We did a project talking about it back then, but I decided to revisit it today with my younger son.

First we recreated part of the drawing using our Zometool set, and then I had my son share his thoughts about the shapes – it is always so fun to hear what kids see when they look at shapes:

Next I had my son talk about the Zometool shape we made:

With school nearly over for the year I was looking for some ideas to explore with my younger son over the summer. I thought some introductory trig ideas might be fun since he saw some basic right triangle trig inn his math class at school this year.

The first thing that came to mind for me was a short exploration of what the functions sin(x) and cos(x) look like. It was fun to hear his ideas about these functions evolve over the course of our discussion this morning.

I started by asking him what he already knew:

After that introduction, I introduced the unit circle and asking him to make a guess as to what the graph of y = cos(x) would look like:

In the last video we looked only at the interval 0 to 90 degrees. Here we made a sketch of y = cos(x) and y = sin(x) from 0 to 360 degrees. It was fun to hear what he thought of these graphs as he was drawing them:

The discussion we had today was really fun and even had a few nice surprises. I’m excited to continue this discussion a bit more over the next couple of weeks.

I left two copies of the puzzle for my son to work through while I was out this morning. For the first run through I asked him to solve the puzzle as it was stated. Here’s his work and his explanation:

For the second run through I asked him to solve the problem assuming that the radius of the circle was X rather than 5. This was first step in what I was hoping would be an interesting algebra exercise. Here he was successfully able to use the quadratic formula even though the equation he found had 2 variables:

For the last part of the project I wanted to see if he could factor the equation he found in the last video. This turned out to be a significantly more difficult challenge, but he figured out how to do it just as we ran out of space on the memory card!

I suspected that the factoring challenge would be more difficult than simply using the quadratic formula, though I didn’t realize how much different it would be. I might try to find some more challenges that involve multiple variables just to get a bit more practice with these ideas.

Last night I asked my younger son what he wanted to do today for a project and he said that he wanted to talk about the permutahedron a bit more. In yesterday’s project we talked about the permutations of the set (1, 2, 3, 4), so today we started by going down to some simpler sets of permutations:

Next we looked at the shape made by the permutations of the set (1, 2, 3). The way my son thinks through this problem shows why I love sharing ideas from math research to my kids.

To wrap up today we dove a little deeper into one of the ideas we talked about yesterday – in the permutations of the set (1, 2, 3, 4) is there a permutation that requires 4 or more flips to get back to the starting point of (1, 2, 3, 4)?

The permutahedron is a really neat shape to explore with kids, and hearing them talk about and think through the shape itself is incredibly fun.

We’ve done lots of projects with Catriona’s puzzles in the past, so just search for “Catriona” and you’ll find them.

My younger son spent some time off camera solving the puzzle and then I asked him to walk through his solution. His solution gets the main idea about tangents and circles, and then computes the radius of the semicircles using the Pythagorean theorem:

Typically when we play with one of Catriona’s puzzles I have my son look through the twitter thread afterwards and find a neat solution. I took a different approach today and showed him how to use similar triangles to get to the answer with slightly less computation:

I really like Catriona’s puzzle. I also think that my son’s explanation is a great example of what kids doing math looks like.

The first example we looked at were the “hexagonal” numbers. Here my son explains what those numbers are and gives a little introduction into the surprising geometric idea that helps understand these numbers:

My son had some difficulty seeing the argument from the pictures in the book, so we tried out a few examples (a few days before this project) using snap cubes. Here’s his explanation of the surprising geometry:

The next thing we looked at in the book were the “tetrahedral” numbers. The book game an amazing proof showing a formula for these tetrahedral numbers. Here he explains this clever proof:

This was a really fun project, and I’m also really happy that this book is teaching my son a bit about reading math books – sometimes even reading and understanding just a couple of pages can take time.

