Lesson 15

Compare Fractions with the Same Denominator

Warm-up: Notice and Wonder: Two More Strips (10 minutes)

Narrative

The purpose of this warm-up is to elicit the idea that the size and the number of unit fractions can help us compare fractions. Students can see that the two diagrams have same-size parts but not how much of one diagram is shaded, prompting them to think about the number of shaded parts. While students may notice and wonder many things about these images, what fractions could be represented by the partially hidden strip is the important discussion point.

Launch

• Groups of 2
• Display the image.
• “What do you notice? What do you wonder?”
• 1 minute: quiet think time

Activity

• 1 minute: partner discussion
• Share and record responses.

Student Facing

What do you notice? What do you wonder?

Activity Synthesis

• “How many parts could be shaded on the top strip? Could less than $$\frac{3}{4}$$ be shaded? Could more than $$\frac{3}{4}$$ be shaded?” (If 2 parts are shaded, that's $$\frac{2}{4}$$, which is less than $$\frac{3}{4}$$. If 3 parts are shaded, that's $$\frac{3}{4}$$. If the whole strip is shaded, that's $$\frac{4}{4}$$, which is more than $$\frac{3}{4}$$.)

Activity 1: Compare Fractions with the Same Denominator (20 minutes)

Narrative

The purpose of this activity is for students to compare two fractions with the same denominator. Students may use any representation to reason about how the size or length of the parts in the two fractions are the same because the denominator is the same, but that there are different numbers of those parts because the numerator is different (MP2). Students are also reminded about the meaning of the symbols > and <.

MLR7 Compare and Connect. Synthesis: Invite groups to prepare a visual display that shows the strategy they used to compare the fractions. Encourage students to include details that will help others interpret their thinking. For example, specific language, using different colors, shading, arrows, labels, notes, diagrams, or drawings. Give students time to investigate each other’s work. During the whole-class discussion, ask students, “What do these representations have in common?” “How are they different?” and “What kinds of additional details or language helped you understand the displays?”

Launch

• Groups of 2
• Display the first problem.
• “Take a few minutes to decide which fraction is greater for each of these pairs.”
• 2–3 minutes: independent work time
• 1–2 minutes: partner discussion
• Share and record responses.
• Display: < and >
• “Let’s refresh our memories about ‘less than’ and ‘greater than’ symbols. How do we read these symbols?” (“is less than” and “is greater than”)
• Display: $$\frac{1}{2} < 1$$ and $$\frac{5}{4} > 1$$
• “How do we read these statements?” (One-half is less than 1. Five-fourths is greater than 1.)
• “What expressions could you write about $$\frac{1}{2}$$ and $$\frac{3}{2}$$ and $$\frac{2}{8}$$ and $$\frac{3}{8}$$ using these symbols?” ($$\frac{1}{2} < \frac{3}{2}$$, $$\frac{3}{2} > \frac{1}{2}$$, $$\frac{2}{8} < \frac{3}{8}$$, $$\frac{3}{8} > \frac{2}{8}$$)
• Share and display responses. Ask students to read aloud each statement that is shared.

Activity

• “Work with your partner to compare the fractions in the next problem and use these symbols. Be sure to explain or show your reasoning.”
• 5–7 minutes: partner work time
• Monitor for students who explain their reasoning with:
• an area diagram
• a fraction strip diagram
• a number line
• written description about the size or length of parts

Student Facing

1. For each pair of fractions, circle the fraction that is greater. Explain or show your reasoning.

1. $$\frac{1}{2}$$ and $$\frac{3}{2}$$
2. $$\frac{3}{8}$$ and $$\frac{2}{8}$$
2. Use the symbols > or < to make each statement true. Explain or show your reasoning.

1. $$\frac{1}{6} \, \underline{\phantom{\frac{1}{1}\hspace{1.05cm}}} \, \frac{4}{6}$$
2. $$\frac{4}{4} \, \underline{\phantom{\frac{1}{1}\hspace{1.05cm}}} \, \frac{5}{4}$$
3. $$\frac{2}{3} \, \underline{\phantom{\frac{1}{1}\hspace{1.05cm}}} \, \frac{1}{3}$$
4. $$\frac{4}{8} \, \underline{\phantom{\frac{1}{1}\hspace{1.05cm}}} \, \frac{6}{8}$$

If you have time: Write in the missing numerator of the fraction to make each statement true. Explain or show your reasoning.

1. $$\frac{1}{2} < \frac{\phantom{0000}}{2}$$
2. $$\frac{6}{4} > \frac{\phantom{0000}}{4}$$
3. $$\frac{4}{3} < \frac{\phantom{0000}}{3}$$
4. $$\frac{5}{8} > \frac{\phantom{0000}}{8}$$

Activity Synthesis

• Select previously identified students to share different representations for comparing fractions with the same denominator.
• Consider asking: “How are these representations alike? How are they different?”

Activity 2: Spin to Win: Same Denominator (15 minutes)

Narrative

The purpose of this activity is for students to practice comparing fractions with the same denominator while playing a game. Students spin a spinner for the numerator of their fractions and then locate and label the fractions on a number line to determine which fraction is greater.

Representation: Access for Perception. To support understanding, begin by demonstrating how to play one round of “Spin to Win.”
Supports accessibility for: Memory, Social-Emotional Functioning

Required Materials

Materials to Gather

Materials to Copy

• Spin to Win Spinner
• Spin to Win Recording Sheet

Required Preparation

• Each group of 2 needs a paper clip for their spinner.

Launch

• Groups of 2
• Give each group a paper clip, colored pencils, a spinner, and a sheet of number lines.
• “Now you will play a game in which you compare fractions with the same denominator. To start, one player will choose a denominator for the first round.”
• “You will each spin for the numerator of your own fraction with that denominator. Then you’ll locate and label your fractions on the same number line and determine whose fraction is greater.”
• “If your fraction is greater, you get to choose the denominator for the next round. The goal of the game is to win as many rounds as you can.”
• Ask the class to choose a denominator (2, 3, 4, 6, or 8). Then spin the spinner and discuss how to represent the fraction on a number line on the recording sheet.

Activity

• 10 minutes: partner work time
• As students work, monitor for students who notice patterns as they play.

Student Facing

In this game, you will record fractions on number lines. Choose a writing utensil in a color different than your partner's so you can tell which fraction is whose on each number line.

1. Each player spins the paper clip. The player who spins the highest number is Player 1.
2. Player 1 chooses a denominator for the first round: 2, 3, 4, 6, or 8.
3. Each player spins for the numerator of their fraction.
4. Each player locates and labels their fraction on the same number line on the recording sheet.
5. The player with the greater fraction wins and picks the denominator for the next round.
6. Repeat for 10 rounds. The player who wins the most rounds wins the game.

Activity Synthesis

• “What kind of number did you want to spin on your turn? Why?” (I wanted to spin a large number because a large numerator means a greater fraction. If you spin a small number, then the small numerator makes a smaller fraction.)

Lesson Synthesis

Lesson Synthesis

“Today we compared fractions with the same denominator.”

“How do you compare fractions with the same denominator? Does your strategy always work?” (I can just look at the numerators to see which is greater. This always works because the whole is split into the same number of parts that are the same size if the denominator is the same, so we just need to think about how many of those parts we have, which is given by the numerator.)