C. Algebra · Linear and Non-Linear Relations

Slope: Measuring Steepness

C4.4 C3.2 A1About 70 minutes

Learning goals

I can find the slope of a line by counting rise and run on a graph.
I can calculate the slope between two points with the formula m = (y₂ − y₁) ÷ (x₂ − x₁).
I can recognize positive, negative, zero and undefined slopes.
I can explain that slope is the rate of change, and interpret it with units in a real situation.

Words to know

slope
A number that measures how steep a line is: rise ÷ run. It is usually called m.

rise
The vertical change between two points. Up is positive, down is negative.

run
The horizontal change between two points. Right is positive, left is negative.

zero slope
The slope of a horizontal line: the rise is 0, so the slope is 0.

undefined slope
The 'slope' of a vertical line: the run is 0, and you cannot divide by 0.

collinear
Points that all lie on the same straight line.

How steep is it?

A ski hill, a roof, a wheelchair ramp and a stair all have a steepness. Carpenters, engineers and ski-hill designers measure steepness with one number: the slope. In this lesson you will see that slope is not new at all. It is the rate of change you have been calculating, seen on a graph.

Slope compares how far something goes up (rise) with how far it goes across (run). A bigger rise for the same run means a steeper slope.

The slope formula

For any two points on a line, (x₁, y₁) and (x₂, y₂):

m = rise ÷ run = (y₂ − y₁) ÷ (x₂ − x₁)

The little numbers ₁ and ₂ are just labels: "the first point" and "the second point". You can choose either point as the first one, as long as you subtract in the same order on the top and on the bottom.

On a graph you can also count: start at one point, count squares up or down (rise) and right or left (run) to reach the other point.

From A(1, 2) to B(5, 8): run 4 to the right, then rise 6 up. Slope = 6 ÷ 4 = 3/2 (or 1.5). Every 2 steps right, the line goes 3 up.

Four kinds of slope

Read lines from left to right, like a sentence:

Positive slope: the line goes up. (m > 0)
Negative slope: the line goes down. (m < 0)
Zero slope: the line is horizontal. The rise is 0, so m = 0 ÷ run = 0.
Undefined slope: the line is vertical. The run is 0, and division by 0 has no answer.

The bigger the number (ignoring its sign), the steeper the line: a slope of −4 is steeper than a slope of 2.

The four kinds of slope, read from left to right.

Worked example

Example 1: Count on the graph

A line passes through (−2, 5) and (4, −1). Find its slope by counting rise and run.

Start at (−2, 5). To reach x = 4, move 6 right: run = +6.
To reach y = −1 from y = 5, move 6 down: rise = −6.
m = −6 ÷ 6 = −1.

m = −1. The line falls 1 unit for every 1 unit to the right.

Worked example

Example 2: Use the formula

Find the slope of the line through P(−3, 4) and Q(5, −2).

Let P be point 1 and Q be point 2: x₁ = −3, y₁ = 4, x₂ = 5, y₂ = −2.
m = (−2 − 4) ÷ (5 − (−3)) = −6 ÷ 8.
Simplify: m = −3/4.
Check the other order: (4 − (−2)) ÷ (−3 − 5) = 6 ÷ (−8) = −3/4 ✓. Same answer.

m = −3/4. The line falls 3 units for every 4 units to the right.

Worked example

Example 3: Slope as a rate of change

A cyclist has gone 10 km after 0.5 hours and 40 km after 2 hours, riding at a steady pace. Find the slope of the distance–time graph and explain what it means.

Points: (0.5, 10) and (2, 40), with time in hours and distance in km.
m = (40 − 10) ÷ (2 − 0.5) = 30 ÷ 1.5 = 20.
The units are km ÷ h, so the slope is 20 km/h.

The slope is 20 km/h: it is the cyclist's speed. On a distance–time graph, slope means speed.

Worked example

Example 4: Which ramp is steeper?

Ramp A rises 50 cm over a run of 600 cm. Ramp B rises 40 cm over a run of 400 cm. Which is steeper?

