Lesson 23 Rates of change

Episode 3

Reasoning Resources: Worksheet 3 for pupils, Worksheet 3 (A2/A3 size) or on OHT
Whole class preparation
It is necessary to let pupils use the words ‘velocity’ (or speed) and ‘acceleration; (rate of change of velocity speed) in the process of connecting up their mathematical meaning with their everyday and scientific meaning. Equally necessary is to realise that the change in velocity is itself increasing continuously. This can be done by discussing how the change increase as the start of the step increases. Ask pupils if they have seen on TV rockets taking off upward very slowly to begin with, and what that means as distance travelled in the first second, second second and third second, etc.

Then also think of a balloon rising. Emphasise that balloons rise steadily while rockets get faster and faster.
Alternating pair and whole class work
Velocity and acceleration are discussed here as examples of the functions in context.

Constant velocity corresponds to a linear function, and is modelled by a balloon having reached constant rising velocity. Newton's 3rd Law says that constant acceleration is produced by constant force on a mass, and is modelled by the rocket climbing initially with constant rocket thrust. Here the function of distance with time is quadratic.
Give out Worksheet 3 to pairs. Pupils should then be able to complete the tables and graphs fairly quickly.

Most of this section can be handled as a whole class discussion with a large version of the graph on the board, and the teacher asking questions on ‘speed of rising; ‘how fast’ and in which the ‘rate of rising’ is itself rising.

Alternatively, when most pupils have plotted their graphs, the class can be asked to clarify the meaning and use of the terms ‘velocity’ and ‘acceleration’ They can then be given about 5 minutes to discuss with their partners how to describe their graphs in terms of these words or their
equivalent in everyday speech, and these can be compared in class discussion.
Whole class sharing/discussion
This part of the lesson could lead to a discussion of instantaneous velocity with more advanced Classes, i.e. the fact that the change of velocity is a continuous one
so that if we have infinite accuracy we will find that the velocity is different in each case, while in practice we assume it is constant on average in small periods.
Begin the discussion with inputs from each group without comment. Bear in mind that as the ideas flow many pupils may challenge each other and your role is one of mediation as the conflict within the graph is drawn out.

Typical ideas are recognitions that: ‘the balloon has risen to a higher height in a shorter time that the rocket', ‘the balloon and rocket do not meet in the air', ‘the graphs are not a picture of how they move in the air’

Many pupils will not have fully grasped the meaning of constant and changing velocity and this activity will highlight that for them and also productively challenge them as to the fundamental nature of graphs.An important consideration in the summary phase will be to include a discussion on the misreading of the two graphs. Visually the straight line can be seen as tracing the trajectory of a rocket shot at an angle, while the curve traces the trajectory of a balloon rising with the wind blowing from left to right! Pupils must clarify to themselves the difference between a graph of a function, here representing the relationship between time and height, and an actual picture of the path objects take.

With most classes a question can be posed on how the change in the rocket height, or the ‘rise, changes according to the start of the single step', e.g. What will the ‘rise’ be if we start the step at 2.5 seconds, or 3.5 seconds? This would hopefully bring to pupils’ attention the continuous, or ‘smooth, nature of the rate of change.

Look at both graphs — what can you say about the differences in steepness?(The rocket graph starts off with smaller step sizes than the balloon graph, but they very quickly become larger. This begins as soon as time is greater than 3.)

With all classes the ‘crossover’ values of the two graphs can be discussed. Higher attaining pupils can explore the crossover values for different linear functions keeping the quadratic function the same. They can tackle questions such as whether there will always be a crossover point, and how to describe the difference between the functions as they both become infinitely large with time.
End of Lesson Reflection
Pupils would benefit from discussing the flow of the lesson, how the tables and the graphs have different usages and where else they have met these ideas before. The intention is for them to have some word labels for some concepts and modes of work.


Thinking Mathematics Lessons Copyright © by Michael Shayer and Mundher Adhami. All Rights Reserved.

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