# A “new to me” demonstration of the difference between Nassim Taleb’s “mediocristan” and “extremistan” thanks to Steve Phelps

Yesterday I saw a really neat tweet from Steve Phelps:

The idea he is studying goes like this:

Select three points uniformly at random inside of a unit square. What is the expected area of the circle passing through those three points?

This question turns out to have a lot of nice surprises. The first is that exploring the idea of how to find the circle is a great project for kids. The second is that the distribution of circle areas is fascinating.

I started the project today by having the kids explore how to find the inscribed and circumscribed circles of a triangle using paper folding techniques.

My younger son went first showing how to find the incircle:

My olde son went next showing how to find the circumcircle:

With that introduction we went to the whiteboard to talk through the problem that Steve Phelps shared yesterday. I asked the boys to give me their guess about the average area of the circle passing through three random points in the unit square. Their guesses – and reasoning – were really interesting:

Now that we’d talked through some of the introductory ideas in the problem, we talked about how to find the area of a circle passing through three specific points. The fun surprise here is that finding this circle isn’t as hard as it seems initially:

Following the sketch of how to find the circle in the last video, I thought I’d show them a way to find the area of this circle using ideas from coordinate geometry and linear algebra – topics that my younger son and older son have been studying recently. Not everything came to mind right way for the boys, but that’s fine – I wasn’t trying to put them on the spot, but just show them how ideas they are learning about now come into play on this problem:

Finally, we went to the computer to look at the some simulations. The kids noticed almost immediately that the mean of the results was heavily influenced by the maximum area – that’s exactly the idea of “extremistan” that Nassim Taleb talks about!

This project is a great way for kids to explore a statistical sampling problem that doesn’t obey the central limit theorem!

I really love the problem that Phelps posted! It is such a great way to combine fascinating and fundamental ideas from geometry and statistics

# Playing with Pascal’s triangle and angles hidden in cubes

In April 2018 I saw a great Numberphile video with Federico Ardila:

The project that we did after seeing that video is here:

Federico Ardila’s Combinatorics aand Higher Dimensions video is incredible

This week my younger son was learning about coordinates in 3 dimensions in his precalculus book and I though it would be fun to revisit some of the ideas about cubes from Ardila’s video.

We started by looking at cubes in 0 to 4 dimensions and discussing how we could see Pascal’s triangle hiding in the cubes:

In the last video we got a little hung up on the 4-dimensonal cube, so for the next part of the project we looked at the coordinates of the vertices of the various cubes to see if that could help us see Pascal’s triangle in the 4d cube.

Next we moved on to looking at the angles made by the long diagonal in the various cubes. This exercise was particularly nice since my younger son has been learning a little trig and my older son has been learning a bit of linear algebra.

For the final part of the project we looked at the 4-d cube. Here are zometool shape isn’t really helping us see the long diagonal. My younger son did a really great job seeing the pattern in the right triangles with the long diagonal. He also noticed the amazing fact that there is a 30-60-90 triangle hiding in a hypercube!

My older son was also able to find the same angle using ideas from linear algebra:

Definitely a fun project. It is fun to introduce coordinates not just for 3 dimensions, but for all dimensions at the same time. There’s also an enormous amount of fun math hiding in the seemingly simple idea of n-dimensional cubes, which makes this project sort of doubly fun!

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# Having the boys learn from Mathologer’s Fermat’s Christmas Theorem video

Mathologer put out a fantastic video last week:

I had the boys watch the full video and come up with two things that they thought were interesting. Here they explained their choices and gave a few thoughts about the video:

My younger thought the approximation of a circle by 1×1 boxes was interesting. Here we talked about that idea and sort of hand waved why the approximation gets good:

My older son thought that the concepts of the “good” and “bad” numbers was interesting. I let my younger son do a lot of the talking when we were talking about sums of squares. It was also interesting talk about why the proof that none of the integers of the form 4n + 3 can be written as the sum of two squares was easy, but the proof that all integers of the form 4n + 1 can be is hard.

