Here we look at fractions on a number line (or 'scale'), where the value of a fraction is give by how far it is to the right of 0. We restrict ourselves to the interval 0–1, ie to fractions less than 1, and we use a thin bar, rather than just a line, to make the representation more accessible to pupils.
TASK 06A: In this task, most pupils will probably realise that ⅟₇ is to the left of ¼, but at first they might feel that they don't have enough information to mark its exact position.
If pupils are stuck on the task, there is a good chance that they will hit upon the necessary insight if given time. The solution (below) is really quite simple - once one sees it! The quarter interval has been divided into 7 equal parts, so we can divide the whole 0–1 interval into 7×4 = 28 such parts, and so ⅟₇ will consist of 28÷7 = 4 such parts.
The task can, of course, also be solved in a more formal manner, by finding common denominators in a rule-based way. A strength of the task is that it can take pupils back to first principles and therefore throw light on a rule where pupils who have lost touch with the relationships on which the rule is based.
TASK 06B: This task can be used if you want to make the above explanation more explicit (by showing all 28 parts) - or if pupils are genuinely stuck on the first task.
TASK 06C: This is a more extreme version of the first task, and can be used to reinforce the ideas that emerged there.
TASK 06D: Here is another variant, but this time the fractions have a common factor. Pupils might initially choose the 4th small mark to the right of 0 for the position of ⅟₁₀, though ⅟₁₀ is clearly not as close to ¼ as that!
TASK 06E: This involves the same fractions as in Task 06A, but this time the desired fraction is to the right rather than the left of the given fraction. This might make the task more demanding; pupils will have to imagine, sketch or construct the necessary marks to the right of the ⅟₇ mark.
TASK 06F: Here is a similar task, with the desired fraction to the right of the given fraction, but with the entire 0–1 interval partitioned into equal parts. This should make the task less demanding, as well as perhaps making it easier to explain precisely what is going on.
