Year 3 maths · Key Stage 2 · Number – fractions
Year 3 fractions: tenths, fractions of amounts and equivalence
In Year 3 children start to treat fractions as numbers in their own right. They count in tenths, find fractions of a set of objects such as 3/5 of 20, place fractions on a number line, spot equivalent fractions with diagrams, compare unit fractions, and add and subtract fractions with the same denominator when the answer is no more than one whole.
Tenths and equivalent fractions, explained
A fraction describes equal parts of a whole. The denominator, the bottom number, tells you how many equal parts the whole has been split into, and the numerator, the top number, tells you how many of them you are talking about. A unit fraction has 1 on top, like 1/4, and a non-unit fraction has more, like 3/4. The curriculum asks only for “small denominators” in Year 3, and schools choose which to use, for example halves, thirds, quarters, fifths, eighths and tenths. The word “equal” matters: a shape cut into four unequal pieces has not been cut into quarters.
Tenths get special attention. A tenth is one of 10 equal parts of a whole, and also the result of dividing a quantity by 10, so 3 ÷ 10 = 3/10. Children count up and down in tenths (1/10, 2/10, 3/10…), reaching 10/10 which is one whole and, in many classrooms, carrying on past it. Tenths link to measures: one tenth of a metre is 10 cm, and one tenth of a centimetre is 1 mm. This is the groundwork for decimals, which children start to write in Year 4.
Fractions of a set of objects are found by sharing. To find 1/5 of 20, share 20 into 5 equal groups and look at one group: 20 ÷ 5 = 4. For a non-unit fraction such as 3/5, take that many groups, so 3 × 4 = 12. Children also meet fractions on the number line, where 1/4 and 3/4 sit between 0 and 1, and a fraction such as 5/4 sits one quarter beyond 1. This is how they come to see a fraction as a number with a size, not just a shaded part of a picture.
Equivalent fractions are different names for the same amount, and Year 3 children find them with diagrams, fraction strips and fraction walls: 1/2 = 2/4 = 4/8, or 1/5 = 2/10. At this stage the curriculum asks them to show equivalence with diagrams rather than with a rule, so that they see for themselves that cutting each part into smaller equal pieces changes the numbers but not the amount. They also compare fractions: with a unit fraction, the bigger the denominator the smaller the piece, so 1/8 is smaller than 1/3, and fractions with the same denominator are ordered by their numerators, so 5/8 is bigger than 3/8.
Finally, children add and subtract fractions with the same denominator within one whole. They treat the denominator as the name of what is being counted: 5 sevenths + 1 seventh = 6 sevenths, written 5/7 + 1/7 = 6/7, and 6/7 − 2/7 = 4/7. Because the parts are the same size, only the numerators change.
Words your child will hear
- Numerator
- The top number of a fraction: how many parts are being counted.
- Denominator
- The bottom number: how many equal parts the whole is split into.
- Unit fraction
- A fraction with 1 as the numerator, such as 1/3 or 1/10.
- Non-unit fraction
- A fraction with a numerator greater than 1, such as 2/3 or 7/10.
- Tenth
- One of 10 equal parts of a whole, written 1/10. Also what you get from dividing by 10.
- Equivalent
- Worth the same amount. 1/2 and 4/8 are equivalent fractions.
- Fraction wall
- A set of stacked bars, each split into a different number of equal parts, for comparing fractions.
What the National Curriculum says
In Year 3, pupils should be taught to:
- Count up and down in tenths; recognise that tenths arise from dividing an object into 10 equal parts and in dividing one-digit numbers or quantities by 10
- Recognise, find and write fractions of a discrete set of objects: unit fractions and non-unit fractions with small denominators
- Recognise and use fractions as numbers: unit fractions and non-unit fractions with small denominators
- Recognise and show, using diagrams, equivalent fractions with small denominators
- Add and subtract fractions with the same denominator within one whole [for example, 5/7 + 1/7 = 6/7]
- Compare and order unit fractions, and fractions with the same denominators
- Solve problems that involve all of the above
Objectives from the National Curriculum in England, © Crown copyright, used under the Open Government Licence v3.0.
A worked example
Find 3/5 of 20.
- Read the fraction. The denominator 5 says to share 20 into 5 equal groups. The numerator 3 says to take 3 of those groups.
- Share 20 into 5 equal groups: 20 ÷ 5 = 4. So one fifth of 20 is 4.1/5 of 20 = 4
- Take 3 groups: 3 × 4 = 12.3/5 of 20 = 12
- Check. The 2 groups left over are 2 × 4 = 8, and 12 + 8 = 20, so all five fifths make the whole amount again.
3/5 of 20 is 12.
Common mistakes and how to help
Calling any shape cut into 4 pieces “quarters”
Children have often been told to “split it into parts” and haven’t yet connected fractions with equal size.
What helps: Cut or fold real shapes. Lay one piece over another to test whether they match. If they don’t, they can’t be called quarters.
Thinking the bigger denominator makes the bigger fraction
With whole numbers 8 is more than 3, so it seems 1/8 should be more than 1/3.
What helps: Share one chocolate bar between 3 people or between 8 people. The more people, the smaller each share. A fraction wall makes the same point visually.
Adding the denominators: 5/7 + 1/7 = 6/14
Children add “top with top and bottom with bottom” because that’s how adding works elsewhere.
What helps: Say it in words: 5 sevenths and 1 seventh is 6 sevenths, just as 5 apples and 1 apple is 6 apples. The name of the part doesn’t change when you count more of it.
Believing fractions stop at one whole
Most pictures show a shape with part of it shaded, so counting 9/10, 10/10 and then 11/10 can feel wrong.
What helps: Draw a number line from 0 to 2 divided into tenths and count along it. 10/10 lands on 1 and 11/10 is one tenth beyond it.
How to help at home
- Tenths on a ruler. A 10 cm strip of paper makes tenths easy to see: each centimetre is 1/10. Ask your child to show 3/10, then count along in tenths, saying “1 tenth, 2 tenths…”.
- Share a handful. Count out 12 raisins or pasta shapes. “Give me 1/3 of them. Now 2/3.” Sharing small objects into equal groups is exactly how fractions of amounts are taught.
- Fold paper into halves, quarters and eighths. Fold three strips of the same length into 2, 4 and 8 equal parts, then lay them side by side. Your child can see for themselves that 1/2, 2/4 and 4/8 are the same length.
- Use a measuring jug. A litre jug marked every 100 ml has ten steps, so 300 ml is 3/10 of a litre. Pour and read the jug together, and ask what fraction is left.
- Ask “which is bigger?”. Would you rather have 1/4 or 1/6 of the same pizza? 3/8 or 5/8? Explaining the choice builds understanding faster than any calculation.
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