Lesson 23 Rates of change

# Episode 1

Reasoning Resources: Worksheet 1
Whole class preparation
The simple algebraic formula gives rise to the table, and the graph is then drawn. Much of the initial discussions can concentrate on the patterns of change through the use of ‘step and rise’ pairs, in which the step is always the same, but the rise may not be. We're going to look at graphs today - starting with a simple graph and moving on to more complex graphs.

Write the function b = 4a on the board asking questions such as: what is the value of b when a is 3 or 10 etc? Introduce Worksheet 1 and make explicit the three headings in the table, and how the values in the ‘change in b' column have to be placed in-between the rows. (If the value for ‘change’ is placed in a row, such as in a spreadsheet on the computer, it must be understood that it is a result of a calculation in two rows.)
Pair work
Help pupils to complete the table and plot the graph. It may be necessary to show pupils how to put in the first ‘step and rise’ pairs.

Pupils could be reminded of earlier experiences of drawing linear graphs, including those in TM7: Which offer shall | take?, TM10: Rectangle functions, TM12: Functions, TM13: Chocolate box, and TM15: Circle functions. The emphasis here, however, is on a particular property of the graph rather than its overall shape.

Ask pupils to look at the ‘step and rise’ for different values of a and b — for example, comparing the rise in the value of b to the step from a = 1 to a = 2
We found that ‘step and rise’ in a graph is more appropriate than ‘change triangles' as this shape dominates and may distract younger pupils from recognising where the change comes from.
Whole class sharing and discussion
Encourage pupils to jot down their ways of explaining the pattern of changes in a and b in the table and on the graph, in order for them to contribute to a class discussion.

• What is the best way to describe the relationship?
• How does the graph help? Can you talk about the graph?
• What can we say about the rise in b for same size steps in a?

Get the class to see that on a linear function graph the ‘rise’ is the same regardless of where the ‘step’ starts from. In classes with a high attainment profile pupils may plot other linear graphs such as: b =2a and b = 6a etc.
• Different descriptions of the graph and the pattern it shows are possible. Even slight variations in wording are useful in pupils recognising their own power.
of generalisation, and that
all ideas are valued.
Notes on the maths
Different wording would bring out that as a increases the rises stay the same for the same size steps, i.e. the graph has a constant steepness. Relate this to the column ‘change in b’ which
is constant.
The simple start is necessary for 13-year-olds. The top 10% of the ability range would handle the whole of this lesson’s agenda in an ordinary instructional manner. Since this will not be the case in most classes, it will be necessary to approach it from the point of view of the pupils’ use of descriptive models — like the tables of changes in successive y-values, before progressing to comparison with other models. Different pupils will get to different points in thinking about their own strategies, so the greater the variety of ideas that can be exchanged in class discussion the better.