An absolutely mind blowing project from James Tanton

I was flipping through James Tanton’s Solve This last night and found a project I thought would be fun. I had no idea!

We’ve done a few paper cutting projects before. For example:

Cutting a double Mobius strip

Tadashi Tokieda’s “World from a Seeht of Paper” lecture

Our one cut project

Fold and cut project #2

Fold and cut part 3

Fold and Punch

and this list also need this incredible project from Joel David Hamkins:

Math for nine year olds: fold, punch and cut for symmetry

So, having see a few folding / cutting project for kids previously the projects about Möbius strip in Chapter 8 of Tanton’s book caught my eye. As I said above, though, I had no idea how cool this project would turnout to to be!

We started with the standard project of cutting a Möbius strip “in half”. What happens here??

Oh, and before getting to the move – most of the movies below have a lot of footage of us cutting out the shapes. I was originally going to fast forward through that, but changed my mind. The cutting part isn’t that interesting at all, but I left it in to make sure that anyone who wants to repeat this project knows that the cutting part (especially with kids) is a tiny bit tricky. You have to be careful!

Now, once we’d done the cut the boys were still a little confused about whether or not the result shape had one side or two. I thought it would be both important and fun to make sure we’d resolved that question before moving on:

With the Möbius strip cutting out of the way we moved on to what Tanton describes as “a diabolical Möbius construction”. All three shapes start as a thin cylinder with a long ellipse cut out. You then cut and twist the strips outside of the ellipse making a Möbius strip-like component of the new shape. Hopefully the starting shapes will be clear from the video.

Try to guess what the resulting cut out shape(s) will look like prior to them getting cut out🙂

The first shape involves putting one half twist into one of the strips left over after cutting the ellipse out of the cylinder.

The second shape also starts as a cylinder with a long ellipse cut out. This time, though, we make put half twists (in the same direction) in both of the long strips that are left over after cutting out the ellipse.

Sorry about the camera being blocked by my son’s head a few times – oops!

The final shape for today’s project is similar to the second shape, but instead of two half twists going in the same direction, they go in opposite directions.

All I can say is wow – what an incredible project for kids. Thanks James Tanton!!

Taking kids through John Baez’s post about the Gyroid

[sorry for no editing on this one – had some computer problems that ate up way too much time. I finished typing with 2 minutes to spare before rushing out the door.]

I saw this neat tweet from John Baez earlier in the week:

You should be able to click through to Baez’s blog post from the tweet, but just in case that isn’t working, here’s the link:

The Butterfly, the Gyroid and the Neutrino by John Baez

I spent the rest of the week sort of day dreaming about how to share some of the ideas in the post with kids. Last night the day dreaming ended and I printed a gyroid that I found on Thingiverse:

The specific gyroid that I printed is here:

Alan Schoen’s Gyroid on Thingiverse by jamesosaurus

This project connects with several of our prior projects on 3d printing (particularly the recent ones inspired by Henry Segerman’s new book) as well as projects on minimal surfaces. Though the list below is hardly complete, here are a few of those projects:

Zometool and Minimal Surfaces

Trying out 4 dimensional bubbles

More Zome Bubbles

Playing with shadows Inspired by Henry Segerman

Playing with more of Henry Segerman’s 3d Prints

Henry Segerman’s flat torus

Using Hypernom to get kids talking about math

So, with that introduction – here’s what we did today.

First we revisited the zome bubbles to remind the kids about minimial surfaces – it is always fun to hear kids describe these complicated shapes:

Next we looked at the Gyroid that I printed last night. This shape is much more complicated than the zome bubbles and the kids sort of had a hard time finding the words to describe it – but we had a similar shape (and I don’t remember why or where it came from) that helped the kids get their bearings:

So, after playing with the blue shape for a bit and seeing some of the symmetry that this shape had (yay!) we returned to the Gyroid. The boys still struggled to see the symmetry in the gyroid (which is really hard to see!) but we made some progress in seeing that not all of the holes were the same:

Finally, we turned to Baez’s article to see the incredibly surprising connection with butterflies and physics. There’s also a fun connection with some of the work we’ve done with Bathsheba Grossman’s work and Henry Segerman’s 3D printing book:

So, a fun project. I love how 3d printing helps open up advanced ideas in math to kids. After we finished the boys kept reading Baez’s article to find the connection with neutrinos – it is really gratifying to see how engaged they were by today’s project!

