## Revisiting Jacob Lurie’s Breakthrough Prize lecture

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 $x^2 + x + 1 = y^3 - y$ 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 $4n + 1$ 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 $4n + 3$ 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 $A + Bi$ where $A$ and $B$ 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.

## Steve Phelp’s 3d pentagon

Sorry that this post is written in a bit of a rush . . . .

I saw a neat tweet from Steve Phelps earlier in the week:

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.

## A good (though tricky) introductory counting problem

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.

## A neat and easy to state geometry problem

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.

## My week with “juggling roots”

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:

(1) The original tweet:

(2) The blog posts:

Sharing John Baez’s “juggling roots” tweet with kids

Sharing John Baez’s “juggling roots” post with kids part 2

Today I got one step closer to a long-term goal

(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:

Dan Anderson made some pretty neat 3d prints, too:

For this blog post I’m going to focus on the 5th degree polynomial $x^5 - 16x + 2$. 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 $x^5 - 16x + 2$ 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 $x^5 - 16x + A$ where $A$ moves around a circle with radius 8 centered at 10 + 0 I in the complex plane. So, one of our polynomials will be $x^5 - 16x + 2$, another will be $x^5 - 16x + (10 - 8i)$, another will be $x^5 - 16x + 18$, 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 $x^2$ 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.

What a fun week this has been!

## Using Evelyn Lamb’s tiling pentagons to talk about lines and shapes with kids

Evelyn Lamb’s latest article about tiling pentagons is incredible:

Math Under My Feet

We used it for a fun project this morning with some 3d printed pentagons. That project is here:

Talking about Evelyn Lamb’s tiling pentagons with kids

Tonight I wanted to show the boys how I made those pentagons. Not the 3d printing commands in Mathematica, but rather just how I described the shape.

I stared by digging in to Wikipedia’s description of the kind of pentagon that Lamb found.

Wikipedia’s page on pentagon tilings

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 🙂

## Matt Enlow and Suzanne von Oy’s geometry problem

Saw a tweet from Matt Enlow today that led to a fun discussion and also a fun project tonight with the kids:

The last tweet in the conversation was a new Desmos activity from Suzanne von Oy showing how the problem worked:

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!

## A neat geometry problem I saw from David Butler

I saw this problem today when it was re-tweeted by Matt Enlow:

It is a little advanced for my younger son, but I still thought it would be fun to turn into a mini project tonight with the boys.

We started by talking through the problem and taking a guess at what we thought the answer was -> Is there enough information to determine the side length of the square?

Although we didn’t really make any progress towards a solution in this initial discussion, I really like the ideas that we talked about. Specifically, I liked how much thought my older son put into how to label the diagram.

In this part of the project we began to discuss how to solve the problem. We found two equations, but had 3 variables. My older son began to think that we weren’t going to find a solution.

In trying to simplify one of our equations my younger son made a common algebra mistake. I spent most of the video slowly showing him how to tell that the algebra he thought was right was actually off.

At the end of the last problem we found an equation that seemed to be a step in the right direction of finding a solution to the problem. In this part of the project we explored that equation.

At the beginning my older son was really confused. I think he’s used to seeing problems where there is always a solution – the open endedness of this problem seemed to leave him puzzled.

We did get our sea legs back, though, exploring a few specific cases. The happy accident was that the two solutions we found to the problem gave us the same perimeter for the square – was a unique solution hiding here?

To wrap up the project we went up to the computer to look at our equation using Mathematica. We’d covered the important mathematical ideas already, but finding some of the exact solutions was going to be a chore and certainly finding the maximum perimeter wasn’t going to be in reach.

Nonetheless, there were a few fun surprises to be found 🙂

## Returning to Patterns of the Universe

As we searched for our old patty paper supply yesterday I ran across our copies of Patterns of the Universe:

We’ve done a few projects with the book before:

Fibonacci, Zome, and Patterns of the Universe

Patterns of the Universe Part 2

I thought doing some mathematical coloring would make for a great project today.

My older son talked about his coloring project first – he chose a section of the book on a random walk dice game. It was fun to hear his thoughts on random walks (and walks constrained like the one in this activity). By coincidence Kelsey Houston-Edwards recently did a video on random walks, so there’s a really nice way to continue the discussion we had here.

My younger son chose a section on “snug bunnies” – a picture of bunnies tiling the plane. He picked this section because he likes tesslations. It was fun to hear him talk about his coloring pattern. At the end we took a look at some Penrose tiles just to extend the tiling idea.

I really enjoyed the project today – the seemingly simple act of coloring led to some really fun discussions and also gave me some ideas for fun follow up projects.

## Returning to inclusion / exclusion

We’d taking a break from our inclusion / exclusion project but I wanted to return to it tonight. I picked a fairly challenging problem from one of my old math books:

How man 5 card poker hands have at least one card from each suit?

I didn’t have any idea how it would go . . .

We started by reviewing the ideas in inclusion / exclusion and the moved on to try to get our bearings in the problem:

Having formed a pretty good plan in the first video, we moved on to tackling the rest of the problem.

I’m really happy with how this went. It is fun to see the boys learning to break complicated problems down into problems that are slightly easier to deal with.