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Stage 9 takes rates further: read distance-time graphs with confidence, meet the speed-time graph and the idea of acceleration, calculate average speed across a whole journey, and interpret the rate from any real-world graph.
Mapped to the Cambridge Lower Secondary Mathematics curriculum framework.
A rate compares two quantities; speed compares distance with time.
In Grade 7 you met the idea of a rate — a comparison of how one quantity changes against another. The word per ("for each") is its signpost: km per hour, dollars per kilogram, litres per minute.
The most familiar rate is speed:
The three quantities — speed, distance, time — are linked, so any one can be found from the other two:
The steepness of a distance-time line is the speed — rise over run.
A distance-time graph shows a journey as a picture: time along the bottom, distance up the side. In Stage 9 you read these more precisely.
The steepness of a line — its gradient — is the speed. You find it by comparing how far the line rises with how far it travels across:
Average speed uses the total distance and the total time — including stops.
Real journeys are rarely at a single speed — they speed up, slow down and stop. To describe the journey overall, you use the average speed:
The key word is total. You use the whole distance and the whole time, including any time spent stopped.
A speed-time graph shows how speed itself changes — its slope is acceleration.
A speed-time graph is a different kind of graph: time still runs along the bottom, but now speed goes up the side.
This makes the lines mean something new:
The steepness of a speed-time line is a rate too — the rate at which speed changes. That rate is called acceleration. If a car's speed rises from to m/s in seconds, the acceleration is metres per second, each second.
Any straight-line graph of two quantities has a rate — its gradient.
The gradient-as-rate idea works for any graph that compares two quantities, not just journeys.
A few examples of rates read from graphs:
For each one, you find the rate the same way: pick two clear points, work out the change up and the change across, then divide:
Rate of change leads on to gradients, linear graphs and compound measures.
The rate-of-change skills you have built in Stage 9 carry straight into harder maths:
If a later graphs topic feels tricky, the rate idea — how much one quantity changes for each unit of another — is almost always underneath it. Keep this guide close.
Sources: Cambridge Lower Secondary Mathematics curriculum framework (Stage 9). Last reviewed 2026-05-19.
Step-by-step solutions for rate of change, written exactly the way a tutor would explain them at the board.
Question
A train travels in hours. Find its speed.
Step-by-step solution
Step 1
Speed is found by dividing the distance by the time taken.
Step 2
Substitute the distance and the time.
Answer
Question
A cyclist rides at for hours. How far do they travel?
Step-by-step solution
Step 1
Rearrange to find distance: multiply the speed by the time.
Question
On a distance-time graph, a line rises over a time of hours. Find the speed.
Step-by-step solution
Step 1
The speed is the gradient of the line — the rise divided by the run.
Question
A car covers in a journey that takes hours in total, including a stop. Find the average speed.
Step-by-step solution
Step 1
Average speed uses the total distance and the total time, including any time stopped.
Question
On a speed-time graph, a car's speed rises from to over seconds. The speed change in m/s is . Find the acceleration.
The important words for rate of change and what they mean — learn these so you can explain your thinking clearly.
A comparison of how one quantity changes against another, such as kilometres per hour.
The rate at which distance is covered: .
Example
Travelling in hours is a speed of .
A graph with time across the bottom and distance up the side, showing a journey as a line.
The steepness of a line, found as the change up divided by the change across.
The total distance of a journey divided by the total time, including any stops.
Example
Covering in hours gives an average speed of .
A graph with time across the bottom and speed up the side, showing how speed changes.
The rate at which speed changes, measured in units such as metres per second, each second.
A speed that does not change — shown by a straight line on a distance-time graph.
Not moving at all. On a distance-time graph this is a flat, horizontal line.
A word meaning 'for each' or 'divided by', the signpost of a rate — as in km per hour.
The change up (rise) and the change across (run) used to work out a gradient.
A measure that combines two quantities, such as speed (distance and time).
The slip-ups students most often make with rate of change — and simple ways to avoid them.
Why it happens
It is easy to forget that time runs in 60s, not in powers of 10.
How to avoid it
Convert minutes to hours first: minutes is hours.
Why it happens
A flat line still looks like part of the journey, so it seems like movement.
How to avoid it
A flat line means the distance is not changing — the object has stopped completely.
Why it happens
Averaging numbers usually means adding and halving, so it feels natural.
How to avoid it
Use total distance divided by total time. Averaging speeds only works if the parts last equal times.
Why it happens
The two graphs look similar, but the up-axis is measuring different things.
How to avoid it
Check the axis label first. On a speed-time graph a flat line means steady speed, not a stop.
Why it happens
The two changes get swapped when setting up the fraction.
How to avoid it
Always put the change up (the rise) on top: .
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In Stage 9 the work goes deeper: you read journeys from graphs, you find an average speed across a journey with several parts, and you meet a brand-new graph — the speed-time graph.
One habit carries through everything here: keep your units consistent. If a speed is in km/h, the time must be in hours. Convert minutes to hours before you calculate.
For the line above, the distance rises over a time of hours, so the speed is km/h.
Reading the rest of a distance-time graph:
Because the gradient is the speed, a distance-time graph lets you read off the speed of every part of a journey just by looking at how steep each section is.
Suppose a journey covers in a total of hours, even though there was a stop in the middle:
Notice the average speed is not the same as the speed during the moving parts — those parts were faster, because no distance is covered while stopped. The average smooths the whole journey into one figure.
A reliable plan: add up all the distance, add up all the time, then divide. Do not average the separate speeds — that gives the wrong answer when the parts last different lengths of time.
Be careful — a speed-time graph reads very differently from a distance-time graph. On a speed-time graph a flat line is moving steadily, not stopped. Always check which graph you are reading before you describe a journey.
A steeper line always means a greater rate — a higher price per kg, a faster filling rate, a better pay rate. And a flat line means the vertical quantity is not changing at all. Whenever you meet a straight-line graph, ask "how steep is it?" — that steepness is the rate.
Step 2
Substitute the speed and the time.
Answer
Step 2
The rise is and the run is hours.
Answer
Step 2
Substitute the total distance and the total time.
Answer
Step-by-step solution
Step 1
Acceleration is the rate at which speed changes — the change in speed divided by the time.
Step 2
Use the speed change in m/s so the units match: the speed rises by .
Step 3
Divide the change in speed by the time of 5 seconds.
Answer