Here we look at fractions (in particular 1/10) on a number line. However, we are only shown a small part of the line, not even the whole interval 0 - 1, so that the emphasis is more on fraction as a number than as part of a whole.
TASK 02A: The task is quite challenging but can be solved in a number of ways. The first thing one might look for is whether pupils realise that 1/15 is less than 1/10 and so will lie to the left of 1/10 on the number line.
The four responses below were posted on Twitter.
Given that 1/15 lies to the left of 1/10 on the number line, one might expect a response to focus on the interval 0 to 1/10. That makes the elegant top-left response particularly interesting, as it first involves locating the fraction 1/5 which lies outside (to the right of) that interval.
In the top-right response, it is not entirely clear whether the statement 1/15 = 1/10 × 2/3 is intended to show how to find the solution or to confirm it. Either way, it is a neat response.
Responses like the bottom-left response will probably occur quite often. It makes explicit use of the common denominators 150 and 30. The former can be found in a routine way (multiply the given denominators) but the use of 30 is far neater.
The bottom-right response illustrates an interesting trend in the UK that started with decimalisation in the 1970s: an increasing number of people generally seem to feel more comfortable working with decimal fractions than with vulgar fractions.
The second part is a reformulation of the first task, though the idea of a fraction of a fraction might be very challenging. The first part is much more accessible. Moreover, the same relation applies since 15 and 10 are both 150 times 1/10 and 1/15. Ten is 2/3 of 15, and 1/15 is 2/3 of 1/10.
We can express the desired relation using the number line representation of the first task, as here:
