Lesson 11

Modeling Exponential Behavior

11.1: Wondering about Windows (5 minutes)

Warm-up

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

Student Facing

Here is a graph of a function \(f\) defined by \(f(x)=400 \boldcdot (0.2)^x\).

Graph of a function on grid, origin O. Horizontal axis, x, from 0 to 40 by 10's. Vertical axis, y, from 0 to 1,000, by 500's. Line crosses  y axis at 0 comma 400, quickly decreasing then running horizontal to x axis.
  1. Identify the approximate graphing window shown.
  2. Suggest a new graphing window that would:

    1. make the graph more informative or meaningful
    2. make the graph less informative or meaningful
    Be prepared to explain your reasoning.

Student Response

Student responses to this activity are available at one of our IM Certified Partners

Activity Synthesis

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.)

11.2: Beholding Bounces (15 minutes)

Activity

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).

Launch

Arrange students in groups of 2–4.

Representing, Conversing: MLR7 Compare and Connect. Use this routine to prepare students for the whole-class discussion. At the appropriate time, ask students to prepare a visual display that shows their mathematical thinking and reasoning for the first question. Invite students to quietly circulate and read at least 2 other posters or visual displays in the room. Give students quiet think time to consider what is the same and what is different about the displays. Next, ask students to return to their group to discuss what they noticed. Listen for and amplify observations for choosing an exponential or linear model. This will help students interpret real-world data and provide justification for choosing an exponential model.
Design Principle(s): Cultivate conversation
Representation: Internalize Comprehension. Begin with a demonstration or video of bouncing a ball to clarify what bounce height means. Highlight connections between representations by using color or annotations to connect the numbers in the table to the symbols in the equation \(y = 1000\boldcdot\frac12^x\).
Supports accessibility for: Conceptual processing; Visual-spatial processing

Student Facing

Here are measurements for the maximum height of a tennis ball after bouncing several times on a concrete surface.

\(n\), bounce number \(h\), height (centimeters)
0 150
1 80
2 43
3 20
4 11
  1. Which is more appropriate for modeling the maximum height \(h\), in centimeters, of the tennis ball after \(n\) bounces: a linear function or an exponential function? Use data from the table to support your answer.
  2. Regulations say that a tennis ball, dropped on concrete, should rebound to a height between 53% and 58% of the height from which it is dropped. Does the tennis ball here meet this requirement? Explain your reasoning.
  3. Write an equation that models the bounce height \(h\) after \(n\) bounces for this tennis ball.
  4. About how many bounces will it take before the rebound height of the tennis ball is less than 1 centimeter? Explain your reasoning.

Student Response

Student responses to this activity are available at one of our IM Certified Partners

Anticipated Misconceptions

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.

Activity Synthesis

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?”

11.3: Which is the Bounciest of All? (40 minutes)

Optional activity

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).

Launch

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. 

Engagement: Provide Access by Recruiting Interest. Leverage choice around perceived challenge. Invite students to select one or two balls to bounce, instead of 3. Chunking this task into more manageable parts may also benefit students who benefit from additional processing time.
Supports accessibility for: Organization; Attention; Social-emotional skills

Student Facing

Your teacher will give your group three different kinds of balls.

Your goal is to measure the rebound heights, model the relationship between the number of bounces and the heights, and compare the bounciness of the balls.

Group of different types of sport balls.
  1. Complete the table. Make sure to note which ball goes with which column.
    n, number of bounces  a, height for ball 1 (cm) b, height for ball 2 (cm) c, height for ball 3 (cm)
    0      
    1      
    2      
    3      
    4      
  2. Which one appears to be the bounciest? Which one appears to be the least bouncy? Explain your reasoning.
  3. For each one, write an equation expressing the bounce height in terms of the bounce number \(n\).
  4. Explain how the equations could tell us which one is the most bouncy.
  5. If the bounciest one were dropped from a height of 300 cm, what equation would model its bounce height \(h\)?

Student Response

Student responses to this activity are available at one of our IM Certified Partners

Student Facing

Are you ready for more?

  1. If Ball 1 were dropped from a point that is twice as high, would its bounciness be greater, less, or the same? Explain your reasoning.
  2. Ball 4 is half as bouncy as the least bouncy ball. What equation would describe its height \(h\) in terms of the number of bounces \(n\)?
  3. Ball 5 was dropped from a height of 150 centimeters. It bounced up very slightly once or twice and then began rolling. How would you describe its rebound factor? Explain your reasoning.

Student Response

Student responses to this activity are available at one of our IM Certified Partners

Anticipated Misconceptions

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.

Activity Synthesis

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.

Writing, Conversing: MLR1 Stronger and Clearer Each Time. Use this routine to help students improve their writing by providing them with multiple opportunities to clarify their explanations through conversation within their group. Give students time to share their response to the question, “Explain how the equations could tell us which one is the most bouncy.” Provide listeners with prompts for feedback that will help their partner add detail to strengthen and clarify their ideas. For example, students can ask their partner, “How did you use the data in the table?” or “Can you explain the rebound factor in your equations?” Next, provide students with 3–4 minutes to revise their initial draft based on feedback from their group. This will help students produce a written generalization for how they can estimate the rebound factor and use successive quotients in a real-world exponential situation.
Design Principle(s): Optimize output (for explanation)

11.4: Beholding More Bounces (15 minutes)

Activity

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.

