VertexMath · MPM2D · Free worksheet

Grade 9 Skills Review

Try each problem first. You don’t need to finish everything.

Core path ~55 min Full sheet ~1 h 45 min

Recommended: K1–K5, then one A and one T problem

K = Knowledge & Understanding · A = Application · T = Thinking · Boss = Challenge

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Quick reference

Slope: $m=\dfrac{y_2-y_1}{x_2-x_1}$ , rise over run, subtracting in the same order on top and bottom. A horizontal line $y=k$ has slope $0$ ; a vertical line $x=k$ has an undefined slope.
From $Ax+By+C=0$ to $y=mx+b$ : isolate $y$ , then divide every term by $B$ . For an intercept, set the other variable to $0$ .
Expanding: every term of one bracket multiplies every term of the other. $(a+b)^2=a^2+2ab+b^2$ , not $a^2+b^2$ .
Pythagorean theorem: $a^2+b^2=c^2$ , where $c$ is the hypotenuse, the longest side, opposite the right angle. To find a leg, subtract: $a^2=c^2-b^2$ .
Map scale $1:n$ : $1$ cm on the map is $n$ cm in reality. $1$ km $=100\,000$ cm.

Show your work on separate paper. Show your work in the dot-grid space.

Core Skills (K)

Slope · Equations of Lines · Expanding · Pythagorean Theorem · Proportions

K1 Reading Slope

a)
Determine the slope of the line through $(-3,4)$ and $(5,-2)$ .
b)
The points in the table lie on a line. Determine its equation in the form $y=mx+b$ .

$x$ $-1$ $1$ $3$ $5$
$y$ $8$ $5$ $2$ $-1$
c)
Determine the equation of the line graphed below. Each grid square is 1 unit.

$x$	$-1$	$1$	$3$	$5$
$y$	$8$	$5$	$2$	$-1$

Stuck? Peek at a hint

Slope is the change in $y$ divided by the change in $x$ . In the table, check how much $x$ goes up between columns.

K2 Forms of a Line

a)
Write $3x-4y+8=0$ in the form $y=mx+b$ .
b)
Find the $x$ - and $y$ -intercepts of $5x-2y=20$ .
c)
Determine the equation of the line through $(-2,7)$ with slope $-3$ , in the form $Ax+By+C=0$ .

Stuck? Peek at a hint

Move the $x$ -term and the constant to the right side, then divide every term by the coefficient of $y$ , sign included.

K3 Distribute Every Term

Expand and simplify.

a)
$-3x(2x^2-5x+4)$
b)
$4(2x-3)-(5x-7)$
c)
$2x(x+5)-3(x^2-2x+1)$

Stuck? Peek at a hint

A minus sign in front of a bracket multiplies every term inside by $-1$ .

K4 Multiplying Binomials

Expand and simplify.

a)
$(x-6)(x+4)$
b)
$(3x-2)^2$
c)
$3(x+2)^2-(2x-1)(2x+1)$

Stuck? Peek at a hint

Write a square as a product, $(3x-2)(3x-2)$ , and expand it before multiplying by any number outside.

K5 Right Triangles & Proportions

a)
A right triangle has legs $6$ cm and $9$ cm. Find the hypotenuse, to the nearest tenth.
b)
Find $x$ in the right triangle below.
c)
Solve $\dfrac{9}{x}=\dfrac{12}{20}$ .

Stuck? Peek at a hint

First find the hypotenuse: it is the side opposite the right angle. If it is the side you know, subtract squares.

Real World (A)

Linear Models · Ladders · Map Scales

A1 Rain Barrel

A full rain barrel drains through a tap at a constant rate. The table shows the volume of water left.

$t\text{ (min)}$	$0$	$4$	$8$	$12$
$V\text{ (L)}$	$180$	$156$	$132$	$108$

a)
Determine an equation for $V$ in terms of $t$ . What do the slope and the $V$ -intercept mean for the barrel?
b)
After how many minutes will 75 L remain?
c)
The equation gives $V=-60$ when $t=40$ . How much water is in the barrel after 40 minutes?

Stuck? Peek at a hint

The volume drops by the same amount every 4 minutes: find the drop per minute.

A2 Ladder Safety

A painter has a $6.5$ m ladder. Safety guidelines say the foot of a ladder should be 1 m out from the wall for every 4 m of height it reaches up the wall. Round answers to the nearest hundredth of a metre.

a)
The foot of the ladder is $1.2$ m from the wall. How high up the wall does the ladder reach?
b)
The painter follows the guideline: the height reached is 4 times the distance $d$ from the foot to the wall. Find $d$ and the height reached.
c)
A gutter is $6.35$ m high, less than the ladder's length. Can the painter reach it safely with this ladder?

Stuck? Peek at a hint

The ladder is the hypotenuse. In (b), the legs are $d$ and $4d$ .

