Introduction

An impetus for producing the tasks in this blog was to find ways that would help younger pupils develop a sound fraction-sense. While creating such tasks, I also hit upon ideas that resulted in tasks that would challenge older pupils and perhaps even allow us, as teachers, to see fractions in a fresh way. So the tasks in this blog are aimed at a wide range of pupils, in terms of age and experience. I hope that the commentary that goes with the tasks will help teachers select tasks that are fitting for their pupils and find ways to modify tasks to make them suitable.

A second impetus came from the observation (including an examination of widely used text books) that we tend to introduce pupils to procedures such as 'finding a common denominator' before they have developed a sound fraction-sense. Such procedures are powerful but that power is fragile, and perhaps even damaging, when pupils don't have sufficient insight into what they are doing. So it is worth continually giving pupils the opportunity to 'play' with fractions, in the course of which they might even construct procedures for themselves, or, just as important, discover the need for them.

I also think we risk inhibiting the development of fraction-sense if we only allow pupils to meet very simple fractions. Of course we want pupils to feel at home with fractions like 1/2, 1/4, 3/4, but do they act as stepping stones, or shackles that hold pupils back? Pupils might benefit from meeting fractions such as 17/19, or even 99/203, at quite an early age. Such fractions might seem unusual, but they are also far more 'ordinary', or generic, than very simple, special fractions like 1/2 and 1/4. [And if it turns out that some pupils can't make any sense of such an encounter, then we can always row back for a while - with no harm done!]

The tasks in this blog are grouped into 20 sets of 5 'weekly' tasks. The tasks within a set are closely related, though they don't have to be used in the order that they appear in the blog. That also applies to the weekly sets themselves. Whatever tasks are chosen, teachers should feel free to modify them to make them more or less demanding.

In collaboration with Laurie Jacques (UCL Institute of Education), I have come across some particularly interesting research* on ordering fractions, including an Australian study involving 11-12 year-olds that provides useful baseline data on what pupils of that age can achieve. The table below, from that study, shows the percentage of pupils in their sample (n=323) who could successfully order various pairs of fairly 'ordinary' fractions.

Table 1: Percentage of grade 6 students choosing appropriately from fraction pairs with appropriate explanation (n=323)

Clarke, D. M., & Roche, A. (2009). Students’ fraction comparison strategies as a window into robust understanding and possible pointers for instruction. Educational Studies in Mathematics, 72, 127-138.

Comparing fractions is a nice activity which can reveal pupils' thinking because it is amenable to a variety of methods, not just on knowing formal procedures. For that reason, I have started with such tasks in Week 1, and have used some of the fraction pairs shown in the above table. However, putting these tasks in Week 1 should not be taken to mean that such tasks are the first fractions tasks that pupils should encounter. And anyway, most pupils using these tasks are likely to have had extensive prior experience of fractions, though perhaps not always of the kind that enhanced their fraction sense!

Fractions can be interpreted in a variety of ways: as parts of a whole, a ratio, an operator, a quotient and a measure. There are some interesting studies that suggest a number line is a particularly effective way of developing the ratio and measurement aspects and we have made extensive use of number lines (sometimes in the restricted form of bars) in our tasks. We have also used contexts, such as sharing pancakes, to bring out the ratio and quotient aspects.

*Informed by such research, Laurie Jacques and I are currently engaged in a small study which involves interviews with small groups of Year 6 and Year 8 pupils on a range of comparison tasks involving 'uncommon' fractions.

Dietmar Küchemann
mietmau@gmail.com