## Exit Ticket 4.docx - Section 3: Closing

# Bunnies and Exponential Equations

Lesson 4 of 26

## Objective: Students will write and solve exponential equations to find missing information in problems involving exponential growth.

#### Closing

*10 min*

The exit ticket questions are based on different students’ questions and comments about setting up these functions. It turns out that these functions provide a particularly rich context in which students can make meaning of negative exponents. The first question is designed to make this learning explicit: at this point, even if students have no idea about what negative exponents mean, they will likely be able to say, or at least understand, that the negative exponent means “2 months ago” and in order to work backwards they need to divide by the multiplier. This is basically enough to define the negative exponent.

The second question relates to problems in which the current amount of bunnies is given and students need to work backwards to find the “original number” of bunnies. Obviously this question is not totally clearly defined, but it brings forth the idea that the function can be written with different starting values. In order to change the starting value but keep the function the same, the exponent needs to change as well. This question is the beginning of a deeper conversation the horizontal shift and the exponent laws, which some students will engage in in future lessons.

The third question is another attempt to get students thinking about the meaning of the negative exponents. If the internet is readily available to students in your classroom, you could easily give them the opportunity to graph the functions on desmos.com/calculator so that they could see for themselves that these functions are the same. Obviously this is just the beginning of the conversation, because then the question is *why* are these functions the same.

To facilitate the closing, I state the questions clearly so that all students understand exactly what I am asking. Before I ask them to write about the questions, I explain a little bit about the context of each question and refer to specific students who brought up this question during the lesson. For instance, I say, “Jonai asked a brilliant question about the starting values and the exponents so I want you to think about the two functions in the second question to see if we can answer her question.”

Then, I ask them to write on their own. After giving them some individual think time, I ask some students to share their answers and I highlight the key ideas myself (taking notes on the projector.) Then I give them more time to add to their ideas before turning in their exit ticket.

#### Resources

*expand content*

- UNIT 1: Linear and Nonlinear Functions
- UNIT 2: Piecewise Functions
- UNIT 3: Absolute Value Functions and More Piecewise Functions
- UNIT 4: Introduction to Quadratic Functions through Applications
- UNIT 5: More Abstract Work with Quadratic Functions
- UNIT 6: Rational Functions
- UNIT 7: Polynomial Functions
- UNIT 8: Exponential Functions
- UNIT 9: Ferris Wheels
- UNIT 10: Circles
- UNIT 11: Radical Functions
- UNIT 12: Cubic Functions

- LESSON 1: Bunnies and Exponential Growth
- LESSON 2: More Bunnies and Exponential Growth
- LESSON 3: Candy Bars and Exponential Decay
- LESSON 4: Bunnies and Exponential Equations
- LESSON 5: Graphing Bunnies
- LESSON 6: Exponential Data Tables
- LESSON 7: Fitting Exponential Functions Given Two Points
- LESSON 8: Matching Exponential Graphs to Equations
- LESSON 9: Exponential Functions Review
- LESSON 10: Exponential Functions Portfolio and Summative Assessment
- LESSON 11: Exponential Functions and Approach Statements
- LESSON 12: Graphing Exponential Functions
- LESSON 13: Matching Graphs of Exponential Functions to their Equations
- LESSON 14: Exponential Function Designs
- LESSON 15: Graph Exponential Functions Review
- LESSON 16: Graph Exponential Functions Summative Assessment and Portfolio
- LESSON 17: Bouncy Ball Investigation
- LESSON 18: Percent Change: Growth and Decay
- LESSON 19: More Percent Changes and Exponential Functions
- LESSON 20: Writing Exponential Functions to Solve Problems
- LESSON 21: Different Time Intervals and Exponential Functions
- LESSON 22: Compound Interest
- LESSON 23: Compound Interest Formula
- LESSON 24: Continuously Compounded Interest
- LESSON 25: Applications of Exponential Functions Review
- LESSON 26: Applications of Exponential Functions Summative Assessment