Multiplication and division structures

Contexts in the inverse of multiplication structure - part 1

What are some of the contexts in which pupils will meet division in the inverse-of-multiplication structure? There are many practical situations in which a set is to be sorted out into subsets of a given size and the question to be answered is, ‘how many subsets are there?’ For example, the head of a school with 240 pupils may wish to organise them into classes of 30 children. How many classes do we need? This is modelled by 240 divided by 30, in other words, how many 30s make 240? Then the teacher with a class of 30 children may wish to organise them into groups of 5 children and asks: how many groups? This is modelled by 30 divided by 5. In other words, how many 5s in 30?

Remember to give pupils plenty of experience of real-life situations involving repeated addition to reach a target and repeated subtraction from a given quantity, making the connection with division. These processes are the basis for the most effective mental and written methods for doing division calculations

Once again, the structure extends quite naturally into the context of money. A familiar question is: How many of these can I afford? This kind of question incorporates the idea of repeated subtraction from a given quantity. For example, how many items costing €6 each can I buy with €150? The question is basically ‘how many 6s can I get out of 150?’ We could imagine repeatedly spending (subtracting) €6 until all the €150 is used up. Similar situations occur in the context of measurement. For instance, the question, ‘how many 150-ml servings of wine from a 750-ml bottle?’ is an example of the inverse-of-multiplication structure of division, in the context of liquid volume and capacity, corresponding to the calculation, 750 divided by 150. In other words, how many 150s make 750? Again the notion of repeated subtraction from a given quantity is evident here. We can imagine repeatedly pouring out (subtracting) 150-ml servings, until the 750 ml is used up.