The program makes the ideas behind Fourier transformations accessible to kids and I decided to share the program with the boys this morning. So, I had each of them play around with it on their own for about 10 to 15 min. Here’s what they thought was interesting. (sorry for all of the sniffing – I’ve got a cold that’s been kicking my butt for the last few days):

(1) My older son who is in 9th grade:

(2) My younger son who is in 7th grade – it is really fun to hear how a younger kid describes advanced mathematical ideas:

I think Swanson’s program is a great program to share with kids – feels like at minimum it would be fantastic to share with kids learning trig.

It inspired me to do a project on prime numbers with the boys. So, I grabbed my copy of Martin Weissman’s An Illustrated Theory of Numbers and looked for a few ideas:

We began by talking about why there are an infinite number of primes:

Next we moved on to taking about arithmetic sequences of prime numbers. There are a lot of neat results about these sequences, though as far as I can tell, they have proofs way beyond what kids could grasp. So instead of trying to go through proofs, we just played around and tried to find some sequences.

I also asked the boys how we could write a computer program to find more and they had some nice ideas:

Next we played with the computer programs. Sorry that this video ran a bit long. As a challenge for kids – why couldn’t we find any 4 term sequences with a difference of 16?

Finally, we looked at Evelyn Lamb’s joke to see if we could understand it!

It is definitely fun to be able to share some elementary ideas in number theory with kids!

I saw an amazing tweet from Craig Kaplan this week:

Tactile is a modern reimplementation of my PhD code for manipulating isohedral tilings, now available as open-source libraries in C++ and Javascript. I've also created a few fun web-based apps for playing with tilings. https://t.co/vnt8hLIPbl

Ever since seeing it I’ve been excited to share the program with the boys and hear what they had to say. Today was that day 🙂

So, this morning I asked the boys to take 15 to 20 min each to play with the program and pick 3 tiling patterns that they found interesting. Here’s what they had to say about what they found.

My older son went first. The main idea that caught his eye was the surprise of distorted versions of the original shapes continuing to tile the plane:

My younger son went second. I’m not sure if it was the main idea, but definitely one idea that caught his attention is that a skeleton of the original tiling pattern seemed to stay in the tiling pattern no matter how the original shapes were distorted:

Definitely a neat program for kids to play around with and a really fun way for kids to experience a bit of computer math!

Yesterday Nervous System in Somerville, MA had an open house and I was lucky to have a few hours free while the boys were at their karate black belt tests. Visiting their shop was absolutely incredible:

At the open house I bought two new puzzles. The boys had seen one previously at Christmas, too. For our project today I’d already wanted something on the easy to talk about / less heavy math side because of the black belt tests yesterday, so talking about the new puzzles was perfect.

We started with the geode puzzle – one of the fun things we talked about was how the boys thought the computer generated the geode shape:

After the introduction to the puzzles, we moved on to talking about the challenge of putting the puzzle together. Favorite line – “once you get started, it gets pretty hard.” Yep!

Next I showed them the latest creation from Nervous Systems – an “infinity puzzle” inspired by the Mobius strip!

I was incredibly lucky to be able to buy one of the infinity puzzles yesterday. So, for the last part of today’s project we did an unboxing:

If you know kids who like puzzles – or you like puzzles! – all I can say is the Nervous System puzzles are absolutely incredible.

3 hours after we finished the project this morning, my younger son had returned to the Geode puzzle:

Solving this problem requires calculus, and trig to even begin to understand how to approach it, but it still seemed like one that would be interesting to talk through with kids. Especially since a Monte Carlo-like approach is going to lead you down a surprising path.

So, I presented this problem to the boys this morning. It took a few minutes for them to get their arms around the problem, but they were able to understand the main ideas behind the question. That made me happy.

Here’s the introduction to the problem:

Next I asked the boys what they thought the answer to this question would be. It was fascinating to hear their reasoning. Both kids had the same guess -> the expected average distance was 1.

Now we went to the computer to see what the average was when we did a few trials. We started by doing 100 trials to estimate the average and then moved up to 10,000 trials.

Next we went to 1 million trials and found a few big surprises including this amazing average:

We wrapped up by discussing how you might get an infinite expected value by looking at the values of Tan(89), Tan(89.9), Tan(89.99), and so on. It was interesting for them to see how individual trials could have large weights, even with large numbers of trials.

Definitely a fun project to show kids, and a nice (though advanced) statistics lessonm too -> What happens when the mean you are looking for is infinite?

One warning – this is not a “popular math” book, it is pretty math heavy. Flipping through the first 1/4 of the book, I really enjoyed the presentation and was once again reminded of the surprising fact that foundational research on basic gambling problems was being done in the late 1950s and early 1960s. An accessible and incredibly interesting account of some of this work can be found in Ed Thorp’s autobiography “A man for all Markets.”

One nice example from the beginning of the book relates to gambling in a 50/50 game called “red and black.” Think of the game as trying to guess the color of a card pulled from a randomly shuffled deck, or just betting on a coin flip. If you want to turn, say, $100 into $1,000 by betting on this game, what is your best strategy?

IF you are interested, a shorter account of this problem (with accompanying practice problems) can be found in this nice summary paper by Kyle Seigrist published by the Mathematical Association of America.

Because this particular gambling problem is accessible to kids, for today’s project I wanted to introduce the idea of 50/50 gambles and ask them what they thought the optimal gambling strategy would be. The specific question is what is the best strategy to follow if you want to try to turn $100 into $1,000?

They had some absolutely terrific ideas. My 6th grade son practically suggested the betting strategy from the Kelly criterion!

Next we turned to the computer to study this game in Mathematica. We looked at some simple betting ideas first. So, if we want to turn $100 into $1,000 in this game, what happens if we bet $100 on each bet? What happens if we bet $50 on each bet?

After seeing the surprising results from the fist set of trials, we looked at the gambling strategies that the boys proposed. First we looked at a version of the strategy that my old son suggested -> basically bet the maximum amount every time (except when you don’t need to bet the max amount to reach $1,000).

Are you more or less likely to turn $100 into $1,000 with this strategy?

Now we checked the betting strategy that my younger son suggested -> bet 1/2 your money each time (except when you don’t need to bet that much to reach $1,000).

The boys had some pretty interesting ideas about what would happen here.

So, definitely a fun project and the result is pretty surprising (at least to me!) -> in 50/50 games your betting strategy doesn’t matter.

My son had an interesting problem on his enrichment math homework this week, and it gave him a lot of trouble this morning:

Interesting problem on my son's enrichment math homework. Solve { x^2 – xy + y^2 = 13 and x – xy + y = -5 }. The section is on solving quadratics. I wonder if the idea they are trying to teach with this problem is u-substitution (u = x + y, v = xy) or something else.

Tonight I thought it would be good to talk through the problem since I think the main idea he needed to solve it was new to him.

Here’s the introduction and some of the ideas he tried this morning:

Next we took a look at the equations on the computer and talked about some of the ideas we saw:

After looking at the graphs of the equations on the computer we came back to the whiteboard to talk about substitution.

Finally, having worked through the introductory part of u-substitution in the last video, I let him finish off the project on his own.

I can’t remember talking through this topic previously, but it was fun. It is always neat to be there when a kid is seeing a math topic for the first time.