D08: Dividing and multiplying fractions by a whole number

Here we look at some relatively simple division (and multiplication) tasks - in that the divisor (or multiplier) is a (small) whole number. One point of interest is whether pupils recognise that they can divide a fraction by operating on its denominator.

TASK 08A: In the first part of this task, the division can readily be performed by dividing the numerator by two or by doubling the denominator. Which will be more common? I imagine more pupils will opt for the former. The second part is likely to be more demanding as the numerator is odd - do the pupils who halved the numerator in the first part, switch to doubling the denominator in the second part? Some pupils might give the answer 8½/25, which could lead to an interesting discussion. Should we allow it?! Can we find an orthodox equivalent?

TASK 08B: This might be thought to be a bit of a trick question.... Careful scrutiny will show that the yellow strip in B is a quarter the size of the strip in A. But some pupils might opt for one third, as this leads to the nice, simple answer, ⅕.

TASK 08C: This is a more explicit version of Task 08B, but does that mean it is easier? It might still be demanding for some pupils to find a way to divide 3/5 by 4. And it looks as though it might be possible to partition the bar into a total 7 of the smaller pieces, leading to the answer ⅟₇ (which would in fact be a very good estimate, as it is very close to the correct answer, 3/20).

TASK 08D: Here each of the three given fifths has been divided by 4. Does that lead to the same result as in the previous two tasks? [This is an example of what Streefland calls French division. If 4 people decide to share 3 pizzas equally, do they get the same amount if the waiter brings the pizzas one at a time or all three at once?]

TASK 08E: Here we are just doubling and halving so it is fairly easy to locate the position of the fractions on the scale, but of course only if pupils realise that we are doubling and halving, and display the resulting fractions the right way round!

TASK 08F: This task is not adhering to the division theme, but it provides a nice challenge.

The first part can be solved by visualising one third of the given interval and moving the arrow on by that amount. Do pupils solve it that way or do they first try to divide the whole strip into 11 equal parts?

In the second part, do pupils realise that the desired fraction is to the right of the given fraction, and that the difference is 'small'!? 

Do any pupils attempt to solve this precisely? We can think of the strip as being divided into 110 equal parts, with 3/11 covering 30 of them and 3/10 covering just 3 more!
Put another way: 3/11 + 1/10 of 3/11 = 3/10.

TASK 08G: Here we are dividing and multiply the given fraction by 3. How readily do pupils realise that? The ruler allows us to locate the desired fractions precisely, although pupils might be thrown by the observation that the green rod is not an exact number of units long. As with earlier tasks, we can find the desired fractions without explicitly dividing the whole rod into equal parts.
 

TASK 08H: Here we sidestep the theme again, but make use of the ruler from the previous task. A relaxing task to round things off, with a moderate challenge offered by rod C.

There are some more tasks involving the ruler in the next section.