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3.2

Activity

25 mins

Comparing Two Different Representations

Standards Alignment

Building On

Addressing

Building Toward

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

Activity

None

Activity Synthesis

Begin with a gallery walk for students to see how other groups answered the same set of questions they did. In small groups, invite students who created a display for the same set of problems to discuss what is the same and what is different about their work and representations on the posters. Here are some questions for discussion:

“What representations did you choose to answer the questions? Why did you pick them?”
“What representation did you not use? Why?”
“How did you decide what scale to use when you made your graph?”

Math Community
Conclude the discussion by inviting 2–3 students to share a norm they identified in action. Provide this sentence frame to help students organize their thoughts in a clear, precise way:

“I noticed our norm “” in action today and it really helped me/my group because .”

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.

Elena babysits her neighbor’s children. Her earnings are given by the equation , where represents the number of hours she worked, and represents the amount of money she earned in dollars.

Jada earns $7 per hour mowing her neighbors’ lawns.
1. Who makes more money after working 12 hours? How much more do they make? Explain your reasoning by creating a graph or a table.
2. What is the value of the rate of change for each situation, and what does each value mean?
3. Using your graph or table, determine how long it would take each person to earn $150.
Clare and Han have summer jobs stuffing envelopes for two different companies.

Han earns $15 for every 300 envelopes he finishes.

Clare’s earnings can be seen in the table.

number of
envelopes money earned
in dollars

400 40

900 90
1. By creating a graph, show how much money each person makes after stuffing 1,500 envelopes.
2. What is the value of the rate of change for each situation, and what does each value mean?
3. Using your graph, determine how much more money one person makes relative to the other after stuffing 1,500 envelopes. Explain or show your reasoning.
Tyler plans to start a lemonade stand and is trying to perfect his recipe for lemonade. He wants to make sure the recipe doesn’t use too much lemonade mix (lemon juice and sugar) but still tastes good.

Recipe 1 is given by the equation , where represents the amount of lemonade mix in cups, and represents the amount of water in cups.

Recipe 2 is given in the table.

lemonade mix (cups) water (cups)

10 50

13 65

21 105
1. If Tyler had 16 cups of lemonade mix, how many cups of water would he need for each recipe? Explain your reasoning by creating a graph or a table.
2. What is the value of the rate of change for each recipe, and what does each value mean?
3. Tyler has 16 cups of lemonade mix to use for his lemonade stand. Which lemonade recipe should he use? Explain or show your reasoning.

number of envelopes	money earned in dollars
400	40
900	90

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

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Lesson 3

Comparing Proportional Relationships

Warm-up

What's the Relationship?

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

Standards Alignment

8.EE.B.5

Instructional Routines

Materials

Activity Narrative

Launch

Activity

Activity Synthesis

Lesson Synthesis

Student Lesson Summary

Have feedback on the curriculum?

Standards Alignment

8.EE.B

Student Task Statement

Student Response

Building on Student Thinking

Are You Ready for More?