Gr 9 Complete Review Packet

8/6/2019 Gr 9 Complete Review Packet

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Grade 9 Academic Math Final Exam Outline

1. Order of Operations (BEDMAS) Adding, subtracting, multiplying & dividing integers & fractions

2. Exponent Laws (3.2/3.3)am

! an= a

m+n am

an

= am!n am( )

n

= amn ab( )

m

= ambm a!m =

1

am

a

b

!

"#$

%&

'm

=

b

a

!

"#$

%&

m

a0 = 1

3. Polynomials Determining degree of polynomial (3.4) Collecting like terms (3.5)

i. When collecting like terms, add terms with identicalvariables (3x2 & 3x are NOT like terms)ii. Only coefficient changes when adding like terms (ie: 2x + 7x = 9x, NOT 9x2)

Addition/Subtraction of Polynomials (3.6) Expanding (Distributive Property) (3.7)

4. Solving Equations Simple (4.1) Multi-step (4.2) Equations with fractions (4.3) Rearranging formulas (4.4) Word problems (numbers, coins, age) (4.5 & additional worksheets)

5.

Analytic Geometry Direct/Partial Variation (5.1/5.2)

i. Direct variation (y = mx, goes through origin)ii. Partial Variation (y = mx + b)

Slope m = y2 ! y1x2

! x1

(5.3/5.6)

First Differences (5.5)i. First differences of a relation are constant if the relation is linear. Thus, if the first differences are

not constant, the relation is non-linear.

ii. An equation is linear if there are notexponents on the variables (ie: y = 2x, x = 3, etc.) Slope y-intercept form y = mx + b(6.1) Standard form Ax + By + C = 0(6.2)

i. A must be positiveii. No fractions or decimals

Graphing using different methodsi. Table of Valuesii. Slope & y-intercept (6.1)iii. x and y intercepts (At x-int, y = 0; At y-int, x = 0) (6.3)


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Parallel and perpendicular lines (6.4)i. Slopes of parallel lines are the sameii. Slopes of perpendicular lines are negative reciprocals

Writing equations of lines given:i. One point & slope (6.5)ii. Two points (6.6)iii. Relationships to other lines (parallel, perpendicular)

Systems of equations (find point of intersection by graphing) (6.7)6. Measurement and Geometry Angle relationships within parallel lines (p.363 & worksheets) Interior and exterior angles of triangles (7.1)

i. The sum of the interior angles of a triangle is 180 oii. The sum of the exterior angles of a triangle is 360oiii. The exterior angle at each vertex of a triangle is equal to the sum of the interior angles at the other two

vertices

Interior and exterior angles of quadrilaterals (7.2)i. The sum of the interior angles of a quadrilateral is 360 oii. The sum of the exterior angles of quadrilateral is 360 o

Angle relations in Polygons the sum of interior angles = 180(n-2) (7.3) Midpoints & Medians of Triangles (7.4)

i. Median of triangle bisects its areaii. A line segment joining the midpoints of two sides of a triangle is parallel to the third side and half as long

Midpoints & Diagonals in Quadrilaterals (7.5)i. Joining the midpoints of the sides of any quadrilateral produces a parallelogramii. The diagonals of a parallelogram bisect each other

Pythagorean Theorem a2 + b2 = c2 (8.1) Perimeter & Area of Composite Figures (8.2) Surface Area and Volume of basic shapes (Formulas will be given to you)

i. Prisms & Pyramids (8.3)ii. Cylindersiii. Cones (8.4/8.5)iv. Spheres (8.6/8.7)

HAPPY STUDYING!

No Graphing Calculators Permitted on Exam


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MPM1D Grade 9 Academic Mathematics

Final Exam Review

Part A Exponents

1. Evaluate:a) 23! b) 4)2(! c) 423 !" d)

33

2e)

3

3

2!"#

$%&

f)2

4

3 !"#$

%&'

g)1

3

2 !"#$

%&'

h)

21

3

4

5

3!!

"#$

%&'

"#$

%&'

i)2

1

23

2

!

!

j) 8 23 ! 2!3"#$

%

&'k)

25

1

!

l)1

2

3

!

m)5

3

2

2!

n)1

2

22

3

!

+

o)

226

5

4

5

4

5

4!"#

$%&

!"#

$%&

!"#

$%&

'p)

123

5

12

5

12

5

12!!!

"#$

%&'("

#$

%&'

"#$

%&'

q)

121

4

3

3

4

4

3!!

"#$

%&'

"#$

%&'("

#$

%&'

r)

023

43

212

!!"#

$$%& '

()*

+,'

()*

+,

-s)

2

12

3

!

