3.NF Comparing Fractions Game

Task

This activity is for pairs of students. You will need a set of fraction cards. On each card, there are two fractions. Your goal is to compare the two fractions and decide: are they equal? If not, which one is bigger?

Here's how to play:

a. Follow these steps with the fraction cards:
i. You and your partner pick a card.
ii. Each of you decides on your own whether the fractions are equal, or if not, which one is greater. Then show each other your answer.
iii. If you agree, take turns explaining how you figured it out. If you disagree, talk it over until you agree.
iv. Pick a new card and do it again.

b. After 10 rounds, write down what methods you and your partner used to compare the fractions.

Notes for Teachers

The goal of this task is to compare fractions and explain your thinking clearly. When fractions are not equal, here are three good ways to compare them:

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

The third method — comparing fractions to a whole — is usually taught in 4th grade. But since understanding what a "whole" means is so important for understanding fractions, it's okay to use this method in 3rd grade too. Teachers can remove cards that use this method if they want to skip it.

There are two sets of cards: one with pictures showing the fractions, and one without pictures. The pictures help students compare fractions visually. Teachers might also ask students to draw their own pictures to explain their thinking, instead of just looking at ready-made pictures. Teachers can also remove cards with equal fractions if they only want students to practice finding which fraction is bigger or smaller.

After the game, it's helpful to talk as a class about which strategies worked best. Drawing pictures or using fraction strips can help compare any two fractions. But using a common numerator or common denominator are especially important math ideas, so make sure to talk about those strategies too.

Three extra resources come with this activity:

• Symbols for "less than," "equal to," and "greater than" (one set per student)
• A set of fraction cards without pictures (one set per pair)
• A set of fraction cards with pictures (one set per pair)

Solution

a. There are four types of fraction pairs students will compare:

i. Fractions with the same numerator.
The denominator tells us how many equal pieces make up the whole — so it tells us how big each piece is. The numerator tells us how many of those pieces we have.
For example, to compare 2/3 and 2/5: since a whole broken into fifths has smaller pieces than a whole broken into thirds, 2/5 is smaller than 2/3.

ii. Fractions with the same denominator.
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 same idea as before: since the denominator is the same, each piece is the same size, but one fraction simply has more pieces than the other.

iii. One fraction is less than 1 whole, and the other is greater than 1 whole.
For example, 2/3 is less than 3/2, because 2/3 is a little less than a whole (which would be 3/3), while 3/2 is more than a whole (which would be 2/2), plus an extra half.

iv. Equivalent fractions, like 1/2 and 2/4.
One way to prove these fractions are equal is to draw a picture. Imagine two same-sized squares. Divide one into 2 equal parts, and the other into 4 equal parts. If you shade the same amount of each square, you'll see that 1/2 and 2/4 take up the same space — so they are equivalent (equal) fractions.

b. Here are some important ideas to learn from comparing fractions:

• If you draw a picture of two fractions, the bigger fraction will have more space shaded in. 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. More pieces means smaller pieces — that's why 1/3 is smaller than 1/2.
• The numerator tells you how many pieces you have. So 3/5 is more than 2/5, because you have one extra piece.
• Fractions are built out of unit fractions (fractions with a numerator of 1), so it's important to understand those first.
• When using picture cards, remember that the wholes being compared must be the same size for the comparison to make sense.
• Equivalent fractions can look different (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.
• In math, patterns show up again and again. Noticing these patterns helps you make guesses, explain your thinking, and prove your math ideas are correct.

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3.NF Comparing Fractions Game
Licensed by Illustrative Mathematics under a Creative Commons Attribution-NonCommercial-ShareAlike 4.0 International License.

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Fraction Card Sets (for cutting out and playing)

• 1/2, 1/6, 2/3, 2/8, 1/4, 1/8
• 3/4, 3/8, 4/6, 4/8, 2/4, 2/3
• 5/8, 5/6, 1/3, 1/2, 2/8, 2/6
• 1/4, 3/4, 1/3, 2/3, 2/6, 5/6
• 6/8, 7/8, 3/6, 6/6, 2/4, 3/4
• 1/8, 3/8, 4/8, 7/8, 4/6, 5/6
• 11/8, 4/6, 3/2, 2/3, 7/8, 4/3
• 7/6, 9/8, 5/4, 3/4, 5/6, 1/2
• 13/8, 11/6, 5/3, 5/8, 2/4, 3/4
• 6/8, 2/6, 1/2, 1/3, 3/4, 2/4
• 2/8, 1/2, 4/8, 1/4, 3/6, 1/2

Comparing Fractions Symbols

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Directions: Cut out each set of "greater than," "less than," and "equal to" circles. Give one set to each player.