## Francis Edward Su: My Favorite Math Fun Facts @ NCTM Boston 2015

Of all of the presentations I attended at NCTM Boston, Professor Francis Edward Su's was the most fun. But, I guess that makes sense. Fun is in the title!

Professor Su teaches math at Harvey Mudd in California. He often teaches calculus to undergrads and recognizes that, for many, that may be the last math course they ever take (that was true for me). To help inspire his students to see the beauty of mathematics, he started sharing "Math Fun Facts" at the start of every class. These are 5-10 minute presentations on a cool idea from math. They are rarely, if ever, directly connected to that day's lesson.

Professor Su's presentation was just 10 of these facts presented to a room full of people who, presumably, already like math. In other words, he had found the right audience.

The facts came from nearly every field of math and beyond. We looked at the pythagorean theorem, combinatorics, card tricks, spherical geometry, and more. There were a few facts that were especially fun that I want to share here.

He asked us to imagine tying a rope at the base of the goalposts on a football field and then tying the other end of the goal post so that there was no slack (exactly 120 yards[3]Don't forget the endzones.). More simply, the rope is so tight that you could not lift the rope off the ground. Next, he said imagine that the rope was actually 1 foot longer than it needed to be. If you stood at midfield and lifted up the rope until there was no slack, how high off the ground would it be? Could you crawl under it? Walk? Drive a truck under it?

Your intuition tells you it would barely come off the ground. Using the pythagorean theorem, however, you can see that it could be lifted 13.43 feet in the air! Professor Su posts these facts on a website. Here is the football field fact. That fact reminded me of a very similar problem in which a string is wrapped around the world and you need to find out how much extra string is needed to lift it 1 foot off the ground over its whole length. He has that fact as well.

My favorite fact was a neat little proof he showed us. First he asked "is it possible to raise an irrational to a rational power and get a rational number?" Of course it is...

$\sqrt2$ is irrational, and 2 is rational

Lots of people were ready to volunteer this answer. So then he asked "is it possible to raise an irrational to an irrational and get a rational?" After the room was silent for a bit, he provided a wonderful proof.

Begin with $\sqrt{2}^{\sqrt2}$. If the result is rational, then it is possible. Otherwise...

If $\sqrt{2}^{\sqrt2}$ is irrational then raise it to$\sqrt{2}$$\big(\sqrt{2}^{\sqrt2}\big)^{\sqrt2}=\sqrt2^2=2$

Therefore either $\sqrt{2}^{\sqrt2}$ or $\big(\sqrt{2}^{\sqrt2}\big)^{\sqrt2}$ is an example of an irrational raised to an irrational power resulting in a rational.

This is a harder proof to explain in writing than it is to explain in person. I love that you don't even need to know if $\sqrt{2}^{\sqrt2}$ is rational in order to see that it is possible.[4]Professor Su told us at the end that$\sqrt{2}^{\sqrt2}$ is in fact irrational.

I definitely appreciated Professor Su's desire to showcase the beauty of mathematics. It is so easy to get locked into the content and standards we need to teach that we sometimes forget to showcase that aspect of mathematics.

The idea of opening class with something akin to Su's "Fun Facts" is an exciting idea that has me thinking about building interesting openers for my class. My school's schedule is a bit funky. Our schedule rotates over six days. On 2 of those days I see my students for 90 minutes, on 3 I see them for 45, and on 1 I don't see them. I never really get that much time though. Our schedule doesn't have travel time and we don't have a bell system.

At the start of 5th grade the students arrive promptly ready to work. As they get older, though, they socialize more and trickle in. I have experimented with 'Do Now's and similar activities in the past, but he 45 minute periods just don't really leave me with enough time to actual do a full lesson afterwards. I need something more desirable. Something they WANT to get there for.

One idea I have is to use Estimation 180. I really like that these images have context clues and sometimes build from day to day. I think this would also allow me to infuse more thinking about units into my teaching. My 5th and 6th graders have very little knowledge about what units are or how they work. This project has been added to my summer to do list (post coming later).