This Week's tasks are about dividing fractions by a whole number. We use bars or number lines to represent the fractions, and the tasks are again about splitting intervals into smaller equal parts. However, this time the resulting parts do not represent unit fractions, as they don't (necessarily) fit neatly into the original bar or the 0 to 1 interval on the number line.
These tasks are likely to be considerably more demanding than last Week's, for many pupils. Although we use diagrams to help pupils 'see' the fractions, visualising the results of the divisions in a precise way is difficult. Whereas it might be relatively easy to 'see', or to accept, that dividing a quarter, say, by 5 results in one 20th, it is much more difficult to 'see' that the result of dividing three quarters by 5 is three 20ths. Yes, dividing 3/4 by 5 results in a smallish 'part' and 3/20 is smallish, but looking at the resulting part, it is difficult to read-off its precise value or to confirm that it is indeed three 20ths. On the other hand, the visual nature of the tasks can help pupils find ways of assessing whether their proposed solutions are of roughly the right size, which is an important aspect of sense-making.
Task 07A: In number line A, the arrow is pointing at one third of 9/30. We can find one third of 9/30 by dividing the numerator by 3 or multiplying the denominator by 3. The former is much more grounded - we can visualise the result on the number line more easily - and it seems likely that more pupils will solve the task in this way. (Do pupils spot that the result, 3/30 or 9/90, is equivalent to 1/10?)
Some pupils' might decide that one third of 9/30 can found by dividing the denominator by 3, rather than the numerator (resulting in 9/10). Others might decide to divide both numbers by 3 (resulting in 3/10), perhaps by applying a half-remembered rule about equivalent fractions. However, given time to reflect, pupils might realise that neither result produces a fraction in the right location.
In line B, neither component of the given fraction is divisible by 3. This might prompt some of the pupils who used the correct strategy of dividing the numerator by 3 in line A, to switch to multiplying the denominator by 3. Others might stick with their strategy and come up with 3⅓/31- can this be written in a more orthodox way?
It might be interesting also to use a purely numerical variant of the task, as in this example (below). In this form, it is of course easy to change the numbers.
Task B: It turns out that the yellow region in Bar B is a quarter of the yellow region in Bar A. However, pupils who see (or assume) that, won't necessarily go on try to divide the 3/5 region in Bar A by 4. Rather, they might well attempt to solve the task in a more grounded way by trying to see how many times the yellow part in B fits into the whole bar. Does it go 4+2=6 times, or 4+3=7? Well, neither quite works! Careful scrutiny will show that it is definitely more than 4+2 and slightly less than 4+3.
You might prefer to use this version of the task (below), where it is made clear that the yellow part of B is a quarter of the yellow part of A. Here it is also fairly easy to see that more than two of the yellow parts in B would be needed to fill the right-most region of the bar and that three would be too many.
This third version of the task (below) can provide a rich discussion point for pupils. Bar B now contains three regions that are each one 5th of the bar and where one quarter of each is tinted yellow. Are these yellow mini-parts equivalent to the yellow part we had before? And what is the size of each of these yellow mini-parts?
Task 07C: This can be solved in a fairly grounded way: a half of 3/5 in 1½/5, which is 3/10; three times 3/10 is 9/10.
Some pupils might try to solve the task in a more visual way and arrive at the correct solution: it looks as though the white region in Bar B might fit 3 times into each of the 3 yellow parts, and so fit into the whole bar 10 times; so the white region is 1/10 of the bar, making the rest 9/10 of the bar.
Task 07D: Here we apply two operations to the fraction 5/12. (The operations can be expressed as ÷3 and ×2.) When we change the order of the operations, the result is the same, 10/36 or 5/18.
Task 07E: This is similar to Task 07D but pupils are asked to focus more on the steps. Here the operations are ÷4 and ×5, with the order reversed for the second set of bars.
