as the factors of an algebraic 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 mathematics will often misinterpret this expression. Frequently assess students’ understanding 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)
D i s cover t h e M at h An s we r s
1. a)
x
x
x
+ + +
+ + +
+ + +
b) There are 3 groups. c) x 2 d) 3x 6 3 (x 2)
2. a)
+
+
+
x x
+
+
+
+
+
x x
+
+
4x 10 2 (2x 5)
256 MHR • M athematics 9: Fo cus on Understanding Te acher ’s Resource
b)
+ 2y 6 2 (y 3)
y +
+
+
y +
+
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.
b) The length is x 3 and the width is 5.
c) 5x 15 5 • (x 3)
3. a)
x x x x x
+ + + + +
+ + + + +
+ + + + +
4. a)
x x
x x
x x x
x x x
x x x x x x
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
x x
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
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)
x x
+
+
+
+
The width is 2 and the length is x 4.
2x 8 2 • (x 4)
+
+
+
+
Chapter 7 • MHR
257
b)
x x
x x
+ +
+ +
+ +
The width is 2 and the length is 2x 3.
4x 6 2 • (2x 3)
c)
x2
x x x x
The width is x and the length is x 4.
x2 4x x • (x 4)
d)
x2
x x x
x2
x x x
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)
x2
x x x
x2
x x x
x2
x2
x x x x x x
x2
x x x
x2
x2
x x x x x x
x2
x x x
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-
258 MHR • M athematics 9: Fo cus on Understanding Te acher ’s Resource
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 factored to further simplify the original algebraic expression.
Examples
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 recognize 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 considering 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 polynomial 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 factor 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
Chapter 7 • MHR
259
expression is greater (has more terms) and is, therefore, expanded. The first expression is factored because it is expressed as a product.
Com m u n i c ate t h e Key I d e a s
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.
Com m u n i c ate t h e Key I d e a s An s we r s
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 common 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 common factors other than 1. The fully factored forms of the above examples are
3x(x 3) and 6(x 2).
Check Your Understanding
Question Planning Chart
Level 1
Knowledge and
Understanding
Level 2
Comprehension of
Concepts and Procedures
Level 3
Application and
Problem Solving
1–4, 6, 7, 12e)
5, 8–10, 12a)-c)
11, 12d), 13–16
Question 10 provides an opportunity for students to develop their skills in preparing study notes. By trading with a classmate, they have the opportunity to learn organizational methods from each other.
Question 11 provides an opportunity for students to apply a unique and powerful 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 discussion vehicle for illustrating this technique, which can be transferred to other subject 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
260 MHR • M athematics 9: Fo cus on Understanding Te acher ’s Resource
number chosen for the variable). It should be pointed out that algebraic expressions
are powerful in their symbolic form because they summarize many different situations simultaneously. Forms of technology (e.g., graphing calculators, The
Geometer’s Sketchpad®) can be useful in illustrating simultaneous situations, or families of solutions, such as those described in this question.
Common Errors
•
A common factor is identified, but not divided out. For example,
5x 15 5(5x 15).
Ask students to use multiplication to verify that the polynomial has been
factored correctly. For example, 5(5x 15) 5x 15.
Rx
•
A polynomial is incompletely factored. For example,
8y2 4y 2(4y2 2y).
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.
Rx
Ongoing Assessment
•
•
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?
•
•
ASSESSMENT
Qu es t i o n 1 2 , p ag e 3 3 7 , An s we r s
a)
x2
x2
x2
x2
x2
x2
x2
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
x2
x2
x2
x2
x2
x x x x
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.
Chapter 7 • MHR
261
e) 85 000 000 m2
A D A P TAT I O N S
BLM 7.2 Extra Practice provides additional reinforcement for those who need it.
Visual/Perce ptual/Spatial/Motor
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 additional 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.
Ex tension
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.
Journal Ac tivity
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
Use this prompt for the journal entry.