Ramp A: m = 50 ÷ 600 = 1/12 ≈ 0.083.
Ramp B: m = 40 ÷ 400 = 1/10 = 0.1.
0.1 > 0.083.

Ramp B is steeper. Ramp A is gentler, which makes it easier for people who use wheelchairs.

Watch out!

Put the y-values on top: slope is rise over run, not run over rise.
Subtract in the same order on top and bottom. Mixing the order flips the sign.
Be careful with negatives: 5 − (−3) = 8.

Just for fun

Why was the vertical line so hard to describe?

Its slope was undefined, and it refused to run.

Hands-on activity

Slope hunt at home

You need: a tape measure or ruler, a pencil, paper. A grown-up should be nearby when you measure stairs.

Stairs: measure the height of one step (rise) and the depth of the part you step on (run) in centimetres. Calculate the slope. Does every step on the staircase have the same slope?
A ramp you make: lean a large hardcover book or a cutting board on a stack of books. Measure the height of the stack (rise) and the horizontal distance from the stack to where the board touches the table (run). Calculate the slope. Add one more book and repeat.
Roll a toy car down each ramp. Which slope makes it roll fastest?

What to look for: indoor stairs usually have a slope less than 1 (the step is deeper than it is tall). Adding books increases the rise, so the slope gets bigger and the car rolls faster.

Your turn

A line has a rise of 6 and a run of 2. What is its slope?

Find the slope of the line through (1, 3) and (4, 9).

Find the slope of the line through (−2, 7) and (2, −1).

Find the slope of the line through (−1, 4) and (5, 4). What kind of line is it?

Find the slope of the line through (3, −2) and (3, 6).

A hiker is at 0 km after 0 h and at 9 km after 2 h, walking steadily. What is the slope of the distance–time graph? What does it mean?

Which line is steeper: slope 3/4 or slope 4/5?

More practice: warm-up

Rise 4, run 8. Slope?

A line goes down from left to right. Is its slope positive or negative?

What is the slope of a horizontal line?

Find the slope through (0, 0) and (2, 10).

Rise −3, run 1. Slope?

More practice: on your own

Find the slope through (−4, −1) and (2, 3).

Find the slope through (0.5, 1.5) and (2.5, 7.5).

A roof rises 2.4 m over a horizontal distance of 6 m. What is its slope?

The slope of the line through (1, k) and (4, 10) is 2. Find k.

A rain barrel holds 600 L at time 0 and 150 L after 30 minutes of draining at a steady rate. Find the slope of the volume–time graph and explain it.

Challenge

Are the points (2, 5), (5, 11) and (9, 19) collinear? Explain.

A line has slope −3/2 and passes through (4, 1). Find two more points on the line, including the point where x = 0.

Hill A drops 120 m vertically over 400 m horizontally. Hill B drops 150 m over 600 m. Which hill is steeper? Show your reasoning.

Check your understanding

Quiz

What is the slope of the line through (2, 1) and (6, 9)?

A vertical line has a slope that is:

What is the slope through (−1, 5) and (3, −3)?

Which slope gives the steepest line?

On a distance–time graph, the slope represents:

Maya calculated the slope through (1, 2) and (3, 8) as (3 − 1) ÷ (8 − 2) = 1/3. What did she do wrong?

For parents and teachers

Big idea: Slope is the rate of change of a linear relation, seen as the steepness of its graph. Expectation C4.4 asks students to connect rates of change and slopes, which this lesson starts and the next two lessons complete.

Common difficulties:

Run over rise instead of rise over run. A memory aid: "you rise before you run" (top before bottom).
Inconsistent subtraction order, which flips the sign.
Confusing zero and undefined slope. Horizontal = 0 (flat floor, no climbing); vertical = undefined (a wall, no running).

How to help: The stair activity is quick and memorable. Please supervise on stairs. Ask your child to predict which of two ramps is steeper before calculating, then check.