In this activity, students examine the successive heights that a tennis ball reaches after several bounces on a hard surface, and they consider how to model the relationship between the number of bounces and the height of the rebound. To do so, they need to determine the growth factor of successive bounce heights. Because some data is provided here, students engage in only some aspects of mathematical modeling. To engage students in the full modeling cycle that includes data gathering, consider asking students to measure the bounce heights of a ball, as suggested in the next optional activity.

Real-world data is often messy, and that is the case for the data provided here. While each successive bounce height is about half of the preceding height, there is variation in the data, with the largest factor being a little more than 0.55 and the smallest a little less than 0.47.

Monitor for students who:

• Decide that an exponential model is not appropriate because the growth factor is different from bounce to bounce.
• Make a rough approximation for the growth factor, for example, observe that each bounce height, , is about half the preceding bounce height: .
• Find and use the growth factor from the first two points: .
• Take an average of the successive quotients: .
• Use graphing technology to generate a regression equation: .

Have students present in this order to support more precise methods for finding a model function to fit data.

Making graphing technology available gives students an opportunity to choose appropriate tools strategically (MP5).

Students may not be comfortable with the data not fitting an exponential function exactly. Remind them that real-world data is messy, so, when modeling, we must do our best to approximate the data. If an exponential model does a good job at approximating the data and showing its general trend, then this is a reasonable model to use even though it does not accurately predict or match all of the data.

Invite previously selected groups to share how they decided whether a linear or exponential model is more appropriate and how they found a model for the data. Sequence the discussion of the strategies by the order listed in the Activity Narrative. If possible, record and display their work for all to see.

Connect the different responses to the learning goals by asking questions such as:

• “Which methods for finding a model are more precise?” (Doing calculations is more precise than estimating, and using technology to fit the data is even more accurate than that.)
• “Use the different models to find the predicted height of the third bounce. How does it compare to the data?”
• “Do most of the data support the conclusion?”
• “How can we explain the third bounce?” (It is lower than predicted, but there could have been an error in measurement, or the tennis ball could have hit something on the floor.)

To follow up on the last question, consider discussing the practical domain in this context. Ask, for instance: “How long could we expect this behavior to continue? Can it go on indefinitely? What is a reasonable domain for our model?” (Due to additional factors from physics like friction, the model will not fit for too many bounces. Probably after 4 or 5 bounces the model will get too far away from real-world data to be useful.)

This activity, designed for an extra class period, gives students an opportunity to gather and analyze data for bounce heights for multiple balls. Each ball should be sufficiently bouncy to allow measurement of at least 4 bounces. Good examples include tennis balls, basketballs, super balls, golf balls, and soccer balls. It will also be important to find a surface that is hard, flat, and level. Any padding will dampen the bounces, and any slant or irregularity on the surface will affect the direction of the bounce. A tiled or concrete floor, or a flat and paved surface outdoors should work. Students will need measuring tapes and may need some practice gathering the data.

Notice how students record the bounce heights. Recording these heights to the nearest inch or centimeter will already be challenging, and anything beyond that is too much precision. This activity is a good opportunity to choose a degree of precision appropriate to the context and to the measuring device used (MP6).

As in the previous task, monitor for how students process their data and decide on an appropriate factor to quantify the bounciness of each ball. Students should now be comfortable with the fact that the data is not exactly exponential but may still choose different ways for deciding on an appropriate exponential decay factor.

Here is a typical rebound factor for several types of balls:

• Tennis ball: 0.5 to 0.6
• Super ball: 0.9
• Basketball: about 0.6

Making graphing or spreadsheet technology available gives students an opportunity to choose appropriate tools strategically (MP5).

Students may struggle to measure the heights of the bounces. Consider allowing phones or other technology that can record a video of the bounces so that it can be replayed in slow motion.

Depending on available time, you may choose to have groups of students prepare a presentation for sharing their findings or simply discuss the data and findings as a whole class.

As in the previous task, highlight different methods for estimating the rebound factor (taking the quotient of two successive values, taking an average of successive quotients, or making a general estimate of successive quotients). Also highlight the inherent inaccuracy of bounce height measurements, which in turn influence how accurately we should report the successive quotients (MP6). Probably no more than one significant decimal digit should be used.

Focus the discussion on the meaning of the rebound factors. Ask questions such as:

• “Does a larger factor mean that the ball is more bouncy or less bouncy?”
• “Does a larger factor mean that the heights are decreasing more quickly or more slowly?”

Emphasize the fact that in a situation modeled by a function, , given by , the number expresses the growth factor. In situations where \(0, a larger value of means that the function decays more slowly (because more of the quantity remains). In terms of comparing the bounciness of the balls, the larger the successive quotients in the table columns, the bouncier the ball is. While all of the balls eventually will lie still on the ground, a larger successive quotient (or rebound factor) means that the bounce heights decrease more slowly.

This activity continues to examine exponential decay in the context of successive ball bounces. Students use the given data to calculate a rebound factor and use it to write a function that models the relationship between number of bounces and bounce heights. They then use the function to answer questions about the ball and its bounces.

Students also think about the domain of the function and address the fact that this is a discrete context. It does not make sense to examine of a bounce, so the graphs should not be continuous.

If students struggle to make sense of the graph, remind them that the horizontal axis depicts the number of bounces, not the height of the ball or time. Although height and time are continuous, the number of bounces is discrete.

Invite students to share how they decided on the bounciness of the balls and to reflect on their reasoning process:

• "How are the data here different from the data in the first activity?" (In this case, the successive quotients are very close to the same value, making the choice of rebound factor more straightforward.)
• "When working with the table, how is calculating the missing value in the row above a given value different from calculating a missing value in the row below?" (To find the missing value in the row above we need to divide by the rebound factor rather than multiply by it.)
• "In the final question, why are both the initial height and the bounciness important?" (The value of is found by multiplying the initial height by the rebound factor times, so both values matter.)