3.NF Comparing Fractions Game

Content Standards: 3.NF.A.3, 3.NF.A.3.d

Task

This activity is designed for pairs of students working together. Each pair needs a set of fraction cards (provided as an attached resource). The goal is to compare the two fractions shown on each card, decide whether they are equivalent (equal in value), and, if they are not, figure out which one is larger.

The activity works as follows:

a. Students work through these steps using the fraction cards:

1. The pair selects a card.
2. Each student decides individually whether the two fractions are equal or, if not, which one is greater. Then the partners reveal their answers to each other.
3. If they agree, they take turns explaining their reasoning aloud. If they disagree, they discuss the problem until they reach agreement.
4. They repeat steps 1–3 with a new card.

b. After completing ten rounds, each pair writes down observations about the methods they used to compare the fractions.

Commentary

The purpose of this task is to compare fractions while pushing students toward explanations that show real conceptual understanding — not just guessing or memorizing rules. When comparing fractions that are not equivalent, students typically rely on one of three strategies:

• Using a common numerator (the top number of a fraction, showing how many equal parts are being counted). For example, thirds are larger pieces than fourths, so two-thirds is greater than two-fourths.
• Using a common denominator (the bottom number, showing how many equal parts the whole is divided into). For example, 1/5 is less than 2/5 because 2/5 contains one extra fifth of the whole.
• Using the whole number 1 as a benchmark, or reference point. For example, 2/3 is less than 5/4 because 2/3 is less than 3/3 (one whole), while 5/4 is greater than 4/4 (also one whole).

The benchmark method — comparing fractions to the number 1 — is technically introduced later, in the 4th-grade standard 4.NF.2. However, since understanding what a "whole" represents is fundamental to understanding fractions at all, it's reasonable for third graders to use 1 as a reference point too. That said, a teacher who wants to avoid this strategy can simply remove any cards where one fraction is greater than a whole and the other is less than a whole.

Two versions of the card set are provided: one includes pictures showing the two fractions being compared, and one does not. The pictures let students make a visual, side-by-side comparison, which is a valuable strategy in its own right. Alternatively, a teacher might have students draw their own pictures as part of explaining their answers, rather than providing the pictures ahead of time. Teachers who want students to focus only on inequalities (cases where the fractions are different) may also remove any cards showing equivalent fractions.

The second question in the task (recording observations) is meant to set up a class discussion once the activity is finished. To help students prepare for that discussion, teachers might suggest that students pay attention to their own strategies while they play, not just afterward. Some methods — like drawing a picture or using fraction strips (physical or drawn strips divided into equal parts) — work for any pair of fractions. Other methods, such as looking for a common numerator or common denominator, are especially important conceptually, and teachers should make sure these come up in the discussion.

Three resources accompany this task: a set of greater-than/less-than/equal-to symbols (one set per student), a set of cards without pictures (one set per pair), and a set of cards with pictures (one set per pair).

Solution

a. Students will encounter four basic types of fraction comparisons:

i. Fractions with the same numerator. The denominator tells you how many equal pieces the whole is divided into — and therefore how large each piece is. The numerator tells you how many of those pieces you have. For example, to compare 2/3 and 2/5: since the whole is split into more pieces for fifths than for thirds, each fifth is smaller than each third. So 2/5 is less than 2/3.

ii. Fractions with the same denominator. For example, 1/3 is less than 2/3, because 2/3 is made up of 1/3 plus one more third — so it's larger. This connects to the same-numerator reasoning above: here, both fractions divide the whole into the same number of equal pieces, but one fraction includes more of those pieces than the other.

iii. One fraction less than 1, and the other greater than 1. For example, 2/3 is less than 3/2, because 2/3 is one-third short of a full whole, while 3/2 is a full whole plus an extra half.

iv. Simple equivalent fractions, such as 1/2 and 2/4. One way to demonstrate that these are equal is with a picture: imagine two squares of the same size. Divide the left square into 2 equal parts and shade 1 of them; divide the right square into 4 equal parts and shade 2 of them. Since both pictures show the same amount of the whole shaded, 1/2 and 2/4 represent the same quantity.

b. Working through these comparisons teaches several important lessons:

• If you draw a picture of two fractions, the larger fraction will show more shaded area. If the fractions are equal, the shaded amounts will match exactly.
• The denominator tells you how many pieces the whole is cut into. Cutting a whole into more pieces makes each piece smaller — which is why 1/3 is smaller than 1/2.
• The numerator tells you how many of those equal-sized pieces you have. So 3/5 is greater than 2/5 simply because it includes one extra piece.
• All fractions are built from unit fractions (fractions with a numerator of 1, like 1/3 or 1/5), so understanding and being able to represent unit fractions is essential.
• When using picture cards, it's crucial that the wholes being compared are the same size — otherwise the comparison isn't valid.
• Equivalent fractions may divide the whole into different-sized pieces, but they always represent the same total shaded amount.
• When a fraction's numerator is greater than its denominator, the fraction represents more than one whole.
• Working through many examples like this reveals patterns — and recognizing patterns helps students form conjectures (educated guesses), support their reasoning, and ultimately prove mathematical claims.