In short: To find the gradient between two coordinates, divide the change in y by the change in x: m = (y₂ − y₁) ÷ (x₂ − x₁). Subtract the coordinates in the same order on the top and the bottom, then simplify the fraction.
The gradient between two points tells you how steep a straight line is and whether it rises or falls. It is a core GCSE Higher maths skill (AQA) that feeds into equations of lines, parallel and perpendicular lines, and rates of change. This guide shows the one formula you need, a worked example with the arithmetic checked, and the traps that cost easy marks.
The reliable method
You are given two coordinates, A (x₁, y₁) and B (x₂, y₂). Follow these steps.
- Label the coordinates. Decide which point is A and which is B. It does not matter which is which, as long as you stay consistent.
- Find the change in y. Subtract the y-values: y₂ − y₁. This is the vertical change, sometimes called the rise.
- Find the change in x. Subtract the x-values in the same order: x₂ − x₁. This is the horizontal change, or run.
- Divide and simplify. The gradient is m = (change in y) ÷ (change in x). Simplify the fraction or write it as a whole number. A positive value rises left to right; a negative value falls.
Keep the subtraction order identical on top and bottom. That single habit prevents most sign errors.
A worked example
Find the gradient of the line joining the points (1, 2) and (5, 14).
Step 1 — label. Let A = (1, 2) so x₁ = 1, y₁ = 2. Let B = (5, 14) so x₂ = 5, y₂ = 14.
Step 2 — change in y.
y₂ − y₁ = 14 − 2 = 12
Step 3 — change in x.
x₂ − x₁ = 5 − 1 = 4
Step 4 — divide and simplify.
m = 12 ÷ 4 = 3
So the gradient is 3. For every 1 unit you move right, the line rises 3 units.
Sanity check. Swapping the labels gives (2 − 14) ÷ (1 − 5) = (−12) ÷ (−4) = 3 — the same answer, which confirms the order of labelling does not change the gradient.
This works because gradient is a ratio: the vertical change measured against the horizontal change. As long as both subtractions run in the same direction, the ratio stays the same.
Common mistakes to avoid
- Mixing the subtraction order. Doing y₂ − y₁ on top but x₁ − x₂ on the bottom flips the sign and turns a positive gradient negative. Subtract in the same order both times.
- Dividing run by rise. The gradient is change in y over change in x, not the reverse. Putting the horizontal change on top gives the wrong number entirely.
- Treating the gradient as a total, not a rate. The gradient is the change per single step in x, not the total height gained across the whole line. If you read it as a total, see [gradient as total change explained](/misconceptions/gradient-as-total-change).
- Forgetting to simplify. Leaving the answer as 12/4 can lose a mark where a simplified or exact value is expected. Reduce the fraction or write the whole number.
- Sign slips with negatives. When a coordinate is negative, use brackets: 3 − (−2) = 5, not 3 − 2 = 1.
Frequently asked questions
How do you calculate the gradient between two coordinates? Subtract the y-values to get the change in y, subtract the x-values in the same order to get the change in x, then divide: m = (y₂ − y₁) ÷ (x₂ − x₁). Simplify the result to a fraction or whole number.
What does the gradient actually tell you? The gradient measures steepness and direction. A gradient of 3 means the line rises 3 units for every 1 unit across. A negative gradient means the line falls as you move right.
Does it matter which point you call A and which you call B? No. As long as you subtract the x-values and y-values in the same order, you get the same gradient. Swapping both points just multiplies the top and bottom by −1, which cancels out.
What is the gradient of a horizontal line? Zero. The y-values do not change, so the change in y is 0, and 0 divided by any number is 0. A horizontal line has no steepness.
What if the change in x is zero? Then the line is vertical and the gradient is undefined, because you cannot divide by zero. Vertical lines are written as x = a number instead.
Practise this
See which slips cost you marks — [take the free diagnostic](/diagnostic). Related: [gradient as total change explained](/misconceptions/gradient-as-total-change).