Lesson 3

Comparing Proportional Relationships

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3.1: What's the Relationship? (10 minutes)

CCSS Standards


Addressing

Warm-up

The purpose of this warm-up is for students to create a graph and a description from an equation, building on their work in the previous lesson. Students decide on a context and then make the graph, scaling the axes appropriately to the situation. Moving between representations of a proportional relationship here is preparation for the following activity where students compare proportional relationships represented in different ways.

Launch

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

Student Facing

The equation \(y=4.2x\) could represent a variety of different situations.

  1. Write a description of a situation represented by this equation. Decide what quantities \(x\) and \(y\) represent in your situation.

  2. Make a table and a graph that represent the situation.

Student Response

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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: Comparing Two Different Representations (25 minutes)

CCSS Standards


Addressing

Routines and Materials


Instructional Routines

Required Materials

Activity

The purpose of this activity is for students to compare two different proportional relationships represented in different ways using the skills they have worked on over the past three lessons. Working in groups, students compare the relationships, responding to questions about their rate of change, which rate of change is higher, and one other situation-based question. Groups make a visual display for their problem set to explain each of their responses and convince others of their accuracy.

Identify groups using a variety of representations to share during the Activity Synthesis.

Launch

Remind students that in previous lessons they identified representations of and created representations for a single proportional relationship. In this activity, they will consider representations of two different proportional relationships and make comparisons between them.

Arrange students in groups of 2–3. Assign to each group (or ask groups to choose) one of the three question sets. Tell groups that 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. Let them know that there will be a gallery walk when they finish for the rest of the class to inspect their solutions’ accuracy.

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

Action and Expression: Internalize Executive Functions. To support development of organizational skills in problem-solving, chunk this task into more manageable parts. For example, present one question at a time.
Supports accessibility for: Memory; Organization

Student Facing

  1. Elena babysits her neighbor’s children. Her earnings are given by the equation \(y=8.40x\), where \(x\) represents the number of hours she worked and \(y\) represents the amount of money she earned.

    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 rate of change for each situation and what does it mean?
    3. Using your graph or table, determine how long it would take each person to earn $150.

  2. 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
    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 rate of change for each situation and what does it 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.

  3. 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.

    Lemonade Recipe 1 is given by the equation \(y=4x\) where \(x\) represents the amount of lemonade mix in cups and \(y\) represents the amount of water in cups.

    Lemonade 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 rate of change for each situation and what does it mean?
    3. Tyler has a 5-gallon jug (which holds 80 cups) to use for his lemonade stand and 16 cups of lemonade mix. Which lemonade recipe should he use? Explain or show your reasoning.

Student Response

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Student Facing

Are you ready for more?

Han and Clare are still stuffing envelopes. Han can stuff 20 envelopes in a minute, and Clare can stuff 10 envelopes in a minute. They start working together on a pile of 1,000 envelopes. 

  1. How long does it take them to finish the pile?
  2. Who earns more money?

Student Response

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Anticipated Misconceptions

Some students may confuse the values for the rate of change of a situation. For example, Lemonade Recipe 1's equation, \(y=4x\), shows that the rate of change is 4 cups of water per cup of lemonade mix. Students may switch these values and think that the rate of change is 4 cups lemonade mix per cup of water. Ask students who do this to explain where in the original representations they see the rate of change. Students may need to list a few values or sketch a graph in order to see their mix-up between the two quantities.

Activity Synthesis

Begin with a gallery walk for students to see how other groups answered the same set of questions they did and how students answered questions about the other two contexts.

Invite groups to share the strategies they used with the various representations. Consider asking groups the following questions:

  • “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?”
  • “Now that you have seen the work of other groups, is there anything about your display you would change if you could?”
Representing and Conversing: MLR7 Compare and Connect. During the gallery walk, invite students to discuss “what is the same and what is different?” about the representations on the posters and then share with the whole class. Look for opportunities to highlight representations that helped students answer the questions and decide which scales to use for the graph. This will help students make connections and describe the usefulness of each type of representation.
Design Principle(s): Optimize output

Lesson Synthesis

Lesson Synthesis

This lesson asked students to take a single piece of information about a proportional relationship, such as an equation, and use what they know about proportional relationships, rates of change, and representing relationships to compare it with a second proportional relationship in context.

Consider asking some of the following questions. Tell students to use, if possible, examples from today’s lesson when responding:

  • “What do you need in order to compare two proportional relationships?”
  • “What type of wording in a problem statement or description of a situation tells you that you have a rate of change?”
  • “How did you decide which representation to use to solve the different types of problems?”

3.3: Cool-down - Different Salt Mixtures (5 minutes)

CCSS Standards


Addressing

Cool-Down

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Student Lesson Summary

Student Facing

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

For example, Clare’s earnings are represented by the equation \(y=14.5x\), where \(y\) is her earnings in dollars for working \(x\) hours.

The table shows some information about Jada’s pay.

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

Who is paid at a higher rate per hour? How much more does that person have after 20 hours?

In Clare’s equation we see that the rate of change (how many dollars she earns every hour) is 14.50.

We can calculate Jada’s rate of change by dividing a value in the earnings column by the value in the same row in the time worked column. Using the last row, the rate of change for Jada is 13.25, since \(490.25\div37=13.25\). An equation representing Jada’s earnings is \(y=13.25x\). This means she earns \$13.25 per hour.

So Clare is paid at a higher rate than Jada. Clare earns \$1.25 more per hour than Jada. After 20 hours of work, she earns \$25 more than Jada because \(20 \boldcdot (1.25) = 25\).