! ""#$%%

&' t) 50 (22 + 3!1)!1 u) 2

!1

5!1+ 5

v) 3!1

4!2

w)

16 !1

2

"

#$

%

&'2

+ 3

3!1+ 2

!2

2. Evaluate if3

1,2,1 ==!= zyx . Leave fractions as fractions. Do not convert to decimals. Reduce

all answers to lowest terms.

a)2

2x b) ( )2

xy c) ( )3

xyz d) ( )4!

xy e)x

z f)22

zxy

g) 1!! x h) 32 !y i) ( )2zy ! j) y2 !x"#$

%

&'

x

k) 252 )5( !! ! yz

3. Simplify. Leave no negative exponents in final answers.

a)53

109

3

6

lk

lkb)

423

24

cba

cbac)

!3y2z"

#$%

&'

4

2y2z2

d)2x

2y

3( )2

2xy( )!1

e) 3x2y0z3( ) 2x!5y3z!1( ) f) 24x4y5 !4( )x!2y"#$

%

&' ( (3)x

!4y2 g)

332

334

25

5

cba

cba

!

!!

h)6x5y!3"

#$%

&'

2

12x!2

y5

i) 2x2y!

"#$

%&

'31

4xy

!"#

$%&

2

'2y'2!"#

$

%&

3

j)!!"

#

$$%

&'

4

5223

4

52

9

45

b

ba

ab

bak)

3

1

1!

!

!

""#

$%%&

'

b

al) 2a!2"

#$%

&'

!23a3"

#$%

&'

!1


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2

m) 403

32

x

x

x

+!

n) 5a3b4( )2

ab( ) o) 7x!3y!4"#$

%

&'8x3y5"

#$%

&'

!2p)

4a2m

3b!2( )

2

8a3m

!4b!6

q) !3xy2z

!3( )2

2x!1yz

3( )!2

!2xyz( )

Part B Polynomials & Solving Equations

1. Multiply/Expand, and simplify.(a) (2x)(2x) (b) (3y)(3y) (c) (-4u)(-4u) (d) yy

3

2

3

2!

(e) xx2

1

2

1! (f) )3)(2( xx (g) )3)(2( 22 xx (h) 6 33 2x x( )

(i) 2 37 7x x+ (j) 2 52 2

x x+ (k) ! +8 103 3

x x

2. Simplify.a)! + ! +2 5 7 82 2 2 2 2 2x y x y x y x y b) 3 7 4 22 2 x xy x xy! + !

c) 2 3 1 4 2 1( ) ( )x x+ ! ! d) 3 2 7 3 5 2 3 22 2 2( ) ( ) ( ) x x x x x x! ! ! + !

e) )372(5 2 !+! mmm f) )43(7 222 mnnmnm +

g) )()54(9 322222 mnnmmnmnnmmn ++!!

h) )523()73( 322322 xyyxxyxyxyyx +!!+! i) 3(2x2 ! 7x + 4) ! 3x2 +5x !1( )! 2x2 ! 7x +1( )"# $% j) 2

36 9 12

1

24 6 62 2( ) ( ) x x x x! + ! ! !

k) 83

9

4

9

163

5

2

4

512 2( ) ( ) x x x x! + !

!

! !

l) ! ! ! ! !2 12

3

44

1

8

1

2

1

4

2 2 2( ) ( )a a a a a

3. Simplify, then evaluate, if: (i) 2,1,1 !=!== pnm (ii)

5

3,

2

1,

3

1=== pnm

a) )(3)(2 pnmnpm ++! b) ( ) 32 !! nm c) p !m( )!2

d) !3m2n+ 2mn!p( )+ 4m2n! 6mn+ 6p( )! 2 m2n! 2mn+ 3p( ) e) m2 ! n2( )2 4. Fill in the blanks.(a) ______312 2 != mnm (b) _________312 2 !"= nmn (c) _____550 223 !"=" pqqp (d) ( ) 126__23 +=+ xx (e) ( ) 1015___3___ +!=! xx

(f) _ _ ( ) _ _ _ 2 1 16 2x x+ = ! + (g)! + = !3 6 62 2 2 4 x y x y x y( _ _ _ _ _ _ )


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Part C Problem SolvingAnswer each of the word problems with a complete algebraic solution. Be sure to include an openingand closing statement and a check.

1) Arlene weighs 25 kg more than her young son. The sum of Arlene's weight and twice her son'sweight is 61 kg. How much does Arlene weigh?

2) The number of girls in a chess club is 3 less than double the number of boys. There are 30 studentsin the chess club. How many girls are in the club?

3) The length of a rectangle is 3cm longer than twice the width. If the perimeter of this rectangle is 96cm, what are the dimensions of this rectangle?

4) The length of a rectangle is 6 cm longer than the width. The perimeter is 50 cm. State thedimensions of this rectangle.

5) Find three consecutive even integers such that the sum of the smallest and twice the second is 20more than the third.

6) Suzanne has $2.80 in quarters and dimes. The number of dimes is 7 less than the number ofquarters. Find the number of each kind she has.

7) Jordie had twice as many nickels as dimes, and three more quarters than nickels. If she had $16.15in total, how many of each coin did she have?

