Grade Nine Math

8/12/2019 Grade Nine Math

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Mary Ward C S S

Mathematics Department

Principles of Mathematics

MPM 1D1

Grade 9 Academic Mathematics

Unit 3

ALGEBRA OPERATIONS

Combining Like Terms

The Distributive Law

Adding and Subtracting Polynomials


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MPM 1D unit 3: Algebra

H. Kopach 2010 Mary Ward C S S

2


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3

Unit 3

Algebra Operations

EXPECTATIONS: By the end of the unit, all students will:

Define the vocabulary used in Algebra.

Recognize and group terms that are alike.

Simplify polynomial expressions by adding or subtracting like terms.

Apply distributive law for multiplication over addition and subtraction.

Define the area of a quadrilateral in terms of a polynomial expression.

Add and subtract polynomials with 2 variables

JOURNAL WRITING:

Please refer to handout

COMPUTER WORK:

The Learning Equation: worksheet handed in separately from the unit work.

TEST:

Follows this unit

ASEESSMENT / EVALUATION

ACTIVITY KTCA TIME

A Math Literacy T, C 0.5 hB Like Terms K, A 0.5 h

C Algebra Tiles - 1 T, C 1 h

D Algebra Tiles - 2 T, C 1 hE Combining Like Terms T, C 1 h

F Distributive Property K, A 1 h

Seminar Activity T, C 1 hG Practice Test K, A 1 h

TLE self check K, A 1 h

Unit Test K, A 1 hTotal estimated time (in school and at home) 9 h +


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WHAT IS ALGEBRA ?

Algebra is the branch of mathematics

in which symbols, usually letters of the

alphabet, represent unknown numbers.

Let us compare the language of Algebra to the English language:

English Algebra

Letters Variables

Syllables Terms

Words Polynomials

Phrases Expressions

Sentences Equations

Paragraphs Problem solving

The origin of word al!ge!bra is

mid-16th century via Italian andmedieval Latin: Arabic al-jabr"the

reuniting," in the title of a treatise

b the mathematician al-Khwarizmi


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VAR IABLE A variable is a letter that represents one or morenumbers.

T E R M A term is a number or a variable, or the product orquotient of numbers and variables.

Some examples of terms are: x!, 4y, , 5

x, yarevariables, 4, are coefficients, and 5 is a constant.

P O L Y N O M I A L An algebraic expression formed when terms areadded or subtracted.These are some examples of polynomials:

5x ( monomial one term)2x + 3y (binomial two terms)2x + 4y + 7z ( trinomial three terms)

L I KE TER MS Terms that have the same variables raised toexactly the same exponents.

These are some examples of like terms:

2xand3x, -4xzand7xz, 4x!and-6x!

U N L IKE TER MS Terms that have different variables, or thesame variable but different exponents.

These are some examples of unlike terms:

3band 4a, 11b!and-9b, 5xy and8xz


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* * *

This is a Frayer Diagram. It is made up of 4 boxes, each with a purpose. The concepttying everything together is in the middle oval. It is used for definitions of

mathematical terms. Here is an example of a Frayer diagram to define a Polygon:

Definition (in your own words)

A simple, closed, plane figure made up

of three or more line segments

Characteristics

Curve doesnt intersect itself

Plane figure(2 dimensional)

three or more line segments

No dangling parts

Examples

Rectangle

Triangle

Pentagon

Hexagon

Trapezoid

Non-Examples

Circle

Cone

Arrow(ray)

Cube

Letter A

A. MATH LITERACY

Use a Frayer Diagram to define each of the following terms:

algebra; polynomials; like terms;

Polygon


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variable; numerical coefficient; constant


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* * *

B. LIKE TERMS

Mathematics 9 MHR Mathpower 9 MHR Principles 9

p 265 # 1, 2 p 305 # 1

p 308 # 5, 6, 7

p 151 # 1 - 4

* * *

COMBINING LIKE TERMS

We will use ALGEBRA TILES to represent terms. The following coloured or shaded tiles

represent positive :

1 x x2 y y2 xy

Definition (in your own words) Characteristics

Examples Non-Examples


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The colours of the algebra tiles do not

matter, but the shapes do

The following tiles (absence of colour) represent negative:

-1 - x - x2 -y - y2 -xy

The Zero Principle can be represented by combining a positive and a negative of the

same shape:

* * *

Example 1State the algebraic for the following Algebra Tiles Display :

So lu t i on : The tiles show : 2x2+ y2+ 3x 2

The AW textbook uses green for positive,

red for negative. The MGH textbook uses

colours for positive: red for unit tile, green

for x, orange for y and grey for xy, whilenegative tiles are blank.


