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National Curriculum Resource Tool

Materials to support teachers and schools in embedding the National Curriculum

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Year 6 - Fractions (including decimals and percentages)

New Curriculum

  • use common factors to simplify fractions; use common multiples to express fractions in the same denomination
  • compare and order fractions, including fractions >1
  • 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, ¼ × ½ = 1⁄8]
  • divide proper fractions by whole numbers [for example, 1⁄3 ÷ 2 = 1⁄6]
  • associate a fraction with division and calculate decimal fraction equivalents [for example, 0.375] for a simple fraction [for example, 3⁄8]
  • identify the value of each digit in numbers given to three decimal places and multiply and divide numbers by 10, 100 and 1000 giving answers up to three decimal places
  • multiply one-digit numbers with up to two decimal places by whole numbers
  • use written division methods in cases where the answer has up to two decimal places.
  • solve problems which require answers to be rounded to specified degrees of accuracy
  • recall and use equivalences between simple fractions, decimals and percentages including in different contexts.

Non-Statutory Guidance

Pupils should practise, use and understand the addition and subtraction of fractions with different denominators by identifying equivalent fractions with the same denominator. They should start with fractions where the denominator of one fraction is a multiple of the other (for example,½ + ⅛ = ⅝ ) and progress to varied and increasingly complex problems.

Pupils should use a variety of images to support their understanding of multiplication with fractions. This follows earlier work about fractions as operators (fractions of), as numbers, and as equal parts of objects, for example as parts of a rectangle.

Pupils use their understanding of the relationship between unit fractions and division to work backwards by multiplying a quantity that represents a unit fraction to find the whole quantity (for example, if ¼ of a length is 36cm, then the whole length is 36 × 4 = 144cm).

They practise calculations with simple fractions and decimal fraction equivalents to aid fluency, including listing equivalent fractions to identify fractions with common denominators.

Pupils can explore and make conjectures about converting a simple fraction to a decimal fraction (for example, 3 ÷ 8 = 0.375). For simple fractions with recurring decimal equivalents, pupils learn about rounding the decimal to three decimal places, or other appropriate approximations depending on the context. Pupils multiply and divide numbers with up to two decimal places by one-digit and two-digit whole numbers. Pupils multiply decimals by whole numbers, starting with the simplest cases, such as 0.4 × 2 = 0.8, and in practical contexts, such as measures and money.

Pupils are introduced to the division of decimal numbers by one-digit whole number, initially, in practical contexts involving measures and money. They recognise division calculations as the inverse of multiplication.

Pupils also develop their skills of rounding and estimating as a means of predicting and checking the order of magnitude of their answers to decimal calculations. This includes rounding answers to a specified degree of accuracy and checking the reasonableness of their answers.

Links and Resources

Teachers should use every relevant subject to develop pupils’ mathematical fluency. Confidence in numeracy and other mathematical skills is a precondition of success across the national curriculum.
Teachers should develop pupils’ numeracy and mathematical reasoning in all subjects so that they understand and appreciate the importance of mathematics.

– National Curriculum in England Framework Document, September 2013, p9

Connections within Mathematics

There are many connections and these need to be discussed with pupils. They need to see that fractions are numbers in their own right, and can, thus, be placed on a number line. They need to understand how fractions are linked to division – using both sharing and grouping. Also the link to multiplication and finding factors, and that fractions express a relationship between 2 groups, e.g. 3 out of these 4, the proportion.

Making connections to this topic in adjacent year groups

Year 5

  • compare and order fractions whose denominators are all multiples of the same number
  • identify, name and write equivalent fractions of a given fraction, represented visually, including tenths and hundredths
  • 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, 2⁄5 + 4⁄5 = 6⁄5 = 1 1⁄5
  • add and subtract fractions with the same denominator and denominators that are multiples of the same number
  • multiply proper fractions and mixed numbers by whole numbers, supported by materials and diagrams
  • read and write decimal numbers as fractions [for example, 0.71 = 71/100]
  • recognise and use thousandths and relate them to tenths, hundredths and decimal equivalents
  • round decimals with two decimal places to the nearest whole number and to one decimal place
  • read, write, order and compare numbers with up to three decimal places
  • solve problems involving number up to three decimal places
  • recognise the per cent symbol (%) and understand that per cent relates to ‘number of parts per hundred’, and write percentages as a fraction with denominator 100, and as a decimal
  • solve problems which require knowing percentage and decimal equivalents of ½, ¼, 1⁄5, 2⁄5, 4⁄5 and those fractions with a denominator of a multiple of 10 or 25.

