Unit 3

Graphs of Proportional Relationships

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Linear Relationships

Lesson 2

Expressions and Equations

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Let's think about scale.

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Unit 3 ● Lesson 2

Learning

Goal

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An Unknown Situation

Unit 3 ● Lesson 2 ● Activity 1

Here is a graph that could represent a variety of different situations.

• Write an equation for the graph.
• Sketch a new graph of this relationship.

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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.

An Unknown Situation

Unit 3 ● Lesson 2 ● Activity 1

• How did you decide how to create the new graph of the relationship?
• Which graph looks steeper to you?

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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.

Card Sort: Proportional Relationships

Unit 3 ● Lesson 2 ● Activity 2

Your teacher will give you 12 graphs of proportional relationships.

• Sort the graphs into groups based on what proportional relationship they represent.
• Write an equation for each different proportional relationship you find.

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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.

Card Sort: Proportional Relationships

Unit 3 ● Lesson 2 ● Activity 2

• What strategies did you use to match the graphs? How were you sure that the graphs matched?
• Take a look at Card A. Do you think this graph looks like y = x? Why or why not?
• What might cause a graph to “not look like” the equation? Explain your thinking.

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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.

Different Scales

Unit 3 ● Lesson 2 ● Activity 3

Two large water tanks are filling with water. Tank A is not filled at a constant rate, and the relationship between its volume of water and time is graphed on each set of axes. Tank B is filled at a constant rate of liters per minute. The relationship between its volume of water and time can be described by the equation v = t, where t is the time in minutes and v is the total volume in liters of water in the tank.

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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.

Different Scales

Unit 3 ● Lesson 2 ● Activity 3

• Sketch and label a graph of the relationship between the volume of water v and time t for Tank B on each of the axes.
• Answer the following questions and say which graph you used to find your answer.
• After 30 seconds, which tank has the most water?
• At approximately what times do both tanks have the same amount of water?
• At approximately what times do both tanks contain 1 liter of water? 20 liters?

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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.

Different Scales

Unit 3 ● Lesson 2 ● Activity 3

• What question can you answer using the second graph that you can’t with the first?
• Is the first graph deceptive in any way?
• Which scale do you prefer?

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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.

Graphs of Proportional Relationships

Unit 3 ● Lesson 2

• Draw a copy of the axis and give a signal when you have finished.
• What type of proportional relationship might have a “steep “line?
• Can you think of a reason we might want to graph this relationship with such a large vertical scale?”

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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 3 ● Lesson 2

• I can graph a proportional relationship from an equation.
• I can tell when two graphs are of the same proportional relationship even if the scales are different.

Learning

Targets

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Different Axes

Unit 3 ● Lesson 2 ● Activity 4

Which one of these relationships is different than the other three? Explain how you know.

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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.

constant of proportionality

Unit 3 ● Lesson 2

In a proportional relationship, the values for one quantity are each multiplied by the same number to get the values for the other quantity. This number is called the constant of proportionality.

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In this example, the constant of proportionality is 3, because 2 • 3 = 6, 3 • 3 = 9, and 5 • 3 = 15. This means that there are 3 apples for every 1 orange in the fruit salad.

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Glossary

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

This slide deck is copyright 2020 by Kendall Hunt Publishing, https://im.kendallhunt.com/, and is licensed under the Creative Commons Attribution-NonCommercial 4.0 International License (CC BY-NC 4.0), https://creativecommons.org/licenses/by-nc/4.0/.

All curriculum excerpts are under the following licenses:

IM 6–8 Math was originally developed by Open Up Resources and authored by Illustrative Mathematics, and is copyright 2017-2019 by Open Up Resources. It is licensed under the Creative Commons Attribution 4.0 International License (CC BY 4.0). OUR's 6–8 Math Curriculum is available at https://openupresources.org/math-curriculum/.

Adaptations and updates to IM 6–8 Math are copyright 2019 by Illustrative Mathematics, and are licensed under the Creative Commons Attribution 4.0 International License (CC BY 4.0).

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Adaptations to add additional English language learner supports are copyright 2019 by Open Up Resources, and are licensed under the Creative Commons Attribution 4.0 International License (CC BY 4.0).

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The Illustrative Mathematics name and logo are not subject to the Creative Commons license and may not be used without the prior and express written consent of Illustrative Mathematics.

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