• Bright, GW et all (1988) Identifying Fractions on number lines, Journal for Research in Mathematics Education. Vol 19, No 3, p215 – 232

  This study investigated the ways students represented fractions on number lines and the effects of instruction on those representations. Two clinical teaching experiments and one large-group teaching experiment were conducted with fourth and fifth graders.

• Wong M & Evans D (2007) Students Conceptual Understanding of Equivalent Fractions, Mathematics: Essential Research, Essential Practice, |Volume 2, Proceedings of the 30th annual conference of the Mathematics Education Research group of Australasia. P824 – 833

  This paper investigates students’ conceptual understanding of equivalent fractions by examining their responses to questions using symbolic and pictorial representations. Five questions from this fraction assessment instrument were analysed. The different types of knowledge used to answer each question were examined and common misconceptions identified. The responses of students with limited general mathematics achievement were compared to those of their more competent peers. The differences that emerged between the two groups in their conceptual understanding of equivalent fractions, were highlighted.

• Davidson S & Askew M (2012) Concept cartoons as a Way to Elicit Understandings and Encourage Reasoning about Decimals in Year 7 in J.Dindyal, L.P. Cheng & S.F.Ng (Eds), Mathematics Education: Expanding horizons (Proceedings of the 35th annual conference of the Maths Education Research Group of Australasia). Singapore: MERGA p218 – 225.

  This paper is located within the research into encouraging learners to reason mathematically and become engaged with concepts rather than just procedures. It reports on a research in progress examining two techniques to be used at the beginning of a sequence of lessons in order to elicit students’ prior knowledge and understanding of topics about to be studied. Two techniques will be used to elicit the students’ prior understandings: Concept Maps and Concept Cartoons. This paper reports on the ideas behind Concept Cartoons and the development of Concept Cartoons for eliciting understandings of decimals. The paper briefly outlines the research into difficulties with decimals before discussing the rationale and design of the cartoons.

• Nrich Primary team. Teaching Fractions with Understanding: Part-whole Concept. Nrich article, available on their website.

  In her studies of children learning about rational numbers, Grace Lopez-Charles has identified some different perspectives through which children see fractions. This article starts looking at some of those perspectives, firstly by considering what happens in situations where learners are trying to create equal shares. From sharing using halves learners move to partitioning using other divisions, such as fifths or sixths. This results in the need to develop understandings of equivalence which is discussed in the second section of this article. Finally, we look at some ways of teaching fractions more effectively that arise from Grace's work.

• Bernard Bagnall. Early Fraction Development. Nrich article, available on their website.

  In this article, Bernard outlines suggestions for tasks which will help young children develop the concept of fractions.

A range of web based interactive programmes, aimed at providing practice and consolidation for pupils exploring equivalent fractions.

Of course, concrete materials and activities are the best way to teach children about fractions and enable them to build strong concepts.

Activity A – interactive programmes

1. Game matching pictures and equivalent fractions

Activity B – Tenths and Hundredths

1. A resource pack of materials including an interactive place value grid to teach tenths and hundredths, activities on key vocabulary and terms, reference material and animations and three differentiated worksheets on tenths and hundredths
2. A series of activities to introduce and use hundredths
3. Make a hundred square of pennies in the classroom and talk about one hundredth of a pound and a tenth of a pound. Consider how, for example, 23p is written as a decimal. How would we write four pound and eighty-two pence? What would it look like in our coins?
4. Interactive matching of decimal notation to a shaded 10x10 grid representation

Activity C – fractions to calculate quantities

1. This Count On resource is no longer available.

Activity D – adding and subtracting fractions

1. Game practising adding two or more fractions with the same denominator
2. Online adding and subtracting fractions activities written as word problems. Some useful ideas.

Activity E – decimal equivalents

1. Excellent ‘splat’ game accessible at several different levels, requiring children to match the fraction (in tenths or hundredths) to the decimal representation
2. ‘The Decifractor’. A flexible resource demonstrating equivalences between fractions and decimals.
3. Pelmanism-style game , matching fractions to decimals.
4. Arcade type game, where ‘Fraction Man’ has to defeat the decimals (by matching them). Note – this does get challenging! find the effect of dividing a one- or two-digit number by 10 and 100, identifying the value of the digits in the answer as units, tenths and hundredths

Activity F – dividing by 10 and 100

1. Interactive Teaching Program – ‘Moving Digits’. A good modelling program to show how digits move when multiplied or divided by 10 or 100.

Activity G – rounding and comparing decimals

1. Rounding decimals activity – uses one and two decimal places.
2. Practise rounding numbers to one decimal place.
3. Spreadsheets ‘Rounding’ and ‘Rounding Decimals’
4. Arrange the decimals in order from lowest to highest.

Activity H – solve problems involving fractions and decimals

1. Pelmanism-style game. Match the coin pictures to the correct fraction of amount.
2. BBC skillswise activity focusing on money amounts as fractions.
3. Fractions of amounts interactive word problems.

Examples of what children should be able to do, in relation to each (boxed) Programme of Study statement

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

Recognise that five tenths (⁵⁄₁₀) or one half is shaded.

Recognise that two eighths (²⁄₈) or one quarter (¼) of the set of buttons is ringed

Recognise that one whole is equivalent to two halves, three thirds, four quarters… For example, build a fraction ‘wall’ using a computer program and then estimate parts.

Recognise patterns in equivalent patterns, such as:

½ = ²⁄₄ = ³⁄₆ = ⁴⁄₈ = ⁵⁄₁₀ = ⁶⁄₁₂ = ⁷⁄₁₄ And similar patterns for ⅓, ¼, ⅕, ⅙, ¹⁄₁₀.

