Grade 9 Math – Final Exam Review

Unit 1 – Outcomes

Determine the square root of positive rational numbers that are perfect squares. o Determine whether or not a given rational number is a square number and explain the reasoning. o Determine the square root of a given positive rational number that is a perfect square. o Identify the error made in a given calculation of a square root. o Determine a positive rational number given the square root of that positive rational number.

Determine an approximate square root of positive rational numbers that are non-perfect squares.

o Estimate the square root of a given rational number that is not a perfect square using the roots

of perfect squares as benchmarks.

o Determine an approximate square root of a given rational number that is not a perfect square

using technology.

o Explain why the square root of a given rational number as shown on a calculator may be an

approximation.

o Identify a number with a square root that is between two given numbers.

Determine the surface area of composite 3-D objects to solve problems.

o Determine the overlap in a given concrete composite 3D object, and explain its effect on

determining the surface area (right cylinders, right rectangular prisms, and right triangular

prisms)

o Determine the surface area of a given concrete composite 3D object.

o Solve a given problem involving surface area.

1. Use the diagram to determine the value of the square root of

1

9.

2. Which numbers below are perfect squares? Explain how you know.

a)

25

121 b) 0.004

3. Without the use of a calculator, determine the value of each square root. Show your workings.

a)

225

49 b) 2.56

4. The area of a square garden is 12.25 m2. (Use a diagram to help you.)

a) What is the side length of the garden?

b) Determine the perimeter of the garden.

5. Using benchmarks, approximate

11.6 to the nearest tenth.

6. Each cube has edge length 1 unit. Determine the surface area of the object.

7. In each triangle, determine the unknown length to the nearest tenth of a unit where necessary.

a) b)

8. The local curling rink is shown in the diagram at the right. It is to be painted.

a) Determine the surface area of the structure.

b) The roofs, window, and door are not to be painted.

The door is 1 m by 2 m and the window is 4 m by 2 m.

Determine the surface area to be painted.

c) If it costs $0.32 / m2, determine the cost of the paint

needed.

Unit 2 – Outcomes

Demonstrate an understanding of powers with integral bases (excluding base 0) and whole

numbers by: representing repeated multiplication using powers, using patterns to show that a

power with an exponent of zero is equal to one, and solving problems involving powers.

o Demonstrate the difference between the exponent and the base by building models of a given

power, such as

o Explain using repeated multiplication the difference between two given powers in which the

exponent and base are interchanged (see above example).

o Express a given power as a repeated multiplication.

o Express a given repeated multiplication as a power.

o Explain the role of parentheses in powers by evaluating a given set of powers, eg

( ) ( )

o Demonstrate using patterns, that is equal to 1 for a given value of a ( )

o Evaluate powers with integral bases (excluding base 0) and whole number exponents.

Demonstrate an understanding of operations on powers with integral bases (excluding base 0) and

whole number exponents.

o Explain, using examples, the exponent laws of powers with integral bases and whole number

exponents:

o Evaluate a given expressing by applying the exponent laws.

o Determine the sum and difference of two given powers and record the process.

o Identify the error(s) in a given simplification of an expression involving powers.

9. Write (-3)(-3)(-3)(-3)(-3) as a power?

10. Write 6 4(8 ) (8 ) as a single power.

11. Write 8 4( 7) ( 7) as a single power.

12. Calculate: 5 4 0( 1) ( 1) ( 1) ( 1)

13. Evaluate: 3 0( 3) 4

14. Evaluate:

42

3

15. Complete the table.

Power Base Exponent Repeated Multiplication Standard

Form

53

4( 2)

10 3

(2 2 2 2 2 2)

16. Evaluate each power: a. 4( 3) b. 43

17. Evaluate each of 25 and 5

2 and explain why they are different.

18. Evaluate each expression. Show your work.

a. 2 4 2(18 3 1) 4 b.

4 47 ( 3) (30 6)

19. Write as a single power and then evaluate.

a. 8 2 610 10 10 b.

