Thursday, October 24, 2013

Tennis Balls

Along with coaching, I captain two men's tennis teams.  It's pretty cool.  We have a team of 12-16 guys and on any given night we play eight: three doubles matches and two singles matches.  We have a fun team and we (surprise, surprise) pretty much always go out and celebrate afterward, win or lose.

And by celebrate I mean go out and drink beer.  Our latest favorite watering hole is the Groveland Tap. Excellent beer selection and a good happy hour menu.

If you're counting, we have five tennis matches per night.  Our home courts are at Fred Wells Tennis & Education Center in St. Paul and they are great hosts.  There are five tennis courts in the back bubble and those are the ones we play on.  Here's an overhead view of the facility in the summer when the bubble isn't up.


(As an aside, it's odd that they aren't oriented north-south, as is the convention in the U.S.)

Put this information about the courts in your back pocket for a minute while I introduce the other part of the story.

As a home captain, I have to provide balls for all five matches. My favorite ball is the Pro Penn.


When we play a match, we need to keep track of our balls. Every match gets a can of three balls and we don't want them to get mixed up with those on adjacent courts. To this end, Penn (and every other ball manufacturer) is helpful. They put numbers on their balls so that they don't get mixed up.


The problem is, Penn only numbers balls 1-4 and we have 5 courts. So two matches will have the same numbers on their ball. I need to keep those balls away from each other.  One possible favorable distribution of balls would look like this:


And this would be an unfavorable distribution of balls, because the 1's on the adjacent courts could get mixed up:


Because it is desirable to let the players open the cans (there's a certain thrill to it, even if you've opened thousands in your life), I as captain have to shake the cans and rotate the balls to see the numbers through the wrapper. It's kind of a pain and the other night as I was rotating the balls so that I could give the right numbers to the right players, I thought,

"Is it worth it? How likely is it really that two consecutive courts will end up with the same ball if I do it completely at random?"

And this is how I introduced combinatorics to my HL Math class.

Wednesday, October 16, 2013

Playing Games, Part 2

I recently blogged about how my Higher Level Mathematics class went to the Brunswick Zone to explore functions. The idea was to have the kids take down information about their transactions.  Step one was putting money onto a card. Step two was using the money on the card to play games and get points (tickets). Step three was redeeming those points for prizes.

On the surface it seemed like a great way to explore composition of functions until I realized...STEPS 2 & 3 ARE NOT FUNCTIONS!

DAMMIT!

After a few deep breaths I concluded that it didn't really matter. Truly, who really cares if anything is a function. The refinement from relation to function seems like something we do for mathematical convenience.

The Lesson

Step 1: Putting money on the card.



Prompt: Name the domain, range, and rule of assignment.

Pretty cool conversation about piecewise functions here.  Also, the range is anything but trivial.  I divided the class into two groups: half assumed they had the cards already. The other half assumed they needed to buy the card.


Please excuse any clumsy notation.

Step 2: Play games and win points.

We explored the games we all played by watching a few videos we took and looking at the pictures we all contributed to our lensmob album. BTW, thanks to Frank Noschese for the great lensmob suggestion!

Prompt: Pick a game and name the domain, range, and rule of assignment.

The only actual functions were from those games that did not distribute tickets.



Because you pay a flat fee and you can get a variable amount of tickets, it complicated my original idea about a explicit conversation about function composition. It is possible, however, that the conceptual gains were greater because they were not functions.

There were a couple of really interesting games. Take, for example, what we called "Ball Drop."


The fact that you can get bonus plays makes the range very tricky to figure out!  (Or maybe not? Think about it.)

Step 3: Turn tickets in for prizes.

The nice thing about this step is that the range no longer includes numbers.


Not much to this part; the heavy lifting has already been done.

All-in-all, it was a fun lesson with the potential to increase students' conceptual understanding of relations and functions.  I was very happy with how it turned out, and will do it again!

Mequals and Hats

I write this with complete deference to www.nixthetricks.com. I will rationalize my use of these dirty, horrible tricks by saying that this notation is completely arbitrary and does little for mathematical conception. I'm a math teacher so I like it, but for the average 10 grade Geometer, I'm not sure these subtle differences in notation have much real meaning.

Mequals

How do you get a kid to recognize that measures are equal and angles are congruent?  MEQUALS!  If you use an EQUALS sign, you need to put an M on the front.  MEQUALS.

Shameless, I know.



Hats

If one of them wears a hat, they all wear hats.  BOOM.