Lesson 14

How Do You Compare Fractions?

Warm-up: Number Talk: Which Whole Numbers? (10 minutes)

Narrative

This Number Talk encourages students to use what they know about the meaning of fractions and about properties of operations to mentally relate fractions that are equivalent to whole numbers.

Launch

• Display one fraction.
• “Give me a signal when you have an answer and can explain how you got it.”
• 1 minute: quiet think time

Activity

• Keep fractions and work displayed.
• Repeat with each fraction.

Student Facing

Find the whole number that each fraction is equivalent to.

• $$\frac{16}{1}$$
• $$\frac{16}{2}$$
• $$\frac{16}{4}$$
• $$\frac{20}{4}$$

Activity Synthesis

• “How did the earlier fractions help you find the whole number for the last fraction?”
• “Did anyone have the same strategy but would explain it differently?”
• “Did anyone approach the problem in a different way?”

Activity 1: Equivalent or Not? (25 minutes)

Narrative

The purpose of this activity is for students to analyze pairs of fractions to determine if they are equivalent. Students may use any representation that makes sense to them. Students will create a visual display and have a gallery walk to consider the different ways of looking for equivalence. Highlight representations such as diagrams and number lines, which will support students as they learn to compare fractions with the same numerator or denominator in this section.

This activity uses MLR7 Compare and Connect. Advances: representing, conversing

Engagement: Develop Effort and Persistence. Invite students to generate a list of shared expectations for group work. Record responses on a display and keep visible during the activity.
Supports accessibility for: Social-Emotional Functioning

Launch

• Groups of 2
• “Decide if these pairs of fractions are equivalent and show your thinking for each one. You can use any representation that makes sense to you.”

Activity

• 3–5 minutes: independent work time
• 2–3 minutes: partner discussion
MLR7 Compare and Connect
• “Work with your partner to create a visual display that shows your thinking about $$\frac{4}{6}$$ and $$\frac{5}{6}$$. You may want to include details such as notes, diagrams, drawings, and so on, to help others understand your thinking.”
• Give students materials for creating a visual display.
• 2–5 minutes: partner work time
• 5–7 minutes: gallery walk

Student Facing

Are these fractions equivalent? Show your thinking using diagrams, symbols, or other representations.

1. $$\frac{1}{2}$$ and $$\frac{1}{3}$$
2. $$\frac{4}{6}$$ and $$\frac{5}{6}$$
3. $$\frac{3}{4}$$ and $$\frac{6}{8}$$

Student Response

If students don’t determine whether or not fractions are equivalent, consider asking:
• “What do you know about these fractions?”
• “How could you use your fraction strips or diagrams to decide if they are equivalent?”

Activity Synthesis

• “What are the different representations we can use to decide if $$\frac{4}{6}$$ and $$\frac{5}{6}$$ are equivalent?” (diagrams, number lines)
• “How did each representation show that $$\frac{4}{6}$$ and $$\frac{5}{6}$$ are not equivalent?” (In the diagram, you can see that $$\frac{5}{6}$$ has more space shaded. On the number line, they are not at the same location.)

Activity 2: Same Fractions, Different Result? (10 minutes)

Narrative

The purpose of this activity is for students to recognize that fraction comparisons are only valid when they refer to the same whole. Previously, students analyzed pairs of fractions to determine if they were equivalent. In doing so, they were likely to have used comparison language, such as “larger or smaller than” or “greater or less than.” In this activity, students encounter a pair of fractions they saw earlier ($$\frac{4}{6}$$ and $$\frac{5}{6}$$) and compare them more explicitly. The student work in this activity uses the number line, but this might also come up with student-drawn diagrams.

In order to interpret Lin’s argument that $$\frac{4}{6}$$ is greater than $$\frac{5}{6}$$ students will need to articulate the meaning of fractions and highlight the fact that the two wholes Lin is comparing are not equal (MP6).

MLR8 Discussion Supports. Synthesis: Revoice student ideas to demonstrate and amplify mathematical language use. For example, revoice the student statement “the number lines are different” as “the size of the whole from 0 to 1 is different.”

Launch

• Groups of 2
• “Han and Lin are comparing $$\frac{4}{6}$$ and $$\frac{5}{6}$$ like you did. Take a minute to look at their work.”
• 1 minute: quiet think time

Activity

• “Talk with your partner about how Han and Lin could get different results.”
• 2–3 minutes: partner discussion
• Monitor for students who notice that the whole is different in Lin’s number lines, which makes her think that $$\frac{4}{6}$$ is greater.

Student Facing

Han says $$\frac{4}{6}$$ is less than $$\frac{5}{6}$$. His work is shown.

Lin says $$\frac{4}{6}$$ is greater than $$\frac{5}{6}$$. Her work is shown.

Why might Han and Lin make different comparison statements for the same fractions?

Activity Synthesis

• Select previously identified students to share how Han and Lin could make different comparison statements for the same fractions.
• “It is important to remember when we are comparing fractions that those fractions need to refer to the same whole.”
• “When the whole from 0 to 1 is the same size, we can see that $$\frac{4}{6}$$ is less than $$\frac{5}{6}$$.”
• “This is true whether we are drawing a number line, using fraction strips, or drawing a diagram.”
• Consider asking, “What did both Lin and Han understand about representing fractions on a number line?” (The number lines need to be partitioned into equal parts. The denominator tells us then number of parts. The numerator tells us how many parts to count to locate a point. Points farther to the right are greater than those to the left.)

Lesson Synthesis

Lesson Synthesis

“Today, we studied pairs of fractions to see if they were equivalent or not. How did you decide if two fractions were equivalent?” (Drew diagrams to see if the parts that represent the fractions were the same size. Represent the fractions on number lines and see if they were at the same location.)

“If the fractions were not equivalent, it means that one of the fractions was greater than the other, and one of the fractions was less than the other. We'll learn more about this in future lessons.”

Draw two number lines (or diagrams) with different lengths (or areas) representing 1 whole. Partition each into 3 parts.

“What might be a problem with comparing $$\frac{2}{3}$$ and $$\frac{3}{3}$$ using these number lines (or diagrams)?” (The 1 whole is not the same size, so we can’t use the number lines (or diagrams) to compare the fractions.)