Ross MackellUK-qualified primary teacher · 10 min read

Your child's homework says: simplify 12/18. They halve both numbers, write 6/9, and close the book. The next day it comes back with a cross. "But I simplified it!"

They did, just not all the way. Equivalent fractions and simplifying are where a lot of children first decide they're "bad at fractions", when really they're only missing one or two ideas. And those ideas matter, because almost everything else in fractions, from adding fractions to comparing them, depends on them.

I'm Ross, a UK-qualified primary teacher with over ten years in the classroom. In this guide I'll show you what equivalent fractions are, how UK schools teach finding and simplifying them step by step, the one rule that stops most mistakes, and a 10-minute practice routine with answers.

What your child needs first

Equivalent fractions are really times tables wearing a disguise. Before they'll make sense, your child needs:

• Secure times tables. Spotting that 12 and 18 are both in the 6 times table is the whole skill. If tables are shaky, start with how to learn times tables fast.
• What the two numbers in a fraction mean. The bottom number (the denominator) is how many equal parts the whole is split into. The top number (the numerator) is how many of those parts you have.

In the national curriculum, children start showing equivalent fractions with diagrams in Year 3, learn families of equivalent fractions in Year 4, write equivalent fractions including tenths and hundredths in Year 5, and use common factors to simplify fractions in Year 6.

What are equivalent fractions?

Equivalent means "equal in value". Look at these three bars. Each one is the same size, cut into a different number of equal pieces:

The shaded part is exactly the same length every time. The pieces are just cut smaller. That's all an equivalent fraction is: the same amount, cut into a different number of pieces.

Pizza works brilliantly at home. Half a pizza is the same amount of pizza whether it's one big half or four small slices out of eight. Most children understand that instantly when food is involved.

The one golden rule

Whatever you do to the top, you must do to the bottom. And you can only multiply or divide.

Multiplying the top and bottom by 2 is like cutting every piece in half. You get twice as many pieces, and you have twice as many of them, so the amount stays the same.

Adding doesn't work. If your child adds 1 to the top and bottom of 1/2, they get 2/3, which is bigger than a half. This is the most common equivalent fractions mistake there is, so it's worth saying the rule out loud every single time.

Finding equivalent fractions step by step

Example 1: 2/3 = ?/12

1. Look at the numbers you know. The denominator has gone from 3 to 12.
2. Work out what happened. 3 × 4 = 12, so the bottom was multiplied by 4.
3. Do the same to the top. 2 × 4 = 8.

So 2/3 = 8/12. Your child can check by simplifying back: 8 ÷ 4 = 2 and 12 ÷ 4 = 3.

Example 2: 15/20 = 3/?

This time the numbers are getting smaller, so we're dividing.

1. The numerator has gone from 15 to 3. 15 ÷ 5 = 3.
2. Do the same to the bottom: 20 ÷ 5 = 4.

So 15/20 = 3/4. Money check: 15p out of 20p is the same as 75p out of £1, and 3/4 of £1 is 75p.

Example 3: tenths and hundredths

Year 5 children need to swap between tenths and hundredths. 7/10 = ?/100. The bottom was multiplied by 10, so the top is too: 70/100. This is exactly why 0.7 and 0.70 are the same number, which helps hugely with adding and subtracting decimals.

Fraction families worth knowing

A few families come up again and again. Knowing them by heart saves your child a lot of work:

Notice the half family: the top is always half the bottom. Spotting patterns like that is real mathematical thinking, and it's worth asking your child, "What do you notice?"

Simplifying fractions step by step

Simplifying is finding the equivalent fraction with the smallest possible numbers. You'll also hear it called "writing a fraction in its simplest form" or "cancelling down".

Example 4: simplify 12/18

Here's the homework question from the start. There are two ways to do it, and both are fine.

Method 1: small steps. Keep dividing by any number that goes into both, until you can't any more.

• 12 and 18 are both even, so divide by 2: 6/9.
• 6 and 9 are both in the 3 times table, so divide by 3: 2/3.
• Nothing goes into 2 and 3 except 1. Done.

Method 2: one big step. Find the biggest number that goes into both. That's called the highest common factor.

• Factors of 12: 1, 2, 3, 4, 6, 12.
• Factors of 18: 1, 2, 3, 6, 9, 18.
• The biggest one in both lists is 6. 12 ÷ 6 = 2 and 18 ÷ 6 = 3, so 2/3.

