Lesson 12 Functions

# Episode 1

Reasoning Resources
Whole class preparation
Show pupils the picture of cogs on Worksheet 1, draw out and develop their understanding of how cogs work, and where they are used, e.g: ‘in a car’, ‘in a windmill', ‘on bikes’, ‘large one will go round slower ...’, ‘the smaller one has a smaller distance to travel until it’s gone all the way round!

Give out Worksheet 1 and write the ordered pairs of values from the worksheet on the board. Explain these can be looked at in terms of an input variable - something you can change — and an output variable - something you measure after fixing the input. Ask them to explain this to each other, which should allow wordings such as ‘in-out’, ‘put-in and put-out’, ‘in a machine, you do something then it does something; or similar. Some pupils will be familiar with the idea of a ‘function machine’ from primary school work.
Pair work
From separate pairs of numbers, to a general relationship.

Inverse relationship is accessible.

Some familiar numbers allow pupil to use ‘stepped correspondence; with adding corresponding numbers to each side of a relationship.
Ask pupils how they can fill in the table on the worksheet to help them make sense of the relation between the values mathematically. Try any ideas offered on the board. When the idea of listing them in numerical order comes up and seems to them a good strategy, give them just 5 minutes to do Worksheet 1.

Most pupils will list the pairs of numbers in ascending or descending order, then notice a pattern or relationship in the results. For example, some may notice that the results relate to the 5 times table.
Whole class sharing and discussion
In science pupils are often given an experiment — for example, the stretching of a spring under an increasing load ~ where a linear functional relationship of the form y = mx is handled and plotted. The straight line graph then gives an implicit model of a simple causal relationship, but the mathematical properties of the relation are rarely explored explicitly. The activity addresses this mismatch. Record pupils’ answers to question 3, on the patterns they see, on the board. Possible answers include: ‘They are all times-across’, ‘They make the 5 times table’, ‘They are like the clock times’

Ask if the numbers can get bigger, and whether the pattern will stop: 'Never!’, ‘Go on forever’. Suggest some large numbers for the input, e.g. 100, and elicit their way of finding the output. (Some classes may be able to work out the inverse: What if the output is 100 turns?) Give attention to pupils who use the adding strategy, who see the best way is to add 5s repeatedly. Some may reason that if an input of 5 gives an output of 25 then an input of 10 will give an output of 50. This is developed in the next episode.