# Difficult and not difficult

As I wrote this morning on twitter:

The outcome isn’t really surprising at all since:

(1) I’ve never taught young kids before,
(2) I’ve never taught elementary math before,
(3) The experience with one kid probably doesn’t translate in any way to the other kid,

and . . . well, I could probably get to (100) without much difficultly.

Still, despite maybe not being intellectually surprising, it still surprises me. All. the. time.

We had friends staying overnight and got started with school a little late this morning. My younger son was starting a new section in his number theory book today – “Units Digits.” We went through a few examples of finding units digits. The problems we did were mainly introductory problems like: find the units digit of 4*23, 214*23, and 492*5137. Then we did the same exercise for a few powers of 4.

With those examples out of the way I wanted to explain how looking at patterns can help you find the last digit of large powers. I thought the large powers would make for a fun movie project so I started off our movie by asking him to find the last digit of $3^{1000}$. He worked through the problem as if we’d been studying this subject for weeks. I expected the transition from finding the pattern to evaluating what the pattern would be at the 1000th step to be much more difficult. The approach that I walked through in the 2nd half of the video is what I was expecting to be talking about (in pieces) for the entire discussion:

While I was working with my younger son, my older son was working on a few old math contest problems. The one linked here gave him quite a bit of trouble:

2003 AMC 10B Problem 5

Here’s the problem w/o the link:

“Moe uses a mower to cut his rectangular 90-foot by 150-foot lawn. The swath he cuts is 28 inches wide, but he overlaps each cut by 4 inches to make sure that no grass is missed. He walks at the rate of 5000 feet per hour while pushing the mower. Which of the following is closest to the number of hours it will take Moe to mow the lawn: (a) 0.75 (b) 0.8 (c) 1.35 (d) 1.5 (e) 3.”

Contributing to my surprise was that first the first time ever last weekend he helped me mow the lawn. Ha!

If it wasn’t 35 degrees and raining this morning, I would have gone out in the back yard with him and mowed for a bit just for context! More to the math points, though, I was genuinely surprised at how much difficulty this problem gave him. Two of the larger struggles came from:

(1) The interaction of the 28 inch wide cut and the 4 inch wide overlap.

And, yes, this part of the problem is gimmicky, no question. However, even drawing a picture of what was going on in this problem was hard for him. We probably talked about it for 15 minutes before he had the “aha” moment of realizing you were essentially moving across the lawn in 2 ft steps.

(2) The transition from knowing you were moving in 2 ft steps to figuring out how long it would take to mow the lawn.

We had a picture on our whiteboard of a rectangle chopped up into a bunch of thin (90 foot long) strips. Maybe the long talk about the first half of the problem sort of used up all of his mental energy, who knows, but this part took a lot longer to work through than I would have guessed. Even when we got to the last bit and just needed to evaluate (75 strips) * (90 ft / strip ) / ( 5000 ft / hour ) he told me that he was worried that we hadn’t solved the problem correctly because the units were wrong.

I don’t mind struggles – in fact I want struggles. I’m just surprised (constantly) when I don’t see them coming. In fact, this morning when I looked at the problems he was going to be working on I was sure this was the one that we’d be talking about:

2003 AMC 10B Problem 7

I wonder how many years it will be until I get good at seeing these struggles ahead of time?

# A great piece on Grothendieck by Ed Frenkel and a nice problem for students interested in math

[note: home sick with some stomach bug for the last two days – sorry for what is surely a bit of a rambling post]

Ed Frenkel published a nice piece in the New York Times today on the life and work of Alexander Grothendieck.

The Lives of Alexander Grothendieck, a Mathematical Visionary

In addition to Frenkel’s perspective on Grothendieck, what caught my attention was an almost off-hand observation about complex numbers that is really fascinating. I know it would have been quite a head scratcher for me in high school so I thought it would be fun to write about. Here’s the comment about the equation $x^2 + y^2 = 1$:

“One can show that the solutions of [$x^2 + y^2 = 1$] in complex numbers are points of an entirely different space; namely, a plane with one point removed.”

Students familiar with the equation $x^2 + y^2 = 1$ probably have only thought about this equation when both of the variables $x$ and $y$ are real numbers (when the solution is the familiar unit circle). The extension to complex numbers is a nice mathematical surprise.

So how can you think about Frenkel’s example? An excellent starting point is Richard Rusczyk’s sample solution for problem #25 of the 2013 AMC 12. The video below is a great way for students to see the power of geometric reasoning with complex numbers:

An approach similar to what Rusczyk outlines above is also a good way to start thinking about Frenkel’s equation. Try a few examples first – if $x = 10i,$ for example, what values of $y$ will satisfy the equation $x^2 + y^2 = 1$ (remember that both $x$ and $y$ are complex numbers)?

Now, if you have a generic value of $x$, what values of $y$ will solve the equation? You’ll find that there are 2 values of $y$ for most values of $x,$ though importantly, not all.

Next is a real geometric leap – if every point $x$ in the complex plane paired with exactly two points in Frenkel’s equation, seems as though the solution to the equation would be equivalent to two copies of the complex plane (possibly glued together in some strange way). Though it is challenging for sure, it is fun to think about what’s different from the situation I just described – in what way is the situation Frenkel describes similar to a plane with a point missing?

Away from this fun example of geometry with complex numbers, it was nice to see Grothendieck’s work described to the public. Another recent article about mathematicians written for the public was Michael Harris’ piece in Slate about the Breakthrough Prizes in math:

Michael Harris on the Breakthrough Prizes in Math

One of Harris’ points caught me off guard:

“Tao—the only math laureate with any social media presence (29,000-plus followers on Google Plus)—was a guest on The Colbert Report a few days after the ceremony. He is articulate, attractive, and the only one of the five who has done work that can be made accessible to Colbert’s audience in a six-minute segment.”

I was surprised to hear that Harris thought that the work of Jacob Lurie, Richard Taylor, Maxim Kontsevich, and Simon Donaldson really could not be made accessible to the public. Surprised enough, actually, to ask Jordan Ellenberg on twitter if he agreed with the statement:

Though his answer was not really a shock, it still disappoints me a little that work of these researchers is so inaccessible to the general public. Hopefully Frenkel, or other mathematics writers, can find a way to bring the beauty of their work to the public. I’d love to know more about Lurie’s work, or any of their work, frankly.

More public lectures like the one Terry Tao gave at the Museum of Math would be great, too. I’ve already done three projects with my kids already based on that lecture. It is amazing for them to be able to learn from Terry Tao!

Terry Tao’s MoMath lecture Part 1 – The Moon

Terry Tao’s MoMath lecture Part 2 – Clocks and Mars

Terry Tao’s MoMath lecture Part 3 – the Speed of Light and Paralax

It would be wonderful if there were more opportunities like Tao’s public lecture to introduce kids to research mathematicians and more article’s like Frenkel’s, too. Despite being home sick, Frenkel’s article made my day today.