I’ve been going through some of the videos of the invited talks from the Joint Math meetings and trying to figure out how to share them with kids. The tallks and our first two projects are here:

Today we looked at Lázló Babai’s talk on graph isomorphisms:

I started by explaining what “graph” meant in this context and then exploring a few examples:

I decided to break the final conversation at the end of the last video into two pieces so that the boys wouldn’t feel rushed talking about the last two graphs:

Before moving on to a pretty challenging example, I decided to show them a connection between shadows and graphs – the conversation here was actually much more fun than I expected!

Next we moved on to a pretty challenging example of two isomorphic graphs. The boys did a nice job showing that these two graphs are indeed isomorphic.

To wrap up we looked at a more complicated example from Babai’s talk. From that example I think the boys were able to see that the graph isomorphism problem can be pretty hard!

This was a fun project. I think that kids really enjoy introductory graph theory. Another fun project we’d done in the past on the subject is here:

I’m definitely enjoying the talks, but also wondering if there are ideas – even small ones – that you can take from the talks and share with kids. My hope is that kids will enjoy seeing ideas and concepts that are interesting to mathematicians.

Today I tried out an idea from Federico Ardila’s talk with my younger son (who is in 6th grade). The idea related to an interesting shape called the “permutahedron.”

We began with a quick explanation of the idea and looked at some simple cases:

Next we moved to building the permutahedron that comes from the set {1,2,3}. At the end of the last video, my son speculated this shape would have some interesting symmetry. We used our Zometool set to build it.

One thing I’m very happy about with this part of the project is that building this permutahedron is a nice introductory exercise with 3d coordinates for kids.

Finally, we talked about the permutahedron that comes from the set {1,2,3,4}. My son had some interesting thoughts about what this shape might look like. Then I handed him a 3d printed version of the shape and he had some fun things to say 🙂

Definitely a fun project for kids, I think. Making the hexagon was fun and also a nice little geometric surprise. Exploring the 3d printed shape was also really exciting – it is always great to hear what kids have to say about shapes that they’ve never encountered before.

Despite most of the talk going over their heads, the boys were really excited after the talk and had lots of different “shuffling” ideas that they wanted to explore.

Since this was going to be a long project, I divided it into two pieces – studying “pick up 52” with my younger son last night and studying shaking the cards in a box with both kids this morning.

The idea of using Shannon Entropy to study how random the shuffles are is something that we explored in these two prior projects:

So, I kicked off the project last night with my younger son. Here are his thoughts about the Diaconis talk and about his “pick up 52” idea

We did 4 trials without re-sorting the cards in between. Here are some quick thoughts about how the deck was getting mixed up between the 2nd and 3rd throws:

After we finished I had my son do a few minutes of riffle shuffling to completely mix up the deck (starting from where the deck was after the 4th pick up 52). While he was doing that I entered the numbers from our throws into a spreadsheet.

The surprise was that even after the first throw the cards were really mixed up. I was even more surprised by this because he basically threw the deck in the air rather than what (to me anyway) is the normal way of throwing cards for pick up 52.

This morning we continued the project with my older son’s idea of putting the cards in a box and shaking the box. Here the introduction to that idea:

Here are some thoughts from my older son after the first mixing. He didn’t think they were all that mixed up. We did a total of 3 more mixings – the 3rd and 4th were off camera.

Finally, we wrapped up by reviewing the numbers for mixing the cards up in the box. The first mixing had more entropy than we thought, and after the 2nd mixing the cards appear to be pretty close to as mixed up as you can get (equivalent to about 10 riffle shuffles, I think).

This was a really fun project. The math you need to describe what’s going on here is much to advanced for kids (and worthy of a math lecture at MIT!), but kids can still have a lot of fun exploring some of the ideas. The seemingly simple idea of how can you measure how mixed up a shuffle is is a pretty interest idea all by itself.

Last night I asked my older son what he what topics were being covered in his math class at school. He said that they were talking about different kinds of numbers -> natural numbers, integers, rational numbers, and irrational numbers. I asked him if he thought it was important to learn about the different kinds of numbers and he said that he thought it was but didn’t know why.

I decided share Jacob Lurie’s Breakthrough Prize lecture with the boys this morning since he touches on the study of different kinds of number systems. The first 12 or so minutes of the lecture are accessible to kids:

Near the beginning of Lurie’s talk he mentions that the equation has no integer soltuions. I stopped the video here to what the boys thought about this problem. It took two about 10 minutes for the boys to think through the problem, but eventually they got there. It was fun to watch them think through the problem.

