GCSE MATHS • HIGHER TIER
Finding Gradient Made Simple
Learn how to measure the steepness and direction of a straight line using rise over run or two coordinates, with clear GCSE examples and checks.
Clear method
Exam practice
Worked proofs
Find the Gradient in Four Moves
1
Choose Two Points
Select two clear points on the same straight line. On a graph, use exact grid intersections whenever possible.
2
Find the Change in y
Subtract the first y-coordinate from the second: y₂ − y₁. This is the vertical change or rise.
3
Find the Change in x
Subtract the first x-coordinate from the second in the same order: x₂ − x₁. This is the horizontal change or run.
4
Divide and Simplify
Calculate gradient = change in y ÷ change in x, simplify the result and check whether its sign matches the line’s direction.
What You’ll Learn
Describe Gradient
Explain gradient as the steepness and direction of a straight line.
Use Rise Over Run
Calculate vertical change and horizontal change accurately from a graph.
Use Two Coordinates
Apply m = (y₂ − y₁)/(x₂ − x₁) while keeping both subtractions in the same order.
Interpret the Sign
Recognise positive, negative and zero gradients from the direction of a line.
Recognise Undefined Gradient
Explain why a vertical line has an undefined gradient.
Check Your Answer
Use the line’s direction and a second calculation to confirm the result is reasonable.
Key Rules
Gradient Formula
m = (y₂ − y₁) ÷ (x₂ − x₁)
Rise Over Run
gradient = vertical change ÷ horizontal change
Positive Gradient
m > 0
Negative Gradient
m < 0
Horizontal Line
m = 0
Vertical Line
gradient is undefined
Worked Proof: Arc Length
Step-by-Step Method
- Choose two clear points on the straight line.
- Find the change in y: subtract the y-coordinates in the same order.
- Find the change in x:subtract the x-coordinates in the same order.
- Divide the change in y by the change in x.
- Simplify the gradient and check whether the sign matches the direction of the line.
More Worked Examples
1
Coordinates (2, 3) and (6, 11)
2
Coordinates (1, 2) and (5, 10)
3
Coordinates (0, 5) and (4, 13)
4
A Rise of 12 and a Run of 3
5
A Fall of 9 and a Run of 3
6
Coordinates (−2, 1) and (4, 7)
- Common GCSE Mistakes
- Reversing rise and run
- Mixing subtraction orders
- Losing a negative sign
- Choosing unclear graph points
- Reading the scale incorrectly
- Calling a vertical gradient zero
- Exam Tips
- Write change in y above change in x before substituting any numbers.
- Label the two points consistently as point 1 and point 2.
- On a graph, draw a large rise-and-run triangle to reduce counting errors.
- Check both axis scales because they may use different intervals.
- Use the line’s direction to predict whether the answer should be positive, negative or zero.
Skills Used in Finding Gradient
Equation of a Line
Gradient and Intercept
Parallel and Perpendicular Lines
Plotting Coordinates
Ready to Practise?
Printable Worksheet
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Interactive Quiz
Test your knowledge with instant feedback and hints.
Exam-Style Questions
Timed questions to build confidence for the real exam.
Watch the Finding Gradient Tutorial
Step-by-step walkthroughs of Finding Gradient with clear methods and exam tips.
Frequently Asked Questions
Gradient measures the steepness and direction of a straight line. It tells you the vertical change for each unit of horizontal change.
Use m = (y₂ − y₁) ÷ (x₂ − x₁). Subtract the coordinates in the same order and divide the change in y by the change in x.
Choose two exact points on the line, preferably at grid intersections. Calculate the vertical change and horizontal change between them, then divide rise by run.
A negative gradient means the line falls as you move from left to right. For example, m = −2 means the line falls 2 units for every 1 unit moved right.
A horizontal line has gradient 0 because its vertical change is zero. A vertical line has an undefined gradient because its horizontal change is zero and division by zero is not defined.
Yes. You may choose either point first, provided both subtractions use the same order. Reversing both the numerator and denominator gives the same gradient.