Discover how steepness is measured mathematically. From road grades to roof pitches, gradient is everywhere.
45 min 5 MCQ + 3 SAQ MAS-LIN-C-01
Think First
Before we begin: A road rises 10 metres for every 100 metres of horizontal distance. Another road rises 15 metres for every 200 metres of horizontal distance. Which road is steeper? Explain your reasoning.
Come back to this after you have worked through the lesson.
1
The Big Idea
Gradient measures how much a line rises or falls for every unit it runs horizontally. It is the single most important property of a straight line. A positive gradient means the line goes up from left to right; a negative gradient means it goes down. A horizontal line has zero gradient, and a vertical line has an undefined gradient because you cannot divide by zero.
Know
The gradient formula $m = \dfrac{y_2 - y_1}{x_2 - x_1}$. The four types of gradient: positive, negative, zero, and undefined.
Understand
How gradient measures steepness and direction. How to extract gradient from the equation $y = mx + c$.
Can Do
Calculate the gradient between two points. Identify the gradient from a linear equation. Interpret gradient in real-world contexts.
2
Lesson Objectives
Calculate the gradient of a line given two points.
Identify whether a gradient is positive, negative, zero, or undefined.
Find the gradient from an equation in the form $y = mx + c$.
Interpret the meaning of gradient in practical situations.
3
Key Vocabulary
Gradient
A measure of the steepness of a line, calculated as rise over run.
Rise
The vertical change between two points ($\Delta y$).
Run
The horizontal change between two points ($\Delta x$).
Positive gradient
The line goes up as you move from left to right.
Negative gradient
The line goes down as you move from left to right.
4
Spot the Trap
Wrong: Reversing rise and run. Calculating $\dfrac{x_2 - x_1}{y_2 - y_1}$ instead of $\dfrac{y_2 - y_1}{x_2 - x_1}$.
Wrong: Confusing horizontal and vertical lines. Horizontal has $m = 0$; vertical has $m$ undefined.
Right: Horizontal: $m = 0$ (no rise). Vertical: $m$ is undefined (run = 0, cannot divide by zero).
5
The Gradient Formula
+5 XP
For two points $A(x_1, y_1)$ and $B(x_2, y_2)$ on a line, the gradient $m$ measures how much the line rises (or falls) for every unit it runs horizontally. It is calculated as rise over run.
$m = \dfrac{y_2 - y_1}{x_2 - x_1}$
rise over run
Label your points
Identify $(x_1, y_1)$ and $(x_2, y_2)$ before substituting. It prevents sign errors.
Rise over run, not run over rise
$\dfrac{y_2 - y_1}{x_2 - x_1}$. The y-difference goes on top. Never reverse them.
Simplify your answer
$\dfrac{6}{4}$ simplifies to $\dfrac{3}{2}$ or $1.5$. Always reduce fractions.
6
Types of Gradient
+5 XP
Gradient tells you both steepness and direction. A positive gradient means the line slopes upward from left to right. A negative gradient means it slopes downward. A horizontal line has zero gradient because there is no rise at all. A vertical line has an undefined gradient because the run is zero and division by zero is impossible.
$|m| = 3$ is steeper than $|m| = 0.5$. The absolute value tells you steepness; the sign tells direction.
Horizontal vs vertical
$m = 0$ means flat. Undefined means vertical. Do not confuse these two special cases.
Order does not matter
$\dfrac{y_2 - y_1}{x_2 - x_1}$ gives the same answer as $\dfrac{y_1 - y_2}{x_1 - x_2}$. Just be consistent.
7
Gradient from an Equation
+5 XP
When a linear equation is written in gradient-intercept form $y = mx + c$, the gradient appears as the coefficient of $x$. If an equation is not in this form, rearrange it by isolating $y$. The number in front of $x$ is always the gradient.
$y = mx + c$. $m$ = gradient. $c$ = y-intercept.
coefficient of x
Isolate y first
Rearrange to $y = mx + c$ form. The number in front of $x$ is the gradient.
Watch the sign
$y = -3x + 2$ has gradient $-3$. The negative sign is part of the gradient.
Verify with two points
Pick any two points on the line and calculate $\dfrac{\Delta y}{\Delta x}$. It should match your $m$.
8
Real-World Gradient
+5 XP
Australian road signs express steepness as a percentage. A 10% grade means the road rises 10 m for every 100 m of horizontal distance, giving a gradient of $m = 0.10$. Wheelchair ramps must not exceed a 1:8 gradient (about 12.5%) to be safe. Roof pitches, ski slopes, and railway lines all use gradient to describe steepness.
10% grade = $m = 0.10$. 1:8 ramp = $m = 0.125$.
steepness everywhere
Percentage = gradient x 100
A 10% grade means $m = 0.10$. Multiply gradient by 100 to get percentage.
Ratio form
1:8 means 1 unit rise for 8 units run. Gradient = $\dfrac{1}{8} = 0.125$.
Safety limits
Wheelchair ramps max 1:8. Roads rarely exceed 10%. Ski runs can be 30%+.
Worked Example 1 — Calculating Gradient
Q.Find the gradient of the line through $A(2, 5)$ and $B(6, 11)$.