5 6

7 1

( 3) ( 3)

( 3) ( 3)

20. Identify and then correct any errors in the student’s work below. Explain how you think the errors

occurred.

22

2

( 4) 3 9 3

16 3( 3)

16 3( 9)

16 27

11

Unit 3 – Outcomes

Demonstrate an understanding of rational numbers by comparing and ordering rational numbers

and solving problems that involve arithmetic operations on rational numbers.

o Order a given set of rational numbers, in fraction and decimal form, by placing them on a

number line.

o Identify a rational number that is between two given rational numbers.

o Solve a given problem involving operations on rational numbers in fraction form and decimal

form.

Explain and apply the order of operations, including exponents, with and without technology.

o Solve a given problem by applying the order of operations with and without the use of

technology.

o Identify the error in applying the order of operations in a given incorrect solution.

21. Sketch a number line and mark each rational number on it. Order the numbers from greatest to least.

–3.1, 5

, 3

–1.2,1

 7

, 0.6

22. Write the rational number represented by each letter as a fraction.

23. In each pair, which rational number is greater?

a) 7.3,  7.2 b) 4 5

,  5 4

c) 1.2,  1.3 d) 10 10

,  13 11

24. Determine each sum or difference.

a) 3 2

5 3

b) 23 1

18 4

c) 4.1 3.5

d) 53.9 19.4

25. A technician checked the temperature of a freezer and found that it was -15.70C. She noted that the

temperature had dropped 7.80C from the day before. What

was the temperature the day before?

26. Determine each product or quotient.

a) 1 3

4 5

b) 5 2

6 3

c) 1 3

2 45 4

d) ( 0.32) 1.6

e) 0.9 ( 1.2)

27. A thermometer on a freezer is set at -5.50C. Each time the freezer door is opened, the temperature

increases by 0.30C. Suppose there is a power outage. How many times can the door by opened before

the temperature of the freezer increases to 50C? Show how you found your answer.

28. Evaluate

a) 0.84 ( 0.5) ( 2.3)

b) 1 3 9 3

2 5 10 5

Diagram 1 Diagram 2 Diagram 3

c) 5.8 3.1 0.5

d) 2 1 1 2

4 4 33 3 6 5

29. Evaluate with a calculator. Round to the nearest hundredth if necessary.

8.6 14.6 5.3 19.4 8.6

2.9 6.3 9.5

Unit 4 – Outcomes

Generalize a pattern arising from a problem-solving context, using a linear equation, and verify

by substitution.

o Write an expression representing a given pictorial, oral or written pattern.

o Write a linear equation to represent a given context.

o Write a linear equation representing the pattern in a given table of values and verify the

equation by substituting values from the table.

o Solve, using a linear equation, a given problem that involves pictorial, oral and written linear

patterns.

o Describe a context for a given linear equation.

Graph a linear relation, analyze the graph, and interpolate or extrapolate to solve problems.

o Describe the pattern found in a given graph.

o Graph a given linear relation, including horizontal and vertical lines.

o Match given equations of linear relations with their corresponding graphs.

o Interpolate and extrapolate the approximate value of one variable on a given graph given the

value of the other variable.

o Solve a given problem by graphing a linear relation and analyzing the graph.

30. In the equation m = 3n -2, determine the value of m when n =5

Use the diagram below for questions 31 and 32.

31. Determine the expression that relates the number of toothpicks (t) to the diagram number (d).

32. Given the diagrams shown, how many toothpicks would be used to construct Diagram 15 ?

33. Which table of values represents a linear relation?

A) C)

B) D)

34. The fee to ride in King Taxi is $4 plus $1.25 for each km travelled. Determine the equation that relates

the total cost C to the distance travelled d.

