In this Week's task we again look at division as quotition (or measurment), but we also interpret it (in what some might regard as a rather laboured way!) as partition (or sharing). We compare the effect of dividing 10 by 3/7 using quotition and partition, and of dividing 4/5 by 3/7 using quotition and partition.
Task 19A: In this task pupils are asked to interpret the expression 10 ÷ 3/7 as "How many three-sevenths are there in 10?". This division is then broken down into three steps, starting with 10 ÷ 1/7.
The first step (How many 7ths are there in 10?') is fairly easy to construe, as can be seen from step 2: there are seven 7ths in 1, so there will be 10 times as many in 10, which gives us 10×7.
In the third step (How many three-sevenths are there in 10?) the divisor 3/7 is 3 times as large as the previous divisor, 1/7, so the number of times it fits into 10 will be one-third of the number of times 1/7 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 will 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.
Task 19B: This task involves the same expression, 10 ÷ 3/7, as in the previous task, but this time we interpret the division as partition or sharing. The resulting statement, "10 is three-sevenths of a share" might seem rather awkward and forced, but the case can be made that it does make sense of a sort!Consider the division 10÷4, where the divisor is a whole number. We can think of this as fitting a story like '10 kg of rice are shared fairly between 4 people. How much does each person get?". This can be rephrased, a little awkwardly perhaps, as "10 kg is 4 shares, how much is 1 share?". Similarly, 10÷³⁄₇ can be phrased as "10 is ³⁄₇ of a share, how much is 1 share?".
Overall, it is fairly straightforward to see 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 and shouldn't be pushed too hard.
Task 19C: Here we revert to the quotitive or measurement interpretation of division, as in Task 19A, but this time the dividend is a fraction as well as the divisor.
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 ⅘ than in 1 - can this be achieved by multiplying?
As before, in the third step (How many three-sevenths are there in ⅘?) the divisor 3/7
is 3 times as large as the previous divisor, 1/7, so the number of times it fits into
10 will be one-third of the number of times 1/7 fits. So it fits
10×7÷3 times.
Task 19D: We now interpret ⅘÷³⁄₇ as partition or sharing: "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, compared to Task 19B where the dividend was a whole number, 10. 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 discern how the numerator and denominator of ³⁄₇ come into play.
Task 19E: Here we summarise in a more formal way how we evaluated ⅘ ÷ ³⁄₇ in Tasks 19C and 19D.
For item 1, it will be interesting to see whether pupils give an explanation involving quotition or partition; or whether they provide a more formal argument such as this: 1÷⅟₇ = 7÷⁷⁄₇ = 7÷1 = 7.
In item 2, it might help to add an extra step, by rewriting ⅘×7 as ⅘×⁷⁄₁. Alternatively, in item 3 it might help to provide the interim expression ⅘×7÷3, though pupils might still find it difficult to accept that this can be written as ⅘×⁷⁄₃.
