# The ‘Big Ideas’ in Teaching and Learning Fractions and the New National Curriculum

# Big Idea V Calculating with Fractions

*What does the research say?*

Most of us have learned the procedures for operating with fractions without seeing or understanding the mathematical structure that underpins these processes. Doing without understanding can lead to being unable to ‘undo’ when things go wrong. There is much research today for example see Rittle-Johnson et al (2001) that supports the interrelatedness of procedural fluency with conceptual understanding and this is equally applicable to calculating with fractions. See Hecht (1998); Hecht, Close, and Santisi (2003); Hecht and Vagi (in press); Rittle- Johnson, Siegler, and Alibali (2001), cited in Siegler et al (2010).

*Where does this ‘big idea’ occur in the new programme of study for mathematics?*

### Y3

- Add and subtract fractions with the same denominator within one whole [for example, 57 + 17 ]

### Y4

- add and subtract fractions with the same denominator

### Y5

- add and subtract fractions with the same denominator and multiples of the same number, for example 15 + 110.

### Y6

- add and subtract fractions with different denominators and mixed numbers, using the concept of equivalent fractions
- multiply simple pairs of proper fractions, writing the answer in its simplest form [for example, 14 × 12 = 18 ]
- divide proper fractions by whole numbers [for example, 13 ÷ 2 = 16 ]

## Potential misconceptions

**Misconception 1: When adding (or subtracting) fractions pupils add (or subtract) both the numerators and the denominators.**

e.g. 57 + 17 = 614 or 57 - 17 = 40

Pupils do not recognise that the denominator indicates the number of ‘parts’ of the same whole and therefore treat the two fractions as 4 ‘whole numbers’ to be added together.

Before performing addition and subtraction of fractions, pupils should experience describing part/ whole relationships verbally and in written form, in the same way that they would describe whole number trios.

e.g. The yellow and purple shaded parts in the shape below represent 25 + 35 = 55 or 1; or the yellow parts are represented by 1 - 35;

or the yellow and purple shaded parts in the shape below represent 25 + 25 = 45

**Misconception 2: Pupils do not see the need to ensure that the denominators are common before performing addition or subtraction.**

As above pupils are insecure in the proportional relationship between the numerator and denominator. Pupils need to understand the addition and subtraction of fractions in familiar contexts in order for them to grasp why the denominators must be the same.

e.g. Anna and Paul had a bar of white chocolate and a bar of milk chocolate. The bars were the same size. They each ate 14 of the white chocolate and 13 of the milk chocolate. What fraction of the chocolate bars did they each eat? What fraction of the chocolate bars did they both eat altogether?

Or… Tom and David had a block of butter to make cakes and cookies with. They needed 12 a block for cookies and 13 of a block for the cakes. What fraction of the butter was left after they made their cakes and cookies?

Try this challenging problem from NRICH, Andy’s Marbles to practice adding and subtracting fractions with different denominators.

How important are the five big ideas and how does each one relate to another / others?

*What representations can be used to support pupils’ developing conceptual understanding of this ‘big idea’ (calculating with fractions)*

## Fraction Cards

Use fraction cards made from this sheet to enable pupils to manipulate representations of unit fractions in order to add and subtract fractions with like denominators or with multiples of the same number.

Watch these video clips from our Supporting the new National Curriculum suite of materials to see how pupils in Y2 are able to add and subtract fractions and mixed numbers with support from these materials.

**Arrays** (Addition and subtraction involving fractions with different denominators)

Use arrays to add fractions with different denominators. The principle behind this is to find an array where both the fractions in the calculation can be identified.

13 + 12 = ?

Find an array that can be shaded as a 13 and a 12. In this case this is a 3 × 2 array. Count the number of parts in each whole (6; each part is 16 ). Count the number of shaded parts.

13 + 12 = 26 + 36 = 56

Watch these video clips from our Supporting the new National Curriculum suite of materials to see how a Y4 class are able to add and subtract fractions with unlike denominators using an array representation.

## Using a bar to multiply a fraction by a fraction (a requirement for Year 6)

Please view the animated PowerPoint slides, multiplication and division of fractions at Key Stage 2.

## Using a bar to divide a unit fraction by a whole number (a requirement for Year 6)

Please view the animated PowerPoint slides, multiplication and division of fractions at Key Stage 2.

## Using a bar to divide a whole number by a fraction (a requirement for KS3)

View the NCETM National Curriculum video: Fractions - Key Stage 2 Springfield Y6 - bar model dividing by fractions

*Please note that although filmed in Year 6 it illustrates how this might be taught in KS3. We are not recommending that it is taught in Year 6.*

To explore fraction strips further Go to this page in the NCETM’s What Makes a Good Resource.

## Fractions Wall for Division

Use a fraction wall to explore a fraction divided by another fraction (A KS3 expectation)

1

14

14

14

14

18

18

18

18

18

18

18

18

13

13

13

16

16

16

16

16

16

112

112

112

112

112

112

112

112

112

112

112

112

15

15

15

15

15

110

110

110

110

110

110

110

110

110

110

E.g. 13 ÷ 14 can found by comparing 13 with 14.

It can be seen that there is one

14 and ‘a little bit’ that fits into

13. The little bit is equivalent

112 of the whole. (from the wall)

But we need to know what fraction of 14 this piece is.

We can find this by comparing 14 to 112 and notice that there are three 112 that are equivalent to 14 of the whole. So 112 is 13 of a 14.

So there is 1 and 13 quarters in 13.

The Powerpoint covers both Multiplication and Division, look at slide 6, Expectations for Division of Fractions Y6.