## Investigation Narrative.docx - Section 2: Investigation and New Learning

# Graphing Bunnies

Lesson 5 of 26

## Objective: Students will be able to interpret an exponential graph in the context of a "bunny problem" and write an exponential given a graph and two points on the function. Students will use negative exponents during this process and interpret a negative exponent as "division" in order to make meaning of this.

#### Warm-Up

*30 min*

Some students will immediately see how to interpret the graph and to find the equation using the same tools as the word problems. Other students will need some coaching about how to do this. The scaffold that seems to help students most is asking them to organize the information into a data table. Then I ask them, “What is the missing information here?” They then are able to see that they need to find the growth factor or multiplier. If the only method that they can come up with to do this is guess and check, that is totally fine. They will be motivated later on to figure out a new method, and many students will write their own equations and figure out how to solve those on their own.

When it comes to the data tables at the bottom of the page, it is very important that students develop a method that does not involve only guess and check. The idea is for them to develop shortcuts. I found that many of my students used their calculators to guess multipliers over and over again, but this shortchanges them from the learning associated with dealing with the fractions. The ideal strategy is to break down the numbers into their prime factors, which makes it easier to identify the multiplier.

Obviously by this point in the unit, all students have figured out that the data tables of exponential functions have constant multipliers, but this warm-up offers the chance to discuss this idea more abstractly. Also, when you ask students to describe their process, they often use the word *difference* when they mean *quotient*. It is a great opportunity to discuss the difference between these words, and it also is a chance to highlight the fact that these functions are not linear.

#### 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