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Perfectly Square

written Wednesday, 12/30/2009

In just over the last decade there’ve been three times when my age has been a perfect power – an integer raised to another integer greater than one. Today it just happened again. In fact, my age today is also triangular. That’s got to be a sign that this will be a good year. And I’d better make the most of it since after this my age won’t be perfectly powerful again until thirteen years from now.
Go ‘head, it’s ma birfday.
I’ve spent most of the day rummaging through old computer files and updating this website to include my various artistic and nerdy interests of the last decade. I’ve intended to find time to do that ever since I started this site several years ago.
I’m gon’ party like it’s my birfday.
Also posted a few more videos to my youtube channel. I haven’t even left my apartment today since grocery shopping early this morning. I’ll find something fun to do tomorrow and pretend that all the evening revelry is for me.
Sip some herbal tea like it’s my birfday.
Another year down, and I just keep getting cooler with age.
Halftime

Winter break marks the halfway mark of the school year. As usual I hustled to finish grading for the quarter before leaving school for the holidays. I appreciate that math is so much more objective than other subjects, but I still agonize over the multitude of subjective decisions that constantly have to be made in the course of teaching. Choosing course content for breadth vs. depth, tilting towards the higher vs. lower achievers, setting a fair grading curve, allocating how much of the grade is determined by homework vs. tests/quizzes vs. exam, trying to handle all the excuses and interruptions in the school day, deciding whether to show any leniency when some students can’t get it together, figuring out how to fairly deal with the students who can’t seem to pass a test no matter how hard they try. For a few students who came for help frequently during lunch or after school, I decided to consider our time together as part of an “alternative assessment” after they continued to perform poorly on the standard tests and quizzes. I hate having to make such exceptions, but there are graduation implications for some of them and I also hate to see their hard work continue to go unrewarded.
Of course this type of malleable teaching practice may very well be part of what led many students to get to their senior year without ever adequately learning fractions or times tables, but since my class marks the end of the road for some students’ math education, this is not the time for me to make a principled stand in the name of academic rigor.
Break time

Got back a couple days ago from a pleasant visit with mom, aunt, & uncle in Tucson. As I like it, the stay was relatively unstructured and much of the time was spent just hanging out. Read for pleasure, which is something I always wish to do during the school year but never make time for. Mom’s tuxedo shorthair kitty finally let me pet him for more than a fleeting second for the first time in the 3+ years that she’s cared for him. Trailer was adopted from the Humane Society back in California (guess what kind of vehicle he was found living under). Abused and neglected in his earlier life, it’s taken a long time for him to learn to trust. The long and gradual journey to earning his acceptance caused me to draw parallels to certain societal ills, and I wondered if there was some broader life lesson to take away from it. Perhaps, but instead of over-thinking it I just relished in rubbing his fat, furry belly.
I spent most of the flight and layover time with one of my favorite childhood puzzle books, Mathematical Bafflers, which compiles some challenging problems that originally appeared in Aviation Week and Electronic News periodicals some time back. As a kid I used to write the date next to problems that I was able to solve. I noted that exactly two decades to the day had passed since I tackled some of them. How humbling it was to find a half-dozen or so problems that I had solved more than half my lifetime ago were stumping me present day. My present superior calculus knowledge was of little use. Most puzzles require clever thinking and problem solving over advanced math skills (Which is exactly what makes them intriguing).
I started to feel rather inferior to my former self and it seriously started to eat away at my ego. Locked in a battle of math smarts with the younger me, I felt a little redemption upon eventually cracking a few problems that I had never previously conquered. Sitting in the Dallas airport, I muttered to my yesteryear self, “Take that, punk.”
Parking Problems

Two weeks ago a number of mainstream news sources including NPR and Fox News reported that a British mathematician had crafted a formula to determine the minimal amount of space needed to parallel park. Skeptical of the flimsy details, a few of my math and science colleagues engaged in some spirited discussion of the geometry required for optimal parallel parking. I worked backwards from the equation provided in one of the news reports and figured out what assumptions the mathematician must have made. With one of the physics teachers in particular, Mr. Collins, we debated the flaws in his assumptions and eventually came to agreement on how the problem should have been addressed. For what seems like such a simple premise as parallel parking, the math quickly became quite complex. Initially I concluded that an algebraic solution was beyond my skills and I resorted to a computer model. Fellow math teacher Mr. Wenstrup (my partner in the AMC rap songs) scoffed at the high-tech approach and wondered what Euclid would have thought of how much time it was taking us to figure this one out.
Finally on Christmas day I tackled it again and ultimately found an algebraic solution. Considerably more complex than the Brit’s equation, I was a bit frustrated that I couldn’t find a way to further simplify my formula. However, I was thrilled that the breakthrough in deriving the formula came from the sine of sums identity that is standard for Trigonometry/PreCalculus courses. I had long found this formula fascinating, yet I’ve never found a situation to use it in real life. This topic is coming up in the 3rd quarter of my PreCalculus class, and I’ll look forward to bringing up the case of parallel parking to illustrate its applicability.

Click image for page with full explanation, computer animation, and algebraic solution to parallel parking problem.

Square. Perfect.

Piddling away my time with computer musings and math puzzles isn’t what I once envisioned for myself, but someone recently asked me if I could find contentment in the way things are currently unfolding. Maybe the answer is yes.
I thought of a few more nerdy projects over the break and I can’t wait to find the time for them once I’m done updating the web site. Rather than resent that this is how I choose to spend my time, I think I’ll try to just enjoy it this year.
Aww yeah, it’s my birfday.