Going back to James Tantons’ amazing Möbius Strip cutting project

James Tanton’s Solve This book is full of incredible math projects to do with kids:

Today we went back to do a project that we’ve looked at a few times before – cutting various versions of a Möbius Strip.

We started with a cylinder just to get going with an easy shape:

Next we moved on to a Möbius strip – but when we made this one we didn’t notice that we’d made a full twist rather than a half twist. So we sort of got two surprises. I’ve included this mistake to show that you do have to be a little careful when you do this project – it is easy to accidentally make the wrong shape:

Here’s the correction and the actual Möbius Strip:

Next was a shape where we had a 2nd little error – this time the tape held the final shape together a bit too much. In our video you can sort of guess what the actual shape will be, but I definitely encourage everyone to try this one out for yourself!

The 4th shape is where things really get interesting. For this and the next two shapes we are making a circular cut through a cylinder with a hole cut out of it. Each of the shapes has a interest set of twists in it – the shape in this video has a single half twist on one of the “arms”:

The 5th shape has half twists (going the same direction) in each of the arms:

The 6th shape has half twists going in different directions in each of the arms:

This really is an amazing project for kids. As our version of the project here shows (accidentally!), you do have to be careful with the preparation, but even with the couple of errors we had, it was still an incredibly fun morning.

Why I love sharing ideas from research mathematicians with my kids

Yesterday we did a project inspired by the great podcast conversation between Steven Strogatz and Federico Ardila here:

That project is here:

https://mikesmathpage.wordpress.com/2021/04/03/revisiting-the-permutahedron-with-my-younger-son-after-listening-to-steven-strogatzs-interview-with-federico-ardila/

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.

Revisiting the permutahedron with my younger son after listening to Steven Strogatz’s interview with Federico Ardila

This week I listened a great conversation between Steven Strogatz and Federico Ardila on the Joy of X podcast:

We have played with the shape a few times before – see these blog posts of you are interested in seeing other ways that kids can explore the shape:

Today we explored the and idea that Ardila discussed in the podcast – finding paths on the permutahedron.

We started by just reviewing what the shape is and what it represents:

Next we tried an easy example of finding a path on the permutahedron going from a random permutation back to the correct order:

For the last part of the project we tried a more complicated scramble of the cards and found that walking back to the unscrambled state would take a minimum of 3 steps:

I love playing with this shape with kids. It is a great way to get them talking about fairly advanced mathematical ideas and also allows them to see a really neat 3d shape that research mathematicians find interesting!

Revisiting a dodecahedron folding into a cube

Back in 2016 Simon Greg showed me an incredible GIF of a dodecahedron folding into a cube:

which he found on this other amazing blog post by Herman Serras:

The Golden Section, The Golden Triangle, The Regular Pentagon and the Pentagram, The Dodecahedron

Today we decided to revisit our old project of making the shapes from our Zometool set. We started by looking at the gif:

Next I had the boys play around with the zometool set to try to make this shape. They worked for 15 min and made these shapes:

Here’s a closer look at the dodecahedron and the folded shape:

Finally, we took a look at one last surprise:

What kids learning math can look like -> working through some ideas in one of Burkard Polster’s books

Yesterday I was given a copy of one of Burkaard Polster’s books (he is the face of the Mathologer series on youtube):

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:

“Archimedes Proof” by Steve Portz on Thingiverse

Finding the coordinates of the vertices of a Tetrahedron and an Octahedron

My younger son is starting to learn about coordinates in 3 dimensions. I thought that spending a little time finding the coordinates of the corners of a tetrahedron and an octahedron would make for a nice project this morning.

We started with the tetrahedron and found the coordinates for the bottom face. Once nice thing about the discussion here was talking about the various choices we had for how to look at the tetrahedron:

Having found the coordinates for the bottom face, we now moved on to finding the coordinates for the top vertex:

Now we moved on to trying to find the coordinates for the corners of the octahedron. Here the choices for how to orient the object are a little more difficult:

Finally, we talked through how we would find the coordinates of the octahedron if we had it oriented in a different way. This was a good discussion, but was also something that confused the boys a bit more than I thought. We spent about 10 min after the project talking through how to find the height. Hopefully the discussion here helps show why this problem is a pretty difficult one for kids:

Sharing Laura Taalman’s slices with my younger son

I had an opportunity to visit Laura Taalman at ICERM today who made me a copy of her latest creation. I couldn’t wait to get home to share it with my younger son:

He had some interesting ideas about what the shape was in the last video. Now I shared where the slices came from and had him explain how Taalman’s creation worked:

This is such a fun way for kids to experience shapes from a different point of view. I’m really excited to see if we can create some similar objects to play with!

Labeling each vertex of a permutahedron is a terrific mathematical exercise for kids

Yesterday we did a fun project exploring a permutohedron:

Last night I thought it might be neat to have the kids try to label the vertices of a permutohedron with the permutations represented by each vertex. Fortunately, it was possible to build a truncated octahedron with the green Zometool struts:

We started out today’s project by talking about the rules for making a permutohedron in different dimensions. Here I used the labeling of the permutation of 3 objects as a base case to make sure the boys understood the directions properly.

Next I had the boys label each vertex of the permutahedron with the permutation of {1,2,3,4} that the vertex represented. Then, they talked about the process of figuring out the right labels.

I’m sorry that the video below runs 10 min, but if you listen to the whole discussion I think you’ll see that seemingly straightforward act of labeling these vertexes is a terrific mathematical exercise for kids.

A morning with the permutohedron

Today we are revisiting an old project on a really neat shape -> the permutohedron:

“A fun shape for kids to explore – the permutohedron

I learned about this shape thanks to Allen Knutson at Cornell – he included a fun pic of a large permutohedron in the comment of the blog post above:

He also pointed me to a 3d print on Thingiverse that we used in the last project and again today:

“Permutahedron” by PFF000 on Thingiverse

So, I started today by having the boys describe the 3d printed shape. We have two versions – a larger one that unfortunately broke a little and a smaller – but in one piece! – version. Here’s what the boys had to say about the shapes:

Next I had the boys read the Wikipedia page on the permutohedron for about 10 min and then we discussed some of the ideas that they thought were interesting:

Finally, we built the 2-D permutohedron and showed how it was embedded in a 3d grid:

Definitely a fun project and it is always great to be able to have kids hold interesting math ideas in their hands!

Sharing Vladimir Bulatov’s Tetrahedral Limit Set with kids

Last week I saw a really neat tweet shared by Alex Kontorovich:

I ended up buying Bulatov’s piece from Shapeways and it came today. Here’s a quick video look at it:

When the kids got home from school I asked them to take a look at it and share their thoughts.

Here’s what my younger son had to say about the shape:

Here’s what my older son had to say: