James Tanton’s incredible Möbius strip cutting project

Today we revisited one of my all time favorite math projects for kids (also **revisiting**) :

Reciting one of my favorite projects ever this morning – @jamestanton ‘s diabolical Möbius constructions. pic.twitter.com/WT6VTggRSj

— Mike Lawler (@mikeandallie) January 28, 2018

We did this project once before, but I don’t think the kids remembered it:

An absolutely mind blowing project from James Tanton

The project is relatively simple to set up – you have strips of paper and make 5 Möbius strip-like shapes.  If the short descriptions below aren’t clear, don’t worry, the videos have the “picture is worth 1,000 words descriptions

(1) An actual Möbius strip
(2) Same set up as making one Möbius strip, but you start with two strips of paper stacked on top of each other,
(3) A cylinder with a long oval cut out and a half twist on one of the strips left over after removing the oval.
(4) A cylinder with a long oval cut out and a half twist (in the same direction) in both of the strips left over after removing the oval.
(5) Same as (4) but the twists are in opposite directions.

Then you cut the shapes. In (1) and (2) you cut completely along the center line. In (3), (4), and (5) you cut around the oval.

What shape are you left with after the cutting?

(1) Cutting a Möbius strip

(2) Cutting two strips of paper folded into a Möbius strip

(3) Cutting a cylinder with an oval removed with one half twist

(4) Cutting a cylinder with an oval removed with two half twists in the same direction

(5) Cutting a cylinder with an oval removed with two half twists in opposite directions

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James Tanton’s penny and dime exercise

I saw a neat tweet from James Tanton yesterday:

"Notice & wonder" in K-8 world helps Ss develop the life skill of developing & refining natural next qus. In HS world let's not quelch NNQs! A curiosity like attached leads to: Change nmbr of pennies? Asymmetrical leapfrogging? 3D? Right-angled leaps? etc. And teaches curriculum. pic.twitter.com/1mpghA5KZc

— James Tanton (@jamestanton) December 30, 2017

After seeing the tweet, I couldn’t wait to do the project today. It turned out to be much more difficult than I was expecting, though. One of the difficulties that really caught me off guard was the time it took to make accurate measurements of the positions. The difficulty there turned what I mistakenly thought was going to be a really quick part of the project into maybe 90% of the project’s time.

Maybe a nice surprise with this project is that it could be a good introductory project for introducing kids to measurement.

So, although I’ll publish all 6 videos from today’s project, the main ideas are in the first and last videos.

Here’s the introduction to the project:

The boys weren’t totally sure what happened the first time around, so we tried again:

After two tries, they still weren’t sure what was going on, so we tried one more time:

After this third try, they had some interesting observations:

One idea they had in the last video was to see what would happen if the 3 points were on a line.

To wrap up, the boys wondered about a few other set ups. I was happy to hear that they were starting to think about new ideas.

I think Tanton’s problem is an absolutely great exercise for kids. I’m sorry that I misjudged the difficulty coming from the measurements, though.

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:

P(a+b) =P(a) + P(b) + kab, P(1)=1
For k=2 get 1,4,9,…sq nmbrs
For k=1 get 1,3,6,…triang nmbrs
For k=0 get 1,2,3,..counting nmbrs
k= -1?

— James Tanton (@jamestanton) July 7, 2017

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!

James Tanton’s counting problem part 2

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

If I roll a die five times how many distinct values should I expect to see?

— James Tanton (@jamestanton) June 1, 2017

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.

Working through a challenging counting problem from James Tanton

I saw a neat problem from James Tanton during the week:

If I roll a die five times how many distinct values should I expect to see?

— James Tanton (@jamestanton) June 1, 2017

A follow up post from Amy Hogan made me realize that I should use the problem for a project:

A couple of my math team students decided to explore this today with a simulation first. pic.twitter.com/ltFBHrqYPl

— Amy Hogan (@alittlestats) June 2, 2017

So, I decided to make looking at this problem our weekend project. Today I wanted to try a few simulations. Tomorrow we’ll count the cases. That case work is pretty challenging, but I think we’ll be able to get through it.

