Friday, July 24, 2009

Updates, ODDS AND EVENS and some Geometry Packing Problems

Enjoying your summer hiatus or as busy as ever? I know that feeling!

1. MathNotations' Third Online Math Contest is tentatively scheduled for the week of Oct 12-16, a 5-day window to administer the 45-min contest and email the results. As with the previous contest, it will be FREE, up to two teams from a school may register and the focus for now will be on Geometry, Algebra II and Precalculus. Several other ideas are running through my head but I need the time to bring them to fruition. If any public, charter, prep, parochial or homeschool (including international school) is interested, send me an email ASAP to receive registration materials: "dmarain 'at' geeeemail dot com."

2. Article - Something to tell your students in September!
Here is the link. The 2nd paragraph says it all:

The top 15 highest-earning college degrees all have one thing in common -- math skills.
3. Silly Instruments for Math Teachers to Play
I always told my students that I'm predominantly left-brained -- analytical, organized, detailed, process-oriented, algebraic -- as opposed to most of my children and my wife who are creative, spatial, mechanical, who see the forest more than the trees. One of my sons is a musician and another is a dancer so we are not always on the same wavelength! So I mentioned to my SAT students that I wish I had a more creative side and perhaps be able to play an instrument, but, in fact, the only thing I can "play" is my iPod! One of my students in the front row immediately responded, "I know an instrument you can play, Mr. M -- the triangle! I congratulated her for the cleverness and told her that maybe I will learn how to play the "cymbals." (the class actually applauded that lame attempt at word play!). In fact, I've read that many famous mathematicians were also musicians, so let us know: Do you play an instrument or are passionate about music or do you have a silly instrument for a mathematician to play?

4. Circle Packing ProblemsLink
Even though I am dominantly left-brained, I still enjoy challenging spatial geometry problems. I find these questions have improved my creativity and my spatial sense and they often involve multi-faceted thinking. Here are a couple of famous 'packing' problems which are accessible to geometry students. More important than solving these is to give our students a sense of the importance of packing problems and the ongoing research in this area. There are still unsolved problems here!

Although you can easily research packing problems on MathWorld and Wikipedia, the diagrams below come from an exceptional website I discovered. The author, Peter Szabo (missing accents), provides diagrams for packing 2-100 circles with accompanying data (radii, density, etc).


The two congruent circles at the left are actually enclosed in a unit square which is not shown.The circles are tangent to each other and to the sides of the square. If these circles have the maximum radius possible, determine the radius.
Note: The indicated square (assume it is a square) is helpful in solving the problem. Trig is not necessary here.

Answer (Yes, I'm providing this since the objective is to discuss the method):
[The following is the diameter, not the radius, of each circle. Thanks to watchmath for correcting this error].


Again, imagine that the three congruent circles at the left are enclosed in a unit square and are tangent to each other and to the sides of the square. If the circles have the maximum radius possible, determine this radius.
Notes: The indicated square again may be helpful to solve this problem. Trig can be used but clever use of special right triangles is preferred.

[The diameter is given below, not the radius. Thanks to watchmatch for correcting this]


Sue VanHattum said...

Hi Dave, I used to feel like you, all leftbrain, little or no rightbrain. But now I think it's just harder to recognize our rightbrain's contributions. It doesn't talk, and we're used to listening to the leftbrain's words. Dreams are our rightbrain 'talking' to us.

Actually, I wrote to say I play the pennywhistle. I'm not great at it, but good enough that people will compliment me. (I played guitar badly for years, and never got good enough to enjoy hearing what I was playing.) Pennywhistle is a real instrument used most in Irish music, but it's even easier than the recorder.

You might want to try it. (They were under $10 the last time I bought one.)

watchmath said...

Each radius that you provided are bigger than 1/2. It means that the unit square can't even contain one single circle?
I found your answer for problem one is the diameter (not radius).

Dave Marain said...

Thanks, watchmath, for correcting my error! Those answers were definitely diameters, not radii. I've corrected it and credited you.

Joshua Zucker said...

On the silly instruments front:

Certainly any mathematician who can count is able to play the Peano.

There's also the famous paper titled something like "Can you hear the shape of a drum?"

Dave Marain said...

To continue the silliness,,,

Didn't Peano invent that "little purple pill?"

And have you ever played Pythagoras' Theremin?

We can't go much lower than this!