volumes of solids. Since solid figures and
volumes are three dimensional, it is
sometimes difficult for the students to visualize a 3-dimensional figure drawn
on a 2-dimensional whiteboard. When I attended an AP Calculus Institute, I came
home with several activities and labs that I could use with my students. Some
were helpful and some were less so, but I was intrigued by the one that had
students calculate the volume of the liquid contained in a classic Coca Cola
bottle. Calculating the volume of a can is pretty simple because cans are cylindrical and their radii remain constant. The volume would be equal to the
area of the base (a circle) multiplied by the height of the can. On the other
hand, a Coca Cola bottle is curvy and while any cross section of the bottle
would be circular, the radius changes at different heights of the bottle. In
Calculus, we learn that we can find this volume by defining the area of the
cross section (a circle) as a function of the radius (written in terms of the
variable x) integrated along the height (x) of the bottle. This is not an easy
idea to convey while drawing functions on the board and trying to simulate the
function rotating around the x-axis. I hoped that having something tangible
like a bottle to demonstrate the concept would be helpful for the students so I
decided to try the activity.
handed out Coca Cola and Diet Coke bottles (they were two different sizes),
markers, tape measures and a worksheet to record data. I explained that we
would be calculating the volume and we spent a little time discussing why
calculating the volume might be difficult. The students then worked with
partners to measure the circumference of the bottles at regular heights
measured from the bottom of the bottle to the height of the liquid inside. They
entered their data into their calculators using lists and then back calculated
the varying radii using the circumference data. This allowed them to graph
their height vs. radius of their bottles and they were able to see an approximation
of the profile of one side of their bottles. They then used a quartic
regression to find an equation that best fit the data graphed. This was
completed on day one.
the lesson on calculating volumes of revolution and then afterwards ask them to
calculate the volumes of their bottles. At the last second I decided to have
them calculate the volumes before I taught the lesson. I am so glad I did this!
We were able to spend some time hypothesizing on how we could use our data to
calculate volume. Eventually we were able to draw a picture of our bottle on
the board correctly aligned with our x and y axes. They were able to see the
relationship of the varying radii of the bottle to the volume of the bottle.
When we finally determined that we were supposed to integrate, the process made
sense to them! When I then taught them the lesson (Using the disk and washer
methods to finds volumes of solids of revolution), they were able to visualize
what was happening and I didn’t have to overcome so many objections to the
process that I normally would hear from the students each year.
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| Actual Volumes: Diet Coke 237mL and Coke 355mL |
overestimated the volumes significantly. We tried to adjust by estimating the
thickness of the glass, which helped, but I still think the students were not
careful enough when measuring the circumferences. In the future, I think I
would give them lengths of string to use to wrap around the bottles and then
have them measure the string with the tape measures. I think the tape measures
were too stiff for them to work with accurately. I also think I would let the students wrestle
with the problem longer before I guided them toward integrating the area of the
cross section. However, even without these minor tweaks, I think the lesson
achieved even more than I had hoped for!