Here is a square.

What fraction of the square is shaded?

Here are five diagrams. Look at each one.
Put a tick (✔︎) on the diagram is exactly ½ of it is shaded. Put a cross (✗) if it us not.

count up and down in hundredths; recognise that hundredths arise when dividing an object by a hundred and dividing tenths by ten

Respond to questions such as:

What does the digit 6 in 3.64 represent? The 4? What is the 4 worth in the number 7.45? The 5?

Write the decimal fraction equivalent to:

two tenths and five hundredths; twenty-nine hundredths; fifteen and nine hundredths.

Continue the count 1.91, 1.92, 1.93, 1.94 ...

Suggest a decimal fraction between 4.1 and 4.2

Know how many 10 pence pieces equal a pound, how many 1 pence pieces equal a pound, how many centimetres make a metre.

solve problems involving increasingly harder fractions to calculate quantities, and fractions to divide quantities, including non-unit fractions where the answer is a whole number

What is one-fifth of twenty-five?

Write the missing number to make this correct.

Mary has 20 pet stickers to go on this page.

¼ of them are dog stickers. ½ of them are cat stickers. The rest are rabbit stickers. How many rabbit stickers does she have?

Match each box to the correct number. One has been done for you.

add and subtract fractions with the same denominator

For example:

½ + ½,  ¼ + ¾,  ⅜ + ⅝,  ⅗ + ⅘ + ⅕,  ⁷⁄₁₀ + ³⁄₁₀ + ⁵⁄₁₀ + ⁸⁄₁₀,  ¾ - ⅓,  ⁶⁄₇- ⁴⁄₇,  ⁹⁄₁₀ + ⁴⁄₁₀, – ³⁄₁₀

recognise and write decimal equivalents of any number of tenths or hundredths

Recognise that, for example:

0.07 is equivalent to ⁷⁄₁₀₀ 6.35 is equivalent to 6 ³⁵⁄₁₀₀

Particularly in the contexts of money and measurement

Respond to questions such as:

Which of these decimals is equal to ¹⁹⁄₁₀₀? 1.9 10.19 0.19 19.1 Write each of these as a decimal fraction: ²⁷⁄₁₀₀ ³⁄₁₀₀ 2 ³³⁄₁₀₀

recognise and write decimal equivalents to ¼; ½; ¾

Know that, for example

0.5 is equivalent to ½, 0.25 is equivalent to ¼, 0.75 is equivalent to ¾, 0.1 is equivalent to ¹⁄₁₀

Particularly in the context of money and measurement.

find the effect of dividing a one- or two-digit number by 10 and 100, identifying the value of the digits in the answer as units, tenths and hundredths

Understand that:

When you divide a number by ¹⁄₁₀₀, the digits move one/two places to the right.

Write a two-digit number on the board. Keep dividing by 10 and record the answer. Describe the pattern.

Respond to oral or written questions such as:

How many times larger is 2600 than 26?

How many £1 notes are in £120, £1200?

Divide three hundred and ninety by ten.

Write in the missing number

round decimals with one decimal place to the nearest whole number

Round these to the nearest whole number. For example:

9.7,   25.6,  148.3

Round these lengths to the nearest metre:

1.5m,  6.7m,  4.1m,  8.9m

Round these costs to the nearest £:

£3.27,  £12.60,  £14.05,  £6.50

compare numbers with the same number of decimal places up to two decimal places

Place these decimals on a line from 0 to 2:

0.3, 0.1, 0.9, 0.5, 1.2, 1.9

Which is lighter: 3.5kg or 5.5kg? 3.72kg or 3.27kg? Which is less: £4.50 or £4.05?

Put in order, largest/smallest first:

6.2, 5.7, 4.5, 7.6, 5.2, 99, 1.99, 1.2, 2.1

Convert pounds to pence and vice versa. For example: Write 578p in £.

How many pence is £5.98, £5.60, £7.06, £4.00? Write the total of ten £1 coins and seven 1p coins (£10.07)

Write centimetres in metres. For example, write: 125 cm in metres (1.25 metres)

solve simple measure and money problems involving fractions and decimals to two decimal places. These are the prices in a shoe shop

How much more do the boots cost than the trainers? Rosie buys a pair of trainers and a pair of sandals. How much change does she get from £50?

A box of four balls costs £2.96. How much does each ball cost? Dean and Alex buy 3 boxes of balls between them. Dean pays £4.50. How much must Alex pay? KS2 Paper B level 3

A full bucket holds 5½ litres. A full jug holds ½ a litre. How many jugs full of water will fill the bucket?

Harry spent one quarter of his savings on a book. What did the book cost if he saved: £8…£10…£2.40…?

Gran gave me £8 of my £10 birthday money. What fraction of my birthday money did Gran give me?

Max jumped 2.25 metres on his second try at the long jump.

This was 75 centimetres longer than on his first try.

How far in metres did he jump on his first try?

This Teachers TV clip from the series Great Lesson Ideas shows a Key Stage Two mathematics lesson designed to reinforce the concept that fractions are different parts of a whole. Children go on to make their own fraction wall.

Part of the Lesson Starters series this video is set in a supermarket, and shows a selection of the fractions and decimals found on the packaging of every day products and asks the students to discuss what they mean.

Adding and subtracting fractions with the same denominator – This video begins by showing the common mistake of adding the numerator and adding the denominator and explaining why this cannot be correct. The correct method is used with a pictorial representation to aid understanding. The video continues with a series of simple addition and subtraction examples.