Level 5: Slope as a rate

Change and motion · level 5 of 8 · rough age 12-13 (a guide for parents; anyone can start anywhere)

In level 4 you found rates of change from tables. In this level the rate becomes something you can see: the slope of a graph. The slope of a distance-time graph is a speed, the slope of a cost graph is a price, and the slope of a speed-time graph is an acceleration. You learn to read the sign of a slope as part of a story, to find the average rate along a curve with a secant line, and to work out how things fall on Earth and on Mars, ending with a surprise: the area under a speed-time graph is the distance travelled.

Lesson	You will be able to
Slope means rate	Find the slope of a graph with its units, and say what it means as a speed, a price or a flow rate
Positive, negative, zero and changing rates	Read the sign of a slope in a story, tell speed from velocity, and describe curves that speed up or slow down
Average rate of change on a curve	Find the average rate between two points of a curve with a secant line, and prove the falling stone's pattern with a picture
Acceleration	Work out acceleration in m/s², read it as the slope of a speed-time graph, and handle speeding up and braking
Falling, on Earth and on Mars	Use \(d = 4.9t^2\) and \(d \approx 1.86t^2\), explain where the 4.9 comes from, and compare falls on two planets
Area under speed-time graphs	Find distances as areas of rectangles, triangles and trapezoids, and see why the area must be the distance

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