Playing with Bathsheba Grossman’s “Hypercube B”

Last week we went to Henry Segerman’s talk at MIT:


During that talk one of the 3d-printed objects he passed around “Hypercube B” by Bathsheba Grossman. I was excited to have the boys play around with this shape a bit more so we ordered a copy from Shapeways:

Hypercube B by Bathsheba Grossman on Shapeways

It turns out that Grossman is also the creator of “Double Zarf” which we ordered previously and my older son discussed here:

Exploring some shapes from Henry Segerman’s new book

Hypercube B arrived in the mail today and I was really interested to hear what the boys had to say about the shape. My older son gets home from school first, so he was the first one to play with it:

My older son recognized the shape from the talk (which was fun). My younger son didn’t right away, but he came to believe that it was a hypercube because of some of the projections that it made (when viewed in the camera lens!). I loved hearing what he had to say about the shape!

In the next couple of days we’ll also play around and see what sorts of (2d) shadows this shape makes. Can’t wait to do that!

Martin Hotham’s spirals

Saw a great tweet from Martin Hotham last night:

I thought it would be fun to let the boys play around with this activity. We’ve actually talked about spirals before a little. First with our Zome set:

Fibonacci Spirals and pentagons with our Zometool set

Then with Anna Weltman’s “loop-de-loops”

Anna Weltman’s Loop-de-loops

and Weltman’s project spiraled us into some of Frank Farris’s work:

Extending Anna Weltman’s loop-de-loops with Frank Farris’s “Creating Symmetry”

The difficulty with the math behind the spirals is that it’ll be a few years before either of the boys encounters trigonometry. However, Hotham’s Desmos programs are so stunning and so easy to use that I’m fine just letting the boys play with them.

My younger son played around for 5 minutes – here’s what he had to say:

Next I let my older son play around – here’s what he had to say when he was done:

We’ve played around with one of Hotham’s creations before – when he created a program based on Ann-Marie Ison’s art:

Extending our project with Ann-Marie Ison’s art”

I love his work and per the law “any sufficiently advanced technology is indistinguishable from magic” I can only assume that he’s some wort of wizard. Can’t wait to see what he comes up with next!

Playing with Shadows – inspired by Henry Segerman

Last night the whole family attended Henry Segerman’s talk at MIT:


Here’s Henry in action:

He’s already inspired a ton of projects, for example:

Playing with more of Henry Segerman’s 3d Prints

Henry Segerman’s flat torus

And this joint work with Vi Hart, Andrewa Hawksley, Henry Segerman, Emily Eifler and Marc ten Bosch got a big reaction from the crowd last night:

Using Hypernom to get kids talking about math

As a follow up to Henry’s talk last night we played with shadows of a cube and a tetrahedron this morning. First up was the tetrahedron – it is really fun to hear what the kids have to say about the shadows.

Also, here’s the “hypercube B” shape that my older son mentions:

Hypercube B on Shapeways

After talking about the tetrahedron shadows we moved on to the cube. Again, it is really fun to hear what the kids have to say about these shapes. Several of the questions I asked about the cube shadows came from Henry’s talk:

Finally, we looked at one of Henry’s prints. We’ve studied this print before (and my younger son even brought it to the talk to show Henry!). I wanted to use this shape to explore the question:

“If two lines come together in the shape, do they have tocome together in the shadow?”

So, a fun talk last night and a fun project this morning. I want to live in a world where all kids have the opportunity see Henry’s work!

Examples of kids and problem solving

Both kids gave nice examples of the problem solving process in the two videos we did last night, so I wanted to highlight those videos with a short blog post.

First up was my younger son. He’s learning algebra this year and has a really nice way of thinking and talking through problems. I love how deliberate he is and how he discovers his own mistakes. The problem that he’s working on here is to find 3 solutions to the equation 3A – 5B = 9.

