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

# Big Idea IV Equivalence

*What does the research say?*

Without an understanding of equivalent fractions pupils will struggle to compare rational numbers and perform operations with fractions. Wong and Evans (2007) described how the pupils in their sample from Y3 to Y5 were able to demonstrate the deepest conceptual understanding when they could link pictorial representations to symbolic representations of fractions and offer alternative equivalent fractions. Nunes et al (2009) discuss how pupils use ‘correspondence’ to solve problems involving equivalent fractions. In their example the pupils had to determine the number of pieces of one then two pizzas to share between three people at a restaurant. The discussion drew out the pupils’ early understanding of equivalence by being able to see that when the second pizza arrived the same portions would be distributed the same way as the first time. This strategy was also observed by Kieron (1993, cited in Nunes et al) when solving non-equivalent fraction problems.

The use of manipulatives to support learning and understanding of equivalent fractions is well documented, one example in the Rational Number Project . Study results indicate that students using manipulatives significantly outperformed students taught using just a symbolic approach. Examples of approaches can be found on the project’s website.

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

### Y2

- Write simple fractions for example 12 of 6 = 3 and recognise the equivalence of 24 and 12

### Y3

- recognise and show, using diagrams, equivalent fractions with small denominators

### Y4

- recognise and show, using diagrams, families of common equivalent fractions

### Y5

- identify, name and write equivalent fractions of a given fraction, represented visually, including tenths and hundredths

### Y6

- use common factors to simplify fractions; use common multiples to express fractions in the same denomination

## Potential misconceptions

**Misconception 1: 23 and 46 do not represent the same fraction because the numerators and denominators are different.**

In order for pupils to understand equivalence they must understand that to compare two fractions they must both be fractions of the same ‘whole’. One of the first equivalences that pupils should explore is the equivalences of one and other whole numbers.

i.e. 1 = 33 2 = 42 etc.

Try this problem from NRICH, Fair Feast which will help pupils experience equivalences of 12.

**Adding the same number to the numerator and dominator will give you an equivalent fraction. e.g. 38 = 49**

Multiplication and division of the numerator and denominator preserve the proportional relationships and produce equivalent fractions but addiing to or subtracting from the numerator and denominator do not preserve the proportional relationship.

Provide pupils with opportunities to observe shapes shaded as equivalent fractions. Encourage pupils to compare by asking “What do you notice?” or “What’s the same, what’s different about these shapes?”

e.g.

Pupils should experience and compare fair sharing situations:

Two chocolate bars shared between four children will provide the same number of bars for each person as one chocolate bar shared between two children. Pupils can explore how many chocolate bars are needed for four children or six children etc.

Or, one pizza shared equally between three children will give the same number of slices as two pizzas shared between six children.

Or, a bag of 40 sweets shared equally between four children will lead to the same number per child as 20 sweets shared equally between two children.

For a more abstract understanding of equivalence try this activitiy from NRICH, Fraction Wall.

Reflect on the opportunities that you have given pupils to experience equivalent fractions in contextual situations.

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

## Arrays

Use arrays to represent equivalent fractions to see them ‘grow’ or ‘shrink’. Watch this slide to illustrate how arrays can support conceptual understanding of equivalent fractions.

## Fraction Cards

Use fraction cards made from this sheet to play exchange games that help build an understanding of equivalences of fractions.

## Fraction Wall

Use a fraction wall to find unit and non-unit fractions that are equivalent.

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

Watch these video clips from our Supporting the new National Curriculum suite of materials to see how a fraction wall can be used as a tool to support the addition of fractions with unlike denominators.

If you go by the area of the cards they are OK? They are not meant to be like a fraction wall (where you might compare vertically) but are to be cut up.