Playing with geometric transformations

This week I bought a book by Greg Frederickson on the recommendation of Alexander Bogomolny:

Though I’ve hardly even scratched the surface of the book, just flipping through it showed me dozens of ideas to share with the boys.

Last night I had them skim through the book to find one idea each that they thought would be interesting to study.

My older son picked a project about hexagrams that we were able to study with our Zometool set (sorry that I forgot to adjust the focus on the camera . . . ):

The second project was one my younger son selected, but luckily he forgot how the dissection worked after picking it out last night. You can see from this part of the project that reconnecting the pieces into smaller shapes is a challenging project for kids even when they know it can be done.

Thanks to Alexander Bogomolny for making me aware of this book. I think we’ll have lots of fun playing around with the ideas we find!


What a kid learning math can look like -> working through a pretty challenging geometry problem

My older son had a really neat geometry problem on his enrichment math homework. The problem is this:

Let A_1, A_2, \ldots, A_n be the vertices of a regular n-gon, and let B be a point outside of the n-gon such that A_1, A_2, B form an equilateral triangle. What is the largest value of n for which A_n, A_1, and B are consecutive sides of a regular polygon?

His solution to this problem this morning surprised me because his starting point was “what happens if n is infinite?”

I asked him to present his solution tonight. It certainly isn’t a completely polished solution, but it is a great example of how a kid thinks about math

Lee Dawson’s dart question is great to share with kids!

Saw a great problem for kids on Twitter today:

I had both of the boys talk through it tonight. Their approaches were a little different.

Here’s what my younger son (in 6th grade) had to say:

Here’s how my older son (8th grade) approached the problem:

This is a great problem to get kids talking about arithmetic and also a little bit of number theory. I really loved hearing the boys talk through it.

Sharing an advanced expected value problem from Nassim Taleb with kids

Earlier in the week I saw this interesting problem posted by Nassim Taleb:

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:

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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?

Packing circles into a square

In the last two weeks – completely by coincidence – I’ve done two fun project with each of the boys related to circle packing.

The first was with my younger son and inspired by a Robert Kaplinsky tweet:

Sharing Robert Kaplinsky’s pipe stacking problem with my younger son

The second was with my older son and inspired by one of the problems on the Iowa State “problem of the week” collection.

The problem was about the limit of the area of circles packed into an equilateral triangle:

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A terrific problem to share with calculus and geometry students from the Iowa State problem collection

Today I wanted to combine the two ideas and look at two ways of packing circles into a square. First I introduced the problem and we looked at the problem of packing circles in a square stacked directly on top of each other. As in the Iowa State problem, we found a surprise in the area covered by the circles as the number of circles approaches infinity:

Now we moved on to the problem of “staggered stacking.” In this video I introduce the problem and let the boys try to figure out why this problem is a little bit harder than the stacking problem from the last video:

Now we began to try to solve the “staggered stacking” problem. Turns out this problem is really tough! There are a lot of things about – the number of circles is much harder to calculate – but we were able to make some progress on some of the easy cases:

Now we tried to calculate how tall the stack of circles is. I think the algebra here is close to the edge of my younger son’s math knowledge. But he does a great job of explaining how to calculate the height. The nice thing is that he remembered the main idea from the project inspired by Robert Kaplinsky:

Finally – we put all of the ideas together. There are a lot of them, but with a little math magic, they all fit together really well!

So, a fun project following the neat coincidence of seeing to other problems related to circle packing recently. I think all of these problems are great ones to share with kids learning geometry and also learning calculus.

Federico Ardila’s “Combinatorics and Higher Dimensions” video is incredible

Yesterday Numberphile published an absolutely amazing new video with Federico Ardila:

The video blew me away – it felt like such a great way to share ideas about higher dimensions with kids.

This morning I shared the video with my older son (my younger son had some school homework that he forgot to do . . . ). Here are his thoughts after seeing the video. One surprise (to me) is that he thought this way of thinking about subsets only works for small sets.

