Multiplication and division structures

Picture the multiplication symbol

Is there a picture that can usefully be associated with the multiplication symbol? It is interesting to note first that the commutative property of multiplication is by no means obvious. Other than by counting the numbers in each picture, we would not immediately recognise that (a) and (b) in Figure 9 above have the same number of counters. So this picture is not especially helpful.

Remember to use rectangular arrays frequently to illustrate and to support your explanations about multiplication, particularly for reinforcing the commutative principle.

But there is one very significant picture of multiplication which does make this commutative property obvious. This is the association of multiplication with the image of a rectangular array.

The image on the right shows some examples of rectangular arrays that correspond to 3 x 5 (or 5 x 3). This is the image of multiplication that we should carry round in our heads, particularly when we want to talk to children about multiplication and to illustrate our discussions with diagrams. This picture really does make the commutative property transparently obvious. We can actually see that 3 sets of 5 and 5 sets of 3 come to the same thing, because the array can be thought of as 3 rows of 5, using vertical rows, or 5 rows of 3, using horizontal rows.

There are other good reasons for strongly associating this image of a rectangular array with multiplication. For example, this idea leads on naturally to the use of multiplication for determining the area of a rectangle. In  (c)  3 x 5 gives the number of square units in the rectangle and therefore determines its area. We can also extend this idea to develop an effective method for multiplying together larger numbers.