## Basketball Bounces, Assessment Variation 1

##### SCREEN I

In science class, some students dropped a basketball and allowed it to bounce. They measured and recorded the highest point of each bounce.

The students’ data is shown in the table and scatterplot. The first data point ($n = 0$) represents the height of the ball the moment the students dropped it.

Bounce Number $n$ |
Measured Height in Inches $h(n)$ |
---|---|

0 | 233 |

1 | 110 |

2 | 46.6 |

3 | 21 |

In this task, you will choose a function to model the data and use the model to answer some questions.

##### SCREEN II

a. Compute the first three values in the last column of the table below.

Bounce Number $n$ |
Measured Height in Inches $h(n)$ |
Factor by which Bounce Height Decreased $h(n-1)\div h(n)$ |
---|---|---|

0 | 233 | |

1 | 110 | [____________] |

2 | 46.6 | [____________] |

3 | 21 | [____________] |

*[Student could choose to open up a scientific calculator.]*

##### SCREEN III

b. Let $n$ be the bounce number and $h(n)$ be the height. Consider the following general forms for different kinds of models where $a$ and $b$ represent numbers:

Model 1 | Model 2 | Model 3 | Model 4 |
---|---|---|---|

$h(n) = a\cdot n+b$ | $h(n) = a\cdot n^2+b$ | $h(n) = \frac{a}{n} +b$ | $h(n) = a\cdot e^{bn}$ |

Which of the models shown is most appropriate to use for the given data?

Given the data above, what are reasonable values for $a$ and $b$ if we want to create a specific model to fit the data? Write the appropriate values in the equation below.

* [Based on the choice students make above, they are given the appropriate template below. Here is the template for the exponential model] *

c. Based on your model, what will be the first bounce with a maximum height that is less than 1 inch?

*[When the student clicks "Next" from this screen, they get a message, "Are you sure you are ready to go on? You cannot change any of your previous answers after you continue from this screen."]*

##### SCREEN IV

d. Mika said,

The model I came up with is $h(n) = 233 \cdot e^{-0.8n}$. I used it to predict that after 50 bounces, the height of the bounces will be less than a thousandth of an inch. It is good to have the model because it would be very difficult to measure such small heights.

What is the best way to characterize Mika's claim?

- Mika's claim is true. The whole point of using models is to make predictions.
- Mika's claim is true but she should give a more precise bound for the height of the ball after 50 bounces because the heights will be much much smaller than one thousandth of an inch.
- Mika is correct that the model predicts that the bounces will all be less than a thousandth of an inch, but in reality the ball will be at rest before it has bounced 50 times.
- Mika is not using the model appropriately. Models can't be used to make predictions past the given data, only between data points.
- Mika is not using the model appropriately. The model doesn't fit the data very well so it can't be used to make predictions that far in the future.
- Mika's claim is not true. The model states that the ball will be at rest before it gets to 50 bounces so the bounce heights will be zero, which is easy to measure.