35. Which of the lines shown on the graph has the formula y + 4 = 0 ?

A) Line A C) Line C

B) Line B D) Line D

36. Describe the graph of the equation x = -3

37. Which line represents the equation x + y = 5

A) Line A C) Line C

B) Line B D) Line D

38. Which equation describes the graph?

A) x + 2y = 4 C) x + y = 4

B) Y = 4x D) 2x + y = 4

x 0 1 2 3 4

y 2 4 6 4 2

x 0 1 2 3 4

y 0 1 4 9 16

x 0 1 2 3 4

y 2 4 6 8 10

x 0 1 2 3 4

y 2 4 7 11 16

x- 6 - 5 - 4 - 3 - 2 - 1 1 2 3 4 5 6

y

- 6

- 5

- 4

- 3

- 2

- 1

1

2

3

4

5

6

A

D

C

B

x-6 -5 -4 -3 -2 -1 1 2 3 4 5 6

y

-6

-5

-4

-3

-2

-1

1

2

3

4

5

6

A

B C

D

x-6 -5 -4 -3 -2 -1 1 2 3 4 5 6

y

-6

-5

-4

-3

-2

-1

1

2

3

4

5

6

x- 8 - 6 - 4 - 2 2 4 6 8

y

- 8

- 6

- 4

- 2

2

4

6

8

Use the graph to the right to answer questions 39-40.

39. Determine the value of y when x = 4

A) 9 C) 3

B) -5 D) 1.5

40. Determine the value of x when y = 7

A) 5 C) -11

B) 3 D) -2

41. A stone if dropped from a bridge. Its speed increases due to the force of gravity. If the speed, s in m/s,

after t second is given by the formula 38.9 ts , what is the speed of the stone at 5 seconds?

A) 0.2041 m/s B) 78.4 m/s C) 49 m/s D) 51 m/s

42. The graph shown represents a linear relation. Determine the value of y when x = 6.

A) 0 C) 1

B) 2 D) 3

43. Look at the pattern and assume it continues.

a) Fill in the table below using the diagrams (complete the pattern of diagrams)

# of Triangles (T) 1 2 3 4 5 6

# of Line Segments (L) 3 5

x-8 -7 -6 -5 -4 -3 -2 -1 1 2 3 4 5 6 7 8

y

-9

-8

-7

-6

-5

-4

-3

-2

-1

1

2

3

4

5

6

7

8

b) Graph the data on the grid below.

c) Find an equation relating T and L.

d) How many line segments would there be if you had:

7 triangles _____ 20 triangles _____ 80 triangles ____

Unit 5 – Outcomes

Demonstrate an understanding of polynomials (limited to degree less than or equal to 2).

o Identify the variables, degree, number of terms and coefficients, including the constant term, of a

given simplified polynomial expression.

o Create a concrete model or a pictorial representation for a given polynomial.

o Write the expression for a given model of a polynomial.

o Match equivalent polynomial expressions given in simplified form.

Model, record, and explain the operations of addition and subtraction of polynomial expressions,

concretely, pictorially and symbolically (limited to degree less than or equal to 2).

o Identify equivalent polynomial expressions from a given set of polynomial expressions, including

pictorial and symbolic representations.

o Model addition and subtraction of two given polynomial expressions concretely or pictorially

and record the process symbolically.

o Apply a personal strategy for addition and subtraction of given polynomial expressions, and

record the process symbolically.

o Identify the error(s) in a given simplification of a given polynomial expression.

Model, record and explain the operations of multiplication and division of polynomial expressions

(limited to degree less than or equal to 2) by monomials, concretely, pictorially, and symbolically.

o Model multiplication and division of a given polynomial expression by a given monomial

concretely or pictorially and record the process symbolically.

o Apply a personal strategy for multiplication and division of a given polynomial expression by a

given monomial.

o Provide examples of equivalent polynomial expressions.

o Identify error(s) in a given simplification of a given polynomial expression.

44. Identify each as a monomial, binomial, or trinomial:

A) 25x

B) 2 2 1x x

C) 2a b

D) 22a b

45. Simplify: 2 2 22 2 5 4 3 2x x x x x x

46. What is the opposite of 24 3 2x x ?

47. Identify the coefficient in each.

A) 2 3x

B) 3 2x

C) 2 3x

D) 2x

48. Write a like term for 23x ?

49. Which is an equivalent polynomial of 2 5 3x x ?

A) 25 3x x

B) 2 5 3x x

C) 23 5x x

D) 23 5x x

50. If 3x , what is the value of 2 2 3x x ?

51. What is the degree of 24 3 5x x ?

52. Which polynomial is modeled by the following, if unshaded represents positive and shaded represents

negative?

