Speed & Distance-Time Graphs

For ages 11–12, teach average speed as a proportional rate linking distance and time, built up from distance = speed × time ('m/s means metres travelled each second') before introducing speed = distance ÷ time. The central threshold is that average speed = TOTAL distance ÷ TOTAL time, including any stationary periods — not the mean of the separate speeds (the mean is only correct when equal time is spent at each speed). Use double number lines and ratio tables to expose the proportional structure and support prediction, connect the calculation to the gradient of a distance-time graph (a single leg's speed is its segment gradient; the whole-journey average is the start-to-end chord gradient), and confront the 'mean of the speeds' and 'ignore the stop' errors directly. A journey-building activity with live, synchronised feedback is the richest approach; a quiz consolidates the procedure; a worked-example reference card scaffolds a single case.

• Calculating average speed as the mean of the individual speeds rather than total distance ÷ total time (Third Space Learning).
• Forgetting to include stationary periods in the total time when finding average speed.
• Reading a value off the vertical axis and calling it the speed instead of finding the gradient.
• Treating the formula triangle as the whole concept — procedural recall that blocks proportional reasoning (Susac et al., Phys. Rev. PER).
• Language slips: 'constant speed' read as 'always moving', 'steady speed' as 'not too fast'; conflating distance with displacement (Oak National Academy).

• Double number line linking distance and time (NCETM) — the central proportional model
• Ratio table for distance:time, scaled by a functional multiplier
• Distance-time graph with gradient = average speed
• The speed-distance-time 'formula triangle' (a memory aid, used with caution, not as a substitute for reasoning)
• Unit unpacking: m/s as 'metres each second', km/h, mph

• Teach distance = speed × time before speed = distance ÷ time; anchor on 'm/s = the number of metres travelled each second' (Oak National Academy).
• Use double number lines and ratio tables (NCETM) to make the proportional relationship visible — e.g. scaling distance and time together to predict how far in a new time — rather than relying solely on the formula triangle, which can short-circuit reasoning.
• Define average speed explicitly as total distance ÷ total time, including stopped periods; give journeys with stops so the 'mean of the speeds' error is confronted.
• Connect the calculation to the graph: average speed of a straight segment IS its gradient; have pupils compute it both ways and reconcile.
• Use precise language — qualify 'constant'/'steady' with a value ('a constant speed of 4 m/s') and always state units; keep distance distinct from displacement.
• Use predict-then-calculate tasks (estimate the speed, then compute) and real contexts (cyclists, cars) to keep the abstraction grounded.