Key Stage 3

  • work interchangeably with terminating decimals and their corresponding fractions (such as 3.5 and 7⁄2 or 0.375 and 3⁄8 )
  • define percentage as ‘number of parts per hundred’, interpret percentages and percentage changes as a fraction or a decimal, interpret these multiplicatively, express one quantity as a percentage of another, compare two quantities using percentages, and work with percentages greater than 100%
  • interpret fractions and percentages as operators
  • express one quantity as a fraction of another, where the fraction is less than 1 and greater than 1
  • use ratio notation, including reduction to simplest form
  • divide a given quantity into two parts in a given part:part or part:whole ratio; express the division of a quantity into two parts as a ratio
  • understand that a multiplicative relationship between two quantities can be expressed as a ratio or a fraction
  • relate the language of ratios and the associated calculations to the arithmetic of fractions and to linear functions
  • solve problems involving percentage change, including: percentage increase, decrease and original value problems and simple interest in financial mathematics
  • solve problems involving direct and inverse proportion, including graphical and algebraic representations

Cross-curricular and real life connections

Learners will encounter fractions in many different real life contexts:

  • When shopping, children can compare prices presented in decimal form. Consider reductions in price when the reduction is given as a fraction (e.g. ‘one third off’) or percentage (‘20% off today’). Sharing the cost of a total bill equally in a restaurant provides a useful context in which to practise estimation of fractions as well as calculating.
  • Fractions skills can be also emphasised when focusing on measurement. Journey times and fuel consumption can be estimated and calculated (e.g. what fraction of the journey do we have remaining?) Measurement of area and perimeter is strongly linked to work with fractions, ratio and proportion; what proportion of the playground needs to be set aside for ball games?
  • When interpreting and evaluating data children will need to use their fraction knowledge. E.g. Half a million people are earning 20% below the minimum wage
  • Issue 11 of the NCETM Primary Magazine provides wonderful links to the work of artist Mondrian, with a focus on fractions and decimal work.

This is a booklet written for teachers of adult learners. It is, of course, a useful guide for teachers of learners of all ages and addresses some of the complexities in teaching and learning fractions, including possible misconceptions. The document also includes some suggestions for activities that would be adaptable for use with children.

Nunes, T., Bryant P., Hurry, J. & Pretzlik, U. (2006) Teaching and learning research briefing, no 13, London, The Institute of Education Published by the Economic and Social research council NCETM

Rationale for teaching fractions, with careful consideration of likely misconceptions

Audrey’s acquisition of fractions - Keijzer, R. and Terwel, J. Audrey’s acquisition of fractions: A case study into the learning of formal mathematics in Educational Studies in Mathematics. Vol 47, Numb4er 1/ May 2001, pp 53 - 73

An account of one child’s learning journey, using a range of models and images

Pumfrey, L. Nrich article, available on their website

Explains the history of fractions, making useful links with the emergence of Egyptian fractions, which focus on unitary fractions, a useful starting point.

Programme of Study statements Activities
A B C D E F G
Use common factors to simplify fractions; use common multiples to express fractions in the same denomination            
Compare and order fractions, including fractions >1            
Associate a fraction with division and calculate decimal fraction equivalents [for example, 0.375] for a simple fraction [ for example, 3⁄8 ]            
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, ¼ × ½ = 1⁄8 ]          
Divide proper fractions by whole numbers [ for example, ⅓ ÷ 2 = 1⁄6 ]            
Identify the value of each digit to three decimal places and multiply and divide numbers by 10, 100 and 1000 where the answers are up to three decimal places            

Activity A : Factors and Multiples Game

This game from Nrich could replace standard practice exercises on finding factors and multiples. In order to play strategically, pupils must start to think of numbers in terms of their factors

Activity B(i) : Rod Fractions

Compare a series of coloured rods and the relationships between them with this clearly presented Nrich activity

Activity B(ii) : Laundry Line

This activity features a laundry line with ‘fraction washing.’ The aim is to hang the fractions on the line in the right order between 0 and 1. Once a game has been completed, you can raise the level of difficulty.

