Key Stage 4 maths · Ratio, proportion and rates of change

GCSE higher: proportion equations and rates of change

This higher-tier topic extends proportion and rates of change. Pupils write equations for direct and inverse proportion, including squares, cubes and roots, interpret the gradient at a point on a curve as an instantaneous rate of change, compare it with an average rate of change found from a chord, and work with general iterative processes in which each output becomes the next input.

Proportion equations and rates of change, explained

Earlier work interprets proportion equations; here pupils build them. The statement “y is proportional to x²” becomes y = kx², and “y is inversely proportional to x” becomes y = k ÷ x. The method is always the same: write the equation with k, substitute the pair of values you are given to find k, rewrite the equation with that number in place, and then use it to answer the question. Relationships with squares, cubes and square roots turn up in geometry and science, where area depends on length squared and volume on length cubed.

Knowing the type of relationship also tells pupils what a change does without calculating. If y is proportional to x², doubling x multiplies y by 4. If y is inversely proportional to x², doubling x divides y by 4. Pupils are expected to reason like this and to recognise the shapes of the graphs of y = kx, y = kx², y = kx³ and y = k ÷ x. The same reasoning answers questions such as how pressure changes when the same force is spread over a larger area.

On a curve the rate of change is different at every point, so pupils distinguish two ideas. The average rate of change between two points is the gradient of the chord joining them. The instantaneous rate of change at a single point is the gradient of the tangent there. For y = x² at x = 2, the chord from x = 2 to x = 3 has gradient (9 − 4) ÷ 1 = 5, but from x = 2 to x = 2.1 it is (4.41 − 4) ÷ 0.1 = 4.1. As the second point moves closer, the answers approach 4, the gradient of the tangent.

In practice pupils draw a tangent with a ruler, make a large right-angled triangle on it, and read the rise and run from the axes, not by counting squares. They also calculate chord gradients numerically and algebraically, and say what the answer means in context: a speed of 12 m/s at a moment, or an average of £8 per day over a week. The units always come from dividing the axis units.

An iterative process repeats a rule, feeding each output back in as the next input. Starting with x₀ = 1 and using the rule “add 1 and take the reciprocal”, the values are 0.5, 0.667, 0.6, 0.625, 0.615, 0.619, and they close in on about 0.618 while alternating above and below it. Not every process settles: the rule “double it” grows without limit. Iteration is also how repeated percentage changes, loan balances and populations are modelled. Using iteration to solve equations is covered in the algebra strand.

Words your child will hear

Constant of proportionality: The fixed number k in an equation such as y = kx or y = k ÷ x.
Directly proportional: One quantity is a fixed multiple of another, or of a power of it, such as y = kx².
Inversely proportional: One quantity is a fixed number divided by another, such as y = k ÷ x.
Tangent: A straight line that just touches a curve at a point, with the same direction as the curve there.
Chord: A straight line joining two points on a curve.
Instantaneous rate of change: How fast something is changing at one exact moment: the gradient of the tangent.
Iteration: Repeating a rule, with each answer used as the next input.

What the National Curriculum says

In Key Stage 4, pupils should be taught to:

Construct equations that describe direct and inverse proportion
Interpret the gradient at a point on a curve as the instantaneous rate of change
Apply the concepts of instantaneous and average rate of change (gradients of tangents and chords) in numerical, algebraic and graphical contexts
Work with general iterative processes

Objectives from the National Curriculum in England, © Crown copyright, used under the Open Government Licence v3.0.

A worked example

y is inversely proportional to the square of x. When x = 4, y = 3. Write an equation connecting y and x, and find y when x = 2.

“Inversely proportional to the square of x” means y = k ÷ x², where k is a constant.
Substitute x = 4 and y = 3: 3 = k ÷ 4². Since 4² = 16, 3 = k ÷ 16, so k = 3 × 16 = 48.
Put k back in: y = 48 ÷ x². This is the equation connecting y and x.
Now use x = 2: y = 48 ÷ 2² = 48 ÷ 4 = 12.
Sense check: x has halved, so x² has quartered, and y should be four times as big. 3 × 4 = 12, which agrees.

y = 48 ÷ x², and when x = 2, y = 12.

Common mistakes and how to help

Treating any rising relationship as direct proportion

Pupils notice that both quantities increase and call it proportional, even for y = 2x + 3.

What helps: In direct proportion the ratio y ÷ x is constant and the graph passes through the origin. Test: when x = 0, y should be 0. For y = 2x + 3, it isn’t.

Setting up the equation in the wrong form

Mixing up “proportional to x²” (y = kx²) with “inversely proportional to x²” (y = k ÷ x²), or squaring the wrong thing, is common in word-heavy questions.

What helps: Say the relationship aloud and test the final equation. If y should fall when x rises, a k ÷ x² form is needed.

Using a chord when a tangent is needed, or a tangent that cuts the curve

Joining two points on a curve is easy to do. A tangent needs a ruler to be laid so that it touches at one point and follows the curve’s direction.

What helps: Pivot the ruler against the curve until it just touches at the point, with the curve staying on the same side of the ruler just before and just after it, and the gaps either side looking equal. Use the chord for averages only.

Reading gradients by counting squares instead of using the axis scales

If the axes have different scales, one square across does not mean the same as one square up, so the gradient is wrong.

What helps: Read the actual values at both ends of a large triangle, divide the rise by the run, and write the units: metres divided by seconds gives metres per second.

How to help at home

Share a job among more people. If tidying a room takes one person 60 minutes, how long for two, three or four people working at the same pace? Compare with the idea of inverse proportion, and talk about why real jobs never quite work that way.
Chase a tangent with a calculator. For y = x², work out the gradient between x = 2 and x = 2.1, then 2.01, then 2.001. The answers 4.1, 4.01 and 4.001 close in on 4.
Sketch a journey. Describe a bus journey with a slow start, a fast stretch and a stop. Draw the distance-time graph together and mark where the speed at that moment is greatest.
Try an iteration button game. Enter 1 on a calculator, then repeatedly take “1 ÷ (1 + the last answer)” using the answer key. Watch the answers settle near 0.618 and ask why they stop changing: at that number, 1 ÷ (1 + x) gives x back again.
Predict before you calculate. Ask what happens to y if y is proportional to x³ and x is trebled. Then check the answer with numbers. Estimating the effect first is a skill that carries across to science.

Builds on

Key Stage 3: Units, scale and fractions of quantities
Key Stage 3: Ratio and sharing in a ratio
Key Stage 3: Percentage change, proportion, compound units