• 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 …
262 MHR • M athematics 9: Fo cus on Understanding Te acher ’s Resource
7.3
Multiply a Monomial by a Polynomial
WA R M - U P
Simplify.
Materials
• algebra tiles
2.
1
a ab(24a)
3
<8a2>
3. (8a4)a
4.
(25)(23)(2)
<23>
5.
7.
6.
8.
(4a2)(2.5a4) <10a6>
(4a7)(5a4) <ⴚ20a3>
1. (39)(34)
Related Resources
• BLM 7.3 Assessment
Question
• BLM 7.3 Extra Practice
Specific Curriculum
Outcomes
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
B13 evaluate polynomial
expressions
B14 demonstrate an
understanding of the
applicability of
commutative, associative,
distributive, identity, and
inverse properties to
operations involving
algebraic expressions
9.
11.
13.
15.
<313>
1 3
a b <ⴚ4a7>
2
(10a3)(15a7) <150a10>
(a3b)(8a2b) <8a5b2>
5
a a6b(12a2) <10a8>
6
(2a)(12a2)(5a2) <120a5>
1
a2 abb(10ab3) <25a2b4>
2
(5)(7.5a2)(2a) <75a3>
10. (18a5)(1.5a6)
<27a>
12. (3.5a)(7a)(2a)
<49a3>
14. a
5 4
3
a b(9a)a a2b
3
5
<9a7>
Multiplic ation
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.
1
1
Distributive Property: 3 12 3 12 12
2
2
Suggested Timing
60 min
36 6
42
Link to Get Ready
Students should have
demonstrated understanding
of all concepts in the Get
Ready prior to beginning this
section.
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
TEACHING SUGGESTIONS
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
Chapter 7 • MHR
263
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.
D i s cove r t h e M at h
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 alternative 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 discussion, 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 comprehensive 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 finding 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.
Ongoing Assessment
•
•
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?
D i s cover t h e M at h An s we r s
1. 2l 2w
2. Both expressions for the perimeter are equivalent, but one is fully factored while
the other is fully expanded.
264 MHR • M athematics 9: Fo cus on Understanding Te acher ’s Resource
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).
Part 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)
x2
x2
b) The length is 2x and the width is x. The total area is 2x2.
Part B
3. a)
x x x x
+ + + +
The length is x 1 and the width is 4. The total area is 4x 4.
b)
y2
y y
The length is y 2 and the width is y. The total area is y2 2y.
Chapter 7 • MHR
265
c)
x2
x x x
x2
x x x
The length is 2x and the width is x 3. The total area is 2x2 6x.
d)
x x
+ +
+ +
+ +
The length is x 3 and the width is 2. The total area is 2x 6.
e)
x2
x
The length is x 1 and the width is x. The total area is x2 x.
f)
y2
y y y y
y2
y y y y
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
Com 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. 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.
Com m u n i c ate t h e Key I d e a s An s we r s
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-
266 MHR • M athematics 9: Fo cus on Understanding Te acher ’s Resource
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.
Examples
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 simplifying more complicated algebraic expressions involving a number of terms. When subtracting 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.
Ongoing Assessment
•
•
•
•
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?
Check Your Understanding
Question Planning Chart
Level 1
Knowledge and
Understanding
Level 2
Comprehension of
Concepts and Procedures
Level 3
Application and
Problem Solving
1–6, 9
7, 8, 10–12, 15–17, 19
13, 14, 18, 20–23
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 students 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 13
and subtract them in question 14.
A type of skill like the distributive property outlined in question 18 is useful in
Chapter 7 • MHR
267
everyday life for estimating costs, etc. It is worth learning.
Common Errors
•
Rx
•
Rx
The monomial is only partially distributed to the polynomial. For example,
2(w 3) 2w 3.
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.
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.
Inter vention
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 happening concretely.
Ongoing Assessment
•
•
•
•
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?
