## A surprise 30-60-90 triangle

Over the last couple of days we’ve done two projects that started from a couple of easy to state questions:

(i) Given some squares with area 1, how do you make a square with area 2?

(ii) Given some squares with area 1, how do you make a square with are 3?

Those project are here:

A neat and easy to state geometry problem

Some simple proofs of the Pythagorean Theorem

Tonight my older son is at a school event. That gave me time to do a fun little extension of these two projects with my younger son.

First I reviewed the original problems:

My son solved the 2nd problem above by making triangles with sides 1, $\sqrt{2},$ and $\sqrt{3}$. For this part of the project I wanted to show him a different triangle that has a side length of $\sqrt{3}$ – a 30-60-90 triangle:

Now – for a little extra fun – we made a Zometool cube. That cube shows that the face diagonal (of a 1x1x1) cube has length $\sqrt{2}$. It also shows that the internal diagonal has length $\sqrt{3}.$

Here’s the surprise – if we extend basically the same geometry to 4 dimensions, we find that the “long” internal diagonal of a 1x1x1x1 cube has length 2, and that there’s a secret little 30-60-90 triangle hiding in the cube!

We did a similar project a few years ago:

Did you know that there is a 30-60-90 triangle in a Hypercube

It was nice to revisit this idea today 🙂

## Sharing Numberphile’s Goldbach Conjecture video with kids

Numberphile released a really nice video about the Goldbach Conjecture today:

I thought it would make an excellent project with the boys even though some of the ideas involving logarithms might be over their head. So, we watched the movie and then talked about some of the ideas that caught their eye.

Next we moved on to the individual ideas. The first one was the chart that David Eisenbud made at the beginning of the video. Drawing and then filling in this chart is a nice little arithmetic activity for a kid in elementary school.

Next we talked about logarithms. I started with an idea I learned from Jordan Ellenberg’s book “How Not to be Wrong” – the “flogarithm”. That idea is to oversimplify the logarithm by defining it to be the number of digits in the number. That simple (and genius) idea really opens the door to kids thinking about logarithms.

With that short introduction I explained what the natural logarithm was and moved on to some of the properties of primes that Eisenbud mentioned in the video (after fumbling with the calculator on my phone for a minute . . . .).

(Also, I noticed watching the video just now that I forgot to divide by 2 at one point – sorry about that.)

Finally, we checked a specific example – how many ways were there to take two primes and add up to 50? This part is about as far away from the complexity of logarithms as you can get – just some nice arithmetic practice for kids.

To warp up I asked them if they knew any other unsolved problems about primes. My older son mentioned something about twin primes. I showed the boys a simple argument (fortunately quite similar to the one Eisenbud gave in the movie for why there are lots of ways two primes can add to be a given even number) for why there ought to be infinitely many twin primes.

I think that kids are going to be naturally curious about primes. The Goldbach conjecture is one of the few unsolved problems that kids can understand. It was fun to share this video with the boys tonight.

## Some simple proofs of the Pythagorean theorem

Yesterday we did a fun project on these two questions:

(1) Given a square with area one, find a way to make a square with area 2,

(2) Given a square with area one, find a way to make a square with area 3.

That project is here (where you can see that part 2 gave both kids a lot of trouble):

A neat and easy to state geometry problem

I decided to revisit a piece of that project today to show them that both of their solutions to part 2 were essentially proofs of the Pythagorean theorem.

We started by reviewing yesterday’s project:

Next we talked about how my younger son’s way of constructing the square with area three can be used to prove the Pythagorean theorem:

Finally, we looked at the slightly different way that my older son constructed the square with are 3. This approach proves the Pythagorean theorem in a different way:

This was a fun couple of projects that came from a really innocuous sounding question.