Here’s part 1 of that discussion:

and part 2:

The next problem that we discussed from the video was Lurie’s reference that all primes of the form can be written as the sum of two squares. I checked that the boys understood the problem and then switched to a problem that would be easier for them to tackle -> No prime of the form can be written as the sum of two squres.

Finally, to finish up, we began by discussing Lurie’s question about whether or not numbers were real things or things that were made up by mathematicians. Then we wrapped up by looking at why 13 is not prime when you expand the integers to include complex numbers of the form where and are integers.

There aren’t many accessible public lectures from mathematicians out there. I’m happy that part of Lurie’s lecture is accessible to kids. It is nice to be able to use this lecture to help the boys understand a bit of history and a bit of why these different number systems are interesting to mathematicians.

The shape sort of stuck in my mind and last night I finally got around to making two shapes inspired by Phelp’s shape. My shapes are not the same as his – one of my ideas for this project was to see if the boys could see that the shapes were not the same.

So, we started today’s project by looking at the two shapes I printed overnight. As always, it is really fun to hear kids talk about shapes that they’ve never encountered before.

Next we looked at Phelp’s tweet. The idea here was to see if the boys could see the difference between this shape and the shapes that I’d printed:

Finally, we went up to the computer so that the boys could see how I made the shapes. Other than some simple trig that the boys have not seen before, the math used to make these shapes is something that kids can understand. We define a pentagon region by 5 lines and then we vary the size of that region.

I’m not expecting the boys to understand every piece of the discussion here. Rather, my hope is that they are able to see that creating the shapes we played with today is not all that complicated and also really fun!

This was a really fun project – thanks to Steve Phelps for the tweet that inspired our work.

My older son is re-working his way through Art of Problem Solving’s Introduction to Counting and Probability. He came across a problem in the review section for chapter 5 that gave him some trouble. I decided to talk through part of it tonight and included my younger son.

My younger son hasn’t been studying any counting lately, so I was expecting the problem to be pretty challenging for him. His work through the first part of the problem is, I think, a nice example of a kid working through a challenging math problem.

The problem is this: How many different ways are there to put 4 distinguishable balls into 3 distinguishable boxes?

The next problem is what was giving my son some trouble: How many different ways are there to put 4 distinguishable balls into 3 indistinguishable boxes?

My younger son struggles with the problem for a bit on this one and then my older son offers his thoughts. What gives my older son a little trouble is the case in which you put 2 balls into one box and 2 into another.

So, after struggling with 2-2-0 case in the last video, we talk about it in a little more depth here. The tricky part is seeing that two cases that don’t look the same are actually the same. It was harder for the boys to see the over counting than I thought it would be. But, we made it!

So, the indistinguishable counting part was pretty confusing to the boys. I think we need to do a few more problems like this to let this particular counting concept sink in.

Heard a neat problem on a math podcast today which basically boils down to this question:

If I give you a square (or a bunch of squares) of side length 1, how can you make a square with area 2?

I thought trying out this question with both of the boys would be pretty fun. Here’s how it went:

(1) My younger son went first

(2) My older son went second

Next I thought it would be interesting to extend the problem a little bit and ask them to try to create a square with area 3. To my surprise this problem was significantly more difficult – the two video below are roughly 9 min each.

(3) My younger son went first:

(4) My older son went second

I was surprised at how much more difficult the 2nd problem was for both kids. I was also surprised that they approached it the same way (my older son wasn’t home when I did the project with my younger son so it really was a coincidence).

Would be fun to find some more problems like this one.

A tweet last week from John Baez made for a really fun week of playing around. I’ve written several blog posts about it already. Here’s the summary to date, I guess:

(3) A video from a comment on one of the posts from Allen Knutson that helped me understand what was going on a bit better:

So, with that as background, what follows are some final (for now at least) thoughts on what I learned this week. One thing for sure is that I got to see some absolutely beautiful math:

Playing with the "juggling roots" and printing some of the structures has been the most fun I've had with math in ages. #math#mathchatpic.twitter.com/4cL583Ix2y

For this blog post I’m going to focus on the 5th degree polynomial . I picked this polynomial because it is an example (from Mike Artin’s Algebra book) of a polynomial with roots that cannot be solved.

So, what do all these posts about “juggling roots” mean anyway?