ASSESSMENT
Qu es ti on 2 0 , p ag e 3 4 5 , An s we r s
a) 10x 20 b) 6x2 20x c) 20x 40, 24x2 80x d) Yes. Each term is twice the original. e) No. Each term is 4 times the original.
A D A P TAT I O N S
BLM 7.3 Assessment Question provides scaffolding for question 20.
BLM 7.3 Extra Practice provides additional reinforcement for those who need it.
Visual/Perce ptual/Spatial/Motor
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
268 MHR • M athematics 9: Fo cus on Understanding Te acher ’s Resource
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 omitted 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 struggle 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.
Ex tension
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 question for your stronger students.
Addit i o n al St u d e nt Tex tb o o k An s we r s
Puzzler
13
4
10
9
15
6
Chapter 7 • MHR
269
7.4
Multiply Two Binomials
WA R M - U P
Evaluate.
Materials
• algebra tiles
Related Resources
• BLM 7.4 Extra Practice
Specific Curriculum
Outcomes
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
B13 evaluate polynomial
expressions
1. 66 79 34 79
2. 7 92
<644>
3.
4.
5.
6.
7.
8.
Link to Get Ready
Students should have
demonstrated understanding
of all concepts in the Get
Ready prior to beginning this
section.
5
4
(48) (48)
<ⴚ48>
9
9
8.7 9.3 (10.3) 9.7 <0>
(1.2 3.8)2 <25>
2
1
12 a 1 b <ⴚ6>
3
3
4.5 (80)
<ⴚ360>
24 (3.5)
<84>
Simplify.
9. 69n 31n
10.
11.
12.
Suggested Timing
60 min
<7900>
13.
14.
15.
<ⴚ100n>
3
1
1
n n n
<n>
8
8
2
1.5n2 6.2n 0.5n2 5.2n
4
1
1
n n
< n>
9
3
9
(82)(85)(811) <84>
(5.5n2)(7n3)(2n) <77n6>
1
a3 n4b(8n3) <26n7>
4
<2n2 ⴙ n>
TEACHING SUGGESTIONS
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)
4263
15
This numerical argument should pave the way for the algebraic treatment that follows.
270 MHR • M athematics 9: Fo cus on Understanding Te acher ’s Resource
D i s cove r t h e M at h
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 representation 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 negative 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 connection between area and dimensions is in recording the partial areas once the rectangle 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
2
7x 12
x
This focuses attention on the coefficients of the x-terms and their connection to the
constant term.
D i s cove r t h e M at h An s we r s
1. a)
x
x
x
+ + + +
+ + + +
+ + + +
x2
x x x x
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 (34).
2. a)
x
x
+ + + +
+ + + +
x2
x x x x
The length is x 4 and the width is x 2.
x2 6x 8 (x 4)(x 2)
Chapter 7 • MHR
271
b)
x
x
x
x
x
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
x2
x x x x
The length is x 5 and the width is x 4.
x2 9x 20 (x 5)(x 4)
c)
x
+ + + + + +
x2
x x x x x x
The length is x 6 and the width is x 1.
x2 7x 6 (x 6)(x 1)
3. a)
x+1
x+2
b) Answers may vary. x2 3x 2
c)
+ +
x
x2
x x
The area is (x 2) by (x 1).
d) x2 3x 2 (x 2)(x 1)
4. a)
x+2
x+5
b) Answers may vary. x2 7x 10
c)
x
+ + + + +
x
+ + + + +
x2
x x x x x
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.
272 MHR • M athematics 9: Fo cus on Understanding Te acher ’s Resource
3
(2 + 3)
2
2
1
(2 + 1)
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 numbers in the following way, x2 (sum of numbers)x (product of numbers).
Com m u n i c ate t h e Key I d e a s
Consider using a think-pair-share approach, particularly with question 2. Get students 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.