Method 1 is where most children start, and it's perfectly good. It just takes a few more steps. That's why 6/9 got a cross: your child took the first step and stopped.

Example 5: simplify 24/36

• Divide by 2: 12/18.
• Divide by 2 again: 6/9.
• Divide by 3: 2/3.

Or, in one step, the highest common factor of 24 and 36 is 12, which gives 2/3 straight away.

How to tell if a fraction is fully simplified

Teach your child to ask one question at the end: "Is there any number, apart from 1, that goes into both?"

A quick checklist helps:

• Are both numbers even? Then you can still divide by 2.
• Do both end in 0 or 5? Then you can still divide by 5.
• Are both in the 3 times table? Then you can still divide by 3.
• Is the top number 1? Then it's already as simple as it gets.

If the answer to all of them is no, and the numbers are small, it's almost certainly in its simplest form.

Watch: the times tables behind it

Because simplifying is really times tables in reverse, the quickest way to get better at fractions is often to tighten up tables. This video shows the order I teach them in, and why there are only 36 facts left to learn once the easy ones are done:

4 common mistakes and how to fix them

Mistake 1: Adding instead of multiplying

Your child turns 1/2 into 2/3 by adding 1 to the top and bottom.

The fix: draw it. Shade 1/2 of a bar and 2/3 of an identical bar. They can see straight away that they're different sizes. Then repeat the rule: "Multiply or divide, top and bottom."

Mistake 2: Stopping too early

12/18 becomes 6/9, and the job feels finished.

The fix: after every step, ask the checklist question: "Is there any number, apart from 1, that goes into both?" Make it a habit to check one more time.

Mistake 3: Only changing one number

To simplify 6/8, they divide the top by 2 and leave the bottom: 3/8.

The fix: get them to write the same instruction on both sides, "÷ 2" next to the top and "÷ 2" next to the bottom, before they calculate anything.

Mistake 4: Thinking bigger numbers mean a bigger fraction

Children often think 4/8 is more than 1/2 because 4 and 8 are bigger numbers.

The fix: back to pizza. Would you rather have half a pizza, or 4 slices from a pizza cut into 8? It's the same amount of pizza.

A 10-minute practice routine

Same three steps I use for every skill: watch, copy, practise.

1. Watch: do one example yourself, out loud. "What happened to the bottom? Times 4. So I do the same to the top."
2. Copy: your child does a very similar one, explaining each step.
3. Practise: three or four on their own, then check each one by simplifying it back.

Only move on when they're getting three in a row right. If they keep getting stuck on spotting what goes into both numbers, spend a few days on times tables instead. Ten minutes, a few times a week, is plenty.

Frequently asked questions

What are equivalent fractions?

Equivalent fractions are fractions that have the same value even though they use different numbers. For example, 1/2, 2/4, 3/6 and 50/100 are all equivalent, because they all represent exactly half of a whole.

How do you find an equivalent fraction?

Multiply or divide the numerator and the denominator by the same number. For example, multiplying the top and bottom of 2/3 by 4 gives 8/12, which is equivalent to 2/3. Never add or subtract, because that changes the value of the fraction.

How do you simplify a fraction?

Divide the numerator and the denominator by a number that goes into both, and keep going until only 1 goes into both. The quickest way is to divide by the highest common factor. For example, 12/18 simplifies to 2/3 by dividing both numbers by 6.

How do you know if a fraction is in its simplest form?

A fraction is in its simplest form when no whole number other than 1 divides exactly into both the numerator and the denominator. For example, 2/3 is in its simplest form, but 6/9 is not, because 3 goes into both 6 and 9.

What year do children learn equivalent fractions?

Children start recognising equivalent fractions with diagrams in Year 3, learn families of equivalent fractions in Year 4 and write equivalent fractions, including tenths and hundredths, in Year 5. In Year 6 they use common factors to simplify fractions.

Sources and further reading

• Department for Education, National curriculum in England: mathematics programmes of study (fractions in Years 3 to 6)
• Oak National Academy, Recognising equivalent fractions (free pupil lesson)
• Oak National Academy, Simplifying fractions
• Ross Mackell, Teacher's Perspective on YouTube