Lesson 27 Accelerating the acceleration

Episode 1

Reasoning Resources: Worksheet 1 on A2/A3 or OHT, Worksheet 1 for pupils
Whole class and pair work
This activity is essentially about metacognition rather than construction. Pupils reflect on the qualitative differences between constant speed and other kinds of acceleration, and where these might apply in life. Remind pupils of the work they did in TM23: Rates of change. Show them the graph for thenrise of the balloon and the rocket, on Worksheet 1. Give out Worksheet 1 and ask:

• Describe in words the difference between the risings of the two different craft, using the words ‘speed’ and ‘acceleration’,
• When you double the time, in each case, how many times does the height they have raised increase?

• Whole class comparison of linear and quadratic functions
The relationship is not simply a sequence. It is possible to calculate the speed at any point in time without having to know the sequence. Many pupils would accept a formula such as: Height = 6 X Time in seconds or H = 6T as an algebraic representation for the balloon movement. Some would accept H =T2 for the rocket.

But algebraic symbolisation is not necessary for the understanding we aim for in this lesson. The very process of thinking about the comparison as comparing two or three different kinds of possible models provides another push towards, or into, formal thinking.
In class discussion mode, keep asking different groups to contribute until you feel that they are using terms such as ‘speed’ and ‘acceleration’ in a way that describes the two graphs. Here we are not after mathematical accuracy; rather they should use everyday words to describe these things in both mathematics and science. For example, for the case of the balloon where the speed stays the same: ‘it means every second they go the same distance’, ‘the graph keeps the same slope’, ‘stays a straight line’, ‘the table shows change of height in equal steps’, ‘like a staircase with all the risers and treads the same’

For the case of the steady acceleration of the rocket, accept explanations such as: ‘It starts slower than the balloon but the change of height gets bigger and bigger’, ‘It makes up for the slow rise in 8 seconds’, ‘If you double the time, the rise per second doubles, while with the balloon the rise doesn't change’.

Pupils may recognise the square function, and you need to get them to clarify what variable is the square of what other variable, i.e. to see the relationship not simply as a sequence, but that this relationship applies for different values of time not noted in the table. For example, at time = 40, the height would be 1600 m.