Thursday, April 5, 2012

Visualizing Volumes

My calculus class is working on volumes of revolution. I have found that success with this particular topic is related to a student's ability to visualize in three dimensions. I always have a few who struggle with that.

Fast forward to last week. I was shopping, and I found these on clearance:

For decorative use only. I am such a rule-breaker.

I traced out the shape of the region I wanted to rotate.

Cut it out. Open it up. Voila!

You have to hold it a certain way or it will sort of fall apart, but I thought it was a nice visual and tactile representation of a volume of rotation.

This one fell apart when I opened it up. I learned that you have to make sure some of the original glue is left intact when you cut out your shape. I tried running a bead of glue along one side, but then it wouldn't open very well.

Another drawback is that the semi-circular shape of the original pieces is somewhat limiting. I am sure these must be made in other shapes? I am going to raid a party store before we get to this topic again next year.

Still, my students were really intrigued to see what would happen as we cut and unfolded each one.

This one turned out really nice:

Happy rotating!


  1. I took my husband's drill to school the day I introduced this. I put a pencil in place of a drill bit, and we taped cardboard cutouts of the regions we were rotating to the pencil. When it spins, you can see the shape. Works great for disks but not for washers. :-)

    Someone in each of my geometry classes asked why there was a drill on the table in the front of the room, so they got a quick demonstration of what they'd be able to do with calculus beyond finding the volumes of cones and cylinders.

    I think the drill would work even better if I used a dowel split in two, and added a little "tab" to the cutouts that could go between the pieces, which would be held together at one end by the drill and perhaps a rubber band at the other end. That might hold them out more rigidly from the axis of rotation. As it was, with just tape, they form "solids" that have a slightly smaller radius than they should, because the outer edge of the cardboard trails behind the inner as it rotates.

    For 2-d representations of solids, have you used Winplot?

  2. This would also be a cool idea for showing the shapes of the atomic orbitals - low cost models! I may try that next year! Thanks for the idea.

  3. The drill idea is GENIUS! I can't wait to try that. Thank you for sharing.

  4. Hi Amy. I'm commenting on "interview questions". Here are some that we feel we can test the waters with:
    * What do you love/not love about teaching and why?
    * What does your typical class look like?
    * Where do you pull your resources from?
    * What are your career goals?
    * If you had 2 different levels of a subject, how specifically would you differentiate?
    * What do you do for fun? To combat stress?
    * We also ask a math questions to see how they would teach something and see if they have a good grasp of the pitfalls and strategies.
    * What are your questions? (a biggie)

  5. love this and the drill idea. could not have been better timing - about to do this with my AP Calc class at the beginning of next week!

  6. I love this and love the drill idea! Perfect timing - I'm starting this next week with my AP Calc class.