8) Sue counted her money and found that her 50 coins, which were nickels, dimes and quarters, wereworth $6.40. There were 8 more dimes than nickels. How many coins of each kind did she have?9) Gordie is 5 years older than twice his brothers age. In ten years, the sum of their ages will be 37

years. How old are they now?10)A father is 3 times as old as his son. Fifteen years ago the father was 9 times as old as his son was

then, How old are they now?11)Liz is currently 3 years older than twice her sisters age. Lizs age in 9 years will be 4 times her

sisters age last year. How old are they now?12)The sum of two numbers is 14. Three times the first number minus the second number is 10. Find

the numbers.


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Part D Lines

1. State the slope of the line y = -3x - 5_________2. State the equation of the y axis._________3. State the slope of the x axis.______.4. State the slope and the y intercept of the line defined by 2x + 3 = y.___________________5. State the x intercept of the line 2x - 4y = 16._______6. Name 2 points on the line 4x - 3y = 24.__________________7. State the equation of the line :

a) through A(-2,5) and B(-4,3)

b) through (4,-3) perpendicular to 4x + 7y 3 = 0.c) with the same x intercept as -4x - 3y = 6, same y intercept as x - 2y + 10 = 0.d) with the same y intercept as 3x y 7 = 0, parallel to 3x + 5y + 1 = 0.e) through (4,-3), parallel to the y axis.

8. Graph these lines using the method of your choice. State the slope for each.a) y 3 = 0 b) 2x + 4y = 0 c) x = 8 d) 3x + y = 5

9. Determine which of the following points are on the line 3x ! 7y = 21 :a) (21,6) b)(31,10) c) (-7,6) d)(-70,-33) e)(18,33/7)


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SOLUTIONS

PART A Exponents

1. a) -9 b) 16 c) -48 d)27

2e)

27

8f)

9

16g)

2

3h)

16

15i)

2

1!j) 63 k)25 l)6 m)

256

1

n)

5

18o)

25

16p)

144

25q)

9

16r) 1 s)

36

1t)

3

13

u)5

52

v)16

3

w)12

2a)2 b)4 c)27

8!d)16

1e) 3 f)

9

4!g) 1 h)

4

1i)

9

25j)5

1k)169

1

3. a) 562 lk b)3

c

a

c)2

8126

zyd) 758 yx e)

3

236

x

zyf) 6218 yx! g)

6

2

5c

a

h)11

123

y

xi)

7416

1

yx

!

j) ba28! k)3

3

b

al)12

a

m) 43x n) 257 9

a b o)7

649 14

x y

p)210 2

am b

q)!9

2

5 3

11

x y

z

PART B Polynomials & Solving Equations

1. a) 24x b) 29y c) 216u d)2

9

4y e) 2

4

1x f) 6

2x g) 6

4x h) 18

5x i)5

7x j) 7

2x k)2

3x

2. a) ! +9 132 2 2 x y x y b) 7 92 x xy! c)! +2 6x

d) 9 202

x x! e) mmm 153510 23 +!! f) 3324 2821 nmnm + g)4233

4635 nmnm +!

h) xyxyyx 982 322 +!! i) 5 33 142x x! + j) 2 3 112

x x! +

k) 821

2

2x x+ + l)

!

+7

4

7

2

2a a

3. a) (i)-11 (ii)6

7b) (i)

27

1(i)

8

27!c) (i)

1

9(ii)

225

16d) (i) 3 (ii)

!59

90

e) (i) 0 (ii)25

1296

4. a)4mn b)-4mn c) 210p d)4 e)-52 f)-8x.-8x g) 2 2 2y x,... h) 3 4 52 2ab b a b,.. ,..

5. a) 1 b) 1 c)25

7d)

22

25

22

25e)

1

4f)10

43g)36


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7a) (The first question is done for you, to show you checking form) LS RS

LS RS x is wrong

= ! ! = + !

= ! ! = !

= ! = !

=

" # =

3 3 4 2 3 2 3 3 3

3 1 6 2 6 3

9 12 3

9

3

( ) ( ) ( )

( ) ( )

7b) correct 7c) correct

8a)3

52 cxy

+= b)

3

4 xy

+= c)

ca

dby

33

3

!

+= or

)(3

3

ca

dby

!

+= d)

57

3

75

3

!!

!

=

x

xor

x

xy

Part C Problem Solving1)37kg

2) 19 girls3) W:15cm, L=33cm

4) W: 9.5cm L: 15.5 cm5) 10,12,14

6) 10 quarters, 3 dimes7) 22 dimes, 44 nickels, 47 quarters

8) 14 nickels, 14 quarters, 22 dimes9) Gordie: 13, Brother: 4

10) son: 20, father 6011) Liz:19, Sister: 8

12) 6, 8

Part D Lines


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Gr 9 Complete Review Packet

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