Day: May 15, 2008

Algebraic Manipulation Is Overrated

An intuition question.

Look at the function below. It may surprise you that it is a constant! For any value of x, the function g will have the same value. I’m wondering, now that you know this, if you can get a sense of why it would be a constant, without (a) using your graphing calculator, or (b) taking the derivative to show that it is 0 [that is what I did, and as a side note, I have to use this on a test or homework next year].

g(x)=\frac{\sin(x)+\sin(x+a)}{\cos(x)-\cos(x+a)}

Can you find some geometric way to see that?

It took me somewhere between a half hour and an hour of playing around to get it. I can post my solution in a couple days, but right now I don’t have the energy to find a program to draw my solution [1]. But let me just tell you: it’s beautiful. You’ll be stunned when you first do it. Yeah, the calculus way tells you it is a constant, but seeing the “why” is still a mystery. The geometric way takes a bit, but whoa nellie, you won’t regret spending the time!

[1] Or maybe I should claim there is no room in the margin! (JK)

Update: I did finally write up my solution. I quickly did something I never have done before: do my work in powerpoint. It worked fine.

Update: Mr. K solved the problem in 3 minutes and found a way to show the geometric solution. Head over to his very excellent blog to see it in all it’s glory.

Update: Besides mine and Mr. Ks, a third and perhaps more elegant solution is up at 11011110.

Of the three, I think I like Mr. K’s visualization best, even though it might not be a proof in the formal sense.

“Professional Development”

Each year, my school provides each teacher with $100 of “professional development” money. I don’t know exactly why they call it that (hence the quotation marks). For things like conferences, online courses, etc., we have a really great fund to tap into. No one I have talked to has ever been denied money from that fund. This $100 is more of a mystery. You have to submit receipts for it, and it needs to be for things relating to school. I could buy school supplies, for example. Professional development? Tenuous.

And, in fact, each year there’s a book fair with tons of books for students and teachers to buy from. It’s a fundraiser for yet another something or the other. I learned that it’s tradition for teachers to never use their money during the year, and during this week in May, pick out $100 worth of books from the fair to count as their “professional development” money. This practice is so institutionalized that you don’t ever have to take out your wallet to get the books; the people running the book fair just write your name down and the total amount you’ve spent on a piece of paper and you’re done.

Streamlined, and sweet. Just the way I like it. [1]

I’m not complaining. How could I complain about this? But I do wonder why this money needs to be couched in terms of “professional development”? (No matter how broadly you look at it, my Martha Stewart books will never be professional development.) My suggestion: why not just call it a “we like you teachers and we want to give you a little pick me up” perk and be done with it?  I like my school. But for some reason, getting a $100 and being it’s told because “we like you teachers and we want to give you a little pick me up” is just so much more satisfying than “professional development.” So I’ll pretend that’s what it’s explicitly earmarked as and go along merrily.

[1] I know you’re wondering… I bought two very smart-looking hardcover Martha Stewart books (“classic” and “new” recipes), Middlesex (Jeffrey Eugenides), and The Secret History (Donna Tartt). I’ve read the Donna Tartt book before. One of the best books I’ve ever read, hands down.