Gradient and Rate of Change: KS3 Maths Explained

The gradient of a straight line measures its steepness — how much the line rises (or falls) for every one unit it moves to the right. Gradient = rise ÷ run. A positive gradient means the line slopes upward left-to-right; a negative gradient means it slopes downward. A horizontal line has gradient zero.

What is gradient and why does it matter?

Gradient tells you the rate of change of one quantity with respect to another. On a distance–time graph, gradient = speed. On a cost graph, gradient = price per unit. Understanding gradient bridges pure maths and real-world contexts throughout KS3, GCSE, and beyond.

The gradient of a straight line is constant — it is the same at every point on the line. This is what makes lines straight. Curves have gradients that change, which is why studying straight lines first is essential.

How do you find the gradient from a graph?

The rise-over-run method:

Choose two points on the line that lie on grid intersections (easy to read exactly).

Draw a right-angled triangle between them: a horizontal step (the run) and a vertical step (the rise).

Gradient = rise ÷ run.

Take care with direction:

If the line goes up from left to right, the rise is positive → positive gradient.

If the line goes down from left to right, the rise is negative → negative gradient.

Worked example 1: A line passes through (1, 2) and (4, 8).

Rise = 8 − 2 = 6. Run = 4 − 1 = 3.

Gradient = 6 ÷ 3 = 2

The line goes up 2 units for every 1 unit it moves right.

Worked example 2: A line passes through (0, 6) and (4, 2).

Rise = 2 − 6 = −4. Run = 4 − 0 = 4.

Gradient = −4 ÷ 4 = −1

The line slopes downward — for every 1 unit right, it falls 1 unit.

How do you calculate gradient from two coordinates?

The gradient formula for two points (x₁, y₁) and (x₂, y₂) is:

Gradient m = (y₂ − y₁) ÷ (x₂ − x₁)

This is the same as rise over run, but in algebraic form. The order of the points does not matter, as long as you subtract in the same order in both numerator and denominator.

Worked example 3: Find the gradient of the line through (−2, 5) and (3, −5).

m = (−5 − 5) ÷ (3 − (−2)) = −10 ÷ 5 = −2

Worked example 4: Find the gradient of the line through (0, −3) and (6, 0).

m = (0 − (−3)) ÷ (6 − 0) = 3 ÷ 6 = 0.5

What do different gradient values mean?

Gradient | Line appearance
Large positive (e.g. 5) | Steep upward slope
Small positive (e.g. 0.2) | Gentle upward slope
Zero | Horizontal (flat)
Small negative (e.g. −0.5) | Gentle downward slope
Large negative (e.g. −4) | Steep downward slope
Undefined | Vertical line (the run is 0)

How does gradient link to y = mx + c?

Every straight line can be written as y = mx + c, where:

m is the gradient.

c is the y-intercept (where the line crosses the y-axis).

If you know the gradient and the y-intercept, you can write the equation immediately. If you know the gradient and one point, substitute to find c.

Worked example 5: A line has gradient 3 and passes through (0, −4). Write the equation.

m = 3, c = −4 → y = 3x − 4.

Worked example 6: A line has gradient −2 and passes through (1, 7). Find c.

y = −2x + c. Substitute (1, 7): 7 = −2(1) + c → c = 9. Equation: y = −2x + 9.

How is gradient interpreted as a rate of change?

The gradient tells you how fast one variable changes relative to another.

On a distance–time graph (distance on y, time on x): gradient = speed in distance per unit of time.

On a cost graph (total cost on y, number of items on x): gradient = cost per item.

On a temperature–time graph: gradient = rate of heating or cooling in degrees per minute.

A steeper gradient always means a faster rate of change. A negative gradient means the quantity is decreasing.

Frequently asked questions

Does it matter which two points I choose when reading gradient from a graph?

No — for a straight line, the gradient is the same between any two points on the line. However, choosing points where the line crosses grid intersections makes the reading more accurate. Avoid choosing points where the line falls between grid lines.

What if the gradient calculation gives a fraction?

A fractional gradient is perfectly valid. A gradient of ½ means the line rises 1 unit for every 2 units moved to the right. Express it as a fraction or decimal — both are acceptable at KS3. A gradient of ⅔ means "rise 2, run 3."

What is the gradient of a horizontal line?

A horizontal line has zero rise for any horizontal distance, so gradient = 0 ÷ run = 0. A horizontal line is described by an equation of the form y = k (a constant). A vertical line has zero run, making the gradient division undefined; it is described by x = k.

How is gradient tested at KS3 compared to GCSE?

At KS3, the focus is on reading gradient from graphs and identifying positive, negative, and zero gradients. At GCSE, you use the gradient formula with two points, interpret m in y = mx + c, find equations of lines, and interpret gradients in context. Both stages start from the same idea: rise ÷ run.

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