Rate of Change

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: $\text{speed} = \frac{\text{distance}}{\text{time}}.$

The three quantities — speed, distance, time — are linked, so any one can be found from the other two:

• $\text{distance} = \text{speed} \times \text{time}$
• $\text{time} = \text{distance} \div \text{speed}$

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.

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: $\text{speed} = \frac{\text{change in distance}}{\text{change in time}}.$

For the line above, the distance rises $60\text{ km}$ over a time of $2$ hours, so the speed is $60 \div 2 = 30$ km/h.

Reading the rest of a distance-time graph:

• A steeper line means a faster speed.
• A flat (horizontal) line means the distance is not changing — the object has stopped.
• A line coming back down means the object is returning towards the start.

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.

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: $\text{average speed} = \frac{\text{total distance}}{\text{total time}}.$

The key word is total. You use the whole distance and the whole time, including any time spent stopped.

Suppose a journey covers $120\text{ km}$ in a total of $3$ hours, even though there was a stop in the middle: $\text{average speed} = 120 \div 3 = 40\text{ km/h}.$

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.

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:

• A flat (horizontal) line means a steady speed — the speed is not changing.
• A sloping line going up means the speed is increasing — speeding up, or accelerating.
• A sloping line going down means the speed is decreasing — slowing down.

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 $0$ to $20$ m/s in $4$ seconds, the acceleration is $20 \div 4 = 5$ metres per second, each second.

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.

The gradient-as-rate idea works for any graph that compares two quantities, not just journeys.

A few examples of rates read from graphs:

• A graph of cost against weight — the gradient is the price per kilogram.
• A graph of water in a tank against time — the gradient is the filling rate in litres per minute.
• A graph of pay against hours worked — the gradient is the pay rate per hour.

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: $\text{rate} = \frac{\text{change in the vertical quantity}}{\text{change in the horizontal quantity}}.$

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.

The rate-of-change skills you have built in Stage 9 carry straight into harder maths:

• Gradients of straight lines — finding the steepness of any line is exactly the rate of change.
• Linear functions — equations such as $y = mx + c$ use the gradient $m$ as the rate.
• Compound measures — speed, density and pressure are all rates you will use again.
• In everyday life — speed limits, prices, wages, mobile-data use and fuel economy are all rates.

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.