53. Which polynomial represents the perimeter of this rectangle?

54. Which multiplication sentence is modeled by the set of algebra tiles? (unshaded is positive and shaded is

negative.)

55. Which polynomial has a constant term of 2?

A) 2 4x

B) 3 3xy

C) 24 1x

D) 5 2x

56. Which is NOT a polynomial?

A) 22 3x x

B) 2

32

xx

C) 2

23x

x

D) 22 3x x

57. Perform the operations indicated and simplify:

a. 2 2 2 2(2 5 3 ) (7 5 4 )x xy y xy y x

b. 2 2( 4 6 3) (2 5)x x x x

c. 2 23(2 4 5 )x xy y

d. 230 18

6

x xy

x

e. )3(4)3(2 xxx

4 x + 3

2 x – 1

RUG

2x

(x+1) (3x+4)

4x

58. A student subtracted 2 2(2 5 3) ( 2 4)x x x x like this. Identify the errors and correct them.

2 2

2 2

2 2

2

(2 5 3) ( 2 4)

2 5 3 2 4

2 5 2 3 4

3 1

x x x x

x x x x

x x x x

x x

59. Two sides of a triangle are 3 4x and 2 5x . If the perimeter is 9 6x , what is the length of the third

side of the triangle?

60. A rectangular rug with dimensions 2 by ( 1)x x is placed in a rectangular room with dimensions

(4 ) by (3 4)x x . What area of the floor is left uncovered?

Unit 6 – Outcomes

Model and solve problems using linear equations of the form:

where a,b,c,d,e and f are rational numbers.

o Model the solution of a given linear equation using concrete or pictorial representations, and

record the process.

o Determine, by substitution, whether a given rational number is a solution to a given linear

equation.

o Solve a given linear equation symbolically.

o Identify and correct an error in a given incorrect solution of a linear equation.

o Represent a given problem using a linear equation.

o Solve a given problem using a linear equation and record the process.

Explain and illustrate strategies to solve single variable linear inequalities with rational

coefficients within a problem solving context.

o Translate a given problem into a single variable linear inequality using the symbols

o Determine if a given rational number if a possible solution to a given linear inequality.

o Graph the solution of a given linear inequality on a number line.

o Generalize and apply a rule for adding and subtracting a positive or negative number to

determine the solution of a given inequality.

o Generalize and apply a rule for multiplying and dividing a positive or negative number to

determine the solution of a given inequality.

o Solve a given linear inequality algebraically and explain the process orally or in written form.

o Compare and explain the process for solving a linear equation to the process for solving a given

linear inequality.

o Compare and explain the solution of a given linear equation to the solution of a given linear

inequality.

o Verify the solution of a given linear inequality using substitution for multiple elements in the

solution.

o Solve a given problem involving a single variable linear inequality and graph the solution.

61. What would you do first to solve the equation 5 9x

62. Solve the following equation for x. 8 4 x

63. Solve 9

8 12

n

64. A number is added to four and the result is doubled to equal twenty-two. The equation is:

A) 44 22n B) 2 4 22n C) 2 4 22n D) 4 2 22n

65. Four times a number decreased by 42 is equal to 54 decreased by 2 times the number. What is the

number?

66. Which problem can be solved using the equation 6 36x x

(A) a number increased by 6 is 36

(B) one number is 36 more than 6 times another number

(C) the sum of a number and 6 more than the number is 36

(D) the sum of a number and 6 times that number is 36

67. Which of the following equations is solved correctly?

A) B) C) D)

2 62

42

2

x

x

x

2 62

82

4

x

x

x

2 62

42

8

x

x

x

2 62

82

16

x

x

x

68. Solve for x: -2x > 64

69. Solve the inequality: 6 5 8 1x x

70. Write the inequality for the following graph:

71. Solve each of the following equations:

A) 2x – 4 = -3x

B)

- 3 = 11

C) 2( x - 2) = -2 (x + 4)

72. Write an inequality for each situation and then graph it.

a) You must be at least 13 years old to watch the movie.

b) The gas tank in a car contains no more than 55 L of gas.