Com m u n i c ate t h e Key I d e a s An s we r s
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.
x+1
x+2
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.
x
x
x
+ + + +
+ + + +
+ + + +
x2
x x x x
So, (x 3)(x 4) x2 7x 12.
Chapter 7 • MHR
273
Examples
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 comfortable 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 binomial 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.
Check Your Understanding
Question Planning Chart
Level 1
Knowledge and
Understanding
Level 2
Comprehension of
Concepts and Procedures
Level 3
Application and
Problem Solving
1, 2, 4–6, 10a), b)
3, 7–9, 10c), d), 11, 13, 15
10e). 12, 14, 16, 17
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 products to the context of the chapter problem. Strongly encourage the use of tiles; however, some students may benefit from using technology such as The Geometer’s
Sketchpad® or drawing the tile representation, especially if the quantity of tiles available is an issue.
Question 12 illustrates a very important point: not all trinomials can be represented 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.
Common Errors
• Students have trouble building their area rectangle.
Rx Follow the strategy illustrated in the Math Tip on page 346 to systematically
build the rectangle.
•
Rx
Negative terms are improperly distributed. For example,
(x 2)(x 3) x2 3x 2x 6.
Review integer operations as needed. Emphasize that 2 is being distributed, not 2.
274 MHR • M athematics 9: Fo cus on Understanding Te acher ’s Resource
•
Rx
Expressions are expanded, but not simplified. For example,
(x 3)(x 5) x2 5x 3x 15.
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.
Ongoing Assessment
•
•
•
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 dimensions 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 symbolically as well as concretely?
ASSESSMENT
Qu es t i o n 1 3 , p ag e 3 5 1 , An s we r
The girl on the right is correct.
x
x
x
x
x
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
x2
x x x x x
A D A P TAT I O N S
BLM 7.4 Extra Practice provides additional reinforcement for those who need it.
Visual/Perce ptual/Spatial/Motor
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.
Ex tension
Questions 15 to 17 are good questions to assign to students who are strong in algebra. 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.
Chapter 7 • MHR
275
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.
Journal Ac tivity
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
Use this prompt for the journal entry.
• To apply the distributive property to (x 6)(x 4), first you …
Addi ti ona l St u d e nt Tex tb o o k An s we r s
Chapter Problem
10. a), b), c) Answers may vary.
x
x
x
x
x
x
x
x
x
x
x
+
+
+
+
+
+
+
+
+
+
+
x2
x x x x x x x x x x
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
x2 30x 200
x
x
x
x
x
x
x
x
x
x
x
x
x
x
x
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
x2
x x x x x x x x x x x x x x x
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
x2 30x 225
276 MHR • M athematics 9: Fo cus on Understanding Te acher ’s Resource
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
x
x
x
x
x
x
x
x
x
x
x
+
+
+
+
+
+
+
+
+
+
+
x2
x x x x x x x x x x x x x x x x x x x x
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
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.
Puzzler
E2
Chapter 7 • MHR
277
7.5
Polynomial Division
WA R M - U P
Materials
• algebra tiles
Related Resources
• BLM 7.5 Assessment
Question
• BLM 7.5 Extra Practice
Specific Curriculum
Outcomes
B9 factor algebraic
expressions with
common monomial
factors, concretely,
pictorially, and
symbolically
B12 find quotients of
polynomials with
monomial divisors
B13 evaluate polynomial
expressions
Suggested Timing
60 min
Link to Get Ready
Students should have
demonstrated understanding
of all concepts in the Get
Ready prior to beginning this
section.
Evaluate.
1. 1 3.
1
5
<5>
1
1
4
2
1
< >
2
2.
2.1 0.5
4.
61
6.
25 23
8.
1.6a7 0.5a2
1
2
<4.2>
<4>
Simplify.
