FRA08: Measuring with intervals

In these tasks pupils are given a pair of fractions on a number line and use the interval between them to determine the position of 0 or of other fractions on the number line.

Task 08A: Here we are given two fractions with the same denominator, 19. This means we can read-off the size of the interval between them, 2/19, and use 19ths as a unit of measure: zero is one 19th to the left of the given fraction 1/19.

Some pupils might feel unsure about fractions with an 'unusual' denominator like 19. If so, you could give them a variant like the one below. It turns out that zero is in the same relative position.

Task 08B: The fractions here are more familiar than 19ths, but they have different denominators, so the numerical size of the interval between them is less obvious.

Note that the section of the number line shown in the diagram is long enough to accommodate the location of 1 as well as 0. This opens up the possibility of using trial and improvement to try to solve the task. For example, a pupil might mark an assumed location for 0 and then step off three of the putative 0 to 1/3 intervals to locate a potential position for 1. Does it look about right? Would four of the putative 0 to 1/4 intervals get us to the same place? 

Interestingly, even the result of a fairly rough guess, such as the diagram below, can prove insightful. The diagram tells us that (in the maths world) not only do three thirds make 1, but that three quarters plus three of the 1/4-to-1/3 intervals must also make 1. In other words, three of the 1/4-to-1/3 intervals make a quarter, and so the true location of 0 is three such intervals to the left of the given 1/4 mark.

It might be helpful, especially when discussing the solution, to have a more 'quantifiable' version of the task, as below. Here we can show and discuss why 0 lies three 'units' (three 12ths) to the left of 1/4 and four units to the left of 1/3.
4×¼ = 4×3×1/12 = 12/12 = 1; 3×⅓ = 3×4×1/12 = 12/12 = 1.

Task 08C: This task should be fairly routine for pupils who have established from the previous task that the interval between 1/4 and 1/3 measures 1/12.

The task gives pupils the opportunity to work with equivalent fractions:
the red arrow shows 2/12 or 1/6;
the green arrow shows 3/12 + 1/24 = 6/24 + 1/24 = 7/24; or 3½/12 = 7/24.

Task 08D: This is a much simpler task: 1/2 clearly lies midway between the two given fractions. Some pupils might be able to go on to justify this by converting the given fractions into 10ths: the fraction midway between 4/10 and 6/10 is 5/10, which is 1/2. Or pupils might describe the midway fraction as 2½/5, which is 5/10, which is 1/2.

This variant is slightly more demanding, in that one can't side-step the use of equivalent fractions this time. The task can be solved by converting the given fractions into tenths (so that the interval between them is seen as 2/10, with 7/10 being 1/10 to the right of 3/5); or by converting 7/10 into 3½/5.

This variant is more demanding still. Some pupils might notice that 1/3 lies slightly to the left of 2/5 (1/3=2/6 is slightly less than 2/5). It can be solved more precisely by converting the fractions into 15ths.

Task 08E: Here it is tempting to think that 1/8 lies the same distance to the left of 1/6 as 1/4 lies to the right. But this is not the case. In a sequence like 1/4, 1/6, 1/8, .... the fractions lie more and more closely together. To add to the potential difficulty, we are not given the position of zero on the number line - indeed, it lies off the page. The task can be solved by converting all three fractions to fractions with the same denominator - 24 would be the lowest sensible denominator. Pupils might do this in stages, by first converting 1/6 and 1/4 into 12ths.

Below is a version of the task where zero does fit on the given part of the number line. The interval between 1/6 and 1/4 (or 2/12 and 3/12) measures 1/12, so 0 lies two such intervals to the left of 1/6. If pupils decide to find the position of zero, this will give them another way of solving the task: 1/8 lies exactly halfway between 0 and 1/4.