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# Rates of Change

Lesson 9 of 11

## Objective: SWBAT interpret slope in real world situations and compare rates of change in various linear functions.

#### Launch

*15 min*

Today's entrance ticket, Bathtub Case, is designed to require my students to access prior knowledge and use it productively. Students must analyze the graph and relate it to the real life situation it depicts. We dealt with these graphs earlier in the course when analyzing the distance v. time functions.

The new wrinkle in this task is the curve between A and B, which is the only segment of the graph where the rate is not constant. Once students complete the task, I will call on a few students to share their stories. I make sure to ask the students the following questions if these are not covered within the stories:

- Does the story start off with an empty tub?
- What does the curved line mean?
- Is the graph at points near B and from E to F vertical? What might have caused it to be nearly vertical near B?
- Explain the difference in slopes between segments CD and GH with respect to rates of change.

#### Resources

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#### Activity

*30 min*

The main activity for today's lesson, Activity Worksheet, asks students to apply their understanding of slope in several different situations. As they work, students analyze different representations of functions and respond to questions about the rate of change described by a function.

It's often hard to get my students to use units of measure in their answers. Today, I will stress the need to use proper units and I tell students that including the units gives meaning to the slope of the graphed situation. I find that relating the units to slope (vertical change / horizontal change) helps students place the correct units in the numerator and denominator (**MP4**, **MP6**).

When students finish the worksheet, we will discuss the meaning of the slope calculation in each problem. And, we will review their answers to the rate of change questions. I expect that my students will be able to verbally express the connections between the graph, the changing slope, and the problem situation.

#### Resources

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#### Closure

*10 min*

The last page of the Activity Worksheet contains a graph that I will use as a closure activity for this lesson. I want each student to work individually to analyze the problem and answer the corresponding questions. Students need to figure which animal, a cat or a mouse is faster. Students should calculate the rates and determine that the cat is faster.

At this point, my students should be able to find both rates of change and answer this problem correctly. I don't expect students to make the mistake of thinking that the line further left, which reaches the top of the graph (20 feet) in less time, is the faster animal. But, I look out for this interpretation as something that I need to address if students are making this interpretation.

The y-intercept question provides an opportunity for us to discuss how the mouse had a head start, and how this shows up in graphs of functions.

#### Resources

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- UNIT 1: Number Sense
- UNIT 2: Solving Linear Equations
- UNIT 3: Relationships between Quantities/Reasoning with Equations
- UNIT 4: Powers and Exponents
- UNIT 5: Congruence and Similarity
- UNIT 6: Systems of Linear Equations
- UNIT 7: Functions
- UNIT 8: Advanced Equations and Functions
- UNIT 9: The Pythagorean Theorem
- UNIT 10: Volumes of Cylinders, Cones, and Spheres
- UNIT 11: Bivariate Data

- LESSON 1: Relations that Function (Part 1)
- LESSON 2: Relations that Function (Part 2)
- LESSON 3: Functions Rule (Part 1 of 2)
- LESSON 4: Functions Rule (Part 2 of 2)
- LESSON 5: Crickets Tell Temperature
- LESSON 6: Linear? Yey or Nay
- LESSON 7: Comparing Linear and Exponential Functions
- LESSON 8: Time-Distance Graphs
- LESSON 9: Rates of Change
- LESSON 10: Sequences as Functions (Part 1 of 2)
- LESSON 11: Sequences as Functions (part 2 of 2)