Unit 2

Two Equations for Each Relationship

Introducing Proportional Relationships

Lesson 5

Expressions and Equations

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Let’s investigate the equations that represent proportional relationships.

Unit 2 ● Lesson 5

Learning

Goal

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Missing Figures

Unit 2 ● Lesson 5 ● Activity 1

Here are the second and fourth figures in a pattern.

1. What do you think the first and third figures in the pattern look like?
2. Describe the 10th figure in the pattern.

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Warm-up

Slides are CC BY NC Kendall Hunt Publishing. Curriculum excerpts are CC BY Open Up Resources, with adaptations CC BY Illustrative Mathematics.

Meters and Centimeters

Unit 2 ● Lesson 5 ● Activity 2

There are 100 centimeters (cm) in every meter (m).

1. Complete each of the tables.

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• What is the relationship between these constants of proportionality?

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Slides are CC BY NC Kendall Hunt Publishing. Curriculum excerpts are CC BY Open Up Resources, with adaptations CC BY Illustrative Mathematics.

Meters and Centimeters

Unit 2 ● Lesson 5 ● Activity 2

• How can we find an equation for each table?
• Where does the constant of proportionality occur in the table and equation?
• What is the relationship between the two constants of proportionality? How can you use the equations to see why this should be true?

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Slides are CC BY NC Kendall Hunt Publishing. Curriculum excerpts are CC BY Open Up Resources, with adaptations CC BY Illustrative Mathematics.

Filling a Water Cooler

Unit 2 ● Lesson 5 ● Activity 3

It took Priya 5 minutes to fill a cooler with 8 gallons of water from a faucet that was flowing at a steady rate. Let w be the number of gallons of water in the cooler after t minutes.

1. Which of the following equations represent the relationship between w �and t ? Select all that apply.
1. w = 1.6t
2. w = 0.625t
3. t = 1.6w
4. t = 0.625w
2. What does 1.6 tell you about the situation?
3. What does 0.625 tell you about the situation?
4. Priya changed the rate at which water flowed through the faucet. Write an equation that represents the relationship of w and t when it takes 3 minutes to fill the cooler with 1 gallon of water.
5. Was the cooler filling faster before or after Priya changed the rate of water flow? Explain how you know.

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Slides are CC BY NC Kendall Hunt Publishing. Curriculum excerpts are CC BY Open Up Resources, with adaptations CC BY Illustrative Mathematics.

Feeding Shrimp

Unit 2 ● Lesson 5 ● Activity 4

At an aquarium, a shrimp is fed gram of food each feeding and is fed 3 times each day.

1. How much food does a shrimp get fed in one day?

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2. Complete the table to show how many grams of food the shrimp is fed over different numbers of days.
3. What is the constant of proportionality? What does it tell us about the situation?

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• If we switched the columns in the table, what would be the constant of proportionality? Explain your reasoning.

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• Use d for number of days and f for amount of food in grams that a shrimp eats to write two equations that represent the relationship between d and f .

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Slides are CC BY NC Kendall Hunt Publishing. Curriculum excerpts are CC BY Open Up Resources, with adaptations CC BY Illustrative Mathematics.

Two Equations for Each Relationship

Unit 2 ● Lesson 5

• Why were we able to write two equations for the proportional relationship between meters and centimeters? What were they? What were the constants of proportionality?
• In the proportional relationship where we knew how long it took to fill a water cooler with a certain amount of water, what were the constants of proportionality? What equations did we determine would represent this situation?
• In each case, what was the relationship between the two constants of proportionality and between the two equations?

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Lesson Synthesis

Slides are CC BY NC Kendall Hunt Publishing. Curriculum excerpts are CC BY Open Up Resources, with adaptations CC BY Illustrative Mathematics.

Unit 2 ● Lesson 5

• I can find two constants of proportionality for a proportional relationship.
• I can write two equations representing a proportional relationship described by a table or story.

Learning

Targets

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Flight of the Albatross

Unit 2 ● Lesson 5 ● Activity 5

An albatross is a large bird that can fly 400 kilometers in 8 hours at a constant speed. Using d for distance in kilometers and t for number of hours, an equation that represents this situation is d = 50t.

1. What are two constants of proportionality for the relationship between distance in kilometers and number of hours? What is the relationship between these two values?
2. Write another equation that relates d and t in this context.

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Cool-down

Slides are CC BY NC Kendall Hunt Publishing. Curriculum excerpts are CC BY Open Up Resources, with adaptations CC BY Illustrative Mathematics.

Lesson Video

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