D16: Locating a fraction on the number line, given the position of one-third

This theme is a direct complement of the previous theme. Instead of locating ⅓, given the position of some other fraction on the number line, this time ⅓ is given and we are asked to find the position of another fraction.

TASK 16A: As in Task 15A, this involves ⅓ and ½.

Again, pupils might work in 6ths and hence use the fact that the gap from two 6ths to three 6ths is half the gap from 0 to two 6ths or ⅓ (first line, below). Or they might locate ⅔ and then place ½ midway between ⅓ and ⅔ (second line, below).

TASK 16B: As in Task 15B, this involves the fractions ⅓ and ¼.

A first step here is to recognise that the desired fraction, ¼, lies to the left of ⅓ - but how far to the left? Some pupils might work in an analytical way, by converting the fractions to 12ths. 

Others might adopt a more empirical approach by putting a mark on the number line somewhat to the left of ⅓. Now let us reproduce the interval between this putative ¼ mark and the ⅓ mark, by adding further marks to the left, till we get close to (or preferably land on) 0. And now let us estimate the number of intervals from 0 to 1 by multiplying the number of intervals from 0 to our putative ¼ by 4 and the number from 0 to ⅓ by 3. Do we get the same result? Of course, this only happens when there are exactly 3 intervals from 0 to ¼ and exactly 4 from 0 to ⅓ [4×n = 3×(n+1) when n=3; 4×3 = 3×4 = 12].

In the diagram below, neither attempt works. In the first attempt we have 4×4½ ≠ 3×5½. In the second attempt we have 4×5 ≠ 3×6.

A related and simpler empirical approach, which can act as a nice (albeit rough and ready) check, is to extend the number line and then to locate the position of 1 using the given position of ⅓ and the estimated position of ¼. Do the resulting positions roughly coincide?

[Note: It is worth making the point that even where pupils' method might not be efficient, or even end in success, as perhaps in the empirical approaches described above, these activities can still give pupils a better feel for fractions. That is the prime purpose of these tasks - it is not primarily about getting 'the answer'.]

As with Task 15B, one could vary the task slightly by using fractions like ⅕ or ⅟₇ instead of ¼.

TASK 16C: As with Task 15C, nice, easy fractions this time. And again, you might also want to try factions like ⅟₉ and ⅟₁₂ in place of ⅙.

TASK 16D: As with Task 15D, this is a bit more challenging again. You might also want to try fractions like ⁴⁄₉, ⅚ and ⁵⁄₁₂ in place of ²⁄₉.

TASK 16E: As with Task 16E, here is a variant where we might be content with a qualitative answer: ¹¹⁄₃₀ is a 'tiny' bit to the right of the given fraction. 

We can of course adopt a more analytical approach: ⅓ is ¹⁰⁄₃₀ and so ¹¹⁄₃₀ is one 10th of the interval between 0 and ⅓ to the right of ⅓:

You might also want to try fractions like ⁹⁄₃₀ or ²¹⁄₆₀, or even ¹⁰⁄₃₁ or ¹⁰⁄₂₉, in place of ¹¹⁄₃₀.