## Saturday, February 23, 2013

### Let's Get Mobius Up In Here

Here is an applet to explore the function $$\phi(z) = \frac{z - \lambda}{1 - z \bar{\lambda}}$$ for $| \lambda | \leq 1$. In the bottom left is the unit disc with Darth Vader super imposed on it. On the top is also the unit disc. For each point $|z| < 1$, the applet determines the color of $\phi(z)$ and and then colors $z$ that color.

The unit disc on the bottom right allows you to adjust $\lambda$. Just click anywhere in the unit disc and $\lambda$ will change appropriately.

Note that this demonstrates that $\phi$ maps the unit disc to itself. Click here for a proof.

## Wednesday, February 20, 2013

### Magic Squares

"Alright settle down, Tom, don't you dare throw that paper airplane. (

*Alright! We have a sub!*)

*You all know the deal, sit down. Mrs. G. left instructions to have you do the Scholastic Math (*

*I hate those!*) but she also said that if I wanted, I could do something else (

*Yeah!*).

"So here's the deal, everyone has to draw this grid on piece of paper...

"...and fill in each of the squares with the numbers 1,2,3,4,5,6,7,8, and 9. (

*Is this like Sudoku?*) It's a little like Sudoku. (

*I hate Sudoku.*) Ok, it's nothing like Sudoku. You have to put the numbers so that all the columns add to 15 and all the rows add to 15. If you're really bright (

*Well that leaves me out.*) then try and get it so that the diagonals add to 15 as well.

"(

*This looks hard*.) It is hard. (

*I suck at math*.) That's ok, math is hard, just keep trying."

Eventually....

"(

*Oh, I think I have it!*) Awesome! Now try for the four by four case.

"(

*What numbers do we use?*) You have to use 1,2,3,4,5,6,7,8,9,10,11,12,13,14,15, and 16. (

*What do they add up to*). I don't know. (

*You're not going to tell us what they add up to?*) Nope. (

*Then it's impossible.*) No it's not. (

*How can we make all the rows and columns add to the same number if you don't tell us what they have to add to?*) Maybe you should try and figure it out. (

*How?*) I don't know. Anyone have any ideas?

"(

*Add up all the numbers and then divide four.*) That's an awesome idea. (

*Can we use calculators?*) No. (

*But...*) No. (

*But...*) No. (

*You're mean.*) Yep. (

*So we have to add up all the numbers by hand?*) That's not how I would do it. (

*How would you do it?*) I would start by adding all the numbers twice:

*Because you're crazy?*) Any other ideas? No? Well, what's 1+16? (

*17*) and what 2+15? (

*17*) and what's 3+14 (

*17*) and what's the pattern here? (

*They're all 17!*) And how many 17s are there? (

*16*)

"(

*So the answer is 17 times 16?*) (

*That's magic!*) Not so quick. Remember that I added all of the numbers twice. (

*So the answer is 17 times 16 divided by 2?*) Yes, that will give you the total. (

*Can we use calculators now?*) Of course not. (

*So we have to do 17 times 16 by hand?*) Not at all. Who can tell me how to simplify this?

"(

*Cancel the 2*) And? (

*17 times 8... is that the answer?*) Well we wanted to divide the answer by 4, remember?

"(

*So it's 17 times 2*) What's that? (*34!*) Fantastic! Alright, somebody figure out the 4 by 4 case. (*And then are you going to make us do the 5 by 5 case?*) Yep."
Epilogue.

Of the three seventh grade classes I taught (about twenty students a piece), three students managed to solve the 4 by 4 magic square. One student skipped recess to work on it. Many students did not take to this assignment. One girl in particular was straight-up pissed. (Things were thrown.) Eventually though, she figured it out and was elated.

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