73. Solve and graph the following inequalities:

A) 3.5x < 1.3x + 6.6 B) x – 4 > 3x + 12 C)

+ 10 ≥ 20

74. A taxicab charges $2.50, plus $1.78 per kilometre. How far is a trip that costs $21.19? Write and solve an

equation to show your solution.

75. Nadia gets paid $1000 per month plus 5% commission on her sales. She wants to earn at least $2200 this

month. Write an inequality to represent this situation, then solve it to determine how much Nadia must sell to

reach her goal. (5%)

Unit 7 Outcomes

Draw and interpret scale diagrams of 2-D shapes.

o Identify an example in print and electronic media, e.g., newspapers, the Internet, of a scale

diagram and interpret the scale factor.

o Draw a diagram to scale that represents an enlargement or reduction of a given 2-D shape.

o Determine the scale factor for a given drawn to scale.

o Determine if a given diagram is proportional to the original 2-D shape and, if it is, state the

scale factor.

Demonstrate an understanding of similarity of polygons.

o Determine if the polygons in a given set are similar and explain the reasoning.

o Draw a polygon similar to a given polygon and explain why the two are similar.

o Solve a given problem using the properties of similar polygons.

o Solve a given problem that involves a scale diagram by applying the properties of similar

triangles.

-3 -2 -1 0 1 2 3 4-5 -4

Demonstrate an understanding of line and rotation symmetry.

o Classify a given set of 2-D shapes or designs according to the number of lines of symmetry.

o Complete a 2-D shape or design given one half of the design and a line of symmetry.

o Determine if a given 2-D shape or design has rotation symmetry about the point at the centre of

the shape or design and, if it does, state the order and angle of rotation.

o Rotate a given 2-D shape about a vertex and draw the resulting image.

o Identify a line of symmetry or the order and angle of rotation symmetry in a given tessellation.

o Identify and describe the types of symmetry created in a given piece of artwork.

o Create or provide a piece of artwork that demonstrates line and rotation symmetry, and identify

the line(s) of symmetry and the order and angle of rotation.

o Determine whether or not two given 2-D shapes on the Cartesian plane are related by either

rotation or line symmetry.

o Identify the type of symmetry that arises from a given transformation on the Cartesian plane.

o Complete, concretely or pictorially, a given transformation of a 2-D shape on a Cartesian plane,

record the coordinates and describe the type of symmetry that results.

o Draw, on a Cartesian plane, the translation image of a given shape using a given translation

rule, such as R2, U3 or label each vertex and its corresponding ordered pair and

determine why the translation may or may not result in line or rotation symmetry.

76. Consider the following similar polygons:

a) Calculate the length of PT.

b) Calculate the length of DE.

c) Draw a reduction of pentagon PQRST with a scale factor of

.

d) Draw an enlargement of pentagon ABCDE with a scale factor of 2.

77. Naomi wants to calculate the height of a tree. She is 1.5 m tall and casts a shadow of 2.5 m. At the same

time, the shadow of the tree is 10.5 m long.

a. Sketch a diagram that can be used to calculate the height of the tree.

b. What is the height of the tree?

78. A diagram of a swimming pool is drawn 3 cm by 2 cm. The actual dimensions of the pool are 60 m by

40m. Determine the scale factor of the drawing.

79. A diagram of a building is drawn with a height of 7.5 cm. If the scale factor is 0.004, what is the actual

height of the building?

