Equivalent Fractions

What are equivalent fractions?

Equivalent fractions are different ways of writing the same amount.

Snap a chocolate bar in half, take one piece, and you are holding 12 of it. Now take an identical bar, snap it into eight, and pick up four of those pieces instead.

Put the two amounts side by side and they are the same size, which they have to be: doubling the number of cuts halves the size of every piece, so taking twice as many of them leaves you exactly where you started. Nobody at the table would care which bar they got.

All four are a half. They are not four amounts that happen to be close, they are one amount with four names, in the same way that a person can be called by a name, a nickname and a title without ever becoming a different person.

The fraction wall

Ask a child to prove that 48 is a half and most will tell you their teacher said so, which is not proof and, more importantly, is not something they can rebuild in a test when they have forgotten it. A fraction wall gives them something to point at instead.

Every row below is the same bar, cut a different number of times. Shade half of each one and the shading stops at the same place every time, which is the whole argument made visible.

Why it works

Drawing a wall for every question is not realistic, so there is a rule, and it is the only one this topic needs.

Most children are handed the rule and never told why it works, which is a shame, because the reason is short and it is the kind of thing that makes a child trust the maths instead of just obeying it.

Multiplying the top and the bottom by 2 is really multiplying the whole fraction by 22, and 22 is one. Anything multiplied by one comes back unchanged. So the fraction has been rewritten, not altered, and the two facts a child needs to hold together are that it looks different and that it is not different.

On a number line

The wall shows equivalence as a length. A number line shows it as a position, and position is the version that matters from Grade 5 onwards, when every question about ordering fractions quietly assumes the child can place them.

Worked examples

Simplifying

The rule works just as well in reverse. Divide the top and the bottom by the same number and the digits shrink while the amount stays exactly where it was, which teachers call 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.

Three common mistakes

Nearly every mark lost on this topic goes to one of the next three, and all three are reasonable ideas rather than careless ones, which is exactly what makes them stubborn.

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.

What to learn next