__Rolling Ball Activity__

## Links to Activity

Rubric for labs ( rolling ball and pyramid box were combined as one project grade)

## Main Concepts

Calculus

Derivatives

Optimization

## Duration

80 minutes in class + 200 min homework or classwork

## Summary of Activity

In this lab, students work in groups to collect data, use technology, and solve an optimization problem. The problem at hand is the following: as you vary the slope of an inclined plane, a ball rolling down will travel a different distance. If the angle of elevation is 0 degrees, the ball will travel “0 meters”. Theoretically, if the angle of elevation is 90 degrees, the ball will also be traveling 0 meters (as it will bounce vertically until rest). Therefore, it is clear intuitively that there should be an angle between them that yields the greatest distance. The task and goal of this lab is very easy to understand. In order to find that distance, students will first measure the distance with various angles. Unsing their graphing calculator or a computer software (Excel, Numbers, loggerpro…) they then look for a good model for their data. By differentiating, they can finally determine what the best angle should be.

## Notes and Insights

We ask the students to solve the optimization problem by hand, that is they need to differentiate their model-function by hand and then solve f ’(x) = 0 by hand too. We tell the students that the model they pick has to be sensible, but what they choose doesn’t impact their grade. Even though a parabola seems most appropriate, some groups go for trigonometric graphs and quartic functions that appear to model their data better. We talk about it with them but it is not the purpose of the project to guess the model correctly. Anyway, ultimately, given all the variables in play, like the un-flat floor, who knows what the real model function is… (In case they pick a difficult function, they can use technology to solve f ’(x) = 0).

It is important to emphasize the results that if the angle is zero or ninety degrees, the distance travelled is zero meters. At KIS we require these values to be included in their data. I am not sure if that is acceptable scientifically, but it does help for the best-fit graph.

I found that using two plastic meter-sticks makes a pretty good inclined plane. You stick them together along the edge and they form a wide V. As long as the student responsible is aware of the fact that he needs to hold the inclined plane in the same way each time. (I have noted that some students who are responsible for holding the inclined plane are proud of being the only ones knowing exactly how to do it. This role can be attributed to a student who needs to build their self-confidence.)

Interesting questions arise, like for example regarding the choice and number of angles and whether to repeat the experiment multiple times for each. Students will wonder if a parabola is complicated enough to be a good fit for the curve. In the end, what the math tells them is the best angle actually yields a distance that is significantly shorter than one of their own measurement. Why and how should they handle it? These questions and many others are worthy of going into their reflection.