# Ed Southall’s geometry problem

A problem that Ed Southall posed on twitter caught fire on the internet last week:

I thought it would be fun to share with the boys for our project today. Here are their initial thoughts on the problem. My younger son’s initial guess at the amount of area shaded was “a little bit bigger than 1/4” and my older son’s guess is was 1/3.

In the last video the boys decided to use coordinate geometry to solve the problem. Here’s that work:

Next I wanted to have them try to solve the problem by folding paper. Studying eometry through folding is an approach that I always want to do more of, but almost never do more of. This tweet from Patrick Honner last week reminded me to fold more:

Here’s how I introduced the idea of approaching this problem through folding to the kids:

The boys folded the patty paper off camera (and without me – I was publishing the previous videos). Here are the ideas that came from the time they spent folding. A bit of a surprise to me was one of the ideas really was a folding idea, and one was really a similar triangles idea that my younger son noticed when he was playing with the paper folding.

So, a fun project on a couple of different levels. The problem is definitely great, and it was also a really nice surprise to see ideas that at first glance seem like similar triangles emerge from playing around with paper folding.

# Geometric tilings inspired by Annie Perkins

Annie Perkins has been sharing some amazing math art on Twitter. Probably as good of an example as any is the work she shared while I was writing up this blog post!

If you go back through her feed you’ll find amazing pic after amazing pic after amazing pic.

I’d been wondering how to share some of these ideas with the boys. We recreated one of the drawings using our Zometool set a few weeks ago:

Making one of Annie Perkins’s drawing from Zometool

After that I saw this tweet and ordered a set of tiles:

As a point of full disclosure, the tiles arrived in a box from the “Post of Iran”. I do not know what the status of sanctions and trade between the US and Iran is right now, so I don’t know if I accidentally tripped over a rule that I shouldn’t have tripped over by ordering these tiles. Hopefully these plastic geometric tiles are not somehow banned in the US right now, but in any case they did arrive.

Today I had the boys make some shapes. The first one we made was from the company’s website:

Next I had the boys create their own shapes. Here is what my older son made and what he had to say about the shape:

Here’s what my younger son made:

My younger son seemed to especially enjoy this project and creating his own pattern. Hopefully there will be more patterns to come!

My younger son is working his way through Art of Problem Solving’s Introduction to Geometry book. He’s been doing almost all of the work on his own – I just check in every now and then.

Today we had a little extra time so we did a project on the section he’s currently studying -> angle bisectors.

We started with some of the basic properties -> why is the intersection point of the angle bisectors the same distance away from every side:

Next we moved on to a slightly harder problem -> what is that distance?

This problem gave him a little trouble. BUT, after a hint to think about how the 1/2 base * height formula for the area of a triangle might help, he made some nice progress:

Finally, I had him work put the ideas we talked about to work in a specific triangle. Here he finds the radius of the inscribed circle in a 3-4-5 triangle:

It was interesting to see him pull some old ideas from geometry in to help understand some of the new ideas he’s learning here. I haven’t looked ahead in the book, but assume that the angle bisector theorem is coming soon. That theorem was really difficult for my older son to grasp, so I’m going to try to work a little more carefully with my younger son in the coming weeks to help him with any difficulty he might have there.

# Sharing some number theory with kids thanks to Jim Propp’s “Who knows two?” blog post

Jim Propp published a terrific essay last week:

Who knows two? by Jim Propp

Yesterday we did a fun project about card shuffling using the ideas from Propp’s post:

Sharing a card shuffling idea from Jim Propp’s “Who knows two?” essay with kids

Today we did a second project for kids based on some ideas from Propp’s post. The topic today was “primitive roots”. Unfortunately this isn’t a topic that I know well and I messed up one explanation in the first video below. Oh well . . . still a really neat idea to share with kids.

So, I started by introducing the concept of primitive roots by reminding them of the 8 card and 52 card shuffles we looked at yesterday (pay no attention to my explanation about powers and mods at the end. It will become clear in the next video that I goofed up that explanation . . . . ):

Now we looked at some examples of primitive roots with small numbers. These simple examples give a nice way for kids to get a little bit of arithmetic practice and also help them see the main ideas in the problem that we are studying.

After working through these smaller examples, we moved to the computer to continue studying the problem. My older son noticed that the examples that seemed take the longest time to work were primes, but not all primes took a long time. That’s exactly the math idea we are looking at here.

Next we made a small change to the program to study all of the odd numbers up to 1,000 all at once. After correcting a little bug we found that the numbers we were looking for were indeed all primes.

We wrapped up be talking about what was known and what wasn’t known about these primitive roots. I was happy that my older son seemed to be particularly interested in this problem.

Definitely a fun project. It is always fun to find unsolved problems that are accessible to kids (and lots of them seem to come from number theory!). We will definitely have to do some follow up projects to explore the ideas here in a bit more detail.

# Sharing a card shuffling idea from Jim Propp’s “Who knows two?” essay with kids

Jim Propp published a terrific essay last week:

Who knows two? by Jim Propp

One of the topics covered in the essay is a special type of card shuffle called the Faro shuffle. We have done a few projects on card shuffling projects previously, so I thought the kids would be interested in learning about the Faro shuffle. Here are our prior card shuffling projects:

Card Shuffling and Shannon Entropy

Chard Shuffling and Shannon Entropy part 2

Revisiting card shuffling after seeing a talk by Persi Diaconis

I started the project by asking the kids what they knew about cards. They remembered some of the shuffling projects and then introducing the idea of the Faro shuffle.

My younger son thought he saw a connection with pi, which was a fun surprise.

We continued studying the Faro shuffle with 8 cards and looked for patterns in the card numbers and positions. The boys noticed some neat patterns and were able to predict when we’d return to the original order of cards!

Next we looked at the paths taken by individual cards. My older son thought that there might be a connection with modular arithmetic (!!!) and the boys were able to find the pattern. I’d hoped that finding the pattern here would be within their reach, so it was a really nice moment when he brought up modular arithmetic.

Finally, we wrapped up by talking about how to extend the ideas to a 52 card deck and calculated how many Faro shuffles we’d need to get back to where we started.

I think that kids will find the idea of the Faro shuffle to be fascinating. Simply exploring the number patterns is a really interesting project, and there’s lots of really interesting math connected to the idea. I’m really thankful that Jim Propp takes the time to produce these incredible essays each month. They are a fantastic (and accessible) way to explore lots of fun mathematical ideas.

# 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 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:

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:

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.