I just finished teaching another Light, Math, and Color minicourse. Twenty students researched a math topic and illustrated it with a stained glass window. Their projects this time around are really spectacular, especially considering these are first tries. (Note: If you hover over a picture, the math topic it illustrates will show up.)
I’ve discovered a nice podcast that is produced by the BBC: In Our Time. Melvyn Bragg hosts a different group of guests every week, depending on the topic being discussed, which can be anything related to the history of ideas. I’ve enjoyed hearing how the Book of Common Prayer came to be, the significance of George Orwell’s Animal Farm, and how radio was invented, among many interesting topics. Melvyn keeps things moving along, and the guests are always very knowledgeable and entertaining.
Of course, I most enjoy the math-related conversations, so for my fellow math teachers here is a list of the programs – at least the ones I’ve found so far – that are devoted to that subject (click on the title to go to that program’s download page):
You can subscribe to the In Our Time podcast via iTunes by clicking here.
I’m teaching Algebra 2 for the first time in many years, so I am recording lots of screencasts for it. I’m putting them into a YouTube playlist, and you can access them all here. So far, I’ve covered completing the square, linear inequalities, radical and quadratic form equations, absolute value equations and inequalities, solving quadratic equations, basic tools of graphing, lines, circles, function basics, graphing functions, transformations, and functions operations and composition. (Whew!)
If you’re really bored and want to binge-watch them, here you are:
One of the hardest type of problem for calculus students to understand is calculating the volume of solids of known cross-sections. It’s hard, because they have difficulty visualizing it.
Last year, I went to the regional NCTM conference here in Nashville, TN, and one of the sessions I attended addressed this exact issue. Nina Chung Otterson was the presenter, and she teaches at The Hotchkiss School in Connecticut. She has her students cut cross-sections of different shapes and apply them to a base area enclosed by two parabolas, y = x^2 – 3 and y = 3 – x^2.
Here’s what the base area looks like, courtesy of Desmos.com’s online function grapher:
In her session, Nina Otterson provided templates that fit the given base area for different shapes: semicircles, squares, and equilateral triangles. I had my students cut the square templates diagonally for isosceles right triangles, and horizontally for rectangles. Students use the templates to cut out a cross-section that fits down the middle of the base area, and six others on each side.
Here are my students in action, cutting out the cross-sections:
Here they are, taping the cross-sections onto the base area:
And here are the finished models:
Once they understood that the thickness of the paper was dx, it was very easy to set up the integrals to calculate the volumes of their models. I’ve never had students grasp the idea behind this type of volume as quickly and as easily as this group did. Building a model using actual cross-sections made all the difference!
When I do this activity next year, I think I’ll glue the base area to foamboard, and have students insert the cross-sections into slits cut into the foamboard. That way, they will stand up straighter and stay evenly spaced.
You can download the templates provided by Nina Otterson here.
Update: I used spray adhesive to glue the base area to some foamboard and cut slits in it with an Xacto knife. Then I carefully slid each cross-section into its appropriate slit. It worked great! Now each cross-section stands nicely spaced and vertical.