80. Plot these points on a grid: A(−2, 4), B(2, 4), C(2, 2), D(−2, 2). For each transformation below:

a. Draw the transformation image

b. Record the coordinates of its vertices.

c. Describe the symmetry of the diagram formed by the original shape and its image.

a) rotation 90° clockwise about point E(0, 3)

b) reflection in the horizontal line passing through (0, 2)

c) a translation 4R, 2U

81. Describe the symmetry of each shape. Draw any lines of symmetry and state the order of rotation and

the angle of rotation symmetry.

a) b)

c) d)

Unit 8 Outcomes

Solve problems and justify the solution strategy using the following circle properties:

o The perpendicular from the centre of a circle to a chord bisects the chord

o The measure of the central angle is equal to twice the measure of the inscribed angle

subtended by the same arc

o The inscribed angles subtended by the same arc are congruent

o A tangent to a circle is perpendicular to the radius at the point of tangency.

Explain the relationship between the tangent of a circle and the radius at the point of

tangency

Explain the relationship between the perpendicular from the centre of the circle and a

chord.

Solve a given problem involving application of one or more of the circle properties.

Explain the relationship between the measure of the central angle and the inscribed

angle subtended by the same arc.

Determine the measure of a given angle inscribed in a semicircle using the circle

properties.

Explain the relationship between inscribed angles subtended by the same arc.

82. A circle with radius 10 cm has a chord with length 12 cm. How far from the centre of the circle is the

chord? Draw a diagram to support your solution.

83. Determine the missing measures:

a. b.

c. d.

e. f.

g h.

Unit 9 Outcomes

Demonstrate an understanding of the role of probability in society.

o Provide an example from print and electronic media, e.g., newspapers, the Internet, where

probability is used.

o Identify the assumptions associated with a given probability and explain the limitations of each

assumption.

o Explain how a single probability can be used to support opposing positions.

o Explain, using examples, how decisions based on probability may be a combination of

theoretical probability, experimental probability and subjective judgement.

Describe the effect of:

o Bias

o Use of language

o Ethics

o Cost

o Time and timing

o Privacy

o Cultural sensitivity

on the collection of data.

Analyse a given case study of data collection, and identify potential problems related to bias,

use of language, ethics, cost, time and timing, privacy or cultural sensitivity.

Provide examples to illustrate how bias, use of language, ethics, cost, time and timing,

privacy or cultural sensitivity may influence the data.

Select and defend the choice of using either a population or a sample of a population to answer a

question.

o Identify whether a given situation represents the use of a sample or population.

o Provide an example of a situation in which a population may be used to answer a question and

justify the choice.

o Provide an example of a question where a limitation precludes the use of a population and

describe the limitation, e.g., too costly, not enough time, limited resources.

o Identify and critique a given example in which a generalization from a sample of a population

may or may not be valid for the population.

Develop and implement a project plan for the collection, display and analysis of data by:

o Formulating a question for investigation

o Choosing a data collection method that includes social considerations

o Selecting a population or sample

o Collecting the data

o Displaying the collected data in an appropriate manner

o Drawing conclusions to answer the question

Create a rubric to assess a project that includes assessment of:

A question for investigation

The choice of a data collection method that includes social considerations

The selection of a population or a sample and justifying the choice

The display of the collected data

The conclusions to answer the question

Develop a project plan that describes:

A question for investigation

The method of data collection that includes social considerations

The method for selecting a population or a sample

The method to be used for collection of the data

The methods for analysis and display of the data

Complete the project according to the plan, draw conclusions and communicate findings

to an audience.

Self-assess the completed project by applying the rubric.

84. Identify a potential problem with each data collection. Justify your answer.

a. Rachel asks: “What costume will you wear for Halloween this year?”

b. Jerome asks local dog kennels: “How often do you walk the dogs in your care?”

c. A dentist sends a questionnaire to her patients 6 months after their last check-up, asking them to

rate the quality of care they received and reminding them to make an appointment for a new

check-up

85. Would you use a population or sample to gather each set of data. Justify your choice.

a. To determine whether the strings of a guitar are in tune.

b. To determine citizens’ opinions about the new traffic laws

c. To test a new recipe for a cookbook