A3 Park Map

A map of a provincial park has the scale $1:25\,000$ .

a)
A portage trail measures $7.4$ cm on the map. Find its real length in kilometres.
b)
On the map, a campsite is $6$ cm east and $4.5$ cm north of the canoe launch. Find the real straight-line distance between them in kilometres.
c)
A camper enlarges the map to 200% on a photocopier. The copy still prints " $1:25\,000$ ". What length does the trail from (a) measure on the copy, what real length does the printed scale give, and what is the true scale of the copy?

Stuck? Peek at a hint

Multiply the map length by $25\,000$ to get real centimetres, then convert to kilometres.

Think It Through (T)

Error Analysis · Counterexamples · Building Lines

T1 Spot the Flaw

A student expands $3(x-4)^2$ :
" $3(x-4)^2=(3x-12)^2=9x^2-144$ "

a)
Evaluate $3(x-4)^2$ and $9x^2-144$ at $x=5$ . What does the result tell you?
b)
Identify the two errors.
c)
Expand $3(x-4)^2$ correctly, and check your answer at $x=5$ .

Stuck? Peek at a hint

For (a), work out the bracket first: $5-4=1$ .

T2 Lines from Clues

a)
A student claims: "Every line has exactly one $y$ -intercept." Give the equation of a line that is a counterexample, and state its slope.
b)
Determine the equation of the line through $(-4,1)$ that has the same $y$ -intercept as $3x-2y+6=0$ .
c)
Determine the equation of the line parallel to $2x+5y=10$ with $x$ -intercept $4$ , in the form $Ax+By+C=0$ .

Stuck? Peek at a hint

Think about which lines never cross the $y$ -axis. For (b), set $x=0$ in $3x-2y+6=0$ .

Boss Level

Optional challenge

Hidden Hypotenuse

A right triangle has legs $a$ and $b$ and hypotenuse $c$ , in centimetres. The legs add up to $17$ cm and the area is $30$ cm².

a)
Find $ab$ .
b)
Find $c$ without finding $a$ and $b$ .
c)
Find $(a-b)^2$ , then the lengths of the two legs.

Stuck? Peek at a hint

Expand $(a+b)^2$ and look for $a^2+b^2$ inside it.

Answer key

Final answers only. Try first, then check.

Core Skills

a) $m=-\frac{3}{4}$
b) $y=-\frac{3}{2}x+\frac{13}{2}$
c) $y=\frac{3}{2}x-1$

a) $y=\frac{3}{4}x+2$
b) $x$ -intercept $4$ ; $y$ -intercept $-10$
c) $3x+y-1=0$

a) $-6x^3+15x^2-12x$
b) $3x-5$
c) $-x^2+16x-3$

a) $x^2-2x-24$
b) $9x^2-12x+4$
c) $-x^2+12x+13$

a) $\sqrt{117}\approx 10.8$ cm
b) $x=15$
c) $x=15$

Real World

a) $V=180-6t$ ; slope $-6$ : the barrel loses 6 L per minute; $V$ -intercept $180$ : it holds 180 L when full, at $t=0$
b) $17.5$ min
c) 0 L: the barrel is empty at $t=30$ min, so the equation only holds for $0\le t\le 30$ .

a) $\sqrt{40.81}\approx 6.39$ m
b) $d\approx 1.58$ m; height $\approx 6.31$ m
c) No: following the guideline, the ladder reaches only about $6.31$ m. Reaching $6.35$ m needs the foot about $1.39$ m from the wall, closer than the $1.59$ m the guideline requires.

a) $1.85$ km
b) $1.875$ km (the map distance is $7.5$ cm)
c) $14.8$ cm; $3.7$ km, double the true $1.85$ km; the true scale of the copy is $1:12\,500$ .

Think It Through

a) $3$ and $81$ : the two expressions are not equal, so the expansion is wrong.
b) 1. The 3 was moved inside the bracket before squaring, so it got squared: $(3x-12)^2=9(x-4)^2$ . 2. The binomial was squared term by term, losing the middle term: $(3x-12)^2=9x^2-72x+144$ .
c) $3x^2-24x+48$ ; at $x=5$ : $75-120+48=3$

a) e.g. $x=3$ : it never meets the $y$ -axis; its slope is undefined. ( $x=0$ also works: it meets the $y$ -axis everywhere.)
b) $y=\frac{1}{2}x+3$
c) $2x+5y-8=0$

Boss Level

Boss

a) $ab=60$
b) $a^2+b^2=289-120=169$ , so $c=13$ cm
c) $(a-b)^2=49$ ; the legs are $12$ cm and $5$ cm

Grade 9 Skills Review

Core Skills (K)

K1 Reading Slope

K2 Forms of a Line

K3 Distribute Every Term

K4 Multiplying Binomials

K5 Right Triangles & Proportions

Real World (A)

A1 Rain Barrel

A2 Ladder Safety

A3 Park Map

Think It Through (T)

T1 Spot the Flaw

T2 Lines from Clues

Boss Level

Hidden Hypotenuse

Answer key

Core Skills

Real World

Think It Through

Boss Level

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