# Lesson 11

Modeling Exponential Behavior

### Lesson Narrative

In this lesson, students study the successive bounce heights of balls, model the relationship between the number of bounces and the bounce heights, and use that model to answer questions about the ball’s bounciness. There are options for how much of the modeling cycle (MP4) students undertake. In one optional activity, students collect data for the bounce heights of different balls, while in two activities the data are provided. In all cases, the data are deliberately “messy” to mirror data students would gather through experimentation, and students are left to decide what kind of model to use. Though the data are not perfectly exponential or linear, an exponential model fits much better.

Once students decide to use an exponential model, they still have to find the parameters that best fit the data while maintaining a reasonable level of accuracy (MP6). Some may just use the quotient of the first two bounce heights, while others may look more closely at all of the bounce heights. As they compare and critique the different models, students construct viable arguments and critique the reasoning of others (MP3).

Technology isn't required for this lesson, but there are opportunities for students to choose to use appropriate technology to solve problems. We recommend making technology available.

### Learning Goals

Teacher Facing

• Choose an appropriate model for a situation when given data.
• Determine graphing windows that would make a graph more informative or meaningful.
• Use exponential functions to model situations that involve exponential growth or decay.

### Student Facing

Let’s use exponential functions to model real life situations.

### Required Preparation

Graphing technology should be available if students request it. Measuring tapes and balls that bounce are only needed if doing the optional activity (Which Is the Bounciest of All?).

### Student Facing

• I can use exponential functions to model situations that involve exponential growth or decay.
• When given data, I can determine an appropriate model for the situation described by the data.

### Glossary Entries

• exponential function

An exponential function is a function that has a constant growth factor. Another way to say this is that it grows by equal factors over equal intervals. For example, $$f(x)=2 \boldcdot 3^x$$ defines an exponential function. Any time $$x$$ increases by 1, $$f(x)$$ increases by a factor of 3.