Illustrative Mathematics Grade 8, Unit 3, Lesson 4: Comparing Proportional Relationships

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Illustrative Math
Grade 8

Lesson 4: Comparing Proportional Relationships

Let’s compare proportional relationships

Illustrative Math Unit 8.3, Lesson 4 (printable worksheets)

Lesson 4 Summary

The following diagram shows how to compare proportional relationships represented in different ways.

Lesson 4.1 What’s the Relationship?

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.

• See Video for Whole Lesson

Lesson 4.2 Summer Jobs

1. Elena and Jada each make money by helping out their neighbors. Elena babysits. Her earnings are given by the equation y = 8.40x, where x represents how many hours she works and y represents how much money she earns.
   Jada earns $7 per hour mowing her neighbors’ lawns.
   a. Who makes more money after working 12 hours? How much more do they make? Explain how you know.
   b. What is the rate of change for each situation and what does it mean?
   c. How long would it take each person to earn $150? Explain or show your reasoning.
2. Han and Clare have summer jobs stuffing envelopes for two different companies.
   Han earns $15 for every 300 envelopes he finishes.
   Clare’s earnings:
   a. Who would make more money after stuffing 1,500 envelopes? How much more money would they make? Explain how you know.
   b. What is the rate of change for each situation and what does it mean?
   c. Who gets paid more in their job? Explain or show your reasoning.
3. Tyler plans to start a lemonade stand and is trying out different recipes 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 y = 4x where x represents the cups of lemonade mix and y represents the cups of water.
   a. If Tyler had 16 cups of lemonade mix, how many cups of water would he need for each recipe? Explain how you know.
   b. What is the rate of change for each situation and what does it mean?
   c. 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.

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?
   • Show Answers

     Together, they finish 30 envelopes in 1 minute. To finish 1000 envelopes, they will take 1000 ÷ 30 = 33 minutes 20 seconds

2. Who earns more money?
   • Show Answers

     In 33 minutes 20 seconds, Han would have stuffed about 667 envelopes and Clare would have stuffed about 333 envelopes.
     If they are not paid for partial amounts then Han will earn $30 and Clare would not be paid.
     It might be better for Han to stuff 600 envelopes and then he will be paid $30 and Clare to stuff 400 envelopes and she would be paid $40.

Lesson 4 Practice Problems

1. A contractor must haul a large amount of dirt to a work site. She collected information from two hauling companies. EZ Excavation gives its prices in a table. Happy Hauling Service gives its prices in a graph.
   a. How much would each hauling company charge to haul 40 cubic yards of dirt? Explain or show your reasoning.
   b. Calculate the rate of change for each relationship. What do they mean for each company?
   c. If the contractor has 40 cubic yards of dirt to haul and a budget of $1000, which hauling company should she hire? Explain or show your reasoning.
2. Andre and Priya are tracking the number of steps they walk. Andre records that he can walk 6000 steps in 50 minutes. Priya writes the equation y = 118x, where y is the number of steps and x is the number of minutes she walks, to describe her step rate. This week, Andre and Priya each walk for a total of 5 hours. Who walks more steps? How many more?
3. Find the coordinates of point D in each diagram:
4. Select all the pairs of points so that the line between those points has slope 2/3

The Open Up Resources math curriculum is free to download from the Open Up Resources website and is also available from Illustrative Mathematics.

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