An intro knot activity for kids

I’ve been doing a lot of thinking and playing with our various 3d printed knots lately. It feels like there are lots of great projects for kids here, but I’m struggling a little to find them.

Last night I tried something pretty simple – take several different versions of the trefoil knot and have the kids try to recreate those versions with a knotted rope.

Here are the shows and some initial thoughts about them from the boys. These knots were designed by Henry Segerman and Laura Taalman:

Next we started trying to make the shapes – first the “easy” ones 🙂

Next we moved on to some more difficult shapes – in particular the 2nd one gave the boys quite a bit of difficulty. Making the connection between these two versions of the trefoil knot isn’t completely straightforward:

I let the boys try to finish making the new knot shape with the camera off. It took a few more minutes. In this video they show how to go back and forth between the two versions:

So, definitely a fun project, but a little more difficult than I expected. We’ll see how difficult playing with the knot with 4 crossings is tomorrow.

Extending our coordinate change project with Desmos

Last weekend we did a fun 3d printing project involving changing coordinates:

screen-shot-2017-02-18-at-10-22-50-am

Here’s the link for that project:

Exploring some fun 3d transformations

A youtube commentator – mxlexrd – made the following Desmos version of one part of the activity after seeing the project:

mxlexrd’s Desmos version of our coordinate change project

Last night I had the boys play with it since I’d be traveling for work today. My younger son went first:

Here’s my older son:

The Desmos version of our activity is really fantastic. Even if the concepts are a little bit too complicated for my kids to understand in detail, I love how easy the program made it to explore the mathematical ideas.

Playing with the Cubeoctahedron

Last night I was flipping through the book I bought to understand a bit more about folding – Geometric Folding Algorithms by Erik Demaine and Joseph O’Rourke:

folding-book

and I ran across a short note on the cubeoctahedron. The boys were taking a short trip today (school vacation week!) and I was looking for a short project to do before they left – folding up the cubeoctahedron seemed perfect.

Making my life much easier was a template on Wolfram’s website:

Wolfram’s folding template for a cubeoctahedron

Here’s what the boys had to say after creating the shape:

After the short discussion about the shape we went upstairs to look at the shape using the F3 program. My idea for the ~10 min discussion here was inspired by a talk by Keith Devlin I saw over the weekend:

I thought that an approach similar to a game with our F3 program would help the boys create the shape.

Here’s how we got started. The F3 program allows us to create a cube and an octahedron. It also allows you to add and subtract shapes. How can we use these 4 ideas to create the cubeoctahderon?

I think the video here really shows what Devlin calls “mathematical thinking.” The conversation here was really fun (for me at least!) since trying to discuss the ideas through equations would be impossible. However, the geometric ideas are accessible to the boys via the F3 program, just as the number theory ideas are accessible to kids through Devlin’s “Wuzzit Trouble” program.

I broke the discussion into two pieces – at the start of the 2nd half of the discussion we are trying to figure out how to – essentially – flip the shape inside out. My son comes up with an idea that was very different than what I was expecting, and it worked 🙂

Playing with 3d printed knots from Mathematica

Yesterday I learned that Mathematica has a wide variety of knots that you can 3d print. We’ve done a few knot projects in the past. Here are 3 of them:

Playing with some 3d printed knots

Dave Richeson’s knotted bubbles project

Exploring Colin Adams’s “Why Knot?”

I thought that actually being able to hold the printed versions of so many different knots in your hand was going to be a game changer for knot projects, though. So, I printed a few as test cases and had the boys look at them.

My older son went first:

My younger son went next – he had a couple of things to say, but wanted to point out some of the knots in Colin Adams’s book, so we cut this video a little short so that we could go find the book:

After finding the book we were trying to match one of the printed knots with the knot in the book that he had wanted to print. The knot he wanted to print had 8 crossings and the one that we thought matched it turned out to have 7. Whoops – we had the wrong knot 🙂 A good accidental lesson that comparing two knots isn’t super easy!

I’m really looking forward to trying more projects with these prints. There are a little over 30 different knots with 8 or fewer crossings. It’ll probably take a week to print them all, but that’ll be a fun collection to have for future knot projects!

Sharing Grant Sanderson’s Fractal Dimension video with kids

A few weeks ago Grant Sanderson published this amazing video about fractal dimension:

I’ve had it in my mind to share this video with the boys, but the discussion of logarithms sort of scared me off. Last week, though, at the 4th and 5th grade Family Math night the Gosper curve fractals were super popular. That made me think that kids would find the idea of fractal dimension to be pretty interesting.

Here are the Gosper curves that Dan Anderson made for us:

Screen Shot 2016-04-05 at 4.54.02 PM

We’ve actually studied the Gosper curve several times before, so instead of just linking one project, here are all of them 🙂

A collection of our projects on the Gosper curve

So, today we started by watching Sanderson’s video. Here’s what the boys had to say about it:

At one point in the video Sanderson makes a comment that fractals have non-integer dimensions. I may have misunderstood his point, but I didn’t want to leave the boys with the idea that this statement was always true. So, we looked at a fractal with dimension exactly equal to 2:

Next we looked at the boundary of the Gosper island. I wanted to show that this boundary had a property that was a little bit strange. I introduced the idea with a square and a triangle to set the stage, them we moved to the fractals:

Finally – to clear up one possible bit of confusion, I looked at a non-fractal. For this shape we can see that the perimeter scaled by 3 and the area scaled by 7. Why is this situation different that what we saw with the Gosper Island?