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* * *

Example 2. Use tiles to illustrate: -2y2+ x 2y + 4

2 negative y2 - tiles (blank squares)

1 x - tile (shaded rectangle)

2 negative y tiles

(blank rectangles, different size from the y)

4 unit tiles (shaded small

squares)

* * *

Example 3. Combine the following, using both Algebra Tiles, and Algebra Expressions:


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So lu t i on : Regroup and use the zero principle

x

2

+ 2x -3

first group

-2x2+ 2x +4

second group


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Regroup,

according to like

terms :

x2-2x2

+ 2x+x

-3+4

Use the zero

principle

-x2

+ 3 x

+1


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C. ALGEBRA TILES - 1

1. Use tiles to model each algebraic expression (Draw the diagram in your notebook)a) x2 +5 xb) 3x

2- 4x

c) 2x2+ 3x + 4

d) x2

-5x 3e) 3x + 5yf) 5x! + 4xy + 3y!

2. Make up 3 algebraic expressions and illustrate using tiles

3. Use tiles to build an area model that has length and width as indicateda) Length =x,width =x + 2b) Length =x + 4, width =xc) Length =x + 2, width =x + 3

4. Use algebra tiles to model and simplify each expression. Draw the diagram in yournotebook.

a) 2x + 3 + 4x + 1b) 5y + 2 3y -1c) 2c2+ 3c + 4c2 4x

D. ALGEBRA TILES -2

AW Mathematics 9 MHR Mathpower 9 MHR Principles 9

p 265 # 3, 4 p 308 # 1 - 4 p 108 # 1 - 5

* * *

SIMPLIFY POLYNOMIALSAn expression is in simplest form when there are no like terms. For example,the expression 5xis in simplest form, but the expression 8x 3x is not.

To S imp l i f y P o l y nom i a l s Mea n s To Co m b i n e L i k e Te rms .

Example 1Simplify the following.

a) 3x + 5y + 2x + 3y b)x!+ 3xy + 2y!+ 4x!+ xy + y!


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So lu t i on :3x + 5y + 2x + 3y

= 3x + 2x + 5y +3y Group like terms

= 5x + 8x Combine like terms

x!+ 3xy + 2y!+ 4x!+ xy + y!

= x!+ 4x!+ 3xy + xy + 2y!+ y! Group

= 5x! + 4xy + 3y

!

Arrange final answer in alphabetical and

descending order.

* * *


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Example 2 Find the perimeter of the polygon. Evaluate forx = 5and y = 3

So lu t i on :The perimeter is the sum of the 6

sides

Start at the top left, and work your

way around, clockwise. There are 6

addends, each in brackets. One

bracket represents one side.

Since there are only addends, removebrackets

Group like terms, working from left

to right

Answer is in alphabetical order

Replace the variables with brackets.

Inside the bracket put the value.

* * *


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Example 3Determine the value of each expression when x = 1, and when x = -1.

a) 9 + 3x b) 4x 9

So lu t i on :When x = 1 When x = -1

9 + 3x 4x 9 9 + 3x 4x 9= 9 + 3(1) = 4(1) 9 = 9 + 3(-1) = 4(-1) 9= 9 + 3 = 4 9 = 9 - 3 = - 4 9= 12 = - 5 = 6 = - 13

* * *

Example 4 Subtract 7x + yfrom6x 3

So lu t i on6x 3 is the minuend, so goes first

7x + y is the subtrahend, so goes secondChange the subtract operation to add.

Change the subtrahend to its opposite

Remove brackets

Group like terms

Combine


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* * *

E. COMBINING TERMS


p 265 # 6,# 7 a, g, d

# 8 a, b, c, d

# 9 a, b, d# 11 a, b, d, e, e, f

# 12 a, b, d

# 13

p 308 # 8, 12, 14# 28, 29, 30

p 311 # 5, 6, 7, 8# 18, 21

p 151 # 5, 6, 7

p 157 # 2 a, e, e

# 4 a, d, e# 5 g

* * *

MULTIPLICATION OF MONOMIALSExample

Multiply the following.

a) (2x) (4x) b) (2a) (7)

So lu t i on :

(2x)(4x) (2a) (7)

= (2) (4) (x) (x) = (2) (7) (a)

= 8x! = 14a

* * *

When multiplying monomials,

first multiply the numerical coefficients,then the variables.


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MULTIPLICATION USING ALGEBRA TILESUse Algebra Tiles to illustrate 2 (x + y +2)

2 (x + y+2)

means 2 groups of (x + y +2)

Group like terms

Answer :

2x + 2y + 4

* * *THE DISTRIBUTIVE PROPERTY You have been invited to meet 3 very

important people. In our North American

culture, when you meet people, you are

expected to shake hands. Your hand shake

must be distributed to everyone.

You (VIP # 1 + VIP # 2 + VIP # 3)

= You shake hands with VIP # 1+ You shake hands with VIP # 2

+ You shake hands with VIP # 3

You have to go through the brackets to shake hands with everyone.

You are the monomia

The three VIP are the

polynomial


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If you do NOT shake hands with someone, it is considered rude!

