Equivalent Fractions
Two fractions can look nothing alike and still be worth exactly the same. Once a child sees why, comparing and adding fractions stops being guesswork.
What are equivalent fractions?
Equivalent fractions are different ways of writing the same amount.
Take a chocolate bar. Snap it in half and take one piece. You are holding 12 of the bar. Now snap an identical bar into eight pieces and take four. You are holding 48 of it.
Same chocolate. More pieces, smaller pieces, same amount in your hand.
Every fraction on that line is a half. They are four names for one amount.
The fraction wall
Ask a child to prove that 48 is a half and most will say their teacher told them so. A fraction wall gives them something better than that. It gives them something to point at.
Each row below is the same bar. Only the number of pieces changes. Shade half of every row, and the shading always stops in the same place.
Why it works
You cannot draw a wall every time, so there is a rule. It is the only rule this topic needs.
The equivalent fraction rule
Multiply the top and the bottom by the same number and the fraction keeps its value.
Divide the top and the bottom by the same number and it keeps its value too.
Do anything else and the amount changes.
Here is the reason behind the rule, and it is worth telling a child rather than hiding it.
Multiplying the top and bottom by 2 is really multiplying the whole fraction by 22. And 22 is just another way of writing 1. Multiplying by 1 leaves a number alone. So the fraction now looks different, but nothing about it has actually changed.
On a number line
The wall shows equivalence as length. A number line shows it as a position. That is the version children need later, when they start comparing and ordering fractions.
Worked examples
Making equivalent fractions
- Choose any whole number to multiply by. Start with 2.
- Multiply both numbers by 2. 2 × 2 = 4 on top, 3 × 2 = 6 underneath, which gives 46.
- Do it again with 3. 2 × 3 = 6 and 3 × 3 = 9, which gives 69.
- Once more with 5. 2 × 5 = 10 and 3 × 5 = 15, which gives 1015.
Finding a missing number
- Start with the pair you can see completely. The bottom went from 8 to 40.
- Ask what 8 was multiplied by to reach 40. Since 8 × 5 = 40, the multiplier is 5.
- The rule says do the same on top. 5 × 5 = 25.
- Check by going backwards. 25 ÷ 5 = 5 and 40 ÷ 5 = 8, so you are back at the fraction you started with.
Checking whether two fractions match
- Simplify each one as far as it will go, then compare what is left.
- For 69, both numbers divide by 3, leaving 23.
- For 812, both divide by 4, also leaving 23.
- Two fractions that reduce to the same thing have to be worth the same.
Simplifying
The rule runs backwards as well. Divide the top and bottom by the same number and the digits get smaller while the value stays put. Teachers call this simplifying, or writing a fraction in its lowest terms.
Try it on 1218:
- Find a number that divides into both 12 and 18. Six works, and it is the biggest one that does.
- 12 ÷ 6 = 2 and 18 ÷ 6 = 3.
- So 1218 becomes 23. Now 2 and 3 share nothing except 1, so there is nothing left to cancel.
If the biggest divisor will not come to mind
Use any common factor you can see and repeat. Halving 1218 gives 69, then dividing by 3 gives 23. Two easy steps land in the same place as one hard one.
Three common mistakes
Almost every mark lost on this topic comes from one of these three. If your child is getting equivalent fractions wrong, look here first.
1. Adding instead of multiplying
Turning 12 into 23 by adding 1 to each number. Half a bar and two thirds of a bar are visibly different amounts, and a child will agree the moment you draw both. Adding does not preserve value. Only multiplying and dividing do.
2. Changing only one of the two numbers
Turning 34 into 38 because the bottom was doubled. Twice as many pieces, but still only three of them taken, so the child ends up with half as much. Both numbers move, and they move together.
3. Assuming bigger numbers mean a bigger fraction
Reading 50100 as larger than 12 because the digits are bigger. They are the same fraction. Whenever this one appears, go back to the wall.
Practice questions
Seven questions, from straightforward to a word problem. One attempt each, and the explanation appears once the answer is in, the same way a real test works.
Check yourself
No sign up. Nothing is sent anywhere, the score stays on this device.
For parents
If your child scored under five, the gap is usually the rule itself rather than the arithmetic. Ask them to say out loud what they did to the bottom number, then what they did to the top. If those two answers are different, you have found the problem, and it is a twenty minute fix with a tutor.
What to learn next
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