Laundry line

Activity B(iii) : Chocolate

Using chocolate bars to compare fractions – a context guaranteed to keep them focused!

Chocolate

Activity C : Fraction strips

The ‘Fractions’ Interactive Teaching Programme, allows the user to divide a strip into equal parts and colour them as needed. Strips can be labelled as a fraction, decimal or percentage. The ratio of parts can also be displayed. Multiple strips can be created to demonstrate equivalence.

Activity D(i) : Clock Faces

Give children the chance to explore clock faces as a way of representing time. Talk about 5 minute sectors of the clock being equivalent to twelfths, ten minute sectors to sixths, fifteen minute chunks to quarters etc…

Use this to support them in adding 20 minutes plus 15 minutes as 1/3 + 1/4

Activity D(ii) : Andy’s Marbles

This challenging activity from Nrich requires children to use their conceptual understanding of fractions, and their sums, to determine the answer to Andy’s marble problem

Marbles

Activity D(iii) – Soccer Shoot Out

Score and save goals by correctly answering fraction based calculations. A range of difficulty levels makes this game easy to adapt.

Football

Activity E(i) : Folding ribbons

Show the children a lengths of ribbon (or string) measuring 3/4 metre in length. Fold it into three parts. What fraction of the ribbon have we found? ( 1/3 ). Measure the folded piece together. How long is it? ( 1/4 m). So, 1/3 of 3/4 or 1/3 x 3/4 = 1/4.

Try this with other fractions and starting lengths.

Activity E(ii) – ratio tables

Ask children to use their multiplication tables to scale recipe quantities up and down to complete the table below. Consider their multiplication of simple fractions for quantities of vanilla extract and chocolate chips.

Ingredients 3
cupcakes		 6
cupcakes		 12
cupcakes		 18
cupcakes		 24
cupcakes
Self- raising flour     130 gm   260 gm
Caster Sugar     100 gm    
Butter     125 gm    
Large Eggs     2    
Vanilla extract     ½ teaspoonful   1 teaspoonful
Chocolate chips     ¾ cupful    

Activity F(i) - Pizzas

This pizza themed fraction resource from the TES can be used in many different ways, and includes several suggestions from the author.

Activity F(ii) – Planetary Wars

Published by BEAM, and hosted within the National STEM Centre E-library, this two-player dice game requires children to find fractional quantities of numbers.

Activity G – Decimal Point Chair

When multiplying or dividing by powers of ten, the key issue to emphasise in many classrooms is that it is not the decimal point which moves, but the digits of the number. Prepare a range of cards to include several zeros, a decimal point and two whole numbers [e.g. 2 and 7]. Give a group of children one card each and ask the child with the decimal point to sit in the middle of a row of chairs, facing the class. Ask the 2 children with the cards with whole numbers to choose a chair to sit on, leaving no empty chairs between them and the decimal point. Explore ways to do this. Then introduce one zero in various places and talk about the numbers created. Add a second zero and do the same. Move on to asking the children to decide what happens to the number created if we multiply or divide by 10, then 100, then 1000 – emphasise that the decimal point must not move. Ask children to consider ways in which they might record this.

For a similar activity visit the National Stem Centre E-library, for the Interactive Teaching Programme ‘Moving Digits’

Please note that you will need to log in to this site in order to access the materials. This is free and simple to do.

