**The inverse-of-multiplication structure for division**

The *inverse-of-multiplication *structure, however, interprets 20 divided by 4 in a completely different way, as shown in the image. Now the question being asked is: ‘How many groups of 4 marbles are there in the set of 20 marbles?’

Both figures 11 and 12 are equally valid interpretations of the division, 20 divided by 4, even though they are answering two different questions:

Asking the question ‘what is the calculation to be entered on a calculator to solve this problem?’, helps to focus the pupils’ thinking on the underlying mathematical structure of the situation and hence to make the connection with division explicit.

**The phrase ‘inverse of multiplication’ underlines the idea that division and multiplication are ***inverse* processes. This means, for example, that since 6 x 9 comes to 54, then 54 divided by 9 must by 6. The division by 9 ‘undoes’ the effect of multiplying by 9. Hence to solve a problem of the form ‘what must A be multiplied by to give B?’ we divide B by A. For example: How many tickets costing €1.50 cent each do I need to sell to raise €90? The calculation that must be entered on a calculator to solve this problem is the division, 90 divided by 1.50.

The actual problems that occur in practice which have this inverse-of-multiplication structure can be further subdivided. First, there are problems that incorporate the notion of *repeated subtraction from a given quantity, *such as ‘how many sets of 4 can I get from a set of 20?’ So the process of sharing out the 20 marbles in the above example can be thought of as *repeatedly subtracting* sets of 4 marbles from the set of 20 until there are none left, counting the number of sets as you do this.

Second, there are those problems that incorporate the idea of *repeated addition* to* reach a target, *such as ‘how many sets of 4 do you need to get a set of 20?’ For example, the question, ‘how many groups of 4 marbles are there in a set of 20 marbles?’ could mean, in practical terms, *repeatedly adding* sets of 4 marbles until the target of 20 is achieved, counting the number of sets required as you do this.