Study of an "unknown curve"

"A chewing gum is stuck to the wheel of a bicycle. As the bicycle rides, the chewing gum traces a curve in the air..."

This activity (pdf here) combines hands-on work, research of information on the internet, and applications of the rectangular method of approximation of integrals (Riemann Sums). It is a good and fun activity to apply Riemann sums in a meaningful way, and it will greatly interest students who are curious about math.

The "unknown curve" referred to is actually the cycloid. It is assumed that students don't know this curve at the beginning of the lesson, or at least not very well.

Using are assigned the task of drawing a curve on a large sheet of paper that is stuck to the wall, by attaching a felt pen (or whiteboard pen etc.) to the edge of a classroom (circular) trashcan, and roll the trashcan in order to create a curve. (This is actually the geometric definition of the cycloid: the curve traced by a point on the rim of a circular wheel as the wheel rolls along a straight line).

Several tasks are then explained. Students need to estimate the area under the arch using Riemann sums, as well as the volume of revolution that is formed by rotating the curve about the "x-axis". They will also investigate online to find the name of the curve, historical background and other properties of the curve.

Given that the area under the curve is three times the area of the generating circle, students will also be able to check the accuracy of their Riemann sum for the area. But finding the area under the curve analytically is beyond the curriculum in AP Calculus AB.

                                              (download the pdf here)

Running and Hopping Activity

This exercise (pdf here) is a variation on a traditional textbook exercise using derivatives, but students have to collect their own data in order to solve the problem. The task is to determine the ideal trajectory if you want to go from one corner of a soccer field to the opposite corner, with the limitation that you can run along the side, but before you cross, your feet will be tied together and you will hop the rest of the way.

Students form groups. One of them will be the runner. The information students need to collect are:
- the velocity of the runner as he runs.
- the velocity of the runner when he hops.
- the time it takes to tie the runners feet.
- the length and width of the soccer field.

The rest is an optimization problem, using the Chain rule.

Notes:
- Students don't necessarily realize that they won't need to run and hop all the way.
- Measurements of the field can easily be found using GoogleEarth for example, rather than physically doing it.
- The time it takes to tie the runner's legs doesn't influence the answer to "what is the quickest path?" since it is a constant.

                                                    (download the pdf)

Rolling Ball Activity

In this lab (download pdf here), 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 should also be travelling 0 meters (as it will bounce vertically until rest).

Therefore, it is clear intuitively that there should be an angle between 0 and 90 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. Using their graphing calculator or a computer program (Excel, Numbers, ...) they then look for a good model for their data. By differentiating manually, they can finally determine what the best angle should be.

Interesting questions arise:

• What angles should they use when collecting data, and should they repeat the experience more than once for each angle?
• What parameters must they be careful about in order for the data to be good (Students need to make sure they don't vary anything but the angle of the inclined plane. The position on the inclined plane where they drop the ball must remain constant, etc.)
• What is the best model? a parabola? a higher-degree polynomial? (I personally don't know...)
• How come the best angle according to my model is less than one of the measurements my group made?

Notes:

• I found that using 2 plastic meter-sticks is a pretty good inclined plane, 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 students who are responsible for holding the inclined plane are very proud of being the only ones knowing exactly how to do it. This role can be attributed to students who need to build their self-confidence.)
• It is important to emphasize the results that if the angle is 0 or 90 degrees, the distance travelled is 0 meters. These should be included in the data when looking for a model

                                                Download the pdf file here

Pyramidal box activity

Pyramidal Box Activity

This activity (download pdf here) is a variation of the "maximize the volume of the box" problems we are used to see in textbooks. It is a hands-on activity, and uses a pyramid shaped box, which leads to some very nice equations that can all be solved without a calculator. It allows students to put their learning of derivatives into practice. Students will have to build a pyramidal box and measure its volume using rice and a graduated cylinder. Then they will compare their result with the volume they find algebraically. Finally, they will apply their knowledge of calculus to find the dimensions of the box of maximum volume.

The activity is designed to use about 60 minutes of class. It can be extended depending on the feedback that is expected from the teacher, from a simple account of results to a complete reflection on the activity and description of the process.

Material to prepare for the class: Sheets of (colored) paper, scissors, glue, some rice, compasses, 2 graduated cylinders (volume 100ml), large container where students can safely pour rice into and out of their pyramids.
                                              download pdf file here

Here are a few pyramids that were created by students, with a hole cut at the top to allow them to pour rice in.