Since $m < 0$, the line slopes downward from left to right.
Answer: $m = -\dfrac{1}{2}$
Worked Example 3 — Extracting Gradient
Q.Find the gradient of $4x - 2y + 8 = 0$.
Solution
1
Rearrange to $y = mx + c$ form:
2
$4x - 2y + 8 = 0$
3
$-2y = -4x - 8$
4
$y = 2x + 4$
5
The coefficient of $x$ is $2$.
Answer: $m = 2$
Brain Trainer
4 quick-fire drills. Beat the clock.
Find the gradient through $(1, 2)$ and $(5, 8)$.
3/2 or 1.5
What is the gradient of a horizontal line?
0
Find the gradient of $y = -4x + 7$.
-4
A ramp rises 30 cm over 240 cm. What is its gradient?
1/8
Practice
5 MCQs and 3 short-answer questions. Target: 80% accuracy.
1.The gradient of the line through $(1, 2)$ and $(5, 8)$ is:
Correct! $m = \dfrac{8-2}{5-1} = \dfrac{6}{4} = \dfrac{3}{2} = 1.5$. Note that D is numerically equal but B is the exact fraction form preferred in mathematics.
Use $m = \dfrac{y_2 - y_1}{x_2 - x_1}$. Subtract the y-coordinates for the rise, and the x-coordinates for the run. Then simplify.
2.A horizontal line has gradient:
Correct! A horizontal line has no rise at all ($\Delta y = 0$), so $m = \dfrac{0}{\Delta x} = 0$.
Think about what happens to $\dfrac{\Delta y}{\Delta x}$ when the line is perfectly flat. Is there any vertical change?
3.The gradient of $y = -4x + 7$ is:
Correct! In $y = mx + c$ form, the coefficient of $x$ is the gradient. Here $m = -4$.
The equation is already in $y = mx + c$ form. The number in front of $x$ is the gradient. Do not confuse this with the y-intercept (7).
4.The gradient of $2x - 5y = 10$ is:
Correct! Rearranging: $-5y = -2x + 10$ → $y = \dfrac{2}{5}x - 2$. The gradient is $\dfrac{2}{5}$.
Rearrange the equation into $y = mx + c$ form. Isolate $y$ by moving the $x$ term and constant to the other side, then divide everything by the coefficient of $y$.
5.A wheelchair ramp rises 30 cm over a horizontal distance of 240 cm. Its gradient is:
Correct! $m = \dfrac{30}{240} = \dfrac{1}{8}$. Note that D is numerically equal but B is the exact fraction form preferred.
Gradient = rise over run. The rise is 30 cm and the run is 240 cm. Divide and simplify the fraction.
6.Find the gradient of the line through $A(-4, 3)$ and $B(2, -9)$. State whether the line slopes upward or downward from left to right.
The line slopes downward from left to right because $m < 0$.
7.A straight line has equation $4x + 2y - 6 = 0$. (a) Rearrange into the form $y = mx + c$. (b) State the gradient and the y-intercept. (c) Verify your gradient by choosing any two points on the line and calculating $\dfrac{\Delta y}{\Delta x}$.
8.Three points are given: $A(1, 2)$, $B(4, 8)$, and $C(7, 5)$. (a) Calculate the gradient of $AB$ and the gradient of $BC$. (b) Explain why $A$, $B$, and $C$ do not all lie on the same straight line. (c) Find a point $D$ such that $A$, $B$, and $D$ are collinear. Show your reasoning.
(b) The gradients are different ($2 \neq -1$), so $AB$ and $BC$ have different directions. Therefore $A$, $B$, and $C$ cannot be collinear.
(c) Any point on the line through $A$ with gradient 2 works. For example, from $B(4, 8)$ with gradient 2: $D = (5, 10)$. Check: $m_{AD} = \dfrac{10-2}{5-1} = \dfrac{8}{4} = 2$ ✓
Review
Consolidate and reflect before moving on.
Stretch Challenge
A line passes through $(2, -3)$ and has gradient $\dfrac{4}{3}$. Find two other points on this line that are exactly 5 units away from $(2, -3)$. (Hint: use the 3-4-5 triangle relationship. If run = 3 and rise = 4, the distance is 5.)
Key Idea
Gradient = $\dfrac{\text{rise}}{\text{run}} = \dfrac{y_2 - y_1}{x_2 - x_1}$. Positive = up, negative = down, zero = flat, undefined = vertical. From $y = mx + c$, the coefficient of $x$ is the gradient.
Common Trap
Reversing rise and run. Remember: y-difference goes on top. Also confusing horizontal ($m = 0$) with vertical (undefined).
Connection
Gradient is the foundation of linear equations. In the next lessons, you will use gradient to find the equation of a line and to determine if lines are parallel or perpendicular.
Interactive: Gradient Explorer — drag two points on a coordinate plane and watch the gradient, distance, and equation update in real time.
Gradient Calculator
Equation Reader
Slope Spotter
Daily Challenge
Find the gradient of the line through $(-2, 4)$ and $(3, -6)$, then find the gradient of $3x + 6y = 12$. Are they the same? Time yourself — can you do both in under 60 seconds?
Printable Worksheets
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