Here’s the introduction to the problem:

Next I had the boys roll 5 dice 50 times and record the numbers they saw each time. Here are those results as well as our first estimate of the answer to Tanton’s question:

Finally we moved on to do a simulation. Normally I would have used Mathematica, but I’ve got a program related to one of our old Goldbach Conjecture videos running right now. This problem was easy enough to deal with on Excel, though, so I looked at a simulation there.

Here’s what they saw in the numbers:

Definitely a nice start to this project. Counting the different cases tomorrow will be a bit more difficult, but we’ll also get to talk about some pretty interesting mathematical ideas like partitions and choosing numbers. I’m excited to see if we are able to get all the way to the end.

A nice problem about primes for kids from James Tanton

Saw a really cool tweet from James Tanton today:

What is the average power of a given prime p among the prime factorizations of the counting numbers?

— James Tanton (@jamestanton) April 26, 2017

Tonight I sat down with the boys to make sure they understood the problem. They noticed that half the numbers would have no powers of two – good start! After that observation they started down the path to solving the problem really quickly. In fact, my younger son thought that we might have a geometric series.

Since we covered a few ideas pretty quickly in the last video, so I stared this part of the project by asking them to give me a more detailed explanation for how they got the 1/2, 1/4, 1/8, . . . pattern in the last video. It turned out to be a little harder for them to give precise arguments, but they did manage to hit the main points which was nice.

At the end of this video my older son was able to write down the series that we needed to add up to solve the problem.

Now that we had the series, we had to figure out how to add it up. My guess was that they’d never seen a series like this, but my older son had a really cool idea almost immediately – rewrite the series!

The boys were able to sum the series in this new form – so yay!

At the end of of the last video my younger son said that he was surprised that the “expected value” wasn’t zero since zero was the most likely value. In this part of the video we talked a bit more about what “expected value” meant.

Once we had that I asked what I meant to be a quick question -> is the expected number of 3’s higher or lower. It turned out to be a longer conversation than I expected, though, because my older son was actually able to write down the answer!

Definitely a fun problem. I think it is fun for kids to see how to add up a series like the one in this project. I also think it is fun for kids to explore some of the basic ideas about primes that pop up in this problem.

As an aside, one other place where I’ve seen the series that came up here is in this post from Patrick Honner:

Proof Without Words: Two Dimensional Geometric Series

His “proof without words” for the sum is this picture – can you see how it works?

Revisiting James Tanton’s Tetrahedron problem

A little over 2.5 years ago I saw this very neat question from James Tanton:

What is surface area of figure formed by all points within a distance 1 from a regular tetrahedron with faces of area 1?

— James Tanton (@jamestanton) June 20, 2014

The question led to a really fun – and also one of our first – 3d printing projects:

James Tanton’s geometry problem and 3d printing

We’ve now got a few more years of 3d printing under our belts and a new program we are using – F3 by Reza Ali – is opening completely new 3d printing ideas for us.

Somewhat incredibly, F3 has a one line command that draws all of the points that are a fixed distance away from a cube. Here’s that beautiful shape:

 

Seeing that command inspired me to revisit James Tanton’s old question. I wasn’t quite able to do it in one line (ha ha – my programming skills are measured in micro-Reza Alis . . . .), but I was still able to make the shape. Here’s how it looked on the screen:

After the boys got home from school we revisited the old project together and used both the old and the new 3d prints to help us describe the shape (sorry for the noise in the background – that’s a humidifier I forgot to turn off):

Maybe because it is one of our first projects ever(!), but I love this problem as an example of how 3d printing gives younger kids access to more complex problems.

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!

One of the most exciting projects we've ever done coming later today – stupid slow internet 😦 . Thanks @jamestanton for the inspiration!! pic.twitter.com/9Rvxw991na

— Mike Lawler (@mikeandallie) October 23, 2016

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!!

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!