5. 68 63
7. 3a4 <65>
1 3
a
2
<6a>
9. 8.5a6 (0.5a3)
11. 4a2 0.2a1
<17a3>
<20a3>
13. 1.1a10 0.1a6
<11a4>
7
15. 9a 0.5a
<18a6>
<28>
<3.2a5>
5
1 9
1
a a2 < a7>
2
5
2
1
12. a4 a3
<6a7>
6
14. 10a7 2.5a5
<4a2>
10.
Division
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 coefficients. It is a division problem!
12a5 3a3 4a2
Try strategies such as the following to help with division.
1
1
1
Think “How many?”: 3 10
How many thirds are there in 3 ?
3
3
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
278 MHR • M athematics 9: Fo cus on Understanding Te acher ’s Resource
4.2 0.5 (4.2 2) (0.5 2)
8.4 1
8.4
1
1
1
1
a 3b a 3b
2
3
2
3
3
1
2
3
2
TEACHING SUGGESTIONS
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.
D i s cove r t h e M at h
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 distributive 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 counters 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 represent 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.
D i s cove r t h e M at h An s we r s
1. a)
x x
x x
x x
6x
2x
3
d) When 6x is divided by 3, the result is 2x. This means that 3 2x 6x.
b) 2x c)
2. a)
x x x
x x x
x x x
x x x
x x x
Chapter 7 • MHR
279
15x
3x
5
When 15x is divided by 5, the result is 3x. This means that 5 3x 15x.
There are three x-tiles in each group.
b)
x2
x2
x2
x2
4x2
2x2
2
When 4x2 is divided by 2, the result is 2x2. This means that 2 2x2 4x2.
There are two x2 tiles in each group.
c)
y + y + y +
+
+
+
There are one y and two unit tiles in each group.
3y 6
3
y2
When 3y 6 is divided by 3, the result is y 2.
This means that 3 (y 2) 3y 6.
3. a)
x x x x
x x x x
b) The other dimension is 2x.
c)
8x
2x
4
d) When 8x is divided by 4, the result is 2x. This means that 4 2x 8x.
4. a)
x x x x x
x x x x x
The other dimension is 2x.
10x
2x
5
When 10x is divided by 5, the result is 2x. This means that 5 2x 10x.
b)
y y
y y
+ +
+ +
+ +
280 MHR • M athematics 9: Fo cus on Understanding Te acher ’s Resource
The other dimension is 2y 3.
4y 6
2
2y 3
When 4y 6 is divided by 2, the result is 2y 3.
This means that 2 (2y 3) 4y 6.
c)
x2
x x
x2
x x
The other dimension is x 2.
2x2 4x
x2
2x
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.
y y
y y
+
+
4y 2
2y 1.
2
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.
So,
y y
y y
+ +
The length of this rectangle is 2y 1 and the width is 2.
Chapter 7 • MHR
281
Com 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. 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
Com m u n i c ate t h e Key I d e a s An s we r s
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.
Examples
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 reasoning, 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 prevent this common error. Also see Common Errors.
282 MHR • M athematics 9: Fo cus on Understanding Te acher ’s Resource
Check Your Understanding
Question Planning Chart
Level 1
Knowledge and
Understanding
Level 2
Comprehension of
Concepts and Procedures
Level 3
Application and
Problem Solving
1–6
7–12
13–15
Questions 8 and 9 illustrate why it is useful to simplify an expression. There are fewer
computational steps required when evaluating simplified expressions than unsimplified ones.
In question 12, encourage students to simplify expressions using the algebraic
techniques learned in this chapter before substituting.
Common Errors
• 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.
•
Rx
•
Rx
•
Rx
Variables are improperly divided. For example,
(x2 2x) x x2 2.
Review exponent laws, as needed. Also remind students that x is the same as x1.
3y 6
y 2.
3
Review operations with integers, as needed. Ask what the equations would
be if the variable was 1.
Signs of terms are mixed up. For example,
6x2 2x
3x.
2x
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.
