# 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.

# 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:

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!

# Sharing James Tanton’s area models with my younger son

I saw a really neat tweet from James Tanton yesterday:

The video in the “setting the scene” link is terrific and I had my younger son watch it this morning. I can’t figure out how to get the video to embed, so here’s a direct link to the page it is on:

James Tanton’s area model lecture

One reason that I was extra excited to see Tanton’s video is that by total coincidence I’d used essentially the same idea with my older son to explain why a negative times a negative is a positive:

But I’d never gone through the same ideas with my younger son – at least not that I can remember. So, for our Family Math project today I had him watch Tanton’s video and then we talked about it.

Here’s what he thought about the video and area models in general – I was really happy to hear that my son liked Tanton’s area models and thought they were a really great way to think about multiplication and quadratics:

Next I had my son walk through the negative / positive area ideas that Tanton used to talk about multiplication. He did a really nice job replicating Tanton’s process. I think this is a great way for kids to think about multiplication:

# 15 (+1 bonus) Math ideas for a 6th grade math camp

Saw an interesting tweet last week and I’ve been thinking about pretty much constantly for the last few days:

I had a few thoughts initially – which I’ll repeat in this post – but I’ve had a bunch of others since. Below I’ll share 10 ideas that require very few materials – say scissors, paper, and maybe snap cubes – and then 5 more that require a but more – things like a computer or a Zometool set.

The first 4 are the ones I shared in response to the original tweet:

## (1) Fawn Nguyen’s take on the picture frame problem

This is one of the most absolutely brilliant math projects for kids that I’ve ever seen:

When I got them to beg

Here’s how I went through it with my younger son a few years ago:

## (2) James Tanton’s Mobius strip cutting exerciese

This is a really fun take on this famous scissors and paper cutting exercise:

You will honestly not believe what you are seeing when you go through Tanton’s version:

Here’s the link to our project:

James Tanton’s incredible mobius strop cutting project

## (3) Martin Gardner’s hexapawn “machine learning” exercise

For this exercise the students will play a simple game called “hexapawn” and a machine consisting of beads in boxes will “learn” to beat them. It is a super fun game and somewhat amazing that an introductory machine learning exercise could have been designed so long ago!

Intro “machine learning” for kids via Martin Gardner’s article on hexapawn

## (4) Katie Steckles’ “Fold and Cut” video

This video is a must see and it was a big hit with elementary school kids when I used it for “Family Math” night:

Here are our projects – all you need is scissors and paper.

Our One Cut Project

Fold and cut project #2

Fold and cut part 3

## (5) Along the same lines – Joel David Hamkins’s version of “Fold and Punch”

I found this activity in one of the old “Family Math” night boxes:

Joel David Hamkins saw my tweet and created an incredible activity for kids.  Here’s a link to that project on his blog:

Joel David Hamkins’s fold, punch and cut for symmetry!

## (6) Kelsey Houston-Edwards’s “5 Unusual Proofs” video

Just one of many amazing math outreach videos that Kelsey Houston-Edwards put together during her time at PBS Infinite Series:

Here is how I used the project with my kids:

Kelsey Houston-Edwards’s “Proof” video is incredible

## (7) Sharing the surreal numbers with kids via Jim Propp’s checker stacks game

Jim Propp published a terrific essay on the surreal numbers in 2015:

Jim Propp’s “Life of Games”

In the essay he uses the game “checker stacks” to help explain / illustrate the surreal numbers. That essay got me thinking about how to share the surreal numbers with kids. We explored the surreal numbers in 4 different projects and I used the game for an hour long activity with 4th and 5th graders at Family Math night at my son’s elementary school.