Useful Resources

Activity B(i) - coloured rods

Activity B(iii) – several bars of chocolate

Activity D(i) – analogue clocks

Activity D(ii) – marbles

Activity E(i) – ribbon/string, tape measures, scissors

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

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

Children should be able to recognise that a fraction such as 5⁄20 can be reduced to an equivalent fraction of ¼ by dividing both numerator and denominator by the same number [cancelling] They should also be familiar with identifying fractions in different units. E.g. what fraction is 20 pence of two pounds? Of four pounds etc…

compare and order fractions, including fractions >1

Children should be able to:

i] Position fractions on a number line; e.g. mark fractions such as 7⁄5 , 11⁄20 , 18⁄12 on a number line graduated in tenths

ii] Answer questions such as: What number is half way between 5 ¼ and 5 ½ ?

iii] Which is larger, ⅓ or ⅖? Explain how you know.

associate a fraction with division to calculate decimal fraction equivalents (e.g. 0.375) for a simple fraction (e.g. ⅜)

Children should be able to find fractions of numbers and quantities;

i] What fraction of £1 is 35p, … 170p ?

ii] Write 23⁄100 of 4 kilogrammes in grams

iii] What fraction of 1 litre is 413 ml?

Convert a fraction to a decimal using known equivalent fractions:

i] ¼ = 0.25

ii] ⅖ = 0.4

add and subtract fractions with different denominators and mixed numbers, using the concept of equivalent fractions

Children should be able to solve practical problems such as;

Chocolate

Here is a chocolate bar.

William eats 3 pieces and Amber eats 2 pieces. What fraction of the chocolate bar remains?

Joe has some pocket money. He spends three-quarters of it. He has fifty pence left. How much pocket money did he have?

multiply simple pairs of proper fractions, writing the answer in its simplest form, (e.g. ¼ × ½ = ⅛)

Children should be able to:

i] Recognise that ¼ of 12, ¼ x 12 and 12 divided by 4 are equivalent

ii] Use cancellation to simplify the product of a fraction and an integer

eg ⅕ x 15 = 3

⅖ x 15 = 2 x ⅕ x 15 = 2x3 = 6

ii] Work out how many ½s in 15, how many ⅖s in 15, how many 2/5s in 1 etc.

divide proper fractions by whole numbers (e.g. ⅓ ÷ 2 = ⅙ )

Children should be able to:

Decide whether they would prefer to share ½ of a pizza with 2 people or ¾ of a pizza with 4 people and explain why.

associate a fraction with division and calculate decimal fraction equivalents (e.g. 0.375) for a simple fraction (e.g. 3⁄8)

Children should be able to:

Explain how much pizza each person would get if they divided 4 pizzas between 5 people, as a fraction and a decimal

identify the value of each digit to three decimal places and multiply and divide numbers by 10, 100 and 1000 where the answers are up to three decimal places

Children should be able to identify the value of each digit in the number 17.036 and multiply and divide this by 10.100 and 1000

multiply one-digit numbers with up to two decimal places by whole numbers

Children should be able to calculate the answer to questions such as;

What is 3.86 multiplied by nine?

use written division methods in cases where the answer has up to two decimal places

Children should be able to calculate 601 divided by 36, to two decimal places

solve problems which require answers to be rounded to specified degrees of accuracy

Children should be able to solve problems such as;

Four friends win £48,623. The money is to be shared equally between them – how much will each person receive?

107 pupils and teachers need to be taken to the theatre. How many 15-seater minibuses will be required?

How many boxes of 60 nails can be filled from 340 nails?

recall and use equivalences between simple fractions, decimals and percentages, including in different contexts.

Children should be able to put a ring around the percentage that is equal to three-fifths;

20% 30% 40% 50% 60%

As well as circle the two fractions that are equivalent to 0.6.

6⁄10 1⁄60 60⁄100 1⁄6

A video with demonstrations of how to find the highest common factor and lowest common multiples

Two videos focusing on adding and subtracting fractions and working with mixed numbers

A lesson in which the teacher makes a clear connection between tenths, hundredths and thousandth and decimal places

Many students have problems visualising fractions abstractly and here we see paper cups being used to concretely represent halves and quarters. Moving the cups between two places, a resources table and a maths table, makes addition and subtraction clear.

A demonstration of how paper folding can be used to support work on fractions.