# Lesson 14

Fraction Comparison Problems

## Warm-up: Number Talk: Multiples of Ten (10 minutes)

### Narrative

The purpose of this Number Talk is to elicit strategies and understandings students have for adding and subtracting multi-digit numbers. These understandings help students develop fluency and will be helpful in later units as students will need to be able to add and subtract multi-digit numbers fluently using the standard algorithm.

When students make adjustments and create multiples of ten for mental addition they are looking for and making use of the base ten structure of numbers (MP7).

### Launch

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

### Activity

• Record answers and strategy.
• Keep expressions and work displayed.
• Repeat with each expression.

### Student Facing

Find the value of each expression mentally.

• $$119 + 119$$
• $$139 + 139$$
• $$159 + 159$$
• $$199 + 199$$

### Activity Synthesis

How did you use multiples of ten, for example 20, 40, and 60 to help add these numbers mentally? (I changed the addends by adding one more to each addend and the subtracting the extra two from the final sum.)

• “Who can restate _____’s reasoning in a different way?”
• “Did anyone have the same strategy but would explain it differently?”
• “Did anyone approach the expression in a different way?”
• “Does anyone want to add on to _____’s strategy?”

## Activity 1: Mystery Fractions (20 minutes)

### Narrative

In this activity, students are given several sets of fractions and some clues about the size of a particular fraction in each set. To identify a fraction that meets certain size requirements or falls within a specified range, students need to use multiple comparison strategies they have learned. For example, they can use comparisons to benchmarks such as $$\frac{1}{2}$$ and 1 to eliminate some fractions, and then use equivalent fractions to compare the remaining ones.

### Launch

• Groups of 3–4
• Read the opening paragraph of the task as a class.
• “There are six sets of fractions in the activity. Each set comes with some clues. Your task is to find one fraction that meets all three clues in each set.”

### Activity

• “Work with your group to find the mystery fractions.”
• “Each group member should start with a different set, and should find at least two mystery fractions before discussing their responses with the group.“
• 5–6 minutes: independent work time
• 7–8 minutes: group work time

### Student Facing

Six friends are each given a list of 5 fractions. They each chose one fraction quietly and wrote clues about their choice. Use their clues to identify the fractions they chose.

 Andre: $$\ \frac{8}{12} \quad \frac{3}{6} \quad \frac{3}{4} \quad \frac{3}{2} \quad \frac{2}{12}$$ less than 1 greater than $$\frac{1}{3}$$ less than $$\frac{2}{3}$$
 Tyler: $$\ \ \frac{2}{6} \quad \frac{2}{2} \quad \frac{2}{4} \quad \frac{2}{3} \quad \frac{2}{5} \quad$$ greater than $$\frac{1}{3}$$ less than 1 less than $$\frac{1}{2}$$
 Clare: $$\ \ \frac{4}{3} \quad \frac{4}{2} \quad \frac{3}{4} \quad \frac{1}{4} \quad \frac{2}{10} \ \$$ greater than $$\frac{2}{8}$$ less than $$\frac{11}{6}$$ greater than 1
 Diego: $$\ \frac{2}{8} \quad \frac{6}{12} \quad \frac{6}{8} \quad \frac{12}{10} \quad \frac{11}{12}$$ greater than $$\frac{1}{2}$$ less than 1 greater than $$\frac{3}{4}$$
 Elena: $$\ \frac{2}{12} \quad \frac{50}{100} \quad \frac{4}{10} \quad \frac{3}{5} \quad \frac{7}{5}$$ greater than $$\frac{2}{10}$$ less than 1 greater than $$\frac{3}{6}$$
 Noah: $$\ \frac{18}{10} \quad \frac{7}{8} \quad \frac{2}{5} \quad \frac{18}{5} \quad \frac{150}{100}$$ greater than $$\frac{1}{2}$$ less than $$\frac{25}{10}$$ greater than $$\frac{8}{5}$$

### Activity Synthesis

• “Which clues helped you eliminate fractions the fastest?” (clues about size relative to 1)
• “What strategies did you use to compare fractions?” (compare fractions to $$\frac{1}{2}$$, 1, or another benchmark, write equivalent fractions to compare two fractions, compare fractions with the same numerator or denominator)
• “Did you ever have to use more than one strategy to compare fractions?” (Yes, two or three were often needed to find the mystery fraction.)