Launch

Representation: Internalize Comprehension. Activate or supply background knowledge about discrete and continuous contexts. The number of bounces is counted every time the ball hits the floor. If a student asks about \(1\frac{1}{2}\) bounces, clarify that would be measuring the time since the ball had been released, not the number of bounces; a “bounce” is the event of hitting the floor and changing direction.
Supports accessibility for: Memory; Conceptual processing

Student Facing

The table shows some heights of a ball after a certain number of bounces.

bounce number height in centimeters
0  
1  
2 73.5
3 51.5
4 36
  1. Is this ball more or less bouncy than the tennis ball in the earlier task? Explain or show your reasoning.
  2. From what height was the ball dropped? Explain or show your reasoning.
  3. Write an equation that represents the bounce height of the ball, \(h\), in centimeters after \(n\) bounces.
  4. Which graph would more appropriately represent the equation for \(h\): Graph A or Graph B? Explain your reasoning.

    A

    Graph of a function on grid, origin O. Horizontal axis, bounce number, Vertical axis, height in centimeters. Solid line crosses vertical axis and moves downwards and to the right. 
 

    B

    Graph of a function on grid, origin O. Horizontal axis, bounce number, Vertical axis, height in centimeters. Dotted line crosses vertical axis and moves downwards and to the right.
 
  5. Will the \(n\)-th bounce of this ball be lower than the \(n\)-th bounce of the tennis ball? Explain your reasoning.

Student Response

Student responses to this activity are available at one of our IM Certified Partners

Anticipated Misconceptions

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.

Activity Synthesis

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.)
Conversing: MLR8 Discussion Supports. Use this routine to help students reflect on their reasoning process in deciding on the bounciness of the balls. Before the whole-class discussion, arrange students in groups of 2. Provide the following sentence frames as partners share their ideas: "In the table, I noticed that….", "I interpreted the successive quotients to mean…." or "The equation helped determine the bounciness by...." This will help students make sense of exponential decay through discussing the ball-bouncing situation with a partner.
Design Principle(s): Support sense-making; Cultivate conversation

Lesson Synthesis

Lesson Synthesis

In this lesson, we explored models to represent successive bounce heights of different kinds of balls. The data we used were rather messy. Invite students to reflect on the process of modeling with such data.

  • “Was the data completely accurate? Why or why not?” (Depending on the tools used, it can be very hard to get exact measurements of rebound heights, and it gets harder the lower the bounces are. Measurement errors are likely.)
  • “We saw different rebound factors for different pairs of successive data points. How do we decide if an exponential model is still appropriate?” (We might want to see if the differences are small enough to treat them as measurement error. In the lesson, they are all within about 0.05 of 0.5, and most of them are close to about 0.54.)
  • “Given the inconsistency, how do we find an appropriate factor to use for our model?” (We might disregard the factor(s) that are very different from the rest, or consider finding an average of the factors.)

If time permits, discuss possible limitations of our models. Ask questions such as:

  • “Do the models we produce work well after 45 bounces? After 78 bounces, or when the rebound height is less than 1 cm?”
  • “Can we rely on the models to be appropriate if we bounce the ball on a different hard surface?”
  • “What level of accuracy should we consider? For example, is 0.5 or 0.54 more appropriate for the rebound factor?”

11.5: Cool-down - Drop Height (5 minutes)

Cool-Down

Cool-downs for this lesson are available at one of our IM Certified Partners

Student Lesson Summary

Student Facing

Sometimes data suggest an exponential relationship. For example, this table shows the bounce heights of a certain ball. We can see that the height decreases with each bounce.

To find out what fraction of the height remains after each bounce, we can divide two consecutive values: \(\frac{61}{95}\) is about 0.642, \(\frac{39}{61}\) is about 0.639, and \(\frac{26}{39}\) is about 0.667.

All of these quotients are close to \(\frac{2}{3}\). This suggests that there is an exponential relationship between the number of bounces and the height of the bounce, and that the height is decreasing with a factor of about \(\frac23\) for each successive bounce.

bounce number bounce height
in centimeters
1 95
2 61
3 39
4 26

The height \(h\) of the ball, in cm, after \(n\) bounces can be modeled by the equation: 

\(\displaystyle h = 142 \boldcdot \left(\frac{2}{3}\right)^{n}\)

Here is a graph of the equation.

Graph of function and data on grid.

This graph shows both the points from the data and the points generated by the equation, which can give us new insights. For example, the height from which the ball was dropped is not given but can be determined. If \(\frac23\) of the initial height is about 95 centimeters, then that initial height is about 142.5 centimeters, because \(95 \div \frac23 = 142.5\). For a second example, we can see that it will take 7 bounces before the rebound height is less than 10 centimeters.