To compare fractions, students need strategies beyond "bigger number means bigger fraction." When denominators are the same, the fraction with the larger numerator is greater (3/5 > 2/5). When numerators are the same, the fraction with the smaller denominator is greater (1/3 > 1/5, because thirds are larger pieces). For fractions with different numerators and denominators, finding a common denominator or using benchmark fractions (0, 1/2, 1) are the standard approaches. Students should use <, >, and = symbols and be able to order sets of fractions.
Use fraction strips, area models, and number lines to make comparisons visual before introducing algorithmic methods. Benchmark reasoning is powerful: "Is this fraction more or less than 1/2?" Practice with carefully chosen pairs that target each strategy (same denominator, same numerator, neither). Avoid teaching cross-multiplication as the only method -- it works but builds no number sense.
You already understand what a fraction means — a numerator counting how many pieces you have, a denominator telling you how many equal pieces make up the whole — and you can create equivalent fractions by multiplying or dividing numerator and denominator by the same number. Comparing fractions is where that understanding gets put to work in real comparisons.
When denominators are the same, comparison is straightforward: 3/7 > 2/7 because both fractions use the same size pieces (sevenths), and 3 of them is simply more than 2 of them. This is like comparing 3 cookies to 2 cookies — the unit is identical, so you just compare the counts. When numerators are the same, you compare denominators — but in reverse: 2/3 > 2/5 because thirds are larger pieces than fifths (cutting a whole into 3 gives bigger slices than cutting it into 5), so 2 large pieces is more than 2 small pieces. The key intuition: smaller denominator = larger piece size = greater total amount when the number of pieces is equal.
When neither numerators nor denominators match, you have two reliable strategies. The first is common denominator: use your equivalent-fractions skill to rewrite both fractions with the same denominator, then compare numerators. To compare 3/4 and 5/6, find a common denominator (12), rewrite as 9/12 and 10/12, and the comparison is immediate. The second is benchmarking: check whether each fraction is less than, equal to, or greater than 1/2. If 3/8 < 1/2 and 5/8 > 1/2, you can conclude 5/8 > 3/8 without any conversion at all.
The most persistent mistake is applying whole-number thinking directly: "1/8 is bigger than 1/4 because 8 > 4." This is wrong because the denominator describes piece size, not piece count — it's an inverse relationship. A fraction is a relationship between two numbers, not two independent numbers. Visualizing both fractions as pieces of the same-sized rectangle or as positions on a number line is the best antidote: when you can see that eighths are smaller slices than fourths, the comparison becomes obvious and the whole-number instinct loses its grip.