## Activity 2: Distances on Foot (10 minutes)

### Narrative

This activity has two purposes: to give students an opportunity to solve fraction comparison problems in context, and to reinforce the idea that two fractions can be compared only if they refer to the same whole. To serve the former, students compare fractional distance measurements. To serve the latter, they investigate fractional measurements in two different units of distance: Chinese “li” and kilometer.

When comparing the distances in the first question, students can rely on a number of familiar strategies. Two of the fractional values are close to 2. Some students are likely to use that benchmark for efficient comparison. For example, they may note that the school and the market are both a little over 1 li from home, the library is more than 2 li, and the badminton club is a little under 2 li.

Focus the synthesis on the last two questions about interpreting fractional measurements in two different units.

MLR8 Discussion Supports. Synthesis: Display sentence frames to support whole-class discussion: “I agree because . . .” and “I disagree because . . . .”
Engagement: Provide Access by Recruiting Interest. Optimize meaning and value. Share information about the unit “li” and help students understand the size of one li by referencing a common point of interest. Invite students to share this knowledge with family members and other teachers.
Supports accessibility for: Visual-Spatial Processing, Social-Emotional Functioning

### Launch

• Groups of 3–4
• “What are some units that we use for measuring distance? Let’s name as many as we can think of.” (Sample responses: inches, feet, miles, meters, kilometers)
• Share and record responses.
• “Today we’ll look at distances measured in ‘li,’ a unit commonly used in China.”

### Activity

• “Take a few quiet minutes to work on the questions. Be prepared to explain your reasoning.”
• “Afterward, discuss your responses with your group and work together to complete the activity.”
• 4 minutes: independent work time
• 4 minutes: group work time
• Monitor for students who:
• attend to the location of 1 whole when representing $$\frac{4}{5}$$ and $$\frac{7}{5}$$ on the two number lines (rather than partitioning both number lines into the same 5 parts)
• recognize that $$\frac{4}{5}$$ and $$\frac{7}{5}$$ in the two cases refer to different wholes and can articulate their reasoning

### Student Facing

In China and some East Asian countries, the unit “li” is used for measuring distance.

Here are the walking distances between the home of a student in China and the places he visits regularly.

• school: $$\frac{7}{5}$$ li
• library: $$\frac{23}{10}$$ li
• market: $$\frac{7}{4}$$ li
• badminton club: $$\frac{23}{12}$$ li
1. Which is a shorter distance from the student’s home:

1. His school or the library?
2. The market or the badminton club?
3. The library or the market?
2. A student in America walks $$\frac{4}{5}$$ kilometer (km) to school. These number lines show how 1 kilometer compares to 1 li.

Which student walks a longer distance to school? Use the number lines to show your reasoning.

3. Explain why we can’t just compare the fractions $$\frac{4}{5}$$ and $$\frac{7}{5}$$ to see which student walks a longer distance.

### Activity Synthesis

• Ask previously selected students to share their responses to the last two questions. Display their number lines, or display the number lines from the activity for them to annotate while they explain.
• Emphasize that just as 1 km is not the same distance as 1 li, $$\frac{4}{5}$$ km is not the same distance as $$\frac{4}{5}$$ li. We can’t compare two fractions that refer to different wholes.

## Lesson Synthesis

### Lesson Synthesis

“Today we used a combination of strategies to help us compare fractions. We also solved fraction comparison problems in a situation about distance.”

Keep students in groups of 3–4. Give tools for creating a visual display to each group.

Assign each group one set of fractions in Activity 1 or the first set of questions in Activity 2.

• For the former: “Create a visual display that explains how you found the mystery fraction for your assigned set of fractions from Activity 1. Your display should list the five fractions, the three clues, and how the chosen fraction satisfies all the clues.”
• For the latter: “Create a visual display that shows your responses to the first set of questions in Activity 2. Your display should show the four walking distances and how you compared them.”

“Include diagrams, notes, and any descriptions that might help others understand your thinking.”

Ask students to display their work around the room.

“Visit the display of at least 2 other groups.”

“At each display, check to see if the reasoning strategies make sense to you. Think about how the reasoning in different displays is alike and how it is different.”

“How are the diagrams, explanations, or calculations that you saw alike? How are they different?”

Share and record responses. Reference the displays that students created to show similarities and differences in their reasoning strategies.