Comparing fractions without a rule to memorise

Ask a group of adults which is larger, 3/8 or 2/5, and many will reach for a rule: "cross-multiply" or "find a common denominator". Some will freeze. A fraction comparison is a very good test of how well a person truly understands fractions, because the answer can be found through reasoning about size, and children who learn to reason this way become far more confident than those who learn only the procedures. This article describes the reasoning strategies that make comparing fractions meaningful, and how to practise them at home.

Written for children aged 8 to 11.

Begin with the size of a fraction

A fraction tells you how big a piece is relative to a whole. Before any comparing, a child needs a mental picture of what a fraction looks like. If your child can draw ¼ of a pizza, or can place ½ halfway along a number line from 0 to 1, they have the beginnings of a sense of fraction size. If that picture is shaky, step back and work with paper strips or drawings (see our article on fair sharing and fractions) before comparing.

Everything that follows depends on one idea that children must truly hold: the more equal parts a whole is cut into, the smaller each part. A fifth is smaller than a quarter, even though 5 is greater than 4.

Strategy 1: same denominator, compare the numerators

When fractions have the same denominator, the pieces are the same size, so the one with more pieces is greater. 5/8 is greater than 3/8, because five eighths is more than three eighths. This is the easiest case and a natural place to begin.

Strategy 2: same numerator, compare the denominators

When the numerators are equal, you have the same number of pieces, but the pieces differ in size. 3/4 and 3/8: three pieces of size quarter against three pieces of size eighth. Quarters are bigger, so 3/4 is greater. Children often find this strategy surprising, and drawing the strips makes it clear. It is a good opportunity to reinforce that a bigger denominator means a smaller piece.

Strategy 3: use benchmarks

A benchmark is a friendly number that helps you judge. The most useful are 0, ½ and 1.

• To compare 3/8 and 4/7, ask whether each is more or less than a half. Half of 8 is 4, so 3/8 (fewer than 4 pieces out of 8) is less than a half. Half of 7 is 3½, so 4/7 (more than 3½ pieces out of 7) is more than a half. One is below a half and the other above, so 4/7 is greater, with no calculation at all. (When both fractions sit on the same side of a benchmark, as 3/8 and 2/5 do, another strategy is needed.)
• To compare 5/6 and 7/8, ask how far each is from 1. 5/6 is missing 1/6; 7/8 is missing 1/8. Since an eighth is smaller than a sixth, 7/8 is closer to 1, and so greater.

The "how far from a whole?" idea is one of the most powerful in comparing fractions, and it makes a good family game: "Which is closer to a whole pizza, 3/4 or 4/5?"

Strategy 4: use equivalent fractions

When no shortcut works, children can find a common denominator, but they are best taught to see why it works, not just to follow steps. To compare 3/8 and 2/5, cut both wholes into the same size of piece: fortieths. 3/8 is the same as 15/40, and 2/5 is the same as 16/40. Now the pieces are the same size, and 16 is more than 15, so 2/5 is greater. Children who have folded paper strips and seen that ½ = 2/4 = 4/8 will understand that making equivalent fractions simply cuts the pieces into smaller, equal parts without changing the amount.

Strategy 5: unit fractions as building blocks

A unit fraction has 1 on top: 1/2, 1/3, 1/4, and so on. All other fractions are built from them: 3/4 is three of the one-fourth pieces. Seeing fractions as counted unit fractions (like counting apples) helps children compare: 3/4 is "three quarters" and 3/5 is "three fifths", and since a quarter is bigger than a fifth, three quarters are more than three fifths.

What children get wrong, and why

Whole-number thinking. Children see 1/8 and 1/6 and think "8 is bigger than 6, so 1/8 is bigger". This is the most common error and it is natural, because up to now bigger numbers always meant more. Counter it with pictures, and with the sharing question: "Would you rather share a pizza with 6 people or 8?"

Comparing differences. A child may say 5/6 equals 7/8 because "each is one piece short". The pieces are different sizes, so the missing amounts are different. Using the strips shows that a sixth is bigger than an eighth.

Over-reliance on procedure. A child who always rushes to a common denominator for 5/9 and 5/7 misses the fact that, with equal numerators, the smaller denominator gives bigger pieces. Encourage them to look at the numbers first and ask: "Is there a quick way to see this?"

Games to practise at home

Fraction strips. Cut strips of paper of the same length and fold or mark them into halves, thirds, quarters, fifths, sixths, eighths. Lay them side by side. Ask "Which is longer, 2/3 or 3/5?" and check by overlaying. After a while, children begin to predict.

Closer to one. Pick two fractions and ask which is closer to a whole, and why. It is quick and needs nothing.

Fraction war. Each player draws a fraction card (or writes one down); the player with the larger fraction wins both. At first, allow strips for checking. Later, challenge your child to explain without them.

Order the fractions. Write five fractions on slips of paper and ask your child to line them up from smallest to largest on a number line from 0 to 1 drawn on the floor with string. Then talk through any disagreement.

Cooking comparisons. "A recipe says ⅔ cup. Do we have more or less than that in this ¾-cup measure?" Real measuring brings fractions to life.

Why reasoning beats rules

Rules are fast and efficient once understood, and there is no reason to hide them. But a rule learned without understanding is easily forgotten, easily misapplied, and gives children no way to tell when an answer is unreasonable. Reasoning strategies work for every pair of fractions, build a sense of size, and prepare children for the more advanced fractions of later years: adding unlike denominators, ratios and proportions.

The next time your child compares two fractions, ask: "How do you know?" and listen for the picture they have in their head. If it shows pieces of different sizes, they are well on the way.

Try it in the app

Practise this at the screen too: Compare fractions, Equivalent fractions. You may also like our articles on fair sharing and fractions and place value.