Building Bridges
Adding, subtracting and multiplying fractions
In last month’s article we looked at some of the complexity that lies under the surface of the seemingly simple notation of a fraction, in particular that a written fraction such as “^{1}⁄_{3}” can represent a standalone number (a point on a number line) as well as specifying the process “take a third of”. Until secondary pupils are comfortable with the standalone number meaning of a fraction, they – understandably – find it hard to understand how it is even possible to carry out arithmetic with fractions, let alone do so confidently and fluently. Numbers are added, not processes; it doesn’t make sense to say that you’re adding the processes “take a third of” and “take a half of”. The standalone number meaning must be secure before moving onto arithmetic; some ideas for embedding understanding of this were discussed in Issue 114.
Because this article comes under the Building Bridges strand, I am going to suggest ways of introducing adding, subtracting and multiplying with fractions that link to other parts of the maths curriculum. These links might suggest starter activities in your lesson(s) on fractions and arithmetic, or “warmup” lessons in advance of them, so that the conceptual understanding of the arithmetic can be developed in advance of the technical procedures that need to be taught and practised.
Adding and subtracting: measurements with different units
If asked “what’s £3 plus 3p?", almost all secondary pupils will know that the answer is not “6” of any unit, and will be able to explain why the answer in fact is £3.03 or 303p. Key to this is converting from one unit (£) to another (p), and this skill can be practised. UK currency soon becomes straightforward, whereas …
… In the country of Ruritania, there are six coins:
1 zollar 2 zags 3 zegs 4 zigs 6 zogs 12 zugs
And then you can ask “How many …
…zigs are worth the same as 5 zigs and 4 zags?
…zegs are worth the same as 5 zegs and 8 zugs?
…zugs are worth the same as 4 zegs and 5 zigs?
and “I go to the shop and buy something costing 2 zollars. I give the shopkeeper 5 zogs and 3 zigs. What change do I get?”
Pictures or actual objects – forged Ruritanian banknotes, for example! – could be useful here, so that pupils can physically exchange 2 zollar notes for 4 zag notes, or draw a line down the middle of a 1 zollar note to give two halfnotes each worth 1 zag.
The transition to arithmetic with fractions comes by modelling the denominator as a unit of measurement, and by writing “3 quarters” rather than “^{3}⁄_{4}”. Then 3 quarters and 2 fifths becomes the same as a money sum, evaluated by converting into the common currency unit of twentieths:
3 quarters and 2 fifths
= 15 twentieths + 8 twentieths
= 23 twentieths
= 1 whole and 3 twentieths.
Again, physical banknotes, which can be drawn on or cut up, will help here, especially to identify the common currency unit. A key advantage of modelling the denominator as a unit – here, of currency but of course all this translates into units of length, mass, etc. – is that it removes the question “do we add the denominators?”: £3 + £5 isn’t ££8, and 3 sevenths + 2 sevenths isn’t 5 fourteenths (or 5 sevenths sevenths). The currency units aren’t added, so the denominators aren’t added; the same argument applies when subtracting.
Multiplying: arrays
Most pupils will be familiar from KS1 and 2 with arrays modelling multiplication:
first represent 3 × 7 as
and then the answer is represented by the number of (small) rectangles created: 21. Units make explicit that this an area model for multiplication: 3cm × 7cm = 21cm^{2}. The area model can then be extended, so that ^{2}⁄_{3} × ^{4}⁄_{5} can be represented as
and the answer is the shaded area out of the total: 8 out of 15. Thinking of an array makes it clear why the algorithm is to multiply the denominators and to multiply the numerators: the product of the denominators is the total number of rectangles in the shape, and the product of the numerators is the number of shaded ones. The extension to the product of three fractions is obvious – and to four and more, sort of!
The strength of this model for multiplication is the strong connection with a model that pupils have seen previously. There are two weak spots, one procedural and one conceptual, which need keeping in mind.
1. This model doesn’t suggest cancelling before multiplying, so the products can be large and unwieldy. Understanding the cancelling of the denominator of one fraction with the numerator of the other has to come from exploration of the inverse relationship between the operations of multiplication and division (the argument that, for example, ^{2}⁄_{3} × ^{3}⁄_{5} = (2 ÷ 3) × (3 ÷ 5) = 2 ÷ 3 × 3 ÷ 5 = 2 × 1 ÷ 5 = 2 ÷ 5 = ^{2}⁄_{5}, because the operations “÷ 3” and “× 3” are an inverse pair equivalent to the operation “× 1”). The risk is that pupils’ development of procedural fluency is hindered if they don’t remember to cancel before multiplying.
2. Strictly, we’re calculating ^{2}⁄_{3} of the width × ^{4}⁄_{5} of the length and saying that the shaded answer is an area that is fraction of the whole area. Conceptually, therefore, the width and length must BOTH be thought of as 1 (cm) – that’s why the picture is a square, not a rectangle:
These weaknesses don’t leave the model fatally flawed, but they might need raising and discussing.
To multiply mixed numbers, we can adapt the grid method used for long multiplication: just as 19 × 23 can be represented and evaluated as
× 
10 
9 

20 
200 
180 

3 
30 
27 


230 
207 
437 
so 2^{2}⁄_{3} × 3^{1}⁄_{2} can be represented and evaluated as
× 
2 
23 

3 
6 
3 × 23 = 2 

12 
2 × 12 = 1 
12 × 23 = 13 




9 13 
 as long as pupils are confident with the individual multiplications: “3 lots of 2 thirds is 6 thirds, which is 2 wholes” would be an explanation that suggested good conceptual understanding. This isn’t as efficient as converting the mixed numbers to topheavy fractions, but the link to prior learning is powerful.
Next month we’ll look at models for division of integers, and then consider how to build a bridge from them to the division of fractions.
Image credit
Page header by PauliCarmody (adapted), some rights reserved
