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

# Big Idea I Recognising Fractions

*What does the research say?*

Pupils need to experience the multiple forms of fractions (see above) to have a comprehensive conceptual understanding. Nunes et al (2006) found that although fractions were introduced in schools in the early years as part/ whole relations, pupils come into school with a varied understanding of fractions in the context of sharing and when primary pupils were provided with problems involving fractions they were more confident and accurate in solving problems involving quotient situations. i.e. fractions as a sharing/ division model than in part/ whole situations despite limited teaching of the former.

Howe et al (2005) observed that pupils did not easily transfer knowledge of fractions in one context, across to another and therefore in teaching, a rich experience of multiple representations and with explicit links made should be provided for deep conceptual understanding.

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

### Y1

- Recognise, find and name a half as one of two equal parts of an object, shape or quantity
- Recognise, find and name a quarter as one of four equal parts of an object, shape or quantity

### Y2

- Recognise, find, name and write fractions 13, 12, 24, 34 of a length, shape, set of objects or quantity

### Y3

- recognise, find and write fractions of a discrete set of objects: unit fractions and non-unit fractions with small denominators
- recognise that tenths arise from dividing an object into 10 equal parts and in dividing one – digit numbers or quantities by 10
- recognise and use fractions as numbers: unit fractions and non-unit fractions with small denominators

### Y4

- recognise that hundredths arise when dividing an object by one hundred and dividing tenths by ten

### Y5

- recognise and use thousandths and relate them to tenths, hundredths and decimal equivalents
- Recognise mixed numbers and improper fractions and convert from one form to the other and write mathematical statements > 1 as a mixed number

[for example, 25 + 45 = 65 = 1 15 ]

## Potential misconceptions

*Misconception 1: Fractions are read as pieces rather than equal part/ whole relationships*

Pupils write the shaded part as 35 . i.e. they see three parts green and five parts white and record these numbers as a fraction.

Pupils identify these parts a thirds.

Read this article from NRICH about developing pupils’ understanding of part/ whole relationships.

*Misconception 2: Fractional pieces have to be congruent (the same shape) to be the same fraction.*

Try this activity:

You need two pieces of A4 paper – two different colours works best.

Fold one piece of paper in half to make an A5 sized piece and overlay it on the unfolded sheet as central as you can to make a frame around the folded sheet. This is what you should see:

What fraction of the A4 sheet is the frame?

How can you reason that both the frame and the central piece are the same fraction of the whole?

Here’s another problem from NRICH that helps to dispel this misconception.

*Misconception 3: Identical fractions of different ‘wholes’ are not the same*

Try these activities

Ask pupils to discuss whether they would like to share equally with a friend, half of a pouch of 10 50p coins or half of a pouch of 8 £1 coins. (The amounts and quantities can be changed according to fluency with calculations). Ask them to explain why.

Ask pupils to discuss whether they would prefer to eat **half** of the cupcake or **half** of the chocolate cake.

Identify the halves, thirds and quarters in this fraction tray.

How do these activities above help to challenge pupils’ misconceptions?

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

## Using Shapes

Use shapes to recognise and name equal parts to develop a conceptual understanding of part/ whole relationships. Use more than ‘one whole’ to explore fractional parts. Use incongruent shapes to identify fractional parts.

Watch these video clips from our Supporting the new National Curriculum suite of materials to see how pupils are developing conceptual understanding through the use of these representations.

## Using a number of discrete objects

Use real life discrete objects to experience fractions as a quotient model and consider how this links to division – identifying half of the cherries is the same as dividing the cherries by 2, identifying a third of the cherries is the same as dividing the cherries by 3 etc.

Use double sided counters to represent fractions of discrete quantities eg what fraction of the counters are red?

### Use continuous quantities

Explore fractions of continuous quantities such as length, mass, capacity and time.

## Fraction Wall

Use a fraction wall with increasing variety of unit fractions to recognise and name fractions. A fraction wall cut into separate strips will allow pupils to lay them side by side and compare fifths with eighths for example.

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

## Cuisenaire Rods/ coloured rods

Use Cuisenaire/ coloured rods to recognise fractions of wholes.

Read this article with accompanying video clips for Class 2 (Y2 and Y3) to see how powerful Cuisenaire/ colours rods can be in developing conceptual understanding of part/ whole relations.

The Howe reference is:

Howe, C, Nunes, T, Bryant, P & Jafri, S (2005) Children’s developing understanding of intensive quantities: similarities and differences from extensive quantities. Poster presented at Annual Conference of Society for Research into Child Development, Atlanta, April 2005.

Cited in: http://www.education.ox.ac.uk/wordpress/wp-content/uploads/2011/04/resbrief_quantities.pdf

Citted in: