What’s my rule?

The diagram on the right shows some examples of this game. In each case the pupils are challenged to say what is the rule that is being used to find the numbers in column B and then to use this rule to find the number in column B when the number in column A is 100. In example (a), pupils usually observe first that the rule is ‘adding 2’. Here they have spotted what I refer to when talking to children as the ‘up-and-down rule’. This is the pattern that determines how to continue the sequence.

Asking what answer do you get when the number A is 100, or some other large number, makes us realise the inadequacy of the sequential generalisation just mentioned. We need a ‘left-to-right rule’: a rule that tells us what to do to the numbers in A to get the numbers in B.

This is the global generalisation. Pupils towards the top end of the primary range can usually determine that when the number in A is 100, the number in B is 201, and this helps them to recognise that the rule is ‘double and add 1’.

Later this can be expressed algebraically. If we use x to stand for ‘any number in column A’ and y to stand for the corresponding number in column B, then the generalization is y = x X 2 + 1, or y = 2x + 1. This clearly uses the idea of letters as variables, expressing generalisations. The statement means essentially, ‘The number in column B is whatever number is in column A multiplied by 2, add one’.

Similarly, in  column 2, the sequential generalisation, ‘add 4’, is easily spotted; more difficult is the global generalisation, ‘multiply by 4 and subtract 1’, although again working out what is in B when 100 is in A helps to make this rule explicit. This leads to the algebraic statement, y = x X 4 – 1, or y = 4x – 1.

In these kinds of examples, where x is chosen and a rule is used to determine y, x is called the independent variable and y is called the dependent variable.

The What’s my rule? game can be used in simple examples with quite young children to introduce them to algebraic thinking through making generalisations in words. Use the game with older, more able pupils to express their generalisations in symbols.