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# Comparing and Contrasting Linear and Exponential Functions

Lesson 4 of 10

## Objective: SWBAT compare and contrast the fit of Linear and Exponential functions to actual data. SWBAT paraphrase complex arguments.

## Big Idea: Students compare linear and exponential functions, and, learn about actual and theoretical data.

*90 minutes*

#### Entry Ticket

*15 min*

For this **Entry Ticket** I have students work on the worksheet from the **Mathematics Vision Project** called Linear and Exponential Functions 4.6 Ready Set Go! (pages 28 and 29 of the Module 4 packet included as a resource in this section).

The intent of the entry ticket is to get students to activate their prior knowledge around calculating the rate of change for different functions and is a great entry point to day's lesson on comparing and contrasting linear and exponential functions. This entry ticket gets students to struggle and persevere as they run into situations modeled by linear and exponential functions (**MP.1)**

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Students then work in groups on the Growing, Growing, Gone activity from the **Mathematics Vision Project Module 4** on Linear and Exponential Functions. The activity is attached as a pdf in this section (Collab Work 1 MVP p27_29.pdf). The module is also posted as a resource in the Exit Ticket section.

For the first 20-minute section of this activity, students focus on creating two models of population growth: a linear model and an exponential model.

If students are struggling I help them with some cues. For example, I might start with the question, "what year are we starting with and how do we use the starting population in linear (y-intercept) and exponential (constant or a) functions?" Another useful cue would be something to the effect of, "What is the rate of change for the two data points we have? Where does the rate of change show up in linear and in exponential functions?"

This activity is a great example of what Dan Meyer talks about in his popular TED talk - here we are peeling back a lot of the typical layers of support and are encouraging students to persevere in problem solving. I really like this activity as an example of both **MP.1 **for the perseverance piece and **MP.4** for the modeling piece.

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I like to change it up a bit at this point in the lesson and show a short video that introduces the idea of linear regression. The reason for showing this video is I want to introduce students to the idea of fitting data to a function as a tool to assess the goodness of fit of a function.

This video might be too much for all classes, but I have used it in my honors section and they followed along just fine as a whole.

After showing the video, I have a class discussion on the connection of this video to the task at hand. Students then are cued to shift back to group work and work on comparing the linear and exponential models in terms of telling the story of world population growth.

As an alternative, teachers can always have a short introduction to the terms **predicted value **and **actual value** as an entry into the task of comparing the "fit" of the linear and exponential models they created to model population growth.

I conclude this section by having a whole class share, where each group shares at least 1 or 2 important discussion points that their group engaged in.

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The next step in this lesson is to have students attend to precision (**MP.6**) by comparing the goodness of fit for the linear and exponential models they created to model world population growth. Once again, I like to allow students to think about what would be helpful strategies to analyze this problem. To guide this discussion I ask students questions like these:

- What kind of information do we need (actual and predicted values)?
- How can we organize this information (table or graph)?

I could simply provide a blank graph or table for students to fill in. I prefer to allow students the opportunity to think about how to approach a problem like this one. That being said, there are legitimate reasons to provide some scaffolding, and one possible **Table: Growing, Growing, Gone** is attached as a resource in this section. For example, I would provide a blank table for student with fine motor difficulties because the time and effort to copy a table is not worth the benefit - in this case I would wait to provide the table until after the class discussion on what we need to solve the problem. Then, I am accommodating for a lower level perceptual skill (fine motor) as a means to better access the higher level concepts (comparing the fit of different models on a set of data) for this activity.

#### Resources

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#### Exit Ticket and Homework

*15 min*

To recap the lesson, I have students complete a **Think-Pair-Share** on the following prompt:

**Which model (linear or exponential) better predicts U.S. population growth between 1910 and 2030? Explain your reasoning. **

This prompt is similar to the prompt on the worksheet, but asks students to compare the models over time rather than focusing on one data point.

After the recap I give students the last few minutes of class to work on the homework - creating a written response to the Think-Pair-Share prompt. Students can complete the **Exit Ticket: Comparing and Contrasting Linear and Exponential Functions **as a graphic organizer to initiate and support the writing process, but for this assignment I am looking for a polished, written response for homework today.

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Material in this lesson was used from the Mathematics Vision Project (Creative Common License):

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- UNIT 1: Thinking Like a Mathematician: Modeling with Functions
- UNIT 2: Its Not Always a Straight Answer: Linear Equations and Inequalities in 1 Variable
- UNIT 3: Everything is Relative: Linear Functions
- UNIT 4: Making Informed Decisions with Systems of Equations
- UNIT 5: Exponential Functions
- UNIT 6: Operations on Polynomials
- UNIT 7: Interpret and Build Quadratic Functions and Equations
- UNIT 8: Our City Statistics: Who We Are and Where We are Going

- LESSON 1: Rewriting Radical and Rational Exponents (Plus Exponents Review)
- LESSON 2: Creating and Interpreting Exponential Functions
- LESSON 3: Constructing Linear and Exponential Functions
- LESSON 4: Comparing and Contrasting Linear and Exponential Functions
- LESSON 5: Pizza, Hot Chocolate and Newton's Law of Cooling: Adding Constants to Exponential Functions
- LESSON 6: The Luckiest Man in the World: Graphing Exponential and Linear Functions
- LESSON 7: Formative Assessment: Modeling Population Growth (A Math Assessment Project Classroom Challenge)
- LESSON 8: Marketing Exponential Functions: A Group Performance Assessment Task
- LESSON 9: Review Lesson on Exponential Functions
- LESSON 10: Writing in Math Classroom, Part 3: Comparing and Contrasting Arithmetic and Geometric Sequences