What is a fraction? Fractions explained for parents

What is a fraction?

A fraction is a number that describes part of a whole, when the whole has been split into equal parts. The word "equal" matters: a cake cut into 4 pieces of different sizes hasn't been cut into quarters.

Here is three-quarters. The whole bar is split into 4 equal parts, and 3 of them are shaded:

Government guidance for teachers asks children to name the whole and the parts every time: "The whole is divided into 4 equal parts. 3 of those parts are shaded." It warns against saying "3 out of 4" without saying what the whole is, because a fraction is always a fraction of something.

Numerator and denominator in fractions

The denominator is the bottom number. It says how many equal parts the whole is split into. The numerator is the top number. It says how many of those parts we're talking about.

Fraction	Numerator	Denominator	In words
1⁄2	1	2	one-half
3⁄4	3	4	three-quarters
5⁄8	5	8	five-eighths
7⁄10	7	10	seven-tenths

A quick way to remember it: the denominator is down at the bottom.

When the numerator and denominator are the same, the fraction is one whole. ⁵⁄₅ means all 5 of the 5 parts, which is 1.

What is a unit fraction?

A unit fraction has a numerator of 1: ¹⁄₂, ¹⁄₃, ¹⁄₄, ¹⁄₁₀. It's one single part of a whole split into equal parts.

A non-unit fraction has a numerator bigger than 1, such as ³⁄₄ or ⁵⁄₈. Every non-unit fraction is made of unit fractions. Teachers call five-eighths "5 one-eighths", because it's 5 lots of ¹⁄₈. Children meet ³⁄₄ in Year 2 as their first non-unit fraction.

Why a bigger denominator means a smaller part

The more parts a whole is split into, the smaller each part is. So ¹⁄₈ is smaller than ¹⁄₄, even though 8 is bigger than 4. Think of one pizza shared by 8 people instead of 4: everyone gets less.

This catches lots of children out. The fraction wall below makes it easy to see.

The fraction wall

A fraction wall stacks rows of the same length, each one whole, split into different numbers of equal parts:

Use it to:

• Compare unit fractions. Find ¹⁄₃ and ¹⁄₅ and see which brick is longer.
• Spot equivalent fractions. Two ¹⁄₄ bricks line up exactly with one ¹⁄₂ brick, so ²⁄₄ = ¹⁄₂. There's more on this in equivalent fractions.
• See what makes a whole. Every row adds up to 1, so ³⁄₃ and ¹⁰⁄₁₀ are both one whole.

Drawing a simple fraction wall together, with a ruler, is a good way to start at home.

Fractions as numbers on a number line

Fractions are numbers in their own right, not just pieces of cake. From Year 3, children place them on a number line between 0 and 1, counting "one-quarter, two-quarters, three-quarters, one". In Year 2 they already count in halves and quarters past 1: 1 ¹⁄₄, 1 ¹⁄₂, 1 ³⁄₄, 2.

Government guidance points out a common slip: on a line from 0 to 1 with 3 marks between, children count the marks and say thirds. It's the parts that count. 3 marks make 4 parts, so the line is in quarters.

Worked example: a fraction of an amount

To find a unit fraction of a number, divide by the denominator. ¹⁄₄ of 12 is 12 ÷ 4 = 3.

For a non-unit fraction, find one part, then take as many parts as the numerator says. Here's ³⁄₄ of 12:

So ³⁄₄ of 12 = 9. This is where the times tables help: if your child knows the 4 times table, they know 12 ÷ 4 at once. See fractions of amounts for more.

Worked example: adding fractions with the same denominator

Children add fractions with the same denominator in Year 3, by counting unit fractions. Five-sevenths and one more seventh make six-sevenths:

⁵⁄₇ + ¹⁄₇ = ⁶⁄₇

It's like 5 apples and 1 apple: 5 sevenths and 1 seventh are 6 sevenths. The denominator stays the same, because the parts are still sevenths. Taking away works the same way, so ⁶⁄₇ − ²⁄₇ = ⁴⁄₇. The page on adding and subtracting fractions goes further.