Definitely a fun topic and I think Sanderson’s video makes the topic accessible to kids even if they don’t understand logarithms. I’m excited to find other fractal shapes to talk about now, too!

Exploring some fun 3d transformations

Today’s project with the boys was exploring some simple (to code!) transformations. The question was how would the shapes change under those transformations.

I started with introducing the idea in 2d. It isn’t necessarily the simplest idea, and I had no intention to go into any details. The basic question I wanted them to think about was this – would a straight line stay straight under this transformation?

Next we looked at a tetrahedron (actually two tetrahedrons) under some similar 3d transformations:

Now for the punch line – what do the same transformations do to an octahedron?

Finally, I wasn’t planning on doing this part, but to clarify some of the ideas from the first part of the project we went up to the computer to show them what the transformations did to a line in 2 dimensions:

So, I think this is a fun way for kids to explore some 3d shapes and also begin to understand a little bit about how algebra and geometry are related

Revisiting James Tanton’s Tetrahedron problem

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

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:

 

F3 Box.jpg

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.

Using 3d printing in the college classroom

I saw a couple of tweets from Steven Strogatz yesterday that got me thinking about how you might use 3d printing in the college classroom:

The last tweet, in particular, made me think that having the 3d print versions of the two shapes would be useful. Before I get too far in to this post, though, I had to throw this post together pretty quickly to be able to fit in an hour of shovelling prior to heading to work! Sorry if it isn’t the most well-written or well-argued post. The main takeaway I want is that I think there are many great uses for 3d printing in the college math classroom.

The topic of 3d printing and calculus is one that I’ve thought about briefly before – see these old posts:

3d Printing and Calculus Concepts for kids

Using 3d printing to explore some basic ideas from calculus

Here are the shapes from the first post linked above – I think they would help students understand ideas like Riemann sums and volume by slicing:

Here are the two 3d shapes from the second Strogatz tweet from yesterday. Unfortunately we lost power in the middle of the night before the print project was complete, but you’ll get the idea. One of the things that comes through immediately in the prints is the difference in size of the two shapes:

Finally, an important shape from advanced algebra – a cube inside of a dodecahedron. This shape appears (and plays an important role) in Mike Artin’s Algebra book:

Dodecahedron

I found it hard as a student to understand the shape solely from the picture. Holding the shape in my hand, though, makes it much easier to see what is going on (I have made the cube slightly larger to highlight it):

So, while I’m sure it is true that learning to draw some of these shapes by hand is useful, I also think that 3d printing can be an important tool to help students see, understand, and experience the same shapes.

Sharing John Cook’s Fibonacci / Prime post with kids

Saw a neat post from John Cook about using a fun fact about the Fibonacci numbers to prove there are an infinite number of primes:

Infinite Primes via Fibonacci numbers by John Cook

Funny enough, we’ve played with the Fibonacci idea before thanks to Dave Radcliffe:

Dave Radcliffe’s Amazing Fibonacci GCD post

That project was way too long ago for the kids to remember, so today we started by just trying to understand what the Fibonacci identity means via a few examples:

Next we looked at the idea from Cook’s post that we need to understand to use the Fibonacci identity to prove that there are an infinite number of primes. The ideas are a little subtle, but I think the are accessible to kids with some short explanation:

We got hung up on one of the subtle points in the proof (that is pointed out in the first comment on Cook’s post). The idea is that we need to find a few extra prime numbers from the Fibonacci sequence since the 2nd Fibonacci number is 1. Again, this is a fairly subtle point, but I thought it was worth trying to work through it so that the boys understood the point.

Finally, we went upstairs to the computer to explore some of the results a bit more using Mathematica. Luckily Mathematica has both a Fibonacci[] function and a Prime[] function, so the computer exploration was fairly easy.

One thing that was nice here was that my older son was pretty focused on the idea that we might see different prime numbers in the Fibonacci list than we saw in the list of the first n primes. We saw quickly that his idea was, indeed, correct.

This project made me really happy 🙂 If you are willing to take the Fibonacci GCD property for granted, Cook’s blog post is a great way to introduce kids to some of the basic ideas you need in mathematical proofs.

Grant Sanderson’s “Fair Division” video shows a great math project for kids

[sorry for a hasty write up – had to be out the door by 8:15 this morning . . . ]

Yesterday I saw the latest video from Grant Sanderson, and it is incredible!

I couldn’t wait to share the “fair division” idea with the boys. I introduced the concept with a set of 8 yellow and 8 orange snap cubes. To start, we looked at simple arrangements and just talked about ways to divide them evenly:

Next we looked at the specific fair division problem. We made a random arrangement of the blocks and tried to find a way to divide the cubes evenly with 2 cuts:

To finish up we looked at a few more random arrangements. Some were a little trick, but we always found a way to divide the cubes with two cuts! We also found an arrangement where the “greedy” algorithm from the 2nd video didn’t work.

After we finished the project I had the boys watch Sanderson’s video and they loved it. So many people are making so many great math videos these days – how are you supposed to keep up 🙂