Lesson 12 Functions

# Episode 3

Reasoning Resources: Tables from Worksheet 1 on board on OHT and pupils' copies, Worksheet 3
Whole class preparation
The focus of this lesson is on generating many-layered pupil discussion. The more ways of looking at the numbers that the pupils come up with, the more each pupil will be helped in his/her thinking at the point where they currently are.

Show the completed table of input and output values from Worksheet 1. Point out that there is only one output value for each input. Explain that a relationship like this is called a function. In this case as the input increases, the output also increases. Ask pupils: If you drew a graph using these values, with input on one axis and output on the other, what do you think it would look like? Record, sketch and discuss their ideas.
Pair and small group work
Some pupils may understand that all the points on the line drawn connecting the plotted values can be meaningful in terms of turns and part turns of the cog. Give out Worksheet 3 (or a modified version) to pairs. Now they can check their ideas. Help pupils with difficulties in plotting the graph of output against input. Some pupils may plot a point off, but will be able to self-correct when shown that it doesn't fit the pattern of the other points.
Whole class sharing and discussion
The generalised number aspect of the activity is aimed at making pupils more familiar with some of the language of algebra.

The steepness of the graph is intuitively addressed. Some pupils may want to look at this as expressing how much bigger the values of one variable are compared with the other.
Discussion should include that the graph is a straight line/goes up at the same rate throughout and that all output values are 5 times the value of the corresponding input. Ask: How is the ‘5 times’ relationship shown on the graph? Encourage pupils to look at the ‘steepness, e.g.’For every 1 across, the graph goes up 5‘

Pupils then predict output values from inputs that have not been plotted, by calculation or using the graph. Ask pairs to share their methods/results with the class. Ask the pupils what they think their results mean in terms of the cog wheels and the number of teeth in each.
In question 4 pupils find some way of expressing this function in terms of generalised number symbols. Record and discuss all suggestions, pointing out equivalent statements.

As an extension pupils may imagine other pairs of cogs with different numbers of teeth, and discuss or plot their graph. They can also deal with situations where the output is known and the question is to find the input.
Making connections across subjects and contexts. End of Lesson Reflection
Ask pupils to share their ideas on how their work in this activity relates to other work they may have done, whether in ratio or graphs, in mathematics or science lessons.