Introduction – What is a fraction?
Understanding fractions is not simply a seamless journey from understanding the concept of whole number. It has been shown, however that primary-age pupils’ knowledge of fractions and division can predict the success in mathematics in later stages of their schooling. (Siegler et al, 2012). So it is important to get the teaching right in primary schools.
The term fraction is commonly introduced to pupils to mean an equal part of a whole. However a fraction can have multiple meanings and unless fractions are understood more broadly there is a risk that pupils will become confused when solving problems involving fractions. So what are the multiple meanings for fractions?
Fractions as a division of a whole into equal parts
This meaning is best explained as a part/ whole situation.
e.g. 15 would mean that the whole is divided into five equal portions and one of those five portions is ‘taken’.
Similarly 25 would mean that the whole is divided into five equal portions and two portions are ‘taken’.
So the denominator indicates the number of equal parts and the numerator indicates the number of those parts to be ‘taken’.
This is typical of the introduction to fractions that young pupils experience. Often through shading whole geometric shapes to represent a given unit or naming parts that are already shaded.
A fraction as a quotient
This meaning for a fraction is contextualised as division by ‘sharing’. So 15 would mean one ‘something’ shared equally among five recipients, which results in each person receiving one fifth. 15 = 1 ÷ 5
25 would mean two ‘somethings’ shared equally among five recipients.
Rather elegantly not only does 25 in this example represent two wholes shared between five recipients, it also represents the proportion that each recipient receives, so 25 = 2 ÷ 5 = 25
An alternative way to look at this is that each person receives one fifth from each one, so because there are two things each person will receive two fifths. This is a clearer visual model and more easily generalisable, so 37 is seen as 3 things shared between 7, so each would receive a 17 from each one, so three sevenths in total.
A fraction as a number
A fraction can also be a number and just like any whole number, can be cardinal or ordinal. Its ordinal position never changes and it will always be placed in the same position on a number line.
12 lies between 0 and 1. 34 comes before one but after 14.
Its cardinal (quantity) value however is dependent on the whole and will therefore vary. For example if Jo spends half of his pocket money and Sam spends one quarter, it may be that the quarter is more than the half. For example if Jo had £2, he spends 50p and If Sam had £10, he spends £2.50.
A Fraction as an Operator
To find a fraction of something is an instruction to operate on the whole. For example to find a half involves the operation of halving. This is equivalent to multiplying by a half or dividing by two. This idea generalises to any fraction, so finding a third is the same as multiplying by a third or dividing by three etc. The requirement to identify or find a fraction is a requirement for Year 1, whereas multiplying by a fraction or dividing a fraction by a whole number is a requirement of Year 6. It is important that the connections are made.