Getting Ready for Grade 10 Math

Complete these exercises to review key concepts from Grade 9 math in Ontario.

You will need both lined paper and graph paper.

Make a real effort to complete each question before opening its answer.

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If you would rather work from paper, use the Print button on this page, or open the handout directly:

Grade 9 Math Review - Getting Ready for Grade 10

A. Exponent laws

Grade 10 connection: you will expand expressions like $(2x+5{)}^2$, and meet zero and negative exponents in $y=2^x$.

Question A1

Evaluate.

a) $2^5$

b) $(-3{)}^4$

c) $-3^4$

d) $5^0$

e) $2^{-3}$

Answers

a) $32$

b) $81$

c) $-81$

d) $1$

e) $\frac{1}{8}$

Question A2

Write as a single power.

a) $3^4\times 3^5$

b) $7^{10}÷7^4$

c) ${\left(5^3\right)}^4$

d) $\frac{2^6\times 2^3}{2^5}$

e) $\left(x^3\right)\left(x^7\right)$

f) $\frac{{\left(y^5\right)}^2}{y^3}$

Answers

a) $3^9$

b) $7^6$

c) $5^{12}$

d) $2^4$

e) $x^{10}$

f) $y^7$

Question A3

Simplify.

a) $\left(4x^3\right)\left(-3x^5\right)$

b) $\frac{18a^7{b}^3}{6a^{2}b}$

c) ${\left(2m^3\right)}^4$

d) ${\left(-3x^{2}y\right)}^3$

e) $\frac{\left(6x^5\right)\left(2x^3\right)}{4x^6}$

f) ${\left(x^4\right)}^3\times (5x{)}^0$

Answers

a) $-12x^8$

b) $3a^5{b}^2$

c) $16m^{12}$

d) $-27x^6{y}^3$

e) $3x^2$

f) $x^{12}$

Question A4

Scientific notation.

a) Write $4500000$ in scientific notation.

b) Write $0.000032$ in scientific notation.

c) Evaluate $\left(3\times 10^4\right)\left(2\times 10^5\right)$. Give your answer in scientific notation.

d) A student wrote $2^3\times 2^4=4^7$. Explain the mistake and give the correct value.

Answers

a) $4.5\times 10^6$

b) $3.2\times 10^{-5}$

c) $6\times 10^9$

d) The bases should not be multiplied. Keep the base and add the exponents: $2^3\times 2^4=2^7=128$.

B. Simplifying expressions

Grade 10 connection: expanding and factoring are the main tools for solving quadratic equations.

Question B1

Simplify by collecting like terms.

a) $5x+3-2x+7$

b) $4a^2-3a+2a^2+5a-1$

c) $3xy-5x+2xy+x$

Answers

a) $3x+10$

b) $6a^2+2a-1$

c) $5xy-4x$

Question B2

Expand and simplify.

a) $4(2x-5)$

b) $-3\left(x^2-4x+2\right)$

c) $2x(3x+7)$

d) $5(x+2)-3(2x-1)$

e) $3(2a-4)+2(a+6)$

f) $x(x+4)-2\left(x^2-3x\right)$

Answers

a) $8x-20$

b) $-3x^2+12x-6$

c) $6x^2+14x$

d) $-x+13$

e) $8a$

f) $-x^2+10x$

Question B3

Which of these expressions are equivalent to $6x-9$? Show how you know.

a) $3(2x-3)$

b) $6(x-9)$

c) $2(3x-4)-1$

d) $9-6x$

Answers

a) equivalent

b) not equivalent ($6x-54$)

c) equivalent

d) not equivalent

Question B4

A rectangle has a length of $x+5$ and a width of $2x$.

a) Write and simplify an expression for its perimeter.

b) Write and simplify an expression for its area.

c) Find the perimeter and the area when $x=3$.

Answers

a) $6x+10$

b) $2x^2+10x$

c) perimeter $28$, area $48$

C. Solving equations

Grade 10 connection: solving linear systems and rearranging to $y=mx+b$ happen in almost every lesson.

Question C1

Solve. Check each answer by substituting it into the equation.

a) $5-2x=17$

b) $7x-4=3x+18$

c) $2(3x+1)-5=4x+9$

d) $\frac{x}{4}+3=7$

e) $\frac{2x-1}{3}=5$

f) $0.5(x+6)=2x-3$

Answers

a) $x=-6$

b) $x=5.5$

c) $x=6$

d) $x=16$

e) $x=8$

f) $x=4$

Question C2

Rearrange each formula to solve for the variable shown.

a) $y=3x-6$, for $x$

b) $2x+5y=10$, for $y$

c) $P=2l+2w$, for $w$

Answers

a) $x=\frac{y+6}{3}$

b) $y=-\frac{2}{5}x+2$

c) $w=\frac{P-2l}{2}$ or $w=\frac{P}{2}-l$

Question C3

A number is tripled and then decreased by $8$. The result is $25$. Write an equation and solve it to find the number.

