3.NF Comparing Fractions Game

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

Task

This activity works best with pairs of students. You'll need a set of fraction cards. On each card are two fractions. The goal is to compare them—decide if they are equivalent (equal to each other) or, if not, figure out which one is bigger.

Here's how to play:

1. The pair of students picks a card.
2. Each student decides on their own whether the fractions are equal, and if not, which one is greater. Then the partners show each other their answers.
3. If they agree, they take turns explaining how they figured it out. If they disagree, they talk it through until they agree on one answer.
4. They repeat these steps with a new card.

After 10 rounds, each pair writes down notes about the different strategies they used to compare fractions.

Teacher Notes

The purpose of this activity is to help students compare fractions while explaining their thinking clearly, showing they truly understand why one fraction is bigger than another—not just guessing.

There are three main strategies students usually use when comparing fractions that are not equivalent:

• Same numerator, different denominator: For example, thirds are bigger pieces than fourths, so two-thirds is bigger than two-fourths.
• Same denominator, different numerator: For example, 1/5 is smaller than 2/5 because 2/5 has one extra fifth of the whole.
• Using "one whole" as a benchmark: For example, 2/3 is less than 5/4 because 2/3 is less than a whole (3/3), while 5/4 is more than a whole (4/4).

The third strategy—comparing fractions to a whole—is usually taught in more depth in 4th grade. But since understanding what a "whole" means is so important to understanding fractions in general, it makes sense for 3rd graders to use this idea too. If a teacher wants to focus only on the first two strategies, they can remove any cards where one fraction is bigger than a whole and the other is smaller.

Some card sets include pictures of the fractions, and others don't. The pictures help students see the comparison, which is very useful. A teacher might want students to draw their own pictures instead, as a way of explaining their reasoning. Teachers can also remove cards with equivalent fractions if they want students to focus only on comparing fractions that are different in size.

After the game, it's a great idea to have a class discussion about the different strategies students used. Drawing pictures or using fraction strips works for every pair of fractions, but strategies like comparing numerators or denominators are especially important for building deep understanding, so make sure those come up in the discussion.

Solution

There are four main types of fraction pairs students will compare:

a. Fractions with the same numerator, but different denominators.
The denominator shows how many equal pieces make up the whole—more pieces means each piece is smaller. The numerator shows how many of those pieces you have. For example, to compare 2/5 and 2/3: since the whole is divided into more pieces for fifths than for thirds, each fifth is smaller than each third. So 2/5 is less than 2/3.

b. Fractions with the same denominator, but different numerators.
For example, 1/3 is less than 2/3, because 2/3 is the same as 1/3 plus one more third—so it's bigger. This connects to the idea above: when the denominator is the same, the whole is cut into the same-size pieces, but one fraction simply has more of those pieces than the other.

c. One fraction smaller than a whole, the other bigger than a whole.
For example, 2/3 is less than 3/2 (one and a half), because 2/3 is missing one third to reach a whole, while 3/2 is a whole plus an extra half.

d. Simple equivalent fractions, such as 1/2 and 2/4.
One way to prove these are equal is by drawing a picture. Imagine two squares of the same size. Divide one square into 2 equal parts and shade 1 of them (showing 1/2). Divide the other square into 4 equal parts and shade 2 of them (showing 2/4). Even though the pieces are different sizes, the same amount of each square ends up shaded—proving that 1/2 and 2/4 represent the same amount.

Key Ideas Students Should Learn

• If you draw a picture of two fractions, the fraction with more shaded area is the bigger fraction. If the fractions are equal, the same amount will be shaded in both pictures.
• The denominator tells you how many pieces the whole is cut into. Cutting the whole into more pieces makes each piece smaller (this 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 more than 2/5 because you have one extra piece.
• All fractions are built from unit fractions (fractions with a numerator of 1), so it's important to understand and picture unit fractions clearly.
• When comparing fraction cards that use pictures, it matters that the wholes being compared are the same size.
• Equivalent fractions may use different-sized pieces, but they represent the same total amount.
• When the numerator is bigger than the denominator, the fraction is greater than one whole.
• Doing this kind of math helps students notice patterns—and noticing patterns helps them make good guesses (called conjectures), support their reasoning, and eventually prove their mathematical ideas.