I hope to return to the more complex questions the boys found interesting in a different project. Maybe next week!

# Sharing the tiles from Cherry Arbor Design with kids

Last week I was really lucky to be able to visit ICERM in Providence and saw some amazing mathematical tiles made by Cherry Arbor Design. Their website is here:

When I got home from that trip I ordered 3 sets of tiles. They arrived today!

Tonight I asked the boys to play around with a set and see what they could make. My younger son chose the Twin Dragon Tiles and played with them for 45 min! Unluckily I had a call that came in at roughly the 2 minute mark of the video below, but we resumed after that call. You can see from the video that he really enjoyed creating all kinds of different shapes:

My older son chose to play around with the Penrose Tiles. These tiles are completely stunning. Here’s his creation and what he had to say about the tiles:

The mathematical tiles and puzzles from Cherry Arbor Design are absolutely beautiful. If you are looking for something fun and math-y to get for someone for a present, definitely check out their selections!

# Sharing a intro calculus idea with my younger son inspired by Steven Strogatz’s Infinite Series appendix

Last week Steven Strogatz released two previously unpublished appendicies for his book Infinite Powers:

My older son and I did a fun project with Fermat’s idea. He’d taken calculus last year and the ideas Strogatz shared made for a really nice calculus review:

Sharing Appendix 1 to Steven Strogatz’s Infinite powers with my son

My younger son is in 8th grade and has not taken calculus. I thought some of the ideas about finding areas under simple curves would be interesting, so I tried sharing some of those ideas this morning.

We started by taking a look at the first page of Strogatz’s appendix and then talked about finding the area under $y = x^n$ for small values of $n$

Now we moved on to the case $n = 2$. He had the really neat idea of thinking that this piece of the parabola might be a quarter circle. That idea made for a great little exploration:

I asked for another idea had he decided to chop the parabola up into rectangles. This isn’t an idea that came out of the blue because we have talked about some intro calculus ideas before. I was still happy to have this idea jump to the front of his mind, though:

Finally, I shared the full Riemann sum calculation with him so that he could see how to arrive at the exact answer of 1/3. This part was not as much an exploration for him as it was just me showing him now to do the sum. I was ok with this approach as there is plenty of time after 8th grade to dive into the details of Riemann sums:

I’m very happy that Strogatz shared these unpublished appendixes. They are yet another great way for kids to see some introductory ideas from Calculus.

# Exploring trig and 2d geometry with 3d printing

This week I’ve been doing a fun 3d printing project with my younger son who is learning trig (from Art of Problem Solving’s Prealgebra book). We have used 3d printing to explore 2d geometry before – see some of the projects here, for example:

3d Printing ideas to explore math with kids

Exploring Annie Perkins’s Cairo Pentagons with kids

Evelyn Lamb’s Pentagons are Everything!

This week I had my son create, code, and then print some simple 2d shapes – the project combines ideas from trig, geometry, and algebra.

Here’s his description of the first shape -> a 3-4-5 triangle:

Here’s the 2nd shape – a 7-6-3 triangle. Creating this shape shows how ideas from introductory trigonometry come into play:

Finally, here’s a regular pentagon that we made yesterday. Unfortunately we made a mistake in the code for the print – mixing up a Sin() and a Cos(), but here is explanation of how to make the shape is correct:

I’d forgotten how useful 3d printing can be as a tool to explore 2d geometry – this week was a happy reminder of how fun those activities can be!

# What a kid learning trig can look like

My younger son is working through Art of Problem Solving’s Precalculus book this year. Today we were working through some problems in chapter 4 on right triangle trig. He had been working through this section on his own while I was traveling last week.

One of the problems struck me as one that would make a nice project – the problem gives two lengths of a triangle plus an angle and asks you to calculate the area.

Since the angle is 30 degrees, his solution to the problem did not really use trig:

Now I changed the angle to 40 degrees. My expectation was that this change would not produce a solution that was all that different – wow was I wrong.