lesson
Gradient describes a rate
1 · Learn
On a distance-time graph, gradient is change in distance divided by elapsed time. A horizontal segment means the recorded distance from the start is unchanged over that interval. Steeper straight segments indicate a greater constant rate in the stated model.
Read units and axes carefully. Distance-time and speed-time graphs represent different quantities; their heights and areas do not have interchangeable meanings.
The supplied piecewise graph is a DIFFERENT journey along a straight path away from the start. Its points are (0 s,0 m), (4 s,20 m), (6 s,20 m), (10 s,50 m), joined by straight segments. For this model the traveller moves away at 5 m/s, waits from 4 to 6 seconds, then moves away at 7.5 m/s. The corners mark changes of behaviour, not a smooth curve.
To find when the traveller reaches 35 m, read horizontally from 35 to the final segment, then down to time: about 8 s. Check by interpolation: the last 4 seconds add 30 m, so adding 15 m takes 2 seconds after t = 6. The whole-journey average speed is 50/10 = 5 m/s, including the wait; it is not the average of the two moving speeds. This interpretation uses the explicitly straight, outward path, unlike an unrestricted distance-from-start record.
2 · Worked example
Distance rises from 20 m at 2 s to 50 m at 8 s. The segment's gradient is 30/6 = 5 m/s.
3 · Your turn
A straight segment runs from (3 s,12 m) to (7 s,28 m). Find its gradient.
Check your answer
(28 - 12)/(7 - 3) = 4 m/s.
4 · Apply your learning
Describe a journey with moving and stationary intervals and state which details a distance graph cannot reveal.
Adult guidance and safety
Require reasoning and a check. Provide accurate graphs, construction instruments and varied practice. Annual KS3 allocation is proposed; do not treat these short models as a complete assessed programme.
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Check a calculation · Year 5 onward
Use this only when your teacher or adult allows a calculator for the task. Show your thinking first. It does not replace mental or written arithmetic and does not mark your lesson answer.
Powers, roots and trigonometry · Year 7 onward
Use only when the task allows a calculator. Choose one operation at a time and record your working. Trigonometry here always uses degrees (DEG), not radians. Inverse sine, cosine and tangent return an angle, not a reciprocal. Keep exact fractions and π in answers when requested.
Approximate numerical results use up to 12 significant digits. Inverse sine returns −90° to 90°, inverse cosine 0° to 180°, and inverse tangent −90° to 90°; these are principal values, not every possible solution of a trigonometric equation.
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