One third...of what? The interpretation of fractions in KS2 and KS3
KS3 pupils often find fractions difficult. Some progress further than others in KS2, so that in your class there could be some who can confidently and accurately add and subtract fractions, some who will have already seen the routines for doing so but don’t remember how to apply them successfully, and some who haven’t yet met these routines. This will change in the future because the operations of arithmetic with fractions (add, subtract, multiply and divide them) is, from now, part of the KS2 programme of study, but nonetheless classes will still span the spectrum of confidence, accuracy and conceptual understanding. The new programmes of study emphasise that teaching must develop
- problem solving
and so it’s not sufficient that pupils only memorise algorithms: a deeper conceptual understanding of fractions must, over time, be embedded.
Pupils find fractions difficult to understand for a good reason: they are! The great minds of 19th century mathematics took some time, great pains and much ink to develop formally and securely the concept of the rational number. At school level, the idea of a fraction is hard because pupils encounter it in one context – as a proportion, a part of a whole – and then, later, start to use it in arithmetic as a number: they first see the fraction 1⁄3 represented as 1 slice of a cake which was first cut into 3 slices, or 1 piece of a chocolate bar which was first broken into 3 pieces, and then later they see 1⁄3 as part of a sum such as 1⁄3 + 3⁄5, apparently in the same way as they calculate 4 + 17. A “part of something” has become a “stand-alone number”: that is a conceptually challenging shift, and pupils are right to be uncertain about it. Of course, the same is true of 4, which is first met by pupils as the concrete result of a count (“4 apples”) and then becomes an abstract number, 4.
Before revising or teaching the arithmetic of fractions, it’s important to address this apparent change of status. Honesty is the best policy: pupils should be told that the written symbol – the mark made with ink on paper – that looks like “1⁄3” represents more than one thing: it can be used to specify the process “take one third of” (as in 1⁄3 x) or it can locate a place on a number line, that place being one third of the way between 0 and 1 if you start at 0 (just as the ink on paper mark “2” can be thought of as locating the place that is twice as far from 0 as from 1 if you start at 0). It can be helpful to show this on a selection of number lines: on each of the following, the red line is located at the number represented by writing “1⁄3”.
The connection between the familiar “take one third of” process and the less familiar “a place on the number line” is made explicit by these diagrams: the standalone number 1⁄3 is placed on the number line one third of the way between 0 and 1, starting at 0. There is a good opportunity here to link division with its inverse operation multiplication. The number line model shows that three pieces of length 1⁄3 will fit (and fill) a gap of length 1, i.e. 3 × 1⁄3 = 1. Necessarily, therefore, 1 ÷ 3 = 1⁄3, and 1 ÷ 1⁄3 = 3.
Calculation such as 2 ÷ 5 = 2⁄5 can be similarly represented: the place which is located two fifths of the way between 0 and 1 is the place that marks the end of one of the five pieces you get when you divide a length 2 into 5 equal parts.
Since 21⁄2 pieces of length 2⁄5 will fill the gap of length 1, this means that 21⁄2 × 2⁄5 = 1. Therefore 1 ÷ 21⁄2 = 2⁄5, and 1 ÷ 2⁄5 = 21⁄2.
Recently, UK teachers observing lessons in Shanghai schools have noticed that every opportunity is taken to emphasise the inverse relationships between the operations + and –, and × and ÷. Whenever a sum such as 7 + 13 = 20 arises, the equivalents 20 – 13 = 7 and 20 – 7 = 13 are elicited from the pupils; similarly 4 × 3 =12 immediately prompts 12 ÷ 4 = 3 and 12 ÷ 3 = 4. This happens from grades 1 and 2 (equivalent to Y2 and Y3), and by grade 5 the pupils have abstracted this into a rule they can state and use: we would translate the characters they write as “one factor = product ÷ other factor”.
As pupils develop this conceptual understanding of 1⁄3 or 2⁄5 representing a stand-alone number and not only specifying the process “take a part of” – as they realise that one third doesn’t have to be one third “of” anything, then the idea of using it in arithmetic becomes much less mysterious. Next month we will consider ways to embed deep understanding of the algorithms of the arithmetic of fractions.
Page header by PauliCarmody (adapted), some rights reserved