This project takes a little bit of prep work just to make sure you understand the game, but it is all worth it when you see the kids arguing about checker stacks with value “infinity” and “infinity plus 1” 🙂

Here is a summary blog post linking to all of our surreal number projects:

Sharing the Surreal Numbers with kids

## (8) Larry Guth’s “No Rectangle” problem

I learned about this problem when I attended a public lecture Larry Guth gave at MIT.  Here’s my initial introduction of the problem to my kids:

I’ve used this project with a large group of kids a few times (once with 2nd and 3rd graders and it caused us to run 10 min long because they wouldn’t stop arguing about the problem!). It is really fun to watch them learn about the problem on a 3×3 grid and then see if they can prove the result. Then you move to a 4×4 grid, and then a 5×5 and, well, that’s probably enough for 80 min 🙂

Larry Guth’s “No Rectangles” problem

## (9) The “Monty Hall Problem”

This is a famous problem, that equally famously generates incredibly strong opinions from anyone thinking about it. These days I only discuss the problem in larger group settings to try to avoid arguments.

Here’s the problem:

There are prizes behind each of 3 doors. 1 door hides a good prize and 2 of the doors hide consolation prizes. You select a door at random. After that selection one of the doors that you didn’t select will be opened to reveal a consolation prize. At that point you will be given the opportunity to switch your initial selection to the door that was not opened. The question is  -> does switching increase, decrease, or leave your chance of winning unchanged?

One fun idea I tried with the boys was exploring the problem using clear glasses to “hide” the prizes, so that they could see the difference between the switching strategy and the non-switching strategy:

Here’s our full project:

Exploring the Monty Hall problem with kids

## (10) Using the educational material from Moon Duchin’s math and gerrymandering conference with kids

Moon Duchin has spent the last few years working to educate large groups of people – mathematicians, politicians, lawyers, and more – about math and gerrymandering.  . Some of the ideas in the educational materials the math and gerrymandering group has created are accessible to 6th graders.

Here’s our project using these math and gerrymandering educational materials:

Sharing some ideas about math and gerrymandering with kids

## (11) This is a computer activity -> Intro machine learning with Google’s Tensorflow playground.

This might be a nice companion project to go along with the Martin Gardner project above. This is how I introduced the boys to the Tensorflow Playground site (other important ideas came ahead of this video, so it doesn’t stand alone):

Our complete project is here:

Sharing basic machine learning ideas with kids

## (12) Computer math and the Chaos game

The 90 seconds starting at 2:00 is one of my all time favorite moments sharing math with my kids:

The whole project is here, but the essence of it is in the above video:

Computer math and the chaos game

## (13) Another computer project -> Finding e by throwing darts at a chess board

This is a neat introductory probability project for kids. I learned about it from this tweet:

You don’t need a computer to do this project, but you do need a way to pick 64 random numbers. Having a little computer help will make it easier to repeat the project a few times (or have more than one group work with different numbers).

Here’s how I introduced the project to my kids:

Here’s the full project:

Finding e by throwing darts

## (14) Looking at shapes you can make with bubbles

For this project you need bubble solution and some way to make wire frames. We’ve had a lot of success making the frames from our Zometool set, but if you click through the bubble projects we’ve done, you’ll see some wire frames with actual wires.

Here’s an example of how one of these bubble projects goes:

And here’s a listing of a bunch of bubble projects we’ve done:

Our bubble projects

## (15) Our project inspired by Ann-Marie Ison’s math art:

This tweet from Ann-Marie Ison caught my eye:

Then Martin Holtham created a fantastic Desmos activity to help explore the ideas:

It is fun to just play with, but if you want to see how I approached the ideas with my kids, here are our projects:

Using Ann-Marie Ison’s incredible math art with kids

Extending our project with Ann-Marie Ison’s art

## (16) Bonus project!!A dodecahedron folding into a cube

This is a an incredible idea from 3d geometry.

We studied it using our Zometool set – that’s not the only way to go, but it might be the easiest:

Here’s the full project:

Can you believe that a dodecahedron folds into a cube?

# Sharing Jim Propp’s base 3/2 essay with kids part 2

I’m going through Jim Propp’s piece on base 3/2 with my kids this week.

His essay is here:

Jim Propp’s How do you write one hundred in base 3/2?

