It Stands to Reason
There has been some discussion among mathematics teachers on Twitter recently about whether the order in which the numbers appear in expressions of products of two numbers should have any significance for pupils, and, if so, what that significance should be. The discussion started with a debate about which of these two images …
… represents the expression ‘3 × 4’, and which represents ‘4 × 3’. It was also seen as a debate about which of those two expressions is represented by ‘3 groups of 4 objects’ and which is represented by ‘4 groups of 3 objects’ – which is equivalent to 4 + 4 + 4 and which to 3 + 3 + 3 + 3? Several contributors to the discussion made comments supporting the view that, since …
… pupils should understand that both ‘3 × 4’ and ‘4 × 3’ can be thought of (pictured) in any of these 8 ways.
Pupils’ thinking about multiplication and division is enriched, and their ability to reason multiplicatively is improved, when they can not only call upon repeated-addition images of multiplication (as above) for support but can also use images involving scaling. So we will look at how scaling-images of multiplication can deepen pupils’ understanding of multiplication and division, particularly when operating with and on fractions.
Here is a task for the start of a learning session about scaling-images and multiplication …
What is the same and what is different about Diagram A and Diagram B?
Ask students to comment in response to this question, and collect responses on the board without comment.
Encourage pupils to discuss their responses.
Look out for discussion of these samenesses:
- in both diagrams both triangles are right-angled triangles
- in both diagrams the vertical side-length of the smaller triangle is the height of one ‘pointing-upwards child’
- in both diagrams the vertical side-length of the larger triangle is the height of four ‘pointing-upwards children’
Look out for discussion of these differences:
- in Diagram B the larger triangle is a scaled-up version of the smaller triangle,
- in Diagram A the larger triangle is NOT a scaled-up version of the smaller triangle,
- in Diagram B the two triangles are similar
- in Diagram A the two triangles are NOT similar
- in Diagram B the number of ‘children with stretched-out legs’ on the large triangle is 4 times the number of ‘children with stretched-out legs’ on the small triangle
- in Diagram A the number of ‘children with stretched-out legs’ on the large triangle is NOT 4 times the number of ‘children with stretched-out legs’ on the small triangle
A right-angled triangle is a useful shape to use in ‘scaling-pictures’ of multiplication. If it is oriented with the non-hypotenuse sides horizontal and vertical, a number to be multiplied (say x) can be represented by the length of the horizontal side. When the triangle is scaled-up (to represent multiplication by a number greater than 1) the length of the horizontal side will be multiplied by the same number as the length of the vertical side is multiplied by. So, if the vertical side-length of the starting-triangle is one-of-something, and the corresponding vertical side-length of the scaled-up triangle is n-somethings, the length of the horizontal side of the scaled-up triangle represents n × x. For example, multiplication of 3 by 2 (2 × 3 = 6) could be represented by scaling-up a right-angled triangle with horizontal side-length 3 units (a ‘child with stretched-out legs’ is one unit) so that the vertical side-length (the height of a ‘pointing-upwards child’) is doubled …
… and 3 × 3 = 9 could be shown by scaling-up the same triangle so that the vertical side-length is tripled:
Using one tiny image (in this example a ‘child with stretched-out legs’) to show each unit on the horizontal side and a different tiny image (in this example a ’pointing-upwards child’) to show the vertical side-length of the starting triangle is an effective aid in helping pupils grasp the basic structure of what happens.
Once pupils understand that whatever the vertical side-length is multiplied by the horizontal-side-length-in-units (the number to be multiplied) is also multiplied by, the ‘tiny images’ can be dropped, as in this image showing multiplying 2 by 3 …
Pupils should be prompted (if they don’t come-up with it themselves) to see that several different products can be represented in one image. For example, products represented in the next image include 1 × 2 = 2, 1 × 3 = 3, (and so on), 2 × 2 = 4, 2 × 3 = 6, 3 × 2 = 6, 3 × 3 = 9, 1.5 × 6 = 9 (vertical side-length scaled-up from 6k to 9k), 1.5 × 4 = 6 (vertical side-length scaled-up from 4k to 6k), 1.5 × 2 = 3, and others that you could challenge some pupils to state.
As pupils gain confidence not only the tiny images, but also most of the horizontal grid-lines, can be dropped, as long as the units are shown by equally-spaced vertical grid-lines. For example, you could ask what products this image can represent:
Challenge pupils to explain why the oblique line removes the need to show all the horizontal grid-lines (they should visualise similar right-angled triangles such as those that were previously explicitly shown).
The scope of the image can be increased by marking the vertical lines in multiples of any number, for example in multiples of 4, as shown here:
If the equal intervals on the vertical axis are numbered consecutively the image becomes a times-table representation:
When ‘slant-lines’ at different angles to the vertical are introduced several times-tables can be represented in one image. For example, in this image …
… the red line shows multiples of 2, the blue line shows multiples of 4, and the orange line shows multiples of 8. Challenge pupils to explain how they show these multiples. You could also ask them what the green line shows. The following image gives many more products, which are indicated in new ways. Pupils could discuss the structure of this image and how they can ‘read-off’ products from it.
Can pupils see in the above image (or a similar one of their own making) examples of general facts about multiplication? Most pupils should be able to find particular examples of the commutative property of multiplication, as shown generally in this image …
… and some may spot examples of the fact that multiplication is distributive over addition, as indicated generally here:
Some pupils may like to extend their images into negative numbers in both directions …
…and confirm relationships between products involving negative numbers, which some pupils might be able to show generally:
A strength of these kinds of image is that pupils can use them to focus on multiplication of numbers between 0 and 1. This image should remind pupils of the starting point for these scaling-images …
… and this example …
… shows that 0.8 × 0.25 = 0.5 × 0.4 = 0.4 × 0.5 = 0.25 × 0.8 = 0.2.
Could your pupils draw a similar diagram that shows some products (of two numbers) equal to 0.4 or 0.6?
Scaling-diagrams that show multiplication of fractions can be very revealing to pupils. This is a typical example …
(click to enlarge)
… and this next image shows the result of multiplying various fractions by 3/4 …
(click to enlarge)
Could your pupils draw a similar diagram to show multiplication of some fractions by, say, 2/2 or 5/8?
Page header by Brian Hillegas (adapted), some rights reserved