The Distributive Property in Algebra works like a handshake in that it must be

d i s t r i bu ted to eve ryone .The monomial on the outside has to got h r o ugh t h e b r a c ke t s .Example 1 Simplify 3(x + 4).So lu t i on :

To "simplify" this, get rid of the brackets. The Distributive Property says

to multiply the 3 onto everything inside the brackets. Draw arrows to

emphasize this:

Multiply the 3 onto thex and onto the 4:

= 3(x) + 3(4)

= 3x + 12

* * *

Example 2 Simplify2(x 4)So lu t i on :

Take the 2 through the brackets. This gives:

2(x 4)

=2(x) 2(4)

=2x + 8

The most common error is to take the 3 through the

brackets to shake hands with x, forgetting to shake

hands with the 4 as well. Draw little arrows to help

you remember to carry the multiplier through onto

everything inside the brackets.

A common mistake with this type of problem is to lose a

"minus" sign somewhere, such as doing

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

Did you notice how the "4" somehow turned into a "4" when

the 2 went through the brackets? That's why the answer ended

up being wrong.

Be careful with the "minus" signs! Until you are confident in

your skills, take the time to write out the distribution, complete


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Example 3 Simplify (x 3)So lu t i on :

Take the "minus" through the brackets. It is helpful to write in the

invisible "1" before the brackets:

1(x 3)

Take a1through the brackets:

(x 3)= 1(x 3)

= 1(x) 1(3)

= 1x + 3

= x + 3

* * *

Example 4Multiply the following using the distributive property

a) 4(3x + 2) b) 3(2y 5)

So lu t i on :4(3x + 2) 3(2y 5)

= 4 (3x + 2) = 3 (2y 5)

= 4(3x) + 4(2) = 3(2y) + 3 (-5)

= 12x + 8 = 6y 15

* * *

Distribute

multiplication over

addition or

subtraction.

Note that "1x + 3" and "x + 3" are

technically the same thing; either

would be a perfectly acceptableanswer.

However"1x + 3" is considered as

not fully simplified


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Example 53x + 2

Calculate the area of the rectangle.

4

3x 2

So lu t i on :4

4(3x +2)

= 4(3x) + 4(2)

= 12x + 8

* * *

To mu l t i p l y a po l y no m i a l b y a mo no m i a l u s e t h e d i s tr i b u t iv ep ro pe r t y t o mu l t i p l y e a ch t e rm o f t h e po l y no m i a l b y t h emo no m i a l .

a b + c) = ab + aca b c) = ab ac

F. THE DISTRIBUTIVE PROPERTY

AW Mathematics 9 MHR Mathpower 9 MHR Principles 9p. 259 #2 , 3, 8, 9

p 260 # 13 19

p 265 #7 e, f; 8 f

# 10 c, e, f

# 11 g, h

# 12 e, f, g, h

p 314 # 1 51 (odd #s only);

# 55- 59;

# 65

p 166 # 1 6

p 167 # 8 a, d; 9 a, f

p 168 # 15 a, b, c

Area Area

= 12x = 8


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* * *

Hand in your unit now. Wait until you have been authorized for the test, then do the following

assignment.

G. PRACTICE TEST


p 298 # 5, 6, 7 p 334 # 1 3;

# 7

# 37 40

# 65, 66

p 336 # 1, 2, 27, 28

p 175 # 14, 15, 16 a,b,c

# 18 a,b

# 19 a, c

# 20 a, b

p 176 # 8, 9

Now do the TLE assignment.

* * *

Extra Practice1. Simplify by using the distributive property

a) 2 (x + 3) b) 3(y 4) c) 6 (7 x) d) 5 (8 + y) e) 3 (n + 2)

f) 2 (n 1) g) -2(x + 3) h) -2 (x 3) i) -4 (-x + 2)

2. Simplify by using the distributive property

a) 3 (x + 5) b) 3(4x + 5) c) 7(3n + 1) d)9 (3b + 4)

e) -5(4x + 2) f) -3 (3x 2) g) 6(3 4a) h)6(3 4a)

i) -2(6x 2) j) -4(x 2) k) -4(-2 8x)


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3. Simplify by using the distributive property

a) -1(5 2x) b) -5(a b) c) -3 (-a + 4c) d) -2(a + 5)


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MPM 1D unit 3: Algebra 24

Answers

1

a)

2x+

6

b)3y12

c)426x

d)

40+

57

e)3n+

6

f)2n2

g)

-2x6

h)-2x+

6

i)4x8

2.

a)

3x+

15

b)12x=1

5

c)21n+

7

d)

27b=3

6

e)-2

0x10

f)

-9x+

6

g)18+

4y

h)1824x

i)

-12x+

4

j)-4x+

8

k)8+

32x

3.

a)

-5+

2x

b)-5a+

5b

c)

3x12c

d)-2a-1

0

e)

20b12c

f)1242a

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Grade Nine Math