Yesterday I gave a workshop at the iMathination Conference in St. Charles,IL. The conference is full of hands-on math workshops for teachers, mostly from the city of Chicago – amazing teachers! I was asked to give a hands-on workshop for pre-calculus/calculus teachers and immediately thought of volume and prepared the workshop. In the hour and 15 minute session, I planned to explore volumes of solids created with known cross-sections with oranges, cupcakes, and ending with making a Styrofoam model for them to take back to their classrooms. If time allowed, ideas for hands-on activities for volumes of solids of revolution, but we didn’t get that far.
What I did not expect was that in my workshop were: 1 Calculus teacher, 1 pre-calc teacher, 1 art teacher, some science teachers, math teachers that did not teach calculus, and a middle school teacher who had never had calculus (and hated geometry)! So, not only were there not a group of Calculus teachers, but I was intrigued why people came to this session. Was it food in the title? Curiosity? Last choice on their list and other sessions were full? No matter what the reason, this was my chance to see if I could make the ideas of calculus accessible to non-math/non-calculus people!!!!!
We started with something they were familiar with – visualizing a deck of cards. They told me that they could find the area of a card. If we put them into a stack, they could see that the deck of cards also had volume, introducing the idea of summing the areas to find the volume.
Each person was given 1/2 of an orange and a knife. I asked them, “if I were to ask you what the base of your orange was, what would it be?” All of them quickly identified the cut side of the orange and placed it on the little cutting mat in front of them. I asked them to make a cut – but not through the center. They looked at the surfaces created and they correctly identified them as semi-circles. So, if you cut your half of an orange into more pieces in the same way and VERY thin, each piece would have an area that could be calculated but together they could be stacked and made 1/2 an orange which was more likely to be identified to have volume. We repeated the idea with a Little Debbie’s chocolate cupcake (no Hostess ones anymore – sad) and again they identified them as having trapezoidal cross-sections. We stopped and reviewed basic area formulas.
We looked at models using the concept of Riemann sums in 3-D to realize that if we cut the slices thin enough and put them back together that the model began to smooth out and look more like the original “orange.” They also realized that the model with thinner slices was getting closer and closer to having the same volume as the orange.
It was time to introduce the calculus. With the help from Audrey Weeks Calculus in Motion files for Geometer’s Sketchpad, we watched the shapes being generated from one side of the circle to another using squares vs. semi-circles. It was easy to visualize the sides of the square going from sides of length zero to 4 (twice the radius) and back to zero, creating the solid with a base of a circle and cross-sections of squares. Next we decoded the notation of a defiinite integral that would calculate the volume of the solid and how it related to what we saw.
The area function A(x) was written in terms of x and is called the integrand. The limits of integration showed where the square started and where it ended up as it moved along the x-axis. The dx represented the infinitely thin thickness of each cross-section, The integral sums those areas calculating the volume of the solid. I realize that it might be a bit simplified but it made the calculus make sense with what they saw. I didn’t teach them how to integrate – just how to set up the integrals. There are sites on the Internet that will do the integration like the Definite Integral Calculator and I think it is more important to know how to set them up and take the mystery out of the notation!
What I saw was awesome. Everyone asked great questions, showing that they were invested in the process, and intent on understanding how the integral was set up to find the volume of the “smoothed out” solid made of infinitely thin cross-sections. The artist was relating the models and math to pottery and was trying to figure out how she could make the math accessible to her students. They all made models of solids with known cross-sections using styrofoam. They chose their cross-sections and I saw them making squares, equilateral triangles, semi-circles, and isosceles right triangles on a circular base.
I will admit, that in the early years of my career, this workshop might have not made my list of workshops because I never saw the usefulness or the fun in calculus – scary to even type that. Even as a math major, Calculus was intimidating. My calc teacher in college was less than helpful (and was fired)! Once I wasn’t afraid of it anymore – with help from my friend Carol, a Calculus grant from HP and Oregon State in 1993, and having to teach Calculus, it became my goal to make sure that others never become afraid of MATH! I believe in the power of math and that all people need to find and use the power too!
The materials needed for this activity:
- 1/2 orange and a cupcake (preferably one with a flat top!)
- 8″ x 8″ or 12″ x 12″ squares of styrofoam – 2 per person: one to cut and one to mount the finished solid on – ours were 3/4″ thick and the home improvement store cut them for me for free! 4′ x 8′ sheets run around $10.
- ruler, compass, cutting surface (cutting mat or board) per person
- knife per person – just make sure to keep track of them in the classroom. Collect them 5 minutes before the bell and make sure you have all of them before they leave class!!!
- glue and/or toothpicks to put the solids together
- wet wipes and paper towels
- optional: tablecloths to cover the surfaces to make clean up easier
I love the dollar store for these materials.