Why do you multiply the top and bottom by the same number?

Because that is what cutting every piece into equal smaller pieces does to the count. Splitting each piece in two doubles how many pieces the whole has and doubles how many of them are shaded, so both numbers get multiplied by two and the amount covered is untouched. Multiplying only one of them changes the amount, which is why it gives a different fraction rather than an equivalent one.

When do children learn equivalent fractions?

Third grade introduces them with pictures and simple pairs like halves and quarters, and fourth and fifth grade extend them to larger numbers and to simplifying. The bars in this game show halves, thirds and quarters, matched to fourths, sixths and eighths, which is the range where the idea is normally built before any rule is applied without a picture.

Why does this matter later?

Because fractions with different bottom numbers cannot be added or compared until they are renamed, and renaming is exactly this skill. Working out one half plus one third means seeing them as three sixths and two sixths first. A child who finds equivalence obvious has removed most of the difficulty from adding fractions before meeting it.

← All games

Same Size, Different Name

Find the fraction that matches.

Grades 3-5 · Equivalent fractions

How to play

1A shaded bar shows a fraction, like one half.
2Choose another fraction that covers the same amount.
3Cutting each piece in two doubles both numbers and changes nothing about the size.
4Build a streak of matches.

Try a few

The bar shows 1/2. Which fraction covers the same amount?
Answer: 2/4
Picture a second cut through the middle of each half. The bar has not changed length and no shading has moved, but every piece is now half the size it was, so there are four pieces instead of two and two of them are shaded. Both numbers doubled together, one because the pieces got smaller and one because the shaded pieces got counted again. That is why 1/2 and 2/4 are two names for the same amount rather than two different amounts.
The bar shows 2/3. Which fraction covers the same amount?
Answer: 4/6
Three pieces with two shaded becomes six pieces with four shaded when each piece is split in two. Two shaded pieces each split in two make four, and three pieces each split in two make six, so 2/3 is 4/6. This one is useful because neither number is one, which stops a child from relying on a half-remembered fact about halves. The reasoning is identical whatever the starting fraction: split every piece the same way and count again.
The bar shows 3/4. Which fraction covers the same amount?
Answer: 6/8
Double both: three shaded becomes six, four pieces become eight, so the answer is 6/8. Say the sentence that goes with it, that the bar is the same bar and only the cutting changed, because a child who can say that will not be shaken when told later that 6/8 simplifies back to 3/4. Scaling up and simplifying down are the same journey in opposite directions, and this game always travels upward, from the fraction with fewer pieces to the one with more.

Why this game helps

That one half and two quarters are the same amount is obvious with a picture and deeply unobvious with numbers, because none of the four digits match. Children who have only met the rule, multiply top and bottom by the same number, tend to apply it to one number and not the other, turning one half into two halves. Seeing the bar stay the same length while the cuts double is what makes the rule sensible rather than arbitrary. Equivalence is also the gate to adding fractions at all, since unlike denominators have to be renamed before anything can be combined.

If your child finds this hard

Both wrong answers this game offers are built from one mistake, changing a single number. When the answer is 2/4, the two other buttons read 3/4, where only the top number moved, and 2/6, where only the bottom moved. That is precisely the error most often recorded for this skill: a child applies the rule to the numerator and forgets the denominator, or the other way around. Make the rule a sentence about pieces rather than digits, that cutting every piece in two changes how many pieces there are and how many are shaded, and both numbers move because both were counted.

A different error is adding where multiplying belongs, which produces claims like one third being the same as two fifths because one was added to each number. It is a reasonable guess, since adding the same amount to two numbers feels fair, and it happens to work for comparing differences, just not for fractions. Draw both. One third shaded next to two fifths shaded is visibly not the same bar, and one drawing settles the argument more quickly than any restatement of the rule.

Fold paper when the pictures on screen stop being convincing. Take two identical strips, fold one in half and shade a half, fold the other into quarters and shade two, then lay them on top of each other and let your child see the shaded regions line up exactly. Each bar in this game carries its fraction printed underneath, so your child sees the picture and the numbers at once, which is helpful for matching but means the paper work still has to happen for the idea to stick away from the screen.

Common questions

Why do you multiply the top and bottom by the same number?: Because that is what cutting every piece into equal smaller pieces does to the count. Splitting each piece in two doubles how many pieces the whole has and doubles how many of them are shaded, so both numbers get multiplied by two and the amount covered is untouched. Multiplying only one of them changes the amount, which is why it gives a different fraction rather than an equivalent one.
When do children learn equivalent fractions?: Third grade introduces them with pictures and simple pairs like halves and quarters, and fourth and fifth grade extend them to larger numbers and to simplifying. The bars in this game show halves, thirds and quarters, matched to fourths, sixths and eighths, which is the range where the idea is normally built before any rule is applied without a picture.
Why does this matter later?: Because fractions with different bottom numbers cannot be added or compared until they are renamed, and renaming is exactly this skill. Working out one half plus one third means seeing them as three sixths and two sixths first. A child who finds equivalence obvious has removed most of the difficulty from adding fractions before meeting it.

Try these next

Free games are a great warm-up. For a full program that finds exactly where your child gets stuck, Nova Academy is a personal math tutor powered by AI for grades 1 to 5.

Start a free trial →