How to compare fractions with your child

"Which is bigger, 2/3 or 3/4?" I asked my daughter one night, expecting a shrug. Instead she stared at the two fractions for a long moment, pointed at 2/3, and said it was bigger because 2 and 3 were "smaller numbers that are closer together." It was a genuinely creative wrong answer, and it told me exactly what had not clicked yet. She was reading four numbers instead of two amounts.

How to compare fractions

To compare two fractions, work through four checks in order and stop at the first one that settles it. If the bottom numbers are the same, the bigger top number wins. If the top numbers are the same, the smaller bottom number wins, because fewer pieces means bigger pieces. If neither matches, ask whether each fraction is more or less than one-half, which sorts most pairs instantly. Only when both land on the same side of a half do you rewrite them with a common denominator and compare the tops. Teaching them in that order means your child reasons out most comparisons and saves the slow arithmetic for the cases that actually need it.

Same bottom number: just count the pieces

This is the one case that already matches a kid's instincts, so start here. When two fractions have the same denominator, the pieces are the same size, and all that differs is how many of them you have. 3/8 and 5/8 are both counted in eighths, so 5/8 is bigger for the same reason 5 cookies beat 3 cookies. Say it out loud with your child: same-size pieces, so just count. This is the one comparison where the bigger number really does win, which is exactly why it is a safe place to begin before the rules start bending.

Same top number: fewer pieces, bigger pieces

Now the first twist. When the top numbers match and the bottoms differ, the fraction with the smaller bottom number is bigger. Compare 3/4 and 3/8. Each one is three pieces, but fourths are bigger pieces than eighths, so three of them is more. The fastest way to make this land is food: three slices from a pizza cut into 4 is obviously more than three slices from a pizza cut into 8. This is the exact idea that trips kids up the most, because a smaller number is producing the bigger answer. If your child insists 3/8 is bigger because 8 is bigger, that is the signal to go back to rebuilding what a fraction actually is, because the comparing rules stand on that foundation.

The one-half benchmark does most of the work

Here is the move that carried us furthest, and it needs no common denominator at all. Teach your child to ask one question about any fraction: is this more or less than a half? Half of the bottom number tells you the halfway top. Is 3/8 more or less than 1/2? Half of 8 is 4, and 3 is less than 4, so 3/8 is under a half. Is 5/8 more or less? 5 beats 4, so 5/8 is over a half.

Once your child can do that quickly, a huge share of comparisons solve themselves. To compare 3/8 and 5/8 by benchmark, you notice one is under a half and the other is over, so the over-a-half one wins, no arithmetic needed. The same trick compares 2/5 and 5/8: 2/5 is under a half (half of 5 is 2.5, and 2 is less), 5/8 is over, done. The benchmark only stalls when both fractions sit on the same side of a half, and that is the cue to use the next method.

When you need a common denominator

When the shortcuts do not settle it, you fall back on the method that always works: rewrite both fractions so every piece is the same size, then compare the tops. For 2/3 and 3/4, multiply the bottoms, 3 times 4, to get 12. Then 2/3 becomes 8/12 and 3/4 becomes 9/12, and now it is obvious that 9 pieces beat 8, so 3/4 is bigger. My daughter's "closer together" theory could not survive seeing 8/12 next to 9/12.

This is the same rewriting step your child already uses to add fractions with unlike denominators, so it is worth pointing out that the work is identical, just put to a different purpose. Comparing asks which rewritten top is bigger; adding asks you to combine them. Teach it as the reliable fallback rather than the first move, so your child does not grind through a common denominator every time when a ten-second benchmark would have done it.

Cross-multiplying, and why to teach it last

There is a slick shortcut where you multiply diagonally: for 2/3 versus 3/4, you compute 2 times 4 and 3 times 3, get 8 and 9, and the bigger product sits over the bigger fraction. It works, and it is fast, and I still teach it last on purpose. A child who learns cross-multiplying before they understand what it is doing just adds one more rule to misremember, and when they flip which product goes where, they have no sense of whether the answer is reasonable. Once the benchmark and common-denominator ideas are solid, cross-multiplying lands as a tidy shortcut for the same thing they already know how to reason out. That ordering, meaning before trick, is the same reason multiplying fractions goes better when "times" is taught as the word "of" first and the procedure second.

Where comparing fractions shows up

Comparing is not a throwaway skill that lives for one unit and disappears. It is the quiet engine behind estimating, behind checking whether an answer is reasonable, and behind ordering a whole list of fractions on a number line. It also shows up constantly on the adaptive diagnostics schools run: fraction questions on these tests often ask a child to place fractions in order or decide whether one is bigger than a benchmark, not just to calculate. If your child brings home a result and you want to read it, it helps to know what a good NWEA MAP math score looks like by grade and how iReady diagnostic scores break down by grade, so a number on a page turns into something you can actually act on.

Practice where fractions already live

The best practice does not look like a worksheet. It looks like deciding who got more. Two kids split a candy bar: one took 2/5, the other took 1/2, who got more? Cooking is full of it: "this recipe wants 2/3 cup and that one wants 3/4, which is more?" Hand your child two measuring cups and let them pour to check. Every time you ask "which is bigger" about real amounts, you are giving reps at the thing a test fraction is quietly measuring. When they get one wrong, do not correct it on the spot. Ask, "show me how you thought about that," and their answer tells you which idea slipped: same-size pieces, bigger-bottom-smaller-piece, or the benchmark. Praise the reasoning, not the speed. A wrong comparison is a map of the next thing to practice, not a verdict on whether your child is a "math kid."

Find out where the gap really is

A single confused homework night is a small, noisy sample, and it is hard to tell from one page whether comparing fractions is an isolated wobble or part of a wider soft spot. It is worth getting an outside read now and then. Test My Kid gives you a free first assessment: an eight-minute adaptive assessment for math and reading, grades 1 to 8, calibrated to the same NWEA MAP and iReady benchmarks schools use. It reports which topics are solid and which are shaky, so you can see whether fractions are the whole story or just the part you happened to notice.

My daughter still slows down before she decides which fraction is bigger. But she slows down to think now, instead of grabbing the smaller numbers because they looked friendlier. The 2/3 bluff worked on her once. It would not work today, and not because fractions got easier. She just finally learned to see the amount, not the four numbers.