Hopefully a picture will be worth 1,000 words:

What we are going to do with our polynomial is vary the coefficients and see how the roots change. In particular, all of my examples below vary one coefficient in a circle in the complex plane. So, as the picture above indicates, we’ll look at all of the polynomials where moves around a circle with radius 8 centered at 10 + 0 I in the complex plane. So, one of our polynomials will be , another will be , another will be , and so on.

The question is this -> how do the roots of these polynomials change as we move around the circle? You would certainly expect that you’ll get the same roots at the start of the trip around the circle and at the end – after all, you’ve got the same polynomial! There’s a fun little surprise, though. Here’s the video for this specific example showing two loops around the circle:

The surprise is that even though you get the same roots by looping around the circle, with only one loop around the circle two of the roots seem to have switched places!

Here’s another example I found yesterday and used for a 3d print. Again for this one I’m varying the “2” coefficient. In this case the circle has a radius of 102:

When I viewed this video today, I realized that it wasn’t clear if 3 or 4 roots were changing places in one loop around the circle. It is 4 – here is a zoom in on the part that is tricky to see:

Next up is changing the “-16” in the x coefficient in our polynomial. Here the loop in the complex plane is a circle of radius 26:

Finally, there’s nothing special about the coefficients that are 0, so I decided to see what would happen when I vary the coefficient of the term that is initially 0. In this case I’m looping around a circle in the complex plane with radius 20 and passing through the point 0 + 0i:

So – some things I learned over this week:

(1) That the roots of a polynomial can somehow switch places with each other as you vary the values of the coefficients in a loop is incredible to me.

(2) The idea of thinking of these pictures as slices of a 3-dimensional space (which I saw on John Baez’s blog) led to some of the most visually striking 3d prints that I’ve ever made. The math here is truly beautiful.

(3) I finally have a way to give high school students a peek at a quite surprising fact in math -> 5th degree polynomials have no general solution.

Kids will use a lot of nice introductory geometric ideas in simply describing the shape:

Next we talked about some of the basic properties of the pentagon. It was a bit of a tricky conversation since my older son knows quite a bit about equations of lines and my younger son really hasn’t seen equations of lines at all. So, for this part I let my younger so do most of the talking.

In this part we talk a bit about coordinates and equations of lines that are parallel to the x and y axis.

At the end we moved to the tricky part – how do we describe the final two lines. Describing these lines is even a little bit harder since we want the two line segments to have the same length. How do we do that?

At the end of the last movie we found a way to make the final two line segments have the same length. Now we needed to write down the equation of those two lines. This part took a while because my younger son was essentially seeing the math ideas here for the first time, but I’m glad we went slowly. He seemed to get a lot out of it.

If you are interested, the Mathematica code to make the pentagon looked like this:

I love using 3d printing to talk about 2d geometry ideas. The conversations that you have about making the shapes are really fun conversations about basic geometric and algebraic ideas. Since you either have the shape made already or are in the process of making the shapes, the conversations are really easy to get going 🙂

I couldn’t wait to try out this problem with the boys tonight. We aren’t (obviously!) going to go into a lot of depth – this isn’t really a problem for 5th graders! But, I thought the boys would have some fun talking about it.

Since the problem is a pretty challenging one for kids to even understand, I started the project by trying to explain the problem carefully.

Next we tried to pick some points at random and then draw some triangles. If there are infinitely many equilateral triangles passing through these 3 points, it ought to be easy to draw one of them, right?

My older son went first. The cool thing for me in both this and the next video was seeing kids experience the problem and struggle with both trying to understand it and trying to solve it. There really is a lot of great geometry for kids here:

Next my younger son gave it a try. His approach was absolutely terrific to watch – I never would have approached the problem the way he did.

Next we went to play with Suzanne von Oy’s Desmos program. We got interrupted by the new puppy in the house across the street between leaving the living room and heading upstairs to play with the program, so I took the first minute of this video to review the problem again.

My older son went first again. He quickly found a picture that didn’t satisfy the conditions of the problem and that threw him for a little loop. Once we got past that, though, he seemed to have a much better understanding of the problem.

My younger son went next and eventually found an arrangement of the points that didn’t work at all. That was actually a really cool surprise ( we’ll deal with that surprise in the next video).

So, we got a wonderful surprise in the last video when we stumbled on an arrangement of the three points that didn’t seem to have any equilateral triangles passing through them.

Talking about what went wrong was a fantastic little surprise and it really made this project for me.

This was a super fun project. Thanks so much to Matt Enlow and Suzanne von Oy for sharing both the problem and the Desmos activity. Math twitter is amazing!