Unit 5, Lesson 3

Comparing Proportional Relationships

Accelerated 7

3.1

Warm-up

10 mins

What's the Relationship?

Standards Alignment

Building On

Addressing

Building Toward

Instructional Routines

None

Materials

None

Activity Narrative

The purpose of this Warm-up is for students to describe a situation presented by an equation using other representations. Students decide on a context and then create a table and a graph, scaling the axes appropriately to the situation (MP2). Moving between representations of a proportional relationship here will be useful in a following activity where students compare proportional relationships represented in different ways.

Launch

Arrange students in groups of 2. Give them 2–3 minutes of quiet work time followed by a whole-class discussion.

Activity

None

Student Task Statement

The equation could represent a variety of different situations.

Describe a situation that can be represented by this equation. What do the quantities and represent in your situation?
Create a table and a graph that represent the situation.

Student Response

Building on Student Thinking

Are You Ready for More?

Activity Synthesis

Invite several students to share their situations and display their graphs for all to see. Ask:

“What does the rate of change represent in this situation?”
“How did you decide on the scale for your axes?”

3.2

Activity

25 mins

Comparing Two Different Representations

Instructional Routines

MLR6: Three Reads

Materials

To Gather

Math Community Chart Tools for creating a visual display

Activity Narrative

In this activity, students consider representations of two different proportional relationships and make comparisons between them. They work in groups to compare the relationships and reason quantitatively to answer context specific questions (MP2). Groups make a visual display for their problem set to explain each of their responses and convince others of their accuracy.

Launch

Math community
Display the Math Community Chart for all to see. Give students a brief quiet think time to read the norms or invite a student to read them out loud. Tell students that during this activity they are going to practice looking for their classmates putting the norms into action. At the end of the activity, students can share what norms they saw and how the norm supported the mathematical community during the activity.

Tell students that the activity today is about summer jobs. Invite students to share their experiences with a summer job or any other experience they may have earning money, such as doing chores for a neighbor or selling handmade items. If desired, update the contexts of the question sets so they are more familiar to students or to better reflect the experiences of the class.

Then arrange students in groups of 2–3. Provide each group with tools for creating a visual display. Assign each group one of the three question sets (or have groups to choose). Tell groups they will make a visual display for their responses to the questions. The display should clearly demonstrate their reasoning and use multiple representations in order to be convincing (MP3). Let them know that there will be a gallery walk when they finish for the rest of the class to inspect the accuracy of their solutions.

If time allows, consider asking groups to complete all three problems and make a visual display for just one.

MLR6 Three Reads. In small groups, tell students to keep books or devices closed. Display only the task statement and two descriptions, without revealing the questions. Say, “We are going to read this information 3 times.”

After the 1st read: “Tell your partner what this situation is about.”
After the 2nd read: “List the quantities. What can be counted or measured?”
For the 3rd read: Reveal and read the questions. Ask, “What are some ways one might get started on this?”

Advances: Reading, Representing

Action and Expression: Develop Expression and Communication. Provide students with alternatives to writing on paper: students can share their learning using digital technology, such as creating a digital presentation or slideshow.
Supports accessibility for: Language, Fine Motor Skills

Lesson Synthesis

The goal is for students to discuss strategies they can use when comparing two proportional relationships. Have students conduct a second gallery walk to see how other groups answered questions about the two contexts they did not make a display for. Here are some questions for discussion:

“What was the same and different about the other problem sets?” (Answers vary.)
“What do you need in order to compare two proportional relationships?” (the rate of change)
“What type of wording in a problem statement or description of a situation tells you that you have a rate of change?” (phrases like "per," "every," and "for each")
“How did you decide which representation to use to solve the different types of problems?” (Answers vary.)

Student Lesson Summary

When two proportional relationships are represented in different ways, we can compare them by finding a common piece of information.

For example, Clare’s earnings are represented by the equation , where is her earnings in dollars for working hours.

The table shows some information about Jada’s earnings.

time worked (hours)	earnings (dollars)
7	92.75
4.5	59.63
37	490.25

If we want to know who makes more per hour, we can look at the rate of change for each situation.

In Clare’s equation, we see that the rate of change is 14.50. This tells us that she earns $14.50 per hour. For Jada, we can calculate the rate of change by dividing her earnings in one row by the hours worked in the same row. For example, using the last row, the rate of change is 13.25 since . This tells us that Clare earns more dollars per hour than Jada.

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time worked (hours)

earnings (dollars)

92.75

4.5

59.63

490.25

number of envelopes	money earned in dollars
400	40
900	90

number of
envelopes

money earned
in dollars

400

900

lemonade mix (cups)	water (cups)
10	50
13	65
21	105

lemonade mix (cups)

water (cups)

105

Lesson 3

Comparing Proportional Relationships

Warm-up

What's the Relationship?

Standards Alignment

Building On

Addressing

8.EE.B

Building Toward

Instructional Routines

Materials

Activity Narrative

Launch

Activity

Student Task Statement

Student Response

Building on Student Thinking

Are You Ready for More?

Activity Synthesis

Activity

Comparing Two Different Representations

Instructional Routines

Materials

To Gather

Activity Narrative

Launch

Lesson Synthesis

Student Lesson Summary

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Audience

Assessment Access

Lesson 3

Comparing Proportional Relationships

Warm-up

What's the Relationship?

Standards Alignment

Building On

Addressing

8.EE.B

Building Toward

Instructional Routines

Materials

Activity Narrative

Launch

Activity

Student Task Statement

Student Response

Building on Student Thinking

Are You Ready for More?

Activity Synthesis

Activity

Comparing Two Different Representations

Instructional Routines

Materials

To Gather

Activity Narrative

Launch

Lesson Synthesis

Student Lesson Summary

Have feedback on the curriculum?

Activity

Student Task Statement

Student Response

Building on Student Thinking

Are You Ready for More?

Activity Synthesis

Standards Alignment

Building On

Addressing

8.EE.5

Building Toward