Sharing Sam Hansen’s “Knotty Helix” podcast with kids

The latest Relatively Prime podcast is fantastic:

The short description from the podcast’s website is:

“Sure DNA is important, some might even claim it is absolutely integral to life itself, but does it contain any interesting math? Samuel is joined by UC-Davis Professor of Mathematics, Microbiology, and Molecular Genetics Mariel Vazquez for a discussion proves conclusively that mathematically DNA is fascinating. They talk about the topology of DNA, how knot theory can help us understand the problems which occur during DNA replication, and how some antibiotics are really pills of weaponized mathematics.”

Since it is only 20 min long, I thought it would be fun to share with the boys. We listened to it in the car when we went out to breakfast this morning. Upon returning home I asked the kids what they thought and what were somethings they learned. Here’s what they had to say:

Next we looked at an interesting process described in the podcast. That process was an example by Mariel Vazquez of how you can go from a link with 6 crossings to two unlinked circles.

The process in the podcast seems simple – maybe even obvious – but I think that the process is actually much more subtle than it seems listening to it.

Here we followed the steps to go from the trefoil knot to the two unlinked circles. I think the ideas we followed here are a great way for kids to explore the process described in the podcast:

The next thing we looked at was the idea that when you cut a loop with an even number of twists in half, the halves would be linked. We took a long strip of paper and gave it 6 twists, taped the ends together, and then cut it down the middle.

I fast forwarded through the taping and cutting part, but forgot to remove the sound. Sorry for the “Alvin and the Chipmunks” middle part of this video.

I think both the podcast and the follow up projects are a great way for kids to explore some math ideas that wouldn’t normally be part of a school curriculum.

We’ve done a few other projects with knots and with paper cutting. Here’s a link to those collections:

Our knot projects

A collection of some of our paper cutting (and folding) projects

It was really neat to hear about how knot theory applies to biology.

Trying to study knots with 5 crossings

We’ve been doing a little bit of work with knots lately. Today we were studying the knot with 5 crossings, and it wasn’t so easy.

I’d guess ahead of time that 5 crossing would be tricky. There’s a lot to keep track of! Even what seems like a simple task – making a knot with 5 crossings out of rope – isn’t so easy. See if you can spot the problem:

The boys didn’t notice the problem with their knot, yet, but the problem quickly became clear when they started playing with it:

So, we started over . . .

Having now made a knot with 5 crossings, we ended the project by trying to determine which of the two knots with 5 crossings that it was. We got a little bit of a surprise when it turned out that we’d actually made the mirror image of one of the knots we printed. That was an accidental good lesson, though – mathematicians consider those two knots to be the same even though they are not always the same.

Exploring the knots with 4 and 5 crossings with kids

In our last project we explored the trefoil knot:

img_1745

That project is here:

An introductory knot activity for kids

Today we moved on to the knots with 4 and 5 crossings. We started off by comparing the trefoil knot with the 4 crossing knot – what is the same? what is different?

Also – the 3 white knots (1 with 4 crossings and 2 with 5) that appear in this projects come from Mathematica’s knot data collection and the red knot (the trefoil knot) was designed by Laura Taalman.

Next I had the boys try to make a knot with 4 crossings out of a rope. It is not as simple as it seems! One nice thing about making these knots out of rope is that they also begin to discover some of the ways you can have crossings that can be undone.

Now we compared the knot we made from the rope to the 3d printed knot with 4 crossings. In the last project we had quite a lot of difficulty comparing the different versions of the trefoil knot. Here, though, comparing the knot in our rope to the 3d printed knot was not too difficult.

Finally, we wrapped up this project by inspecting and comparing the two knots with 5 crossings. It is very interesting to hear what kids see in these two knots, and also fun to hear their ideas for how you might determine that these two knots are actually different.

Definitely a fun project. I really like exploring these knots with the kids. It makes me wonder if there is a way to go to the next level and help them understand some of the ideas that help you understand when two knots are different?

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.

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!