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On the Rebound (Statistics)
Objective:  Students will write the equation of an exponential function using data from an experimental situation.
Prior Knowledge:  Students should be able to transform exponential data in order to write a linear regression, and then be able to transform the linear regression to fit exponential data. This experiment is an example of exponential decay. The ideal equation to describe this situation is:
y = abx
a is the starting height of the ball
b is percent of the previous height the ball reaches on each bounce
x is the number of bounces
Materials needed:
    • a ball that bounces fairly well (tennis ball, golf ball, or super ball)
    • Meter stick
    • TI-83 (or later) graphing calculator 
Group size: 3 people
Time required: one 50-minute class period
1. Tape a meter stick to a wall so that you can measure the height of your ball at the top of successive bounces.
2. Hold the ball less than a meter above the ground, close to the meter stick and record the starting height (at the bottom of the ball).
3. Drop the ball and record the height of the bottom of the ball on the rebound of at least 5 bounces. Record your data in a table like the one below. Do at least 3 trials, starting from the same height. This is probably going to take several trials to get 3 that work!
Number of Bounces
Height of Ball
Trial 1
Height of Ball
Trial 2
Height of Ball
Trial 3
Average Height
(Start Height)
(Start Height)
(Start Height)
(Start Height)
4. Enter the Number of Bounces in List1 of your calculator and the Average Height of the ball after each bounce in List2¬.
5. Test to see if this an exponential equation by dividing each of your successive y values by the previous y value. 
Do you get approximately the same ratio? 
If so, what is it?
Does this ratio show exponential growth or decay?
6. Now make a scatter plot of your data in your calculator. 
Does the graph appear exponential? 
How do you know?
We now need to find the equation of the function that fits this data the best.
7. Since we believe this to be an exponential regression, take the log of each of the Average Heights and enter them into List3.
8. Make a scatter plot of log (Average Height) against the Number of Bounces. Are the points fairly linear?

9. Find the linear regression between Number of Bounces and log (Average Height).
 Linear regression: log   = _________
 Correlation: _________
10. Graph your linear regression (in Y1) on the scatter plot of the transformed data from #8. How well does it fit the data?

11. Construct a residual plot for log. To do this, find the difference between the actual values of log(Average Height) and the predicted values from your regression equation. Enter these values into List4. The command to have your calculator do this for you is: L4 = L3 – Y1(L1). Now make a scatter plot of L1 and L4. What does this  residual plot show?
12. The inverse operation of the logarithmic function is raising 10 to a power. To transform the regression equation to fit your data, raise 10 to the power of the left side of your equation and 10 to power of the right side of your equation. This will eliminate the log on the left side of your equation. 
y = __________
13. Simplify the equation from #12 so that it is in y = abx form:
y  = __________
How close is a to your start height?
How close is b to the ratio you found in #5? 
14. Enter this function into Y2 on your calculator (turn off the equation in Y1). Change your scatter plot from the residual plot to the original scatter plot of the actual data. How well does your exponential regression fit your data?
Follow up questions:
1. If your first bounce is from a different height, does this equation still work to predict the rebound heights of each bounce? Try dropping your ball from several different heights (change a in your equation) to support your answer.

2. Will this equation work for another group’s ball? Try it and see.
3. Will this equation work for a different type of ball? Why or why not?
4. Why did we use the average of each height rather than every one of the individual measures?
5. Enter each of the individual measures into your calculator, and graph your equation through the points. How well does your equation fit the data?
6. Use the calculator to find the exponential regression for all of the data.  How close is it to the exponential regression you found in # 12? 
How well does it fit the data?

C Adams

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