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# Modeling Exponentials Using Logarithms

Lesson 8 of 11

## Objective: SWBAT use natural logarithms to develop exponential models and make predictions using exponential models.

*50 minutes*

Using pages 2-7 of the flipchart I am going to talk with students about the semi-log graph and how we can use this linear function to find the equation of the exponential function it was derived from. Using pages 5-6 I am going to show students the general form to find the equation of the exponential function. I will leave this slide up as students work on their activity.

This topic is a bit complex and really not in my curriculum. However, I think it is interesting and useful. So this is a lesson that I may cut if I find I am pressured for time during this unit. Due to the complexity of the task and my number one enemy (time!) I may also give a bit more guidance through the beginning of the worksheet. I don’t want my students getting bogged down with figuring out the calculator and not have time to explore the mathematics…

The most important part of this lesson that students should walk away with is the ability to make predictions using exponential models.

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#### Modeling: US Population

*35 min*

Students should now work in their teams on the US population worksheet. In some classes, I may model how to do the first section, Colonial Days and Early America. If I do, I will lead the class through parts *a-h*. I will then give the class a few minutes to complete the 3 application questions before reviewing them as a class. Students should complete the remainder of the handout in their teams.

As students complete today’s activity many mathematical practices are present. Students will surely be **making sense of the problem and persevering in solving it (MP1). **Students will need to be able to make predictions using their exponential models. They will **reason abstractly and quantitatively (MP2) **in order to do this. Students should also be **attending to precision (MP6) **as they answer questions. And most importantly, students should be improving on their ability to **model with mathematics (MP4).**

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#### Closure: 3 W’s

*5 min*

At the conclusion of the lesson, I will present page 8 of the flipchart and have students do a 3 W’s Closure. On a sheet of notebook paper, students should respond briefly to the three questions:

*What did you learn today? *

*Why do you think we learned this? *

*Where do you think you could this in real life?*

Each student will turn in his/her response before they leave. I am really curious to see how students respond to the third question. If students are able to accurately apply this to something in their lives or their imagined future, then I will be excited about the class's understanding of using logarithms to develop exponential models.

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- UNIT 1: Basic Functions and Equations
- UNIT 2: Polynomial Functions and Equations
- UNIT 3: Rational Functions and Equations
- UNIT 4: Exponential Functions and Equations
- UNIT 5: Logarithmic Functions and Equations
- UNIT 6: Conic Sections
- UNIT 7: Rotations and Cyclical Functions
- UNIT 8: Cyclical Patterns and Periodic Functions
- UNIT 9: Trigonometric Equations
- UNIT 10: Matrices
- UNIT 11: Review
- UNIT 12: Fundamentals of Trigonometry

- LESSON 1: What Is A Log?
- LESSON 2: Graphing and Shifting Logs
- LESSON 3: Discovering Log Rules (because logs rule!)
- LESSON 4: Applying Log Rules
- LESSON 5: Puzzling Log Equations
- LESSON 6: Solving Exponential Equations Using Logs
- LESSON 7: Experts on Exponential Equations
- LESSON 8: Modeling Exponentials Using Logarithms
- LESSON 9: Speed Dating with Logarithms
- LESSON 10: Logarithmic Equations Test Review
- LESSON 11: Logarithmic Equations Unit Test