Terms resulting in 1 or 1 are omitted. For example,
Inter vention
In this section, a reminder of the law of exponents pertaining to division may be of
benefit.
x5
x53 x2 , x 0
x3
Ongoing Assessment
•
•
•
Are students able to divide both concretely, using algebra tiles, and symbolically?
Are students dividing all terms by the divisor?
Are students keeping in mind both the laws of exponents and the integer
rules of division?
Chapter 7 • MHR
283
ASSESSMENT
Qu es ti on 1 3 , p ag e 3 5 6 , An s we r s
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.
A D A P TAT I O N S
BLM 7.5 Assessment Question provides scaffolding for question 13.
BLM 7.5 Extra Practice provides additional reinforcement for those who need it.
Visual/Perce ptual/Spatial/Motor
Pair or group students heterogeneously across learning styles (i.e., pair a strong symbolic 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 diagram 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.
Ex tension
Question 14 is a good challenge for students who are doing well with algebra in symbolic 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-dimensional 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
Use this prompt for the journal entry.
• The advantages of using polynomial division to simplify an algebraic
expression are … For example …
284 MHR • M athematics 9: Fo cus on Understanding Te acher ’s Resource
7.6
Apply Algebraic Modelling
WA R M - U P
Related Resources
• BLM 7.6 Assessment
Question
• BLM 7.6 Extra Practice
Specific Curriculum
Outcomes
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
Evaluate.
1. 4 (63)
<252>
1
2. 2 (60)
<ⴚ150>
2
1
2
3. 4 <14>
3
3
4. 4.4 0.5
<8.8>
Simplify.
5. (53)(55)
<58>
6. a3
7.
8.
9.
10.
11.
12.
13.
14.
15.
1 2
a b(10a) <35a3>
2
(1.5a5)(22a5) <33a10>
7
a a6b(8a4) <ⴚ7a10>
8
(3.5a)(9a2)(2a) <63a4>
1
a a4b(6a5) <3a>
2
5
4 41 <46>
1
a3 a <4a2>
4
3.2a9 0.5a4 <6.4a5>
1
1
5 a5 a1 <11a6>
2
2
5
1 7
1
a a3 < a4>
4
5
4
Suggested Timing
60 min
Link to Get Ready
Students should have
demonstrated understanding
of all concepts in the Get
Ready prior to beginning this
section.
Chapter 7 • MHR
285
TEACHING SUGGESTIONS
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 simplifying an algebraic expression (although students will have no way to suspect this
until the trick is actually performed).
D i s cove r t h e M at h
The purpose of the activity is to show how algebra can be used to model and simplify 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 variable, such as n.) When students have the entire expression written out, they will discover 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.
D i s cover t h e M at h An s we r s
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.
Examples
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 algebra 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.
Com m u n i c ate t h e Key I d e a s
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-
286 MHR • M athematics 9: Fo cus on Understanding Te acher ’s Resource
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., salespersons, professional athletes, etc.).
Com m u n i c ate t h e Key I d e a s An s we r s
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.
Check Your Understanding
Question Planning Chart
Level 1
Knowledge and
Understanding
Level 2
Comprehension of
Concepts and Procedures
Level 3
Application and
Problem Solving
1, 2
3, 11
4–10, 12–15
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.
Ongoing Assessment
•
•
•
Are students able to translate between contextual and symbolic representations? 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?
ASSESSMENT
Qu es t i o n 1 3 , p ag e 3 6 1 , An s we r s
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.
Chapter 7 • MHR
287
b) 17a 14s c) $331
A D A P TAT I O N S
BLM 7.6 Assessment Question provides scaffolding for question 13.
BLM 7.6 Extra Practice provides additional reinforcement for those who need it.
Visual/Perce ptual/Spatial/Motor
If students are struggling with how the magic trick works, you can provide them with
3(2n 50) 100
all, or part, of the algebraic expression
. Have them discuss and
2
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.
Ex tension
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
w
help interpreting what
means and how to use it).
