##
* *Reflection: Intervention and Extension
Exponential Models Day 2 of 2 - Section 4: Homework

The extension problem, #11, previews exponential decay. I have found that students struggle with exponential decay more than they do with exponential growth. This sample Student Assignment shows one common mistake. The first 20% off takes the zombies from 100 to 80. This student then takes continues to take 20 zombies off each night thus creating a linear pattern rather than an exponential one. The next day, when we went over the assignment, I presented this on the board along with the correct answer and asked the students to talk with their partner to determine the correct solution. We then discuss it as a class. This strategy is a great way to determine the correct solution to a problem while encouraging students to discuss mathematics.

*Intervention and Extension: Previewing Exponential Decay*

# Exponential Models Day 2 of 2

Lesson 4 of 15

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

#### Warm Up and Homework Review

*10 min*

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 resource video specifically explains this lesson’s Warm Up- Exponential Models Day 2 which asks students to find the inverse of an exponential function.

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

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

*37 min*

This is the second day of a two day lesson. I have included this lesson, which has some Algebra 1 content, as part of the Common Core transition. In the previous lesson, students used a zombie apocalypse scenario to model an exponential function. Today we introduce additional scenarios building the model beyond a=1 in f(x) =a(b^{x}). This may be a big conceptual jump for some students so I suggest if scaffolding is needed that you build slowly and help students recognize the similarities (with repeated reasoning) to the first model (**Math Practice 8**). I also direct students to consider the initial value in the original problem and make the connection to where this shows up in the function.

Through the lesson, several models come up including f(x) = 2(2)^{x} and f(x) = 2^{x+1}. I encourage the students to build each function from the original scenario which is then discussed as a class (**Math Practice 7**). Presenting and discussing several examples helps reinforce student understanding.

We finish the lesson by formalizing the model into f(x) =a(b^{x}). I ask the students to identify what the "a" and "b" mean given a contextual scenario. We also look at the repetitive nature of "b" in an exponential model. for example 3(2)^{4} means (3)(2)(2)(2)(2). Whenever looking at a generalized formula, I share my enthusiasm for algebra as a means to represent an amazing amount of data in a very small space. This model represents ALL exponential functions.

#### Resources

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#### Exit Ticket

*3 min*

The Exit Ticket asks the students to create a zombie scenario off of a function model. This will let you know that where your students truly understand the contextual significance of f(x) = a(b)^{x}.

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The assignment first asks students to describe some exponential scenarios given a function. The next portion asks students to graph three simple exponential models by making a table. Since our scenarios never dealt with negative exponents, I than ask the student to extend the graphs in this direction. This is all in preparation for the next lesson which will study the graphs of exponential functions. The final three problems ask students to write functions to model a scenario and use the function to answer some questions (**Math Practice 2**).

#### Resources

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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