Since I didn’t think I did a great job communicating the main ideas in Apostol’s proof yesterday, I wanted to try again today. First we started with a review of the main ideas:

Next we tried to take a look at the proof through a slightly different lens -> folding. I learned about this idea yesterday thanks to Paul Zeitz. It takes a bit of time for my son to see the idea, but I really like how this approach helped us understand Apostol’s proof a bit better:

Finally, to really drive home the idea, I asked my son to see if he could see how to extend the proof to show that the square root of 3 is irrational. We were down to about 3 min of recording time, unfortunately, so he didn’t finish the proof here, but you can see how a kid thinks about extending the ideas in a proof here:

So, as I was downloading the first three films, my son continued to think about how to use the ideas to prove that the square root of three was irrational. And he figured it out! Here he explains the idea:

I’m definitely happy that we took an extra day to review Apostol’s proof. It feels like something that is right on the edge of my son’s math ability right now, and I think really taking the time to make sure the ideas could sink in helped him understand a new, and really neat idea in math.

I thought it would be fun to share this proof with my younger son since the geometric ideas in it are both surprising and super interesting. Unfortunately this one didn’t go nearly as well as I’d hoped. I missed a good idea that he had and got caught up in a few details that weren’t that important. Oh well, even after 10 years of doing these projects, I don’t have a good feel ahead of time for how they’ll go.

That said, here’s what we did. I started by having him walk through what is probably the most common proof that the square root of 2 is irrational:

Next we looked at Apostol’s proof and talked about some of the geometric ideas, and I just 100% missed that he was absolutely on the right track:

Now we took a look at an algebraic approach to the problem using the Pythagorean theorem. This part also didn’t go as well as I hoped and I might revisit it tomorrow just to make sure that these algebraic ideas made sense:

Finally, we came back to the geometric ideas since I realized that he was on the right track. Unfortunately I spent way too much time at the end of this part on a minor point. But hopefully the main geometric idea that we talk through in the first half of this video came through ok.

It is always disappointing when these projects don’t go quite as planned – I definitely want to push the “try again” button on this one.

I had the boys work on the problem on their own and then talk through their progress.

My older son went first – his solution is along the same lines as most of the solutions in Catriona’s twitter thread, though is reasoning is pretty interesting to hear:

My younger son went next. He wasn’t able to find the solution on his own, but was able to get there while we talked about his work. I’m sorry that I forgot the camera was zoomed in on the paper here. I do zoom out a little over half way through. Hopefully the words are clear even if some of the work is off screen:

At the end of the last video my son had worked through the main idea of the problem. Here he finishes the solution and talks about what he liked about the problem:

As usual, having the boys work through one of Catriona’s puzzles made for a great project. I really liked the algebra / geometry combo that this problem had as I think that was great practice for my younger son. I also think the more intuitive solution my older son had shows how mathematical intuition develops as kids get older.

This week I was doing a fun overview of higher dimensions with my younger son and we finished by playing around with the famous “central sphere” problem. The earliest reference I know to the problem is in Chapter 11 (“Spheres and Hyerspheres”) of Martin Gardner’s Collossal Book of Mathematics. There the problem is attributed to Leo Moser. I learned about the problem from hearing Bjorn Poonen discuss it with a student.

There are some nice youtube videos about the problem – this one by Kelsey Houston-Edwards is the one I watched with my son this week (the whole video is terrific, but I’m starting it around 6:15 when the discussion of the central sphere problem begins):

Grant Sanderson also has a video about the problem that I’m going to have my son watch later today:

I started the project last night by introducing the problem in 2 dimensions – just like both videos do. My younger son is in 9th grade and calculating the radius of the inner circle is a problem he was able to solve:

Next we moved to 3 dimensions and again my son was able to find the radius of the central sphere and also guess at the formula for the radius of the central sphere in any dimension:

Now we tackled two well-known questions:

(i) Is there a dimension where the radius of the central sphere is larger than the radius of the spheres in the smaller boxes?

(ii) Is there a dimension where the diameter of the central sphere is larger than the side length of the large box?

Finally, off camera we investigated the amazing question that as far as I know is due to Bjorn Poonen:

Is there a dimension where the volume of the central sphere is larger than the volume of the box?

We had to work in Mathematica and work with logarithms because some of the numbers in the calculations got so large. This video is a summary of what we found (and also several interruptions from our cat):

Definitely a fun problem – higher dimensions sure are strange 🙂