Maths

Understanding slope as rate of change

By the Edny editorial team
Published 7 October 2026

A graph can show how far someone has walked. Its slope tells you how quickly that distance is changing.

Suppose a walker covers 120 metres in two minutes at a steady pace. For every minute, the distance increases by 60 metres. On a graph with time along the bottom and distance up the side, the slope is 60 metres per minute.

Slope becomes easier to understand when you give it that sentence: “The vertical quantity changes by this much for each unit of the horizontal quantity.”

Build a walking graph

Here is an imaginary walk, chosen to keep the arithmetic simple. The walker starts at the beginning of a straight path and moves at a constant speed.

• At 0 minutes, the total distance walked is 0 metres
• At 1 minute, it is 60 metres
• At 2 minutes, it is 120 metres
• At 3 minutes, it is 180 metres
• At 4 minutes, it is 240 metres

Label the horizontal axis “Time, minutes” and the vertical axis “Total distance walked, metres”. Plot those five points and join them with a straight line. The line rises steadily from left to right.

This is a simplified model. A real person might speed up, slow down or stop. The straight line represents the constant pace we have assumed for this example.

Calculate the slope between two points

Choose the points at 1 minute and 3 minutes. Their coordinates are (1, 60) and (3, 180).

The time change is 3 − 1 = 2 minutes. The distance change is 180 − 60 = 120 metres.

Divide the distance change by the time change:

Slope = 120 metres ÷ 2 minutes = 60 metres per minute.

You may also see this written as “rise over run”. Rise means the change in the vertical quantity. Run means the change in the horizontal quantity. On this graph, the rise is 120 metres and the run is 2 minutes. Neither word means you must physically climb or run.

The general formula is slope = (y₂ − y₁) ÷ (x₂ − x₁). Keep the points in the same order in both subtractions, and make sure the horizontal change is not zero.

Why the units belong in the answer

“Slope = 60” leaves out something important. Sixty what?

Here it means 60 metres for every minute. If you changed the time axis to seconds, the very same walk would have slope 1 metre per second. The person has not changed speed; the units have changed.

The graph's shape on the page also depends on the axis scales. Stretching the picture vertically makes the line look steeper without making the walker faster. Read the labelled values and calculate the rate rather than judging speed from the angle alone.

A flat section tells a different story

Imagine that after four minutes the walker rests for two minutes. During the rest, time increases from 4 to 6 minutes, but total distance stays at 240 metres.

The slope over that section is (240 − 240) ÷ (6 − 4) = 0 metres per minute. The graph is horizontal because no extra distance is being covered.

A total-distance-walked graph cannot go down as time passes: walking back still adds to the distance travelled. A different graph, showing distance from the starting point, could slope down when someone returns towards the start. Always read what the vertical axis measures.

Two mistakes worth checking

First, use changes between the points, rather than dividing whichever two coordinates look convenient. A walker might already be 100 metres from a marker when a timer starts. The starting value and the rate are different pieces of information.

Second, counting grid squares only works if you account for their values. If one vertical square is 20 metres and one horizontal square is one minute, a rise of three squares over two squares means 60 ÷ 2 = 30 metres per minute.

For a straight line, any two distinct points give the same slope. For a walk with changing pace, the calculation between two times gives the average rate over that interval.

Three questions to try

1. Total distance rises from 80 to 200 metres between minute 2 and minute 5. What is the average rate?

Answer: 40 metres per minute. The distance change is 120 metres and the time change is 3 minutes. Divide 120 by 3.

2. A walker stays at 300 metres of total distance from minute 5 to minute 7. What is the slope?

Answer: 0 metres per minute. Time changes by 2 minutes; distance changes by zero. The graph has a horizontal section.

3. At a steady 50 metres per minute, how much extra distance is covered in 4 minutes?

Answer: 200 metres. Multiply the rate by the time: 50 × 4. This is additional distance; a starting reading would be added separately.

Explain it in one sentence

Choose a section of a distance–time graph and finish: “For every minute, the distance increases by ___ metres.” Then check your sentence against two points.

Parents can see how Edny works: spoken explanations on a shared whiteboard. AI explanations can be mistaken, so check important work.