And the our first project using that essay is here:

Sharing Jim Propp’s base 3/2 essay with kids – Part 1

Originally I wanted to have the kids read the essay and give some of their thoughts for part 2, but I changed my mind on the approach this morning. Instead I asked each of them to answer the question in the title of Propp’s essay -> How do you write 100 in base 3/2?

Propp points out in his essay that his approach to base 3/2 via chip firing / Engel machines / exploding dots is not what mathematicians would normally consider to be base 3/2. The boys are not aware of that statement, though, since they have not read the essay yet.

Here’s how my younger son approached writing 100 in base 3/2. The first video is an introduction to the problem and, from knowing how to write numbers like 100 (in base 10) in other integer bases.

I think the first 3 minutes of this video are interesting because you get to hear his ideas about why this approach seems like a good idea. The remainder of this video plus the next two videos are a long march down the road to discovering why this approach doesn’t work in the version of base 3/2 we are studying:

So, after finding that the path we were walking down led to a dead end, we started over. This time my son decided to try to write 100 as 10×10. This approach does work!

Next I introduced the problem to my older son. He also started by trying to solve the problem the same way that you would for integer bases, though his technique was slightly different. He realized fairly quickly (by the end of the video, I mean) that this approach didn’t work:

My older son needed to find a new approach, and he ended up finding an idea different from my younger son’s idea to find 100 in base 3/2. His idea was to use chip firing:

I thought that today’s project would be a quick reminder of how base 3/2 works (at least the version we are studying). That thought was way off base and was completely influenced by me knowing the answer! Instead we found – by accident – a great example of how to explore a challenging problem in math. Sometimes the first few things you try don’t work, and you have to keep trying new things.

Definitely a fun morning!

# Sharing Jim Propp’s base 3/2 essay with kids

Jim Propp’s essay on base 3/2 is fantastic:

Here’s a direct link to his blog post in case the twitter link doesn’t work:

Jim Propp’s How do you write one hundred in base 3/2?

and here are links to our two prior base 3/2 projects:

Fun with James Tanton’s base 1.5

Revisiting James Tanton’s base 3/2 exercise

I’m hoping to have time to spend at least 3 days playing around with Propp’s latest blog post. Today we had 20 min free unexpectedly in the morning and I used that time to introduce two of the ideas. They haven’t read the post, yet, but instead I started by having them watch Propp’s short video about the binary Engel machine:

After watching that video I had the boys recreate the idea with snap cubes on our white board. Here’s that work plus a few of their thoughts on the connection with binary:

Next I challenged the boys to draw the base 3/2 version of the machine. After they did that we counted to 10 in base 3/2 and talked about what we saw:

I was happy that the boys were able to understand the idea behind the base 3/2 Engel machine. With the work from today giving them a nice introduction to some of the ideas in Propp’s essay, I think they are ready to try reading the essay tomorrow. It’ll be interesting to see what ideas catch their eye. Hopefully we can do another short project on whatever those ideas are tomorrow morning.

# Exploring induction and the pentagonal numbers

Yesterday we did a fun project based on this tweet by James Tanton:

That project is here:

Exploring a neat problem from James Tanton

During the project yesterday we touched on mathematical induction and also on the pengatonal numbers. Today I wanted to revisit those ideas with slightly more depth.

We started with a quick review of yesterday’s project:

Now we looked at a mathematical induction proof. The example here is:

$1 + 3 + 5 + \ldots + (2n - 1) = n^2$

(the nearly camera ran out of batteries, that’s why this part is split into two videos)

Here’s the 2nd part of the induction proof after solving the battery problem:

To wrap up the project we went to the living room to build some shapes with our Zometool set. The Zome shapes really helped the boys make the connection between the numbers and geometry.

The boys really liked this project. In fact, my younger son spent the 30 min after we finished making the decagonal numbers 🙂

# Exploring a neat problem from James Tanton

I didn’t have an specific project planned for today and was lucky enough to see a really neat problem posted by James Tanton:

I didn’t show the tweet to the boys because I thought finding the patterns would be a good exercise for kids. We started with the k = 0 case. This case is also good for making sure that kids understand the basics of functions required to explore this problem:

Next we looked at the k = 1 case.