Next up was my older son. The problem he’s working on is an old Mathcounts problem, and it is pretty challenging:

What fraction of the first 100 triangular numbers are divisible by 7?

His work is a nice example of, for lack of a better phrase, the discovery process. Initially he does not see how to solve the problem, but I love his path to the solution.

After he finished I showed him two other approaches to solving the problem, just to help him see how a few other ideas in math can connect to this problem:

I wanted to share these examples to show that problem solving in math isn’t all about speed. A slow, deliberate process is a great way to get to the solution of a problem.

James Tanton’s candy dividing exercise

Yesterday we watched the “tie folding” part of James Tanton’s latest video:

The video led to a great project with the boys last night:

James Tanton’s tie folding problem

The boys knew from the video that the method could also be applied to sharing candy. Since we didn’t watch that part of the video I was wondering if the boys could figure out the connection on their own. Here’s the start:

Next we tried an example to see what would happen if our initial guess was a big over estimate of 1/3 of the Skittles:

Since we were struggling with our second time through the procedure in the last video, I thought it would be fun to try to be more precise in how we split the piles. That extra precision did lead to slightly better results.

So, a really nice math activity. It was really fun to see the procedure work when we couldn’t be totally sure we were actually dividing the piles in half. Such a great project for kids.

James Tanton’s tie folding problem

Saw a great new video from James Tanton today about folding a tie. The kids had spent yesterday hiking in New Hampshire and were a little tired, but Tanton’s project made for a perfect little afternoon project.

I’ll present the videos in the order that we did them, so Tanton’s video is the third one below. Showing his video later in the project will also give you a chance to think through the problem without spoilers.

Anyway, here’s how we started -> what do you have to do to fold a tie in half?

I was super happy with how the introductory problem went because at the end of the last video my older son said that he thought folding the tie into thirds would be hard. Well . . . that’s exactly what we are going to try to figure out!

Next we watched Tanton’s video. He talks about both folding ties and sharing candy, but for today at least we are just focused on the tie folding part:

Now we tried to replicate Tanton’s procedure. My 5th grader had a little bit harder of a time understanding the procedure than my 7th grader did, but they both eventually got it.

At the end we talked about why they thought the procedure worked.

So, a super fun project and a really easy one to implement, too. So many potential extensions, too – might be neat to see how kids approached folding into 5 parts after seeing Tanton’s video, for example.

Thanks for another great project, James!

What I was hoping for with the boys and math

When I started making math movies with the boys my goal was to show other kids what kids doing math can look like. There are examples everywhere of adults doing math, so kids can see those examples with no problem. There aren’t nearly as many examples of what it looks like when kids work through problems, though.

So, 5 years into it we are all pretty comfortable in front of the camera and my younger son – just by luck – is making exactly the videos that I was dreaming about in the beginning.

Below are the last two ones we’ve made. They show him working through algebra problems. Nothing fancy, nothing speedy, but really nice work through the problems. I love the way he thinks through problems and think that other kids might enjoy these examples showing what a kid doing math can look like.



A fun project from Art Benjamin and the Museum of Math

Yesterday Art Benjamin gave a talk at the Museum of Math. One neat tweet from from the talk was this one:

It is a pretty neat problem and I thought it would make a fun project for the boys today. I didn’t show them the tweet, though, because I wanted to start by exploring the numbers with increasing digits:


Next we tried to figure out what was going on. My older son wanted to try to study the problem in general, but then my younger son noticed a few things that at least helped us understand why the sum should be divisible by 9.


For the third video we started looking at the problem in general. The computations here tripped up the boys a bit at first, but these computations are really important not just for this problem but for getting a full understanding of arithmetic in general.


For the last part of the project we looked at two things. First was returning to a specific example to make sure that we understood how borrowing and carrying worked. Next we applied what we learned to the slightly different way of multiplying by 9 -> multiplying by 10 first and then subtracting the number.


After the project I quickly explored Dave Radcliffe’s response to MoMath’s tweet:

It took a bit of thinking for the boys to see what “works in any base” meant, but they did figure it out.

I love this Benjamin’s problem – it makes a great project for kids!