Answer

$3n-8=25$, so the number is $11$.

D. Graphing lines

Grade 10 connection: slope, $y$-intercepts, and parallel and perpendicular lines are the base of analytic geometry, and moving a line prepares you for moving parabolas.

Question D1

State the slope and the $y$-intercept of each line.

a) $y=2x-5$

b) $y=-\frac{3}{4}x+1$

c) $y=6$

d) $3x+y=8$

e) $2x-4y=12$

Answers

a) slope $2$, $y$-intercept $-5$

b) slope $-\frac{3}{4}$, $y$-intercept $1$

c) slope $0$, $y$-intercept $6$

d) slope $-3$, $y$-intercept $8$

e) slope $\frac{1}{2}$, $y$-intercept $-3$

Question D2

On grid paper, graph each relation. Use a table of values, or the slope and $y$-intercept.

a) $y=3x-2$

b) $y=-\frac{1}{2}x+4$

c) $x+y=5$

d) $x=-3$

e) $2x-3y=6$

Answers

a) $y=3x-2$

The line y = 3x - 2 on a grid from -6 to 6, through the points (0, -2) and (1, 1)

b) $y=-\frac{1}{2}x+4$

The line y = -1/2 x + 4 on a grid from -6 to 6, through the points (0, 4) and (2, 3)

c) $x+y=5$

The line x + y = 5 on a grid from -6 to 6, through the points (0, 5) and (5, 0)

d) $x=-3$

The vertical line x = -3 on a grid from -6 to 6

e) $2x-3y=6$, which is $y=\frac{2}{3}x-2$: $x$-intercept $3$, $y$-intercept $-2$.

The line 2x - 3y = 6 on a grid from -6 to 6, through the points (0, -2) and (3, 0)

Question D3

Find the slope of the line through each pair of points.

a) $(1,2)$ and $(4,11)$

b) $(-2,5)$ and $(4,1)$

c) $(3,-1)$ and $(3,6)$

d) $(-5,4)$ and $(2,4)$

Answers

a) $3$

b) $-\frac{2}{3}$

c) undefined (vertical line)

d) $0$ (horizontal line)

Question D4

Write the equation of each line in the form $y=mx+b$.

a) slope $4$, $y$-intercept $-7$

b) slope $-2$, through $(3,1)$

c) through $(2,3)$ and $(6,11)$

d) the line in the graph below

e) the line in the table below

Graph for part d:

A line on a grid from -6 to 6 falling from left to right through the points (0, 3) and (4, -3)

Table for part e:

$x$	$y$
$0$	$5$
$1$	$8$
$2$	$11$
$3$	$14$

Answers

a) $y=4x-7$

b) $y=-2x+7$

c) $y=2x-1$

d) $y=-\frac{3}{2}x+3$

e) $y=3x+5$

Question D5

Parallel and perpendicular lines.

a) Which of these lines are parallel? $y=2x+3y=3x+2y=2x-54x-2y=6$

b) Write the equation of the line that is parallel to $y=-3x+4$ and has a $y$-intercept of $-2$.

c) Preview of Grade 10: on one grid, graph $y=2x$ and $y=-\frac{1}{2}x$. Measure the angle where they cross. What do you notice about the two slopes?

Answers

a) $y=2x+3$, $y=2x-5$ and $4x-2y=6$ (all have slope $2$)

b) $y=-3x-2$

c) They cross at $90^{\circ }$ (perpendicular). The slopes $2$ and $-\frac{1}{2}$ are negative reciprocals: their product is $-1$.

Question D6

Describe how the graph of $y=2x$ changes to become each line (translation, reflection or rotation).

a) $y=2x+3$

b) $y=-2x$

c) $y=5x$

Answers

a) translated up $3$ units

b) reflected in the $y$-axis (or the $x$-axis)

c) rotated about the origin, so it is steeper (either idea is enough)

E. Linear relations in real situations

Grade 10 connection: you will write and solve linear systems to compare real plans and prices.

Question E1

Maple Movers charges a $120 flat fee (a fixed amount) plus $85 per hour.

a) Write an equation for the total cost, $C$, in dollars, for a move that takes $h$ hours.

b) How much does a 4.5-hour move cost?

c) A family has $800 to spend on movers. How many hours can they afford?

d) What does the slope mean in this situation? What does the $C$-intercept mean?

