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

# Big Idea III Comparing and Ordering Fractions

*What does the research say?*

Comparing and ordering fractions depends on a conceptual understanding of a fraction and its place on a numberline and therefore its position in relation to other fractions and whole numbers on that numberline. As mentioned above, identifying a fraction’s position on a numberline is far more problematic than positioning whole numbers since an understanding of the relationship between the numerator and denominator is required. (Nunes et al, 2009).

Nunes et al replicated an experiment carried out by Hart et al (1985; cited in Nunes et al, 2009) where they asked Y4 and Y5 pupils to compare 37 with 57 and 35 with 34. The majority of pupils were able to compare the fractions with like denominators accurately by treating the numerators as whole numbers. However the fractions with like numerators were not easily compared and only 20% of pupils were able to answer accurately. These results compare with the similar experiments across the world and they cite many examples in the text.

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

### Y3

- compare and order unit fractions, and fractions with the same denominators

### Y5

- compare and order fractions whose denominators are all multiples of the same number

### Y6

- compare and order fractions, including fractions >1

## Potential misconceptions

**Misconception 1: The larger the denominator the bigger the portion.**

Provide plenty of opportunity for pupils to share quantities and portions among different sized groups.

e.g. if we shared this bag of Jelly Babies with five people how many would each get? If we shared this same bag of jelly babies with six people instead would each person get more or fewer than if there were five people?

If we shared this pizza between four people what fraction would each person have? What if it was shared between five people? Would each piece be larger or small than the piece when shared between four people?

If we needed to cut this piece of ribbon equally for some girls’ ponytails what would happen to the length of the pieces, the more girls we have who need ribbon?

We have one hour to have turns on the P.E. equipment. Esme is in a group of four will she have more or less goes on the equipment than her friend Paul who is in a group of five?

How confident are you in identifying whether 38 is larger or smaller than 25. What strategies might you use?

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

## Fraction strips

Use a strip of A4 paper and a paper clip to practice comparing and ordering fractions along an imaginary number line. Ask questions like “show me a half”; “show me a quarter”, “show me a third”, “show me two-thirds”. Use the strip to represent different parameters: 0 to 1; 0 to 2; 1 to 2 etc. Use the strip to represent 1L, £1, £2, 1m etc to represent and compare fractions of continuous quantities.

## Fraction Wall

Use a fraction wall to compare unit and non-unit fractions. Use the < > to compare fractions. Ask pupils to generalise about how to estimate if a fraction is greater than or less than a half, quarter, three quarters and other fractions.

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14

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18

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13

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16

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112

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15

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110

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110

## Cuisenaire or coloured rods

Use Cuisenaire or coloured rods to compare different coloured rods in relation to a given rod.

Read this article and watch the videos of Class 3 (Y4 and Y5) comparing fractions using Cuisenaire or coloured rods to see how powerful they can be in developing pupils’ conceptual understanding of proportional relationships.