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.
Lesson overview
 11.1 Warmup: Wondering about Windows (5 minutes)
 11.2 Activity: Beholding Bounces (15 minutes)

11.3 Optional Activity: Which is the Bounciest of All? (40 minutes)
 Includes "Are you Ready for More?" extension problem
 11.4 Activity: Beholding More Bounces (15 minutes)
 Lesson Synthesis
 11.5 Cooldown: Drop Height (5 minutes)
Learning goals:
 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.
Learning goals (student facing):
 Let’s use exponential functions to model real life situations.
Learning targets (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.
Required materials:
 A collection of balls that bounce
 Measuring tapes
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?).
IM Algebra 1, Geometry, Algebra 2 is copyright 2019 Illustrative Mathematics and licensed under the Creative Commons Attribution 4.0 International License (CC BY 4.0).
The Illustrative Mathematics name and logo are not subject to the Creative Commons license and may not be used without the prior and express written consent of Illustrative Mathematics.