3
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 students to explore this approach.
Te chnology Adaptations
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.
Journal Ac tivity
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 …
288 MHR • M athematics 9: Fo cus on Understanding Te acher ’s Resource
Chapter 7 Review
WA R M - U P
Simplify.
Materials
• algebra tiles
1. 2.4a 11.6a
Related Resources
• BLM 7R Extra Practice
Suggested Timing
60 min
<14a>
2.
2
5
5 a2 3 a2
7
7
<9a2>
3. 4.7ab 18.6ab 3.7ab
<19.6ab>
4. 14a 28a 15a
<a>
1
1
a 17b 4 a 18b
2
2
6. (72)(7)(75)
<78>
1
8. a a4b(12a5)
<4a9>
3
10. (4.5a2)(11a3)(2a2)
<99a7>
1
12. (2a4) a a2b
<8a2>
4
1
1
14. a3 a6b a a2b
<10a4>
3
3
5. 3
<a ⴚ b>
<15a2>
7.
(1.5a)(10a)
9.
(2ab2)(11a4b1)
11. (35) (32)
<22a5b>
<37>
13. (1.4a9) (0.5a)
<2.8a8>
15. (2a2) (0.5a2)
<4a4>
TEACHING SUGGESTIONS
Using the Chapter Review
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 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.
ASSESSMENT
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?
A D A P TAT I O N S
Have students use BLM 7R Extra Practice for more practice.
Chapter 7 • MHR
289
Chapter 7 Practice Test
TEACHING SUGGESTIONS
Using the Prac tice Te st
Materials
• algebra tiles
Related Resources
• BLM 7PT Chapter 7 Test
Suggested Timing
60 min
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, diagrams, 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 algebraic 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.
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
ASSESSMENT
After students complete the Practice Test, you may wish to use
BLM 7PT Chapter 7 Test as a summative assessment.
290 MHR • M athematics 9: Fo cus on Understanding Te acher ’s Resource
A D A P TAT I O N S
Visual/Perce ptual/Spatial/Motor
•
Let students give their answers verbally either in an interview setting or a
recording.
L a nguage/Memor y
•
Allow students to refer to their personal math dictionaries, journals, index
card files, or notes.
Chapter 7 • MHR
291
Chapter 7 Chapter Problem Wrap-Up
1. Introduce the problem.
2. Remind students that they have worked on the chapter problem during chapter
3.
4.
5.
6.
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.
Clarify the assessment criteria by reviewing BLM 7CP Chapter Problem WrapUp Rubric with students.
Brainstorm with students on how to approach the problem.
Allow students time to work on the problem, either individually or in a group.
Students should prepare separate reports.
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.
Overview of the Problem
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 problem, if possible.
ASSESSMENT
Use BLM 7CP Chapter Problem Wrap-Up Rubric to assess student achievement.
High Scoring S a mple Response
Refer to the Chapter Problem Wrap Up Answer for a level 4 response.
Criteria for a High Scoring Response
•
•
•
•
•
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 dimension 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.
What Distinguishes Lowe r Scoring Responses
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.
292 MHR • M athematics 9: Fo cus on Understanding Te acher ’s Resource
•
•
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 generalizations using some representation; just cannot finish.
Chapte r Problem Wrap-Up, page 365, Answers
1. Total area Product of new dimensions
(x 14)(x 12)
x2 26x 168
•
+
+
+
+
+
+
+
+
+
+
+
+
x2
x x x x x x x x x x x x x x
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
•
x + 12
x
x
x
x
x
x
x
x
x
x
x
x
x + 14
2. Total area Product of new dimensions Check:
(x 14)(x 12)
(100 14)(100 12)
114(112)
12 768
The area of the park is 12 768 m2.
3. Total area x2 26x 168
2208 x2 26x 168
Total area x2 26x 168
(100)2 26(100) 168
10 000 2600 168
12 768
Chapter 7 • MHR
293