## Exponential Models - Section 2: Modeling f(x) = b^x

# Exponential Models Day 1 of 2

Lesson 3 of 15

## Objective: Students will be able to write exponential functions to model "real life" scenarios.

## Big Idea: Aarrghhh...Brains...Zombies...and Exponential Functions! Discover Exponential Functions in this Zombie themed Lesson.

*50 minutes*

I include **Warm ups** with a **Rubric** as part of my daily routine. My goal is to allow students to work on **Math Practice 3** each day. Grouping students into homogeneous pairs provides an opportunity for appropriately differentiated math conversations. The Video Narrative explains this lesson’s Warm Up- Exponential Models which asks students to evaluate an exponential function.

I also use this time to correct and record the previous day's Homework.

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#### Modeling f(x) = b^x

*40 min*

This is the first of a two-day lesson. The goal of this lesson is to allow students to build the model f(x) = a(b)^{x} using the scenario of a zombie apocalypse. The lesson begins with this scenario: *They are here... Each night every zombie will infect a new person*. I ask the students to estimate the number of nights it would take infect the whole room. We then do a physical demonstration of this idea. I am the first zombie and “infect” one student who stands up in their seat. This is charted on the PowerPoint slide and on the students’ papers. We model each night by having each “zombie” infect someone else. This activity gives the students a physical model that they sets a strong foundation for the remainder of the lesson.

The next step is to write a function to model this scenario (**Math Practice 4**). The amount of scaffolding here will depend on the students. One possible step could be to point out the repeated reasoning (**Math Practice 8**). For example, on the third night there are 1^{.}2^{.}2^{.}2 and then extend it to 1^{.}2^{3}. I usually walk around the room and call on someone who I saw had the correct solution. If there are multiple models, I call on several students. Once the model is on the board, I either point out or have the students identify how each part of the function is represented in the scenario.

The students then use the model that they created to figure out some questions about the scenario. I prefer to allow the students to try problems like these one at a time, discussing as we go. I highlight any different strategies that students use to find the solutions. The final question is not as easy to them as the first two. They will have to extend their table. This may cause a mild amount of struggle, which is good (**Math Practice 1**). It can also lead to some great discussions (**Math Practice 3). **

The scenario then changes to each zombie infecting two people and then three each night. Depending on the class, we may do a whole group physical demonstration again or I pass out baggies of tokens so they still have a physical model available if they need it. The goal is that the students notice a pattern and come up with each model .

My hope is to get the students beyond the need of the chart to find the functions (**Math Practice 5**). Some students will already be there but some may need to continue using a chart for a while. I encourage but not force a student to write the pattern without the chart.

If there is time, we continue onto the next portion of the lesson which asks students to extend to the model f(x) = a(b)^{x}. I have included this into the PowerPoint just in case.

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#### Exit Ticket

*5 min*

I use an exit ticket each day as a quick formative assessment to judge the success of the lesson.

The Exit Ticket asks the students to describe the scenario given a simple exponential model.

#### Resources

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*Responding to Deborah Edwards*Thanks for the feedback! I like this lesson as well. | 7 months ago | Reply

*Thank you Ms. Jamison. I have used this scenario for three years now after finding your lesson on this site. It grabs their attention like nothing else. As we go through the whole unit, they always go back to the zombie story and apply it to another situation for the initial value, growth or decay rate and exponent. Deb Edwards | 7 months ago | Reply*

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- UNIT 1: Modeling with Expressions and Equations
- UNIT 2: Modeling with Functions
- UNIT 3: Polynomials
- UNIT 4: Complex Numbers and Quadratic Equations
- UNIT 5: Radical Functions and Equations
- UNIT 6: Polynomial Functions
- UNIT 7: Rational Functions
- UNIT 8: Exponential and Logarithmic Functions
- UNIT 9: Trigonometric Functions
- UNIT 10: Modeling Data with Statistics and Probability
- UNIT 11: Semester 1 Review
- UNIT 12: Semester 2 Review

- LESSON 1: Rational Exponents
- LESSON 2: Real Number Exponents
- LESSON 3: Exponential Models Day 1 of 2
- LESSON 4: Exponential Models Day 2 of 2
- LESSON 5: Exponential Functions
- LESSON 6: Exponential Decay Functions
- LESSON 7: Simplifying Logarithms
- LESSON 8: Exponential and Logarithmic Equations
- LESSON 9: Logarithmic Functions
- LESSON 10: Exponential Growth and Interest Day 1 of 2
- LESSON 11: Exponential Growth and Interest Day 2 of 2
- LESSON 12: Natural Logarithms
- LESSON 13: Exponential and Logarithmic Functions Review Day 1
- LESSON 14: Exponential and Logarithmic Functions Review Day 2
- LESSON 15: Exponential and Logarithmic Functions Test