Lesson 23 Rates of change
|Reasoning||Resources: Worksheet 2|
|Whole class preparation|
|As in Episode 1, write the function b =a ? on the board and ask for the value of b for different values of a.|
|Mathematics research suggests that it helps to approach rates of change by first tabulating how successive values of the independent variable are changed at a constant rate. If the function is linear, then the differences are constant, whereas if it is of power two, the differences themselves increase at a constant rate. Better understanding of this is likely to be achieved by pupils seeing how the tabulated numbers correspond to visually accessible changes on the graph.||Pupils should repeat the process for Episode 1 for the quadratic function on Worksheet 2. They may need more time to present their ideas on the quadratic function. If possible, ask pupils for phrases in natural language which fits the graph such as:'‘the amount of change is itself changing; ‘the increase is increasing steadily’, or even ‘accelerating.'|
|Whole class sharing and discussion|
|Can you talk about this graph? How is this graph different to the last one?
Get the class to see how the ‘rise’ on the graph of a quadratic function is dependent on the starting point of the ‘step’ Then have them formulate sentences on the comparison between the two graphs. Recognition that the ‘change is changing’ could be confusing, so it is helpful to give them time to focus upon this and to articulate it in their own way.
Pupils may wish to come out and annotate the graph at the front as they realise how different the two algebraic functions actually look.