D14: Two fractions on a number line

Here we are given two fractions on a number line and asked to find or name a third fraction on the line. A nice feature of these tasks is that it is relatively easy to devise variants and radically to alter the level of difficulty.

TASK 14A: We start with a rather special fraction or number: zero.

This is quite a challenging task as the number line is unmarked, except for the two given fractions. Some pupils will approach this in quite an intuitive or 'qualitative' way and perhaps simply mark 0 'somewhere' to the left of the given fractions. Others might use measurement - for example, if the two given fractions are 1 cm apart, 0 will lie 3 cm to the left of 1/4. (Why?)

Another challenge stems for the fact that the number 1 does not fit on the given segment of the number line, so we can't relate the fractions directly to a visible 'whole'. You might want to modify the task by presenting pupils with a longer line or by having the two given fractions closer together.

TASK 14B: Here we have marked the number line in equal intervals so that it is possible to locate 0 without measuring. The distance between adjacent marks represents 1/12. (Why?) So 0 will be 3 marks to the left of 1/4.

TASK 14C: Pupils with a good 'fraction sense' might see immediately that the red arrow is pointing to 1/6 since it lies halfway between 0 and 1/3. On the other hand, if they have adopted a more formal/analytic approach and have converted the given fractions to 12ths, they might solve the task by counting backwards: 1/3 is 4/12, 1/4 is 3/12, the red arrow is at 2/12, which they might then convert to 1/6.

The green arrow point to the fraction midway between 3/12 and 4/12. This might lead pupils to see the faction as 3½/12 which in turn can be thought of as 7/24. Alternatively, pupils might convert the given fractions to 24ths, so that the desired fraction is the one midway between 6/24 and 8/24.

Some pupils might suggest the desired fraction is given by 1/3½, leading to 2/7. Sadly this doesn't quite work, but why?! The fraction is close to the correct fraction: it can, for example, be thought of 7/24½ which is only slightly less that 7/24.

Pupils might find it instructive to name some of the other given marks on the number line and to examine how the various fractions relate to each other. For example, they might notice that the fraction 2 marks to the right of the given fraction 1/3 is 6/12 or 1/2 and that it is twice as far from 0 as the given fraction 1/4. Similarly, the next fraction, 7/12, is twice as far from 0 as the green arrow.

TASK 14D: Here we have varied the task by changing the given pair of fractions. Other things being equal, the task is less challenging because the given fractions have the same denominator.

 

Pupils with a good fraction sense might see immediately that 1/2 lies midway between the two given fractions. We can confirm this by changing the given fractions into 10ths: the fraction midway between 4/10 and 6/10 is 5/10 which is 1/2. Or this midway fraction might be seen as 2½/5, which is 1/2 because the denominator 5 is twice the numerator 2½ or because the fraction is equivalent to 5/10.

TASK 14E: This task is fairly straightforward: asking pupils for 10ths is likely to prompt them to convert the given fractions to 10ths, namely 4/10 and 6/10, so the desired fraction will be halfway between 6/10 and the next mark to the right which represents 8/10.

TASK 14F: Pupils might be able to get a good approximation to the answer if they have the insight to realise that 1/3 is less than 2/5. However, finding the exact location might be more challenging. How do we relate 3rds and 5ths?

Some pupils might opt for the mark midway between 0 and 3/5 but that gives 3/10 which is slightly too small.

An effective way to work with 3rds and 5ths is to convert fractions to 15ths. 3/5 is 9/15 and 2/5 is 6/15. We want 5/15 which is 1/3 of the way from the 2/5 mark to the next given mark to the left, the 1/5 mark: