## An attempt to share some Katherine Johnson’s math ideas from Hidden Figures with my son

For the last few months I’ve been daydreaming about ways to share some of the math from the movie Hidden Figures with kids. As part of that prep work I found one of Katherine Johnson’s technical papers on NASA’s website:

NASA’s Technical Note D-233 by T. H. Skopinski and Katherine G. Johnson

As you’d expect, there’s a lot of trig, calculus and spherical geometry. I like finding ways to share the work that mathematicians do with kids, but this work is pretty technical and I wasn’t getting any great ideas.

Then my son had a homework problem from his Precalculus book that made me think it was time to stop daydreaming and just try something. Here is that problem, which is a completely standard law of cosines problem:

The problem reminded me of one of the equations for an ellipse used in the Technical Note. One surprising thing is that the equation of an ellipse in polar coordinates is that is is a rational function in $\cos{\theta}$.

So, I drew an ellipse and showed my son that equation.

One of the neat things about the Technical Note is that the solution to some of the complicated trig equations were found by an iteration method. The specific ideas for solving those equations are too advanced for kids, so I decided to show my son a different (and really simple) iteration method that converges to a well known number:

After that introduction to iteration methods, I decided to jump to a second and slightly more complicated example -> solving x = 3*x*(1 – x).

The ideas in the iteration method we use here can be explored purely geometrically:

Next we went upstairs to the computer to see some of the ideas we just talked about. The first idea was the polar coordinate equation for an ellipse:

Now we played with the second dynamical system -> solving x = 3*x(1-x).

By the way, the ideas here are incredibly fun to explore (especially seeing when this method converges and when it doesn’t), but the details of this method wasn’t really the idea here. I just wanted to show him what an iterative method looks like.

Finally, I showed him the actual paper and pointed out some of the parts we explored. Sorry that this film didn’t come out as well as I’d hoped, but you can view the paper from the first link in this post:

This was a fun project – even if it wasn’t planned really well. Showing some of the math behind Hidden Figures I hope helps motivate some of the topics that my son is studying right now. It will be fun to return to a second Hidden Figures project when he is studying calculus.

## A fun math surprise with a 72 degree angle.

We’ve been talking a lot about 72 degree angles recently. Yesterday’s project was about a question our friend Paula Beardell Krieg asked:

Paula Beardell Krieg’s 72 degree question

In that project we learned that a right triangle with angles 72 and 18 (pictured below)

Is nearly the same as a right triangle with sides of 1, 3, and $\sqrt{10}$

Today I wanted to show the boys a neat surprise that I stumbled on almost by accident. The continued fraction expansion for the cosine of the two large (~72 degree angles) are remarkable similar and lead to the “discovery” of a 3rd nearly identical triangle.

We got started by reviewing a bit about 72 degree angles:

Now we did a quick review of continued fractions and the “split, flip, and rat” method that my high school teacher, Mr. Waterman, taught me. Then we looked at the continued fraction for $1 / \sqrt{10}$:

Now we looked at the reverse process -> given a continued fraction, how do we figure out what number it represents? Solving this problem for the infinite continued fraction we have here is a challenging problem for kids. One nice thing here was that my kids knew that they could do it if the continued fraction had finite length – that made it easier to show them how to deal with the infinitely long part.

Finally, we went to the computer to see the fun surprise:

Here’s that 3rd triangle:

I love the surprise that the continued fractions for the cosine of the (roughly) 72 degree angles that we were looking at are so similar. It is always really fun to be able to share neat math connections like this with kids.

## Exploring Newton’s method with kids

Yesterday we had about a 30 min drive and I had the boys open up to a random page in this book for a few short discussions in the car:

There were some fun topics that were accessible for kids, but then Newton’s method came up. Ha ha – not really drive time talk 🙂

It did seem like it could be a fun project, though, so I took a crack at it today. The goal was not computation, but mainly just the geometric ideas. Here’s how we got started:

Next I asked the boys if they could find situations in which Newton’s method wouldn’t work as nicely as it did in the first video. They were able to identify a few potential problems:

Now I had both kids draw their own picture to play out what would happen when you used Newton’s method to find roots. I think there’s a lot of ways to used the exercise here to help older kids understand ideas about tangent lines and function generally. I mostly let the kids play around here, though, and the results were actually pretty fun:

Finally, we went to Mathematica to see some situations in which Newton’s method produces some amazing pictures. Here we switch from real-valued functions to complex valued functions. Since I wasn’t going into the details of now Newton’s method works, rather than using some easier to understand code, I just borrowed some existing code from here:

The page from A. Peter Young at U.C. Santa Cruz that gave me the Newton’s method code for Mathematica

The boys were amazed by the pictures. For example, (and this is one we looked at with the camera off) here’s a picture showing which root Newton’s method converges to depending on where you start for the function $f(z) = z^3 - 2z + z - 1$:

Definitely a fun project. Even if the computational details are a bit out of reach, it is fun to share ideas like this with kids every now and then.

## Sharing Tim Gowers’s nontransitive dice talk with kids

During the week I attending a neat talk at Harvard given by Tim Gowers. The talk was about a intransitive dice. Not all of the details in the talk are accessible to kids, but many of the ideas are. After the talk I wrote down some ideas to share and sort of a sketch of a project:

Thinking about how to share Tim Gowers’s talk on intransitive dice with kids

One of the Gowers’s blog posts about intransitive dice is here if you want to see some of the original discussion of the problem:

One of Tim Gowers’s blog posts on intransitive dice

We started the project today by reviewing some basic ideas about intransitive dice. After that I explaine some of the conditions that Gowers imposed on the dice to make the ideas about intransitive dice a little easier to study:

The next thing we talked about was 4-sided dice. There are five 4-sided dice meeting Gowers’s criteria. I thought that a good initial project for kids would be finding these 5 dice.

Now that we had the five 4-sided dice, I had the kids choose some of the dice and see which one would win against the other one. This was an accessible exercise, too. Slightly unluckily they chose dice that tied each other, but it was still good to go through the task.

Now we moved to the computer. I wrote some simple code to study 4-sided through 9-sided dice. Here we looked at the 4-sided dice. Although it took a moment for the kids to understand the output of the code, once they did they began to notice a few patterns and had some new ideas about what was going on.

Having understood more what was going on with 4-sided dice, we moved on to looking at 6-sided dice. Here we began to see that it is actually pretty hard to guess ahead of time which dice are going to perform well.

Finally we looked at the output of the program for the 9-sided dice. It is pretty neat to see the distribution of outcomes.

There are definitely ideas about nontransitive dice that are accessible to kids. I would love to spend more time thinking through some of the ideas here and find more ways for kids to explore them.

## 3D Printing Paula Beardell Krieg’s dissected cube shapes

I’ve been thinking about exposing the boys to math through 3d printing lately. Today I decided to explore making Paula Beardell Krieg’s cube shapes with them. Here’s the exploration the boys did back in March when we first got them:

Even though we’ve played a bit with these shapes before I still thought that thinking through these yellow and pink shapes would be a fun challenge. The project turned out to be a tiny bit harder than I thought it would be, but it still was a nice conversation.

We started by first looking at the three pyramids that can come together to make a cube and continued by looking at what happens when you slice those shapes in half.

In the last video the boys were thinking about trying to describe these shapes by describing the lines that formed the edges. At the beginning of this video I told them that this particular approach was going to be tough since they didn’t know how to write equations of lines in 3 dimensions.

So, I had them continue to search for properties of the shapes that they could describe.

The boys were still struggling to find some ideas about the shape that went beyond the lines on the boundary, but we kept looking.

My older son hit on the idea that the shape was made from “stacking squares on top of each other.” We spent the rest of the video exploring that idea.

Now that we had the idea about stacking squares we went to Mathematica to try to create the shape. It took a few steps to move from the ideas about the squares to generating the code for the shape. We didn’t get all the way there during this video, but we did figure out how to make a cube.

Unfortunately I had to end the video since the camera was about to run out of memory.

While I was getting the videos off the camera the boys worked on how to change the cube shape to the pyramid shape. It was a good challenge for them and they got it. We talked about that shape for a bit and then moved on to the challenge of creating the “pink” and “yellow” shapes that Paula Beardell Krieg created from paper.

We had a little bit of extra time today and it was fun to walk through this challenging problem. I think creating shapes to 3d print is a really fun way to motivate math with kids. Can’t wait to use the printed shapes in a project tomorrow!

## Sharing Kendra Lockman’s Desmos activity with my son

I saw this tweet from Kendra Lockman yesterday:

It looked like a fun activity to try, so we spent 20 minutes this morning going through it. It was nice to year what my son son thinking about fractions throughout the activity. The first 4 videos below show his work and the last is some quick thoughts from him on the morning:

Part 1:

Part 2:

Part 3:

Part 4:

Here’s his summary of the activity:

## Intro “machine learning” for kids via Martin Gardner’s article on hexapawn

Last month I had the nice surprise of finding Martin Gardner’s book The Colossal Book of Mathematics at the Omaha Public Library’s book sale:

I’ve been flipping through the book and thinking about how to share some of the ideas with the boys.  Chapter 35 – “A Matchbox Game-Learning Machine” really caught my attention.  In particular, Gardner’s discussion of the game “hexapawn” inspired me to try this introductory “machine learning” idea with kids.

I had the boys read the (approximately) 2 pages on the game and the approach and then we talked through the game to make sure they understood it:

Next we started playing. We were very lucky to have a coffee table that allowed us to easily show the 24 cases and their snap cubes. This video shows the first two times through the game. I hope that it shows that playing through the game is something that is accessible to kids:

The next part shows 3 more turns of the game. My main reason for showing these three turns is so you can see some of the parts that kids find challenging. I think these parts are a big part of what makes sharing Gardner’s idea with kids so fun. The pattern matching and the general walk through the game keeps their attention while they learning about machine learning.

Next we played for a while with the camera off. After a while the kids (and the computer) learned something:

Next we played a bit more with the camera off and the before long the “computer” learned to win the game every time. Amazing!

In the last 3 min of this video the boys talk about some of the things that they learned in this project.

This is one of the most fascinating projects that we’ve ever done. It does require a bit of set up and probably a bit more careful supervision than usual to make sure that the kids don’t go down the wrong path, but wow is there a lot to learn here. I think that opening the door for kids to see how computers / machines might “learn” is an amazingly valuable lesson.

## 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 short continued fraction project for kids

I woke up this morning to see another great discussion between Alexander Bogomolny and Nassim Taleb. The problem that started the discussion is here:

and the mathematical point that caught my eye was the question -> which positive integers are close to being integer multiples of $\pi$?

One possible approach to this question uses the idea of “continued fractions.” I learned about continued fractions from my high school math teacher, Mr. Waterman, who taught them using C. D. Olds’s book.

So, today I stared off by talking about irrational numbers and reviewing a simple proof that the square root of 2 is irrational:

Next we talked about why integer multiples of irrational numbers can never be integers. This I think is an obviously step for adults, but it took the kids a second to see the idea:

Now we moved on to talk about continued fractions. I’m not trying to go into any depth here, but rather just introduce the idea. I use my high school teacher’s procedure: split, flip, and rat 🙂

We work through a simple example with $\sqrt{2}$ and also see that the first couple of fractions we see are good approximations to $\sqrt{2}$.

With that background work we went up to use Mathematica to explore different aspects of continued fractions quickly. One thing we did, in particular, was use the fractions we found to find multiples of $\sqrt{2}$ that were nearly integers.

Finally, we wrapped up by using continued fractions to find good approximations to $\pi$, $e$ and a few other numbers.

Definitely a fun project, and one that makes me especially happy because of the connection to Mr. Waterman. Hopefully the boys will want to play around with this idea a bit more tomorrow.

## Playing with 4d Toys

Quick post tonight because I’m running out to dinner . . . .