Answers

a) $C=85h+120$

b) $502.50

c) $8$ hours

d) Slope: the cost goes up $85 for every hour (the rate of change). $C$-intercept: the $120 flat fee, charged even before any time is worked.

Question E2

Speedy Moves charges a $200 flat fee (a fixed amount) plus $60 per hour.

a) Write an equation for Speedy Moves’ total cost.

b) On the same axes, graph the cost equations for both companies, from $0$ to $8$ hours.

c) Find the point of intersection algebraically.

d) Explain what the point of intersection means for a family choosing a mover.

e) A move takes 6 hours. Which company is cheaper, and by how much?

Answers

a) $C=60h+200$

b) See the graph below.

Cost in dollars against time in hours from 0 to 8, with the line C = 85h + 120 for Maple Movers and the line C = 60h + 200 for Speedy Moves crossing at the marked point (3.2, 392)

c) $(3.2,392)$

d) At $3.2$ hours (3 h 12 min) both companies charge $392. Maple Movers is cheaper for shorter moves; Speedy Moves is cheaper for longer moves.

e) Speedy Moves ($560 compared with $630), by $70

Question E3

Water drains from a pool at a steady rate.

Time, $t$ (min)	$0$	$5$	$10$	$15$	$20$
Water, $W$ (L)	$12000$	$11250$	$10500$	$9750$	$9000$

a) Is the relationship linear? How do you know?

b) What is the rate of change? Include units.

c) Write an equation for the water remaining, $W$, after $t$ minutes.

d) When will the pool be empty?

Answers

a) Yes: the first differences are constant ($-750$ L every 5 minutes).

b) $-150$ L/min

c) $W=-150t+12000$

d) after $80$ minutes

Question E4

Is each relationship linear or non-linear?

a) A phone plan costs $25 per month plus $0.10 per text.

b) Money in a bank account earning compound interest.

c) The area of a square compared with its side length.

d) The cost of gas at $1.60 per litre.

Answers

a) linear

b) non-linear

c) non-linear

d) linear

F. Right triangles and proportions

Grade 10 connection: the Pythagorean theorem and proportions lead to the length formula, similar triangles and trigonometry.

Question F1

Find the unknown side length of each right triangle.

a) The legs are 9 cm and 12 cm. Find the hypotenuse.

b) The hypotenuse is 13 m and one leg is 5 m. Find the other leg.

Answers

a) 15 cm

b) 12 m

Question F2

At the same time of day, a 1.8 m tall student makes a 2.4 m shadow and a tree makes a 16 m shadow. How tall is the tree?

Answer

12 m

Curriculum connection

A3.1

expand and simplify second-degree polynomial expressions [e.g., $(2x+5{)}^2$, $(2x-y)(x+3y)$], using a variety of tools (e.g., algebra tiles, diagrams, computer algebra systems, paper and pencil) and strategies (e.g., patterning);

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B1.1

solve systems of two linear equations involving two variables, using the algebraic method of substitution or elimination (Sample problem: Solve $y=\frac{1}{2}x-5$, $3x+2y=-2$ for $x$ and $y$ algebraically, and verify algebraically and graphically);

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B1.2

solve problems that arise from realistic situations described in words or represented by linear systems of two equations involving two variables, by choosing an appropriate algebraic or graphical method (Sample problem: The Robotics Club raised $5000 to build a robot for a future competition. The club invested part of the money in an account that paid 4% annual interest, and the rest in a government bond that paid 3.5% simple interest per year. After one year, the club earned a total of $190 in interest. How much was invested at each rate? Verify your result.).

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B1.3

identify the relationship between the slopes of parallel and perpendicular lines, and use this relationship to solve related problems;

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B1.4

develop the formula for the slope of a line (i.e., $m=\frac{y_2-y_1}{x_2-x_1}$), and use this formula to determine the equations of lines, given information about the lines (e.g., a graph of a line, a table of values, the coordinates of two points);

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B1.5

represent the equations of lines in different forms (e.g., $y=mx+b$, $Ax+By+C=0$, $Ax+By=D$) and translate between these forms, as appropriate for the context.

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C1.3

solve problems involving similar triangles in realistic situations (e.g., shadows, reflections, scale models, surveying) (Sample problem: Use a metre stick to determine the height of a tree, by means of the similar triangles formed by the tree, the metre stick, and their shadows.).

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C2.2

determine the measures of the sides and angles in right triangles, using the primary trigonometric ratios and the Pythagorean theorem;

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