In Theme D18 we looked at how the two interpretations of division (measurement and sharing, or quotition and partition) apply when dividing by a fraction. In particular, we considered whether they threw light on the formal division rule of 'multiply by the reciprocal of the divisor' (flip-and-multiply). I raised the possibility that it might sometimes (or for some students) be more illuminating to think in terms of general fractions rather than using specific fractions.
In this theme I stick with specific fraction divisions, namely 10÷³⁄₇, ⅘÷³⁄₇ and ⅖÷³⁄₇. How well might the specific numbers/fractions work? In other words, how easy is it to see them as generic? Does their effectiveness differ for the two interpretations of division and for the three chosen dividends (10, ⅘ and ⅖)?
Division as measurement (quotitive division)
For the division 10÷³⁄₇, the measurement interpretation can be thought of as asking, 'How many three-sevenths are there in 10?'. We can solve this in stages.
First we can ask, 'How many 7ths are there in 10?'. This is fairly easy to construe: there are seven 7ths in 1, so there will be 10 times as many in 10, which gives us 10×7.
Next we can ask, 'How many three-sevenths are there in 10?'. Three-sevenths is 3 times as large as one-seventh, so the number of times it fits into 10 will be one-third of the number of times one-seventh fits. So it fits 10×7÷3 times.
Overall, it is fairly easy to see, but also to see why, we get the desired answer by multiplying 10 by the denominator of the divisor ³⁄₇, and by then dividing by the numerator. It follows, though this might be a step too far for some pupils, that we can rewrite our division 10÷³⁄₇ as 10×⁷⁄₃, where ⁷⁄₃ is the reciprocal (or multiplicative inverse) of ³⁄₇.
[There is a further step one could take, for this and the other divisions, of scrutinising the actual answer. For this division we have 10×⁷⁄₃ = 23⅓. We can think of 23 as telling us that there are 23 lots of ³⁄₇ in 10. But what about the ⅓? Is that referring to 10, to 1, or to ³⁄₇? This is an important issue, but one that I am not going to focus on here.]
Let's now consider the division ⅘÷³⁄₇. This time the measurement interpretation can be thought of as asking, 'How many three-sevenths are there in ⅘?'. We can apply the same steps as before. However, one of these is less easy to construe this time.
We can again start with the simple and powerful idea that there are seven 7ths in 1; however, some pupils may well balk at the next step, namely that there will be ⅘ as many
7ths in ⅘, giving us ⅘×7. Previously we multiplied 10 by 7 rather than ⅘ by 7, which fits the common sense notion that 'multiplication makes bigger'; this time we can see that there will be fewer 7ths in our dividend than in 1 - can this be achieved by multiplying?
Our third case is the division ⅖÷³⁄₇. For the measurement interpretation we can apply the same steps as before. However, the dividend and divisor are much closer together than in the previous two examples, so pupils might struggle at first to get a good sense of what is going on. How does ³⁄₇ go into something almost the same size, indeed that is slightly smaller? This situation is eased somewhat by the first step, if pupils manage to take it, since we start with single sevenths rather than three sevenths. From then on, it is likely that the steps will be as meaningful (or not) as in the previous example.
In sum, I would argue that the use of specific numerical examples to illustrate the measurement interpretation of division, can enable pupils to get a sense of why the 'flip-and-multiply' rule works. However, it is likely that it is easier to see what is going on when the dividend is substantially larger than the divisor, and larger than 1.
Division as sharing (partitive division)
Consider a division such as 10÷4, where the divisor is a whole number. We can think of this as fitting a story like '10 kg of potatoes are shared fairly between 4 people. How much does each person get?". Let us rephrase this, a little awkwardly, as "10 is 4 shares, how much is 1 share?". We can then interpret 10÷³⁄₇ as "10 is ³⁄₇ of a share, how much is 1 share?".
Again, we can solve this in stages, as in the diagram below. As with the measurement interpretation, a powerful first step is to consider one seventh. If 10 provides ³⁄₇ of a share, then ¹⁄₇ of a share will be one third of 10, which we can write as 10÷3. In turn, seven sevenths, or a whole share, will be 7 times that: 10÷3×7.
Overall, it is fairly straightforward to see, and probably easier than for the measurement interpretation of division, that we arrive at the answer by dividing the dividend 10 by the numerator of ³⁄₇ and then multiply by the denominator. So we can rewrite our division 10÷³⁄₇ as 10×⁷⁄₃, where ⁷⁄₃ is the reciprocal (or multiplicative inverse) of ³⁄₇. But again, this might be a step too far for some pupils.
We can interpret ⅘÷³⁄₇ as "⅘ is ³⁄₇ of a share, how much is 1 share?". Here it is less easy to get a sense of what the answer might be, compare to the previous example. This might prevent some pupils from knowing what to do - even though we can carry out the same steps as before and, once started, they are probably just as easy to visualise and to appreciate how the numerator and denominator of ³⁄₇ come into play.
We can interpret our final division, ⅖÷³⁄₇, as "⅖ is ³⁄₇ of a share, how much is 1 share?". This is probably no more difficult to make sense of and solve than the previous division, in contrast to when the divisions are interpreted as measurement.
