Understanding the structure of multiplication (either consciously or subconsciously) is a fundamental factor in understanding mathematics. Multiplicative relationships lie at the heart of much secondary-age mathematics, indeed, multiplication turns up everywhere. Fractions, decimals, percentages, ratios, proportions, gradients, sequences (and many other topics I’m sure you can think of) all demand multiplication at some level.
Can you think of a story in some context for 15 + 3?
Easy, I expect. Most of you will say something like ‘I have 15 sweets and I get 3 more’. It’s similarly simple for subtraction, ‘I have 15 sweets and I give 3 away’ and division, ‘I have 15 sweets and I share them between 3 people’, or ‘I have 15 sweets and I put them in groups of 3’.
Multiplication requires a little more thought. Students find this difficult, but with a little patience might come up with, ‘I have 3 bags of sweets with 15 sweets in each. I have 45 sweets altogether’.
There are generally thought to be five or six models of multiplication (depending on whose research you look at), and the example above can be described as an ‘allocation’ or ‘rate’ model. In the progression below, I will be considering the ‘array’ or ‘area’ model.
Many of us first encounter multiplication as repeated addition, and this feels like an important stage in understanding that when we multiply, we are combining the same thing many times. Typically, early learners graduate from equal-sized jumps on number lines to arrays, probably using physical counters and arranging them into rectangular shapes, like the example below:
Typically, the next stage would be to draw around these counters, so that we can move towards a grid:
Then removing the counters altogether:
This is the point at which we are moving from an array to an area. It’s clear to see what the area of this rectangle (or this representation of a multiplication) is: 15 squares.
You may have your own views on whether that represents 3 x 5 or 5 x 3 (safe in the knowledge that the value is the same). I tend to call it 3 x 5, but I’m not sure that there is a right or wrong way.
As learners get more sophisticated, we can start to remove the grid lines and just focus on the area of the rectangle from the side lengths:
One of the misconceptions that can arise at this stage (which needs addressing) is a failure to see area as two-dimensional. It’s an idea that most of us never think of, being one of those things ‘we just know’, but it is an integral part of the structure of multiplication. Also key is understanding that area dimensions meet at right angles.
Assuming this is understood, learners go on to use particular models associated with area when they start to multiply bigger numbers. Typically they are taught to ‘partition’ a two-digit number when multiplying by a single-digit number, taking advantage of the fact that multiplication is distributive over addition.
The calculation 7 x 27 might be modelled as:
This can be written as 7(20 + 7)
It’s a short step to expanding something like 3(x+7) using the same model.
The difference is that in the numerical version we choose how to partition 27 (20 + 7); in the algebraic version the partitioning (x+7) has been done for us because we don’t know the value of x
We might even choose to explain how to multiply out a pair of brackets by choosing a numerical example first, such as 26 x 47:
Which follows the same rules as (x+6)(x+7):
Making explicit for students the idea that algebra is generalised arithmetic (i.e., follows the same rules as numbers do) in this way helps with structural understanding of both multiplication and algebra.
My experience of teaching it this way, and dropping back a stage to a numerical example in times of difficulty, is that it definitely helps students understand what’s going on when presented with some awkward-looking brackets to expand.
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Nick Asker is an independent Mathematics Advisor. He started teaching mathematics in 1982 and has taught in a variety of settings including secondary schools and all-age special schools. Nick regularly leads workgroups for the National Centre for the Excellence in Teaching of Mathematics, and is a tutor on their PD lead programme. Nick has also led teacher education programmes in Africa and the Middle East.
