This week I’m going to share math projects based on items you can purchase from small businesses (in the US) who make amazing math-related products. The first project is based on the tiles from Cherry Arbor Design:

Today we did a 3d printing project revisiting an angle sum that we’d looked at last week -> arctan(1/2) + arctan(1/3).

We started by reviewing how to approach the sum using complex numbers:

Next my older son explained a geometric way to approach the problem:

Now we went to Mathematica to create the 4 triangles using the RegionPlot3D function. It is a nice geometry exercise to have kids describe the boundary of a simple 2d object:

At the end of the day I had my younger son use the shapes to assemble the 3×2 rectangle and describe how this arrangement showed that the original angles added up to 45 degrees:

I like using 3d printing to help kids see math in a different way. The problem today was originally inspired from a section on complex numbers in Art of Problem Solving’s Precalculus book. It was nice to be able to use it to explore a little bit of 2d geometry, too.

This morning I had the boys each choose a chapter that they found interesting and we walked through the proof that Polster in that chapter.

My younger son went first – the idea he found interesting was dividing up a square in different ways. Here’s the introduction and an explanation of two of the four ideas:

Here’s how we finished up the last two proofs:

Next my older son found a neat proof relating the volume of a sphere to the volume of a cylinder and a cone. He struggled a little bit to understand the proof, but the struggle that takes place in this and the next video is a great way to see how kids learn and think about math.

Just so there’s no confusion, the formula he derives for the area of the slice of the cylinder / cone we are looking at isn’t right. He’ll discover the mistake and correct it in the next video.

Here we find the second formula that we need to show how the volume of a sphere relates to the volume of the cylinder / cone combination.

Finally, we revisited an old 3d print that we had showing the relationship between the volume of a sphere, cylinder, and a cone. The print is designed by Steve Portz and is on Thingiverse here:

This week my son had two pretty neat homework problems – one from his math class at school and one from Art of Problem Solving’s Precalculus book. I thought it would be a nice and easy project today to go back and review these two problems.

The first one was a geometry problem from his school math homework:

The second is a problem about complex numbers from Art of Problem Solving’s Precalculus book:

Last week I saw this really neat tweet from Tom Ruen:

The projections are closely related to quasicrystals as well. The Penrose tilings come from projections of a 5-cube (which stack infinitely in 5D like cubes in 3D). This shows all the rhombic dissections of a decagon.https://t.co/hmk3Vt0fHnhttps://t.co/knff5ywvn9pic.twitter.com/PRClCio2Hu

Yesterday my younger son and I talked through the decagons after building them from our Zometool set. Today we talked about the projection of the 5d cube.

Here are his initial thoughts:

My son was interested in comparing this 5d cube shape to a shape that we’d built previously. So we got that shape and continued the comparison. We also talked a bit about where else the number 5 appeared in the 5d cube and in our shape:

I’m so happy to have seen the conversation that Nalini Joshi got started on Twitter last week. We’ve had two super fun projects so far inspired by it!

Earlier in the week I saw a really neat twitter thread that had this post:

The projections are closely related to quasicrystals as well. The Penrose tilings come from projections of a 5-cube (which stack infinitely in 5D like cubes in 3D). This shows all the rhombic dissections of a decagon.https://t.co/hmk3Vt0fHnhttps://t.co/knff5ywvn9pic.twitter.com/PRClCio2Hu

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