Fractions in Year 1 to Year 4

This is what the national curriculum sets out for each year:

• Year 1: a half and a quarter of an object, shape or quantity, as one of 2 or one of 4 equal parts.
• Year 2: ¹⁄₃, ¹⁄₄, ²⁄₄ and ³⁄₄ of a length, shape, set of objects or quantity. Writing simple fractions such as ¹⁄₂ of 6 = 3, and knowing that ²⁄₄ and ¹⁄₂ are the same.
• Year 3: tenths, unit and non-unit fractions with small denominators, equivalent fractions shown with diagrams, adding and subtracting fractions with the same denominator within one whole, and comparing and ordering unit fractions and fractions with the same denominator.
• Year 4: families of equivalent fractions, hundredths, harder fractions of amounts such as ³⁄₅ of 20, adding and subtracting fractions with the same denominator (including answers bigger than 1), and the decimals for a quarter, a half and three-quarters.

There's more on each year in the Year 2, Year 3 and Year 4 guides.

Fractions in Year 5 and Year 6

In Year 5 and Year 6, children use what they know to do more with fractions:

• Comparing and ordering fractions with different denominators.
• Improper fractions and mixed numbers, such as ⁷⁄₄ and 1 ³⁄₄.
• Multiplying (or "timesing") fractions. In Year 5, by whole numbers: ²⁄₅ × 3 = ⁶⁄₅. In Year 6, a fraction by a fraction. See multiplying fractions and dividing fractions.
• Reducing a fraction. In Year 6, children use common factors to write a fraction in its simplest form, so ⁶⁄₉ becomes ²⁄₃. See simplifying fractions.
• Fractions as percentages. In Year 5, children learn that per cent means parts per hundred. To turn a fraction into a percentage, make it into hundredths: ¹⁄₄ = ²⁵⁄₁₀₀, which is 25%. See fractions, decimals and percentages and fractions to decimals.

Common mistakes to watch for

• Unequal parts. A shape cut into 4 pieces of different sizes doesn't show quarters.
• Thinking a bigger denominator means a bigger fraction. ¹⁄₁₀ is smaller than ¹⁄₂.
• Adding the denominators. ¹⁄₄ + ¹⁄₄ is ²⁄₄, not ²⁄₈. The parts are still quarters.
• Forgetting the whole. ¹⁄₂ of 10 and ¹⁄₂ of 100 are both halves, but they're very different amounts.
• Counting marks instead of parts on a number line.

How to help at home

• Share things out fairly. Cut a sandwich into halves or quarters, and ask whether the parts are equal.
• Use fraction words for amounts. "Half of your 8 grapes is 4." "A quarter of an hour is 15 minutes."
• Fold paper. Fold a strip in half, then in half again, and count the quarters.
• Ask both ways. "What fraction is shaded?" and "Can you shade ²⁄₃?"
• Keep the times tables going. Finding fractions of amounts is mostly division, so table facts make it quick.

Fraction questions to try

1. A pizza is cut into 8 equal slices. Tom eats 3. What fraction did he eat?
2. In ⁵⁄₆, which number is the denominator?
3. Which is bigger, ¹⁄₃ or ¹⁄₅?
4. What is ¹⁄₅ of 20?
5. What is ²⁄₃ of 12?
6. What is ²⁄₉ + ⁵⁄₉?

Answers

1. ³⁄₈ (8 equal slices, and he ate 3 of them)
2. 6 (the whole is split into 6 equal parts)
3. ¹⁄₃ (fewer parts, so each part is bigger)
4. 4 (20 ÷ 5 = 4)
5. 8 (one-third of 12 is 4, and two-thirds is 2 × 4 = 8)
6. ⁷⁄₉ (2 ninths and 5 ninths make 7 ninths)

For everything else children learn in these years, see the KS1 maths and KS2 maths guides.