Having the boys work through a brilliant puzzle from Catriona Shearer

We returned from our vacation to Iceland yesterday. I didn’t have anything planned for a math project this morning, but fortunately ran across this fantastic puzzle from Catriona Shearer:

After seeing this puzzle my plan for the morning was to share it with the boys, see what they had to say, and then see what ideas they had for solving it.

Here’s their initial reaction. Their first idea was to try a few different configurations to test out the idea that the configurations didn’t change the area:

Now they solved for the area when the two squares had the same size:

The next idea they pursued fascinated me – they wanted to solve the puzzle using the assumption that the big square had twice the side length of the smaller square. Eventually this idea is going to lead to a big surprise!

It took a minute to get going with the algebra, but then they began to make progress. Seeing this progress happen live is why I love working through math problems with my kids

Sorry this video ends so abruptly – the memory card in the camera filled up.

Downloading the movies and clearing the memory card gave the boys a few minutes to think a bit more about the problem. When we restarted filming they had a plan. The algebra work was a little tricky for my younger son, I think, but we made it through and showed that the total area of the squares in this configuration was the same as the area in the last one.

They realized that their solution wasn’t a full solution to the problem, but I’m really happy with the work they did. After we finished with this last video I showed them the full solution off camera.

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Sharing the “lonely runner” problem with kids

I saw a really neat unsolved problem shared on twitter this morning:

I thought it would be a really fun problem to talk through with the boys – especially since kids can definitely say something about the n = 2 and n = 3 case.

Here’s the introduction to the problem:

After the introduction we talked about the n = 2 case.

Now we moved on to the n = 3 case. The boys had an interesting idea on this one that caught me a little off guard. The neat thing is that they were able to come up with a pretty good hand waving argument that in the general case the three runners would eventually form an equilateral triangle.

After that neat hand waving argument from the last video, we tried to find a more precise argument for solving the n = 3 case. Solving it in general was just a bit out of reach, but they did find an argument for why at least one runner would become lonely.

Since we ended up pretty close to the proof of the general case for n = 3, I explained the last step after we finished with the last video. I think this is a really nice problem for kids to play around with and I think that lots of young kids will find the ideas in this problem to be really fascinating.

Revisiting James Tanton’s paper cutting exercise

This week I’m going to be giving a talk at the math camp at Williams college. The talk this year is going to be based on an amazing paper cutting project that I learned from James Tanton’s book Solve This:

As that tweet from 2016 suggests, we’ve looked at these paper cutting ideas before:

An Absolutely Mind-Blowing project from James Tanton

Today I had both kids try out the project with two shapes. One purpose of today’s project was to remind me of the rough paper size we need to do this project (folding an 8 1/2 x 11 inch sheet of paper into thirds – so roughly 3×11 inch strip – worked pretty well). But I was also interested to see what the kids thought of the shapes because the results are so surprising!

Also, the snoring in the background is our dog – lol 🙂

(1) Older son shape 1:

(2) Older son shape 2:

(3) Older son shape 3:

(4) Older son shape 4:

Btw – Solve This is an amazing book. I see several used copies on Amazon right now, and I can’t recommend it enough!

Following up on our “angles in Platonic solids” project

Yesterday we did a fun project on angles in Platonic solids:

Talking about Angles in Platonic Solids

We ended up getting a really neat comment from Allen Knutson on that project. He said:

“You should look for the three orthogonal golden rectangles in an icosahedron! They’re easy to see in a Skwish toy.”

My older son was working on a different math project today, so I had my younger son build an icosahedron out of zome and look for those rectangles. Here’s what he had to say after building the shape:

During his description he found a second rectangle. So, off camera, he filled in that rectangle and then had a bit more to say:

So, thanks to Allen Knutson for the comment that inspired this project, and thanks (as always!) to Zometool for making it so easy to get kids talking about math!

Talking about angles in Platonic solids

My younger son wanted to do a Zometool project today and since my older son is currently learning about the dot product, I thought it would be fun to talk about angles in some platonic solids.

This idea turned out to be one that was better in my mind than it was in practice – ha! – but it was still a nice project even though it got a bit messy.

We started by talking about angles in a cube:

Next we moved to the octahedron:

Here we go through the steps to calculate the angle between two faces in the octahedron:

Finally, we wrap up by looking at the fun surprise that a hypercube has a 30-60-90 triangle hiding in it! My younger son got a little confused about how to find the lengths of some of the vectors we were looking at, so we went slow. It is really fun to see how some relatively simple ideas let you explore hard to visualize objects like a 4-dimensional cube!

Playing with an amazing program on “Scissors Congruence” shared by Francis Su

I saw an incredible tweet from Francis Su yesterday:

After exploring the program a little bit last night I thought it would be really fun for the boys to play with it this morning. So, I showed them the basics of how the program works and had them each play around for 10 min. Here are their thoughts:

Younger son (in 7th grade):

Older son next (in 9th grade):

I am really happy that this program won an NSF award – what an incredibly fun way to share an advanced math topic with everyone!

Intro to Linear Algebra

Having finished a single variable calculus class with my son this school year, I’ve been thinking about what to do next. Probably the next step is going to be linear algebra and we’ve been watching a few of Grant Sanderson’s “Essence of Linear Algebra” videos to get a feel for the subject.

Today I wanted to have a short and introductory talk about vectors with my son, and I had two goals in mind. The first was to show some ideas about (for lack of a better phrase) thinking in vectors rather than thinking in coordinates. The seconds was just sort of a fun introduction to the dot product.

So, I started with a simple introduction to vectors that he’s seen a bit of via the Grant Sanderson series:

Finding a vector representation for the 2nd diagonal of the parallelogram we’d drawn was giving him some trouble, so we took a deeper dive here. I’ve always thought that the equation for the 2nd diagonal was non-intuitive, so I gave him plenty of time to make mistakes and work through the ideas until he found the answer:

Finally, I did a simple introduction to the dot product and we calculated the angle – or the cosine of the angle – between a couple of vectors as a way to show how some ideas from linear algebra help solve seemingly complicated problems:

So, next week I’m having him watch a few more of Grant’s videos while I’m away on a work trip. We’ll get going on linear algebra the week after that.