A Simple Way To Teach Polar Coordinates

The courtyard outside my building is laid in a circular pattern, which makes it a perfect setting to teach polar coordinates:   I took my Precalculus class outside and stood on the polar axis. I assigned an r and theta-value to each student. They then had to start at the pole and figure out where their given polar coordinates would position them. It was a great way to incorporate physical movement with math. They even understood what happens when r is negative! To wrap things up, we worked out what the equations theta = pi/4 and r = 6 would look like, and we compared them to their rectangular counterparts. Then, back inside to apply our new knowledge to some problems!

Do You Have The Stomach For A Geometry Puzzle?

I’ve been reading Clifford Pickover’s excellent and entertaining The Math Book. Its topics are arranged chronologically, with each entry just one page long and paired with a beautiful illustration. It’s the perfect book to keep by your easy chair, where you can pick it up and have a quick read.

One of the entries is about Archimedes’ Stomachion, which is pictured below. If you print it and cut out the pieces, there are 17,152 different ways you can arrange them into a square!

I constructed the one below using GeoGebra, but for a geometry class that is learning the classic straightedge and compass constructions, it would be a nice exercise to draw one using those tools. The only constructions necessary are midpoint of a segment and parallel line through a point.

Get out your scissors and have fun!

Build a Classical Music Library for Less Than $22!

Just when I think the digital age can’t get any better, something like this comes along. Amazon is offering collections of music by some of the greatest classical composers for $1.99 each. These are very nice performances, and each collection contains 100(!) tracks. There are 11 albums, so for an investment of $21.89, you can build the foundation for an excellent classical music library.

Amazon stores your music on their cloud server, so you can stream it any time you want from any device that can access the web. Or, you can download tracks to play on any mp3 player. Here are links to each composer’s collection: Bach, Beethoven, Chopin, Debussy, Grieg, Handel, Mozart, Schubert, Schumann, Tchaikovsky, and Vivaldi. Here’s a link to a 100-track sampler.

The Wheels On The Bus Go Round and Round…

A quick post today about a lesson in precalculus that went a lot better than I expected (and isn’t it nice when that happens!). The objective was to understand linear velocity and angular velocity, and the differences between them.

I like to reduce as much as possible the number of formulas students memorize, so when we discuss this topic, I try to get them to use unit cancellation to arrive at the desired result. We talked a little about how to convert rpm to rad/sec, and mph to rpm. Then I sent them out in teams of four to the parking lot, armed with rulers, to figure out the rpms necessary for the wheels of cars traveling at 35 mph.

I didn’t tell them what to use the rulers for, so they had to figure out for themselves that they needed to calculate their car’s wheel circumference. When they returned to the classroom, they got right to work, and wrestled with the proper setup for their expressions. Eventually, every team got a good answer, and they didn’t use the same process to arrive at their result (which is great!). Here’s one team’s work:

Comparing the teams’ different results led to a nice discussion on how wheel size affects the rpms needed to roll at 35 mph, and why cars need differential gears. Merely getting my students out of the classroom and moving for about ten minutes really energized them, and made them interested in figuring out the answer to an admittedly simple problem. Whenever possible, I need to incorporate movement and outside activity into my lessons, even if it’s only for a few minutes.