Lesson 11

This warm-up prompts students to think about how changing the graphing window influences what you can see and understand about an exponential function.

Invite students to share some suggestions for a graphing window that are more helpful and those that are less helpful. Ask them to explain their reasoning.

Help students understand that, as a general rule, the \(x\) values to show for a graph are usually determined by the quantity we are interested in studying. The \(y\) values need to be selected carefully so that:

• Data of interest shows up on the graph.
• Interesting trends (of increase or decrease) are as visible as possible.

If time permits, discuss:

• “Why does the graph show such a sharp decrease between \(x\) values of 0 and 2 and then start to flatten out?” (The decay factor is 0.2, which means that whenever \(x\) increases by 1, it loses 0.8 of its quantity and keeps only 0.2. That decay is more apparent when the quantity is larger. As it gets smaller, and given the scale of the graph, it is harder to see the change. For example, when \(x\) increases from 0 to 1, \(f(x)\) decreases by 320, because \((0.8) \boldcdot 400=320\). But when \(x\) increases from 2 to 3, \(f(x)\) decreases by \((0.8) \boldcdot 16\) or 10.8, which is a much smaller drop.)

In this activity, students examine the successive heights that a tennis ball reaches after several bounces on a hard surface and 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. Monitor for students who try the following approaches in deciding whether a linear or exponential model is more appropriate for modeling the data:

• Use the table of values and look at successive differences.
• Use the table of values and look at successive quotients.
• Plot the points and observe the general trend in bounce height.
• Plot the points using graphing technology and use the technology to generate a line or curve of best fit.

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 the way students choose to deal with this variation, which affects the model they consider appropriate. They may:

• 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 \(h\) is about half the preceding bounce height: \(h = 150 \boldcdot \left( \frac{1}{2} \right)^n\).
• Find and use the growth factor from the first two points: \(h = 150 \boldcdot \left( \frac{8}{15} \right)^n\).
• Take an average of the successive quotients: \(h = 150 \boldcdot (0.53)^n\).
• Use graphing technology to generate a regression equation: \(h = 150.389 \boldcdot (0.527)^n\).

Select students who use these strategies to share during the discussion. Encourage those who do not think an exponential model is appropriate to look for an exponential model that best fits the given data.

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

Arrange students in groups of 2–4.

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.

Ask students to share how they decided whether a linear or exponential function is more appropriate for modeling the data. Once the class concludes that a linear model is less suitable, select previously identified students to share how they determined a growth factor to use for their models and the resulting equations. Sequence the strategies being presented as shown in the Activity Narrative.

Connect the discussion about growth factors to the data and what the factors tell us about whether the tennis ball satisfies the bounce regulations. Ask questions such as:

• “Do most of the data support the conclusion?”
• “How can we explain the third bounce?” (E.g., it is too low, but there could have been a mis-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?”

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 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).

Arrange students in groups of 3 or 4. Give each group a measuring tape and a ball. Explain to students that their job is to determine the rebound factors of several balls by gathering data on their rebound heights. They then need to use mathematics to model the relationship between the number of bounces and the height of a bounced ball.

Students should drop them all from the same height. Consider letting students realize this on their own. 

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 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 means 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 \(f\) given by \(f(n) = a \boldcdot b^n\), the number \(b\) expresses the growth factor. In situations where \(0<b<1\), a larger value of \(b\) 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, i.e.,it does not make sense to examine \(\frac{2}{3}\) 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.)
• In the final question, why are both the initial height and the bounciness important? (The value of \(h\) is found by multiplying the initial height by the rebound factor \(n\) times, so both values matter.)