Next we looked at the k = 2 case and then my younger son made a really fun little conjecture 🙂

At the end of the last video my younger son thought that the k = 3 case might produce the pentagonal numbers. I had to look up those numbers ( 🙂 ) while the camera was off, but I found them and we checked:

We ended by looking at Tanton’s challenge problem -> what happens when k = -1? I had the boys take a guess and then we looked at the first few terms and the boys were, indeed, able to solve the problem!

The boys had a lot of fun playing around with this problem and I was really excited they found a different pattern than the one Tanton was asking for!

# Connecting yesterday’s probability project with a few old 3d geometry projects

In yesterday’s project we were studying a fun probability question posed by Alexander Bogomolny:

That project is here:

Working through an Alexander Bogomolny probability problem with kids

While writing up the project, I noticed that I had misunderstood one of the
geometry ideas that my older son had mentioned. That was a shame because his idea was actually much better than the one I heard, and it connected to several projects that we’ve done in the past:

Learning 3d geometry with Paula Beardell Krieg’s Pyrmaids

Revisiting an old James Tanton / James Key Pyramid project

Overnight I printed the pieces we needed to explore my son’s approach to solving the problem and we revisited the problem again this morning. You’ll need to go to yesterday’s project to see what leads up to the point where we start, but the short story is that we are trying to find the volume of one piece of a shape that looks like a cube with a hole in it (I briefly show the two relevant shapes at the end of the video below):

Next we used my son’s division of the shape to find the volume. The calculation is easier (and more natural geometrically, I think) than what we did yesterday.

It is always really fun to have prior projects connect with a current one. It is also pretty amazing to find yet another project where these little pyramids show up!

# James Tanton’s counting problem part 2

Yesterday we looked at a really neat problem James Tanton posted last week:

That project is here:

Working through a challenging counting problem from James Tanton

Our first look at the question involved some dice rolling and a computer simulation. Today we are going to look at an exact solution to the problem. That solution involves studying all of the different things that can happen when you roll 5 dice. It turns out that there are 7 different patterns that can happen, and these patterns related to the ways you can write 5 as the sum of positive integers.

(1) 5 different numbers, which I’ll represent as 1 + 1 + 1 + 1 + 1

(2) 3 different and 2 the same -> 1 + 1 + 1 + 2

(3) 2 different and 3 the same -> 1 + 1 + 3

(4) 1 different and 4 the same -> 1 + 4

(5) 1 different and 2 pairs -> 1 + 2 + 2

(6) 1 pair and 1 triple -> 2 + 3

(7) All numbers the same -> 5

For today’s project we’ll count the number of ways that each of these 7 patterns can occur. We know that the total number of arrangements is 7,776, so that’s going to help us make sure we have counted correctly.

Here’s the introduction to the problem and to the approach we are going to take today:

Now we began to count some of the arrangements. In this video we count the number of dice rolls in (1), (4), and (7) above:

Now we moved on to some of the more challenging arrangements. Here we looked at (6) and (2) above:

Now we looked at case (5). This case proved challenging because dealing with the 2 pairs caused a little confusion between over counting and under counting. But, after looking at the cases carefully we did manage to get to the answer.

At this point we had only one case left -> (3) from above. But, the counting practice that we’d had up to this point helped this case go pretty quickly.

Finally, we added up our numbers and checked that we’d found all 7,776 cases. We did!

The one thing left to do was to count the different numbers that we saw in each case and find the average. I’d done that ahead of time just to save a bit of time in the movie. Our final answer was (27,906) / (7,776) or about 3.588. The exact answer was (happily!) very close to the two estimates that we had found in our simulations yesterday.

I love Tanton’s problem. It is a great estimation problem as well as a great counting problem. We might do one more project tonight on yet a different way to solve the problem using Markov chains:

Looks like a fun idea – I’ll be thinking about how to talk through this approach with the kids during the day today.