Next I had him label the corners of a cube using the method that Ardila discusses in the Numberphile video. Sorry that the labels on the tape didn’t show up so well, but I think the idea still comes through:

Next – we talked about two versions of the hypercube that we’ve looked at in the past. Then I asked him to pick one of those hypercubes and label the verticies using the ideas from the Numberphile video. I really believe that Ardila’s idea lets kids experience a 4 dimensional cube in a completely new way:

If you know kids who are interested in understanding the 4th dimension, have them watch and they play around with Ardila’s video. It is completely amazing!

The “standing a cube on a corner” ideas we talked about in the 2nd and 3rd videos come from this video from Kelsey Houston-Edwards. Here’s her video and the two projects that we did from it – using the ideas from her video in combination with the ideas in Ardila’s video give kids an amazing look at the 4th dimension

Kelsey Houston-Edwards’s hypercube video is incredible

One more look at the hypercube

Exploring a 13-14-15 triangle

My younger son was struggling with one of the challenge problems from Art of Problem Solving’s Introduction to Geometry book today. I hadn’t done a project with him this week, so I thought it would be fun to turn talk about this problem in detail.

Here’s the introduction to the problem and some of his initial thoughts:

He had a pretty good idea about how to proceed by the end of the last video. Here we started to work through the algebra. There is one simplification that helps get to the solution fairly quickly – it took him a little while to find it. Once he did, though, the solution came quickly.

Now all that was left was a little arithmetic and checking that the answer we found was correct.

At the end he gives a nice summary of the problem.

It turns out that I did a previous project on the 13-14-15 triangle with my older son:

Mr. Honner’s 13-14-15 triangle and a surprising unsolved problem

Amazing how useful this triangle is!

A terrific problem to share with calculus and geometry students from the Iowa State problem collection

This is another problem from the Iowa State problem collection:

Yesterday I looked at one of the number theory problems with boys:

Some of the Iowa State “problems of the week” are great to share with kids

Tonight I tried another terrific (though very challenging) problem with my older son:

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Here’s are his initial thoughts about the problem:

Now we rolled up our sleeves a bit and started to solve the problem. His first thought about what to do was to try to solve the problem with one inscribed circle and then with three inscribed circles:

The problem with three inscribed circles was giving him trouble so we moved on to a new movie and sort of started over on the three circle problem. While he was re-drawing the picture he was able to see how to make some progress:

Finally, having solved the problem with three circles, he moved on to solving the problem in general and found the surprising answer:

I really like these problems. Obviously not all of them are going to be accessible to kids, but the ones that are accessible are really amazing treasures!

Some of the Iowa State “problems of the week” are great to share with kids

Saw a tweet about a really nice collection of problems yesterday:

As I clicked through a few of the problems last night, I thought that several (though definitely not all!) would be nice ones to share with kids. When my older son got home from school today I asked him about the problem from January 8th, 2018:

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Here’s what he had to say – it is a really nice solution:

Next we moved on to the 2nd problem from January 2018:

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He walked away before we could start solving this one . . . . 🙂


When my younger son got home from school I asked him the same question. His work shows what a kid thinking through a math problem can look like:

One that didn’t go so well – talking about knight’s tour problems with the boys

Saw a neat tweet from Joel David Hamkins at the end of last week:

I thought it would be fun to talk through the knight’s tour problems with the boys today and end by showing them the infinite problem. I ran into trouble almost immediately when we began to talk about the tours on the 3×3 and 4×4 boards. The difficulty they had explaining was a big surprise to me. We ended up talking about the 4×4 problem for almost 30 min.

Tonight I sat down with each of them and asked them to talk me through the problem and explain why the knight’s tour on the 4×4 board was impossible. You can see that my older son (in 8th grade) was able to explain the problem pretty well, but my younger son (in 6th grade) still really struggled.

Here’s what my older son had to say:

Here’s what my younger son had to say:

Definitely a much harder problem for kids than I thought. Hopefully will have some time during the week to explore this and maybe a few other tour problems with them.