Dividing fractions: how to share 71⁄5 apples between the proverbial 22⁄5 children
In Issue 115 we considered how to build bridges from pupils’ understanding of addition, subtraction and multiplication in the context of integers to that of fractions. Let us now consider division, and then division by (non-integral) fractions.
Many pupils first meet division at primary school in the context of “equal-sharing” problems: “There are 3 children at a party and 12 sticky buns are shared equally between them. How many buns does each child get?” Glossing over the incongruity of this scenario – it’s not much of a party if there are only 3 guests, and in the modern day of healthy eating those children will be lucky to get anything more than half an organic low-GI oat and quinoa muesli bar – there is also a conceptual weakness in the “equal-sharing” model for division: how does it extend to non-integer dividends, divisors and quotients? Is there a model that describes 12 ÷ 3 = 4, 12 ÷ 5 = 2.4 and also 12 ÷ 2.4 = 5?
Secondary teachers, especially KS3 teachers, gain a lot from talking with their schools’ feeder primaries, for example knowing which models for division their pupils have met. It is likely that your pupils will also have met the “equal-grouping” model: “12 ÷ 3” is represented by taking groups of 3 objects (apples, buns, muesli-bars) away from 12 of the objects, until none are left. Because this can be done 4 times, we say that 12 ÷ 3 = 4. There’s a conceptual link that can be made here between division being thought of as repeated subtraction (12 – 3 – 3 – 3 – 3 = 0) and multiplication (the inverse of division) being thought of as repeated addition (the inverse of subtraction).
Thinking of the division 12 ÷ 3 as an activity, the “equal-sharing” algorithm is “give a bun to child A, a bun to child B, a bun to child C, then give another bun to child A, another bun to child B, another bun to child C, and so on”; the “equal-grouping” algorithm is “take 3 of the buns to make group A, then take 3 to make group B, then another 3 to make group C, and so on”. The “equal-sharing” model of 12 ÷ 3 = 4 can be verbalised as “12 divided BETWEEN 3 (recipients)”, whereas the “equal-grouping” model can be verbalised as “12 divided INTO 3’s”. Thinking of an array of 3 rows by 4 columns as a representation of 3 × 4 = 12, the “equal-sharing” model of division tells us that there are 4 dots in each of the 3 rows, and the “equal-grouping” model tells us that there are 3 dots in each of the 4 columns:
A pupil can use the “equal-grouping” model to describe 12 ÷ 2.4 = 5 (assuming the objects are, like apples / buns / muesli-bars, divisible into smaller pieces, so that you can put 22⁄5 objects in each group), but it’s harder to use it to model 12 ÷ 5 = 2.4: the pupil has to work out what to do with the last 2 objects, having first taken away 5 and then another 5.
A third, and we would argue more powerful, model of division is to think of 12 ÷ 3 = 4 as of fitting 3’s into 12, and being able to fit 4: for example, the activity of fitting 4 equal sticks each of length 3cm into a gap of 12cm:
The power of this “equal-fitting” model is that it extends naturally to non-integer dividends, divisors and quotients, and also that it bridges back to very early numeracy work. Consider how manipulatives such as Cuisenaire rods would help all pupils, especially hitherto low-attainers, have secure access to examples such as the following, and from there be able to reason about the underlying concepts.
- We model 12 ÷ 22⁄5 as fitting sticks of length 22⁄5cm into a gap of 12cm. We use exactly 5 sticks to do so. We interpret this as 12 ÷ 22⁄5 = 5
- We model 12 ÷ 5 as fitting sticks of length 5cm into a gap of 12cm. When we do so, we use 2 sticks and then need to use a bit – a fraction – of a 5cm stick to fit into the remaining 2cm gap: we need 2⁄5 of a stick. We interpret this as 12 ÷ 5 = 22⁄5
- We model 12 ÷ 12⁄5 as fitting sticks of length 12⁄5cm into a gap of 12cm. To do this, we can start with 4 sticks of length 3cm fitting into the gap, and then halve each stick into two pieces each of length 12⁄5cm. This will give us twice as many sticks, so that now 8 sticks fit the gap. We interpret this as 12 ÷ 12⁄5 = 8
- We model 12 ÷ 3⁄10 as fitting sticks of length 3⁄10 cm into a gap of length 12cm. To do this, we can start with 4 sticks of length 3cm fitting into the gap, and then chop each stick into ten pieces each of length 3⁄10 cm. This will give us ten times as many sticks, so that now 40 sticks fit the gap. We interpret this as 12 ÷ 3⁄10 = 40 … I won’t draw the picture!
- To determine 120 ÷ 3⁄10, we can start with 4 sticks each of length 3cm fitting into a gap of 12cm. When the gap becomes ten times bigger, then we need ten times more sticks, which we interpret as 120 ÷ 3 = 40. Now each stick is chopped into ten pieces each of length 3⁄10 cm, so that now we have ten times as many sticks. Therefore 400 sticks fit the gap. We interpret this as 120 ÷ 3⁄10 = 400.
- Clearly, 12 ÷ 3 will equal the same as 24 ÷ 6: making both the gap and the sticks twice as big won’t change the number of sticks needed. The step from here to having a secure conceptual understanding of equivalent fractions is one that most pupils will be able to make and explain.
The “sticks and gaps” model gives pupils the language to talk about what they observe when considering scale factor changes to the dividend and divisor in a calculation A ÷ B:
- if the dividend (the numerator) A increases by multiplication by a scale factor then this can be modelled as the gap getting bigger, and so the answer to the original division, the original quotient A ÷ B, which we model as the number of sticks we originally needed, gets bigger by the same scale factor i.e. AC ÷ B ≡ (A ÷ B) × C;
- if the divisor (the denominator) B is multiplied by a scale factor > 1 then this can be modelled as sticks being glued together to make bigger sticks, and so the original quotient, the number of sticks we originally needed, gets smaller by the same scale factor i.e. A ÷ BC ≡ (A ÷ B) ÷ C;
- if the denominator B is divided by a scale factor > 1 then this can be modelled as the sticks being broken into smaller sticks, and so the original quotient gets bigger by the same scale factor i.e. A ÷ (B ÷ C) ≡ (A ÷ B) × C;
- if the numerator A is divided by a scale factor > 1 then this can be modelled as the gap getting smaller, and so the original quotient gets smaller by the same scale factor i.e. (A ÷ C) ÷ B ≡ (A ÷ B) ÷ C.
Putting this altogether, pupils develop deep conceptual understanding of the algorithm for dividing a number N (which can be an integer or a non-integral decimal or fraction) by a non-integral fraction c⁄d: they understand that making the sticks of length c smaller by a factor of d (i.e. breaking each stick into d equal pieces) means that d times as many of them are needed compared to the original amount needed (which would be N ÷ c). Put formally:
N ÷ c⁄d ≡ N ÷ (c ÷ d) ≡ (N ÷ c) × d ≡ N ÷ c × d ≡ N × d ÷ c ≡ N × d⁄c
though pupils need to be confident with the order of operations to follow, or create, this argument.
Once pupils understand why the denominator d of the divisor has a multiplicative effect on the dividend, procedural fluency is much more likely to be developed and embedded.
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