Multiplication and division structures

The scaling structure for multiplication - part 1

The scaling structure is a rather more difficult idea. It is an extension of the augmentation structure of addition. That structure referred to addition as meaning that we increase a quantity by a certain amount. With multiplication we also increase a quantity, but we increase it by a scale factor. So multiplication by 10 would be interpreted in this structure as scaling a quantity by a factor of 10.

Teaching Point

The key language to be developed in the scaling structure of multiplication includes: scaling, scale factor, doubling, trebling, so many times bigger than (longer than, heavier than, and so on), so many times as much as (or as many as).

Does 3 x 5 means ‘3 lots of 5’ or ‘5 lots of 3’?

Which of the pictures in the image would we connect with 3 x 5? If we were really pedantic, I suppose we would have to say that, strictly speaking, 3 x 5 means ‘5 lots of 3’.

(a) 3 sets of 5; (b) 5 sets of 3;

You start with 3, and multiply it by 5. That is, you reproduce the three, five times in all, as illustrated in Figure 9. But I would prefer to let the meaning of the symbol be determined by how it is used. It seems to me that, in practice, people use the symbols 3 x 5 mean both ‘3 lots of 5’ and ‘5 lots of 3’, in other words, both the examples shown in the image. It is ok therefore to let 3 x 5 refer to either (a) or (b).  And the same goes for 5 x 3.  One symbol having more than one meaning is something we have to learn to live with in mathematics – as well as being a feature which makes mathematical symbols so powerful in their application.