Lesson 12 Functions

Episode 2

Reasoning Resources: Table from Worksheet 1 on board or OHT or pupils' copies, Worksheet 2
Whole class preparation
Demonstrating ways of working with arrows and loops.

.

With pupils’ help, extend the table on Worksheet 1 up to input 15. Using clear linking lines and loops, show a couple of examples of doubling the input, then looking at what happens to the output. Also show a tripling strategy by arrowed loops from number to number.

Ask pupils to describe in their own words what is happening to the input, then the output, in each case, Press pupils who use the words ‘the same’ to explain more precisely that the same action has to be performed on the output as on the input. Ask: If the arrow this way means doubling, what would it mean if the direction of the arrow were the other way?
Looking at ratio. Unlikely to be assimilated fully by the pupils until they carry it out for themselves.
Pair and small group work
Extracting examples from the table of values allows a look at proportion problems in a fresh way. Give out Worksheet 2. Pupils now identify similar patterns in their own tables. Tell them that their task is to find examples and pairs of values in their table for question 1.This will help them to answer in words question 2 on the worksheet.
Recognising the two relationships: across and down. Recognising the inverse relationships.

Working with different values across and down.

An alternative way of solving proportion problems to the procedure of ‘cross- multiply and divide!' It links intuition and sense of size of numbers to formal work.
Whole class sharing and discussion
Collect different phrases from pupils’ answers to question 2, writing them on the board. Take two pairs of paired values from the table separately and show that there are two relations: one across, the X 5, and the other down, the X 2. If any of the four numbers in this double relationship is missing, we can find it. Ask pupils to explain how. Show with another example that this also applies to tripling.
Extension
Some classes could consider other proportion questions, first with easy then with harder relationships. Pupils should recognise that they can use either relationship, normally the easiest one, and that some relationships may not be in whole number multiplication.

Note that often some pupils lapse into additive relationship while in all cases the ‘across-relation is ratio, or multiplicative. The same relation must be kept for the second pair of values in each four.To ensure the focus on the ratio relation, more than one pair can be given with another with a missing value, e.g.