##
* *Reflection: Developing a Conceptual Understanding
Exponential Models Day 2 of 2 - Section 2: Modeling f(x) = a(b^x)

This set of lessons was so fun to write and even more fun to teach. One of my biggest 'Aha' moments in aligning my curriculum to the Common Core has been how well students conceptually grasp topics when they are presented in context. This is a switch in many mathematics classrooms. Usually, students are taught a skill and then it is applied to a real-life context. By switching this, I have found that a wider range of students really understanding the principles behind mathematical topics. I remember hearing people say this early in my career and being very skeptical. The key to success is that the presentation of the scenario is more open ended than your classic two trains going in opposite directions word problem. That is also the challenge, to find worthy rich scenarios.

In today’s lesson, the students take the basic scenario, developed in the last lesson, and add an initial amount other than one. I did a short physical demonstration of the model f(x)=2(2)^{x} with my students, like yesterday, as this gave a physical representation of what happens in an exponential model. One student and I each ‘infected’ one person to represent the first night, the four of us infected one person to represent the second night, and so on. From this point we talked about them conceptually but the physical model really set the idea.

The one place that students struggled was the idea that f(x)=3^{x}, for example, represents one original zombie with 2 victims totaling 3 zombies by the night’s end (Student Notes). Often, students wouldn’t include the original zombie as part of the three. A reminder of the physical demonstration did a lot to overcome this issue.

*Developing a Conceptual Understanding: Building Transformations Through Context*

# 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