## 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.

## Sharing Grant Sanderson’s “Pi and Primes” video with kids part 2:

Grant Sanderson’s latest video explaining a connection between pi and prime numbers is absolutely fantastic:

This video is sort of at the edge of what kids can understand, but it was fun to explore a few of the ideas with them even if understanding 100% of the video was probably not realistic. Our project on the first 10 min of the video is here:

Sharing Grant Sanderson’s Pi and Primes video with kids part 1

Also, we did a project on a different approach to the problem Sanderson is studying previously:

A really neat problem that Gauss Solved

I intended to divide our study of Sanderson’s video into three 10 minute sections, but the second 20 minutes was so compelling that we just watched it all the way through. After watching the last 20 min a 2nd time this morning I asked the kids what they found interesting. The three topics that they brought up were:

(i) The $\chi$ function,

(ii) The formula for $\pi / 4$, and

(iii) Factoring ideas in the Gaussian integers

Following the introduction, we talked about the three topics. The first was factoring in the Gaussian integers. We talked about this topic in yesterday’s project, too.

Next we talked about the $\chi$ function. I had no idea how the discussion here was going to go, actually, but it turned out to be fantastic. The boys thought the function looked a lot like “remainder mod 4”. Why it does look like that and why it doesn’t look like that is a really neat conversation with kids.

Finally we talked through the formula that Sanderson explained for $\pi / 4.$ It probably goes without saying that Sanderson’s explanation is better than what we did here, but it was nice to hear what the boys remembered from seeing Sanderson’s video twice.

I love having the opportunity to share advanced math with kids. I don’t really have any background in number theory and probably wouldn’t have tackled this project with out Sanderson’s video to show me the path forward. It really is amazing what resources are on line these days!

## Sharing Grant Sanderson’s “Pi and Primes” video with kids. Part 1

[This one was written up pretty quickly because we had to get out the door for some weekend activities. Sorry for publishing the un-edited version]

Grant Sanderson has a new (and, as usual, incredible) video on “Pi hiding in prime regularities”:

By coincidence, we’ve done a project on this topic before:

A really neat problem that Gauss Solved

The old project is based on Chapter 8 from this book:

Sanderson’s new video is pretty deep and about 30 min long, so I’m going to break our project on his video into 3 pieces. Today we watched (roughly) the first 10 min of the video. Here’s what the boys took away from those 10 min:

The first topic we tackled today was how to write integers as the sum of two squares. This topic is the starting point in Sanderson’s video and the main point of the project from the Ingenuity in Mathematics project. We explored a few simple examples and, at the end, talked about why integers of the form 4n + 3 cannot be written as the sum of two squares:

Next we turned our attention to the complex numbers and how they came into play in (the first 10 min of) Sanderson’s video. My focus was on the Gaussian Integers. In this part of the project we talked about (i) why it makes sense to think of these as integers, and (ii) how we get some new prime numbers (and also lose a few) when we expand our definition of integers to include the Gaussian Integers:

To wrap up I mentioned the topic from the prior project. The question there is something like this -> since counting the exact number of ways an integer can be written as the sum of two squares is tricky, can we say anything about how to write an integer as the some of two squares?

Turns out you can, and that the average number of different ways to write a number as the sum of two squares is $\pi$. Pretty incredible.

[and, of course, I confused an $n$ and $n^2$ in the video 😦 Looking at the prior project will hopefully give a better explanation than I did here . . . . ]

I’m always excited to go through Grant Sanderson’s video with the boys. He has an amazing ability to take advanced ideas and make them accessible to a wide audience. Sometimes making the topic accessible to kids requires a bit more work – but Sanderson’s videos are a great starting point.

## Fawn Nguyen’s incredible Euclidean Algorithm project

Fawn Nguyen recently published an incredible blog post about a project related to the Euclidean Algorithm that she did with her students:

Fawn Nguyen’s “Euclid’s Algorithm

Fawn’s projects are usually very easy to do right out of the box, and this one is especially easy since you can just start with her pictures. So, we just dove in.

You’ll see from the comments my kids had that Fawn really has made using this blog post effortless:

Next I asked them to make their own shapes. They built the shapes off camera and then we talked about them.

At the end I asked them when they thought a shape would require 1x1x1 cubes.

After hearing their thoughts about relatively prime numbers at the end of the last video I asked them to make a shape that wouldn’t require 1x1x1 cubes to finish. Here’s what they made and why they thought it would work:

Such a fun project. Fawn’s work is so amazing. I love using her posts with my kids.

## A fun calculus problem for kids – playing with derivatives and absolute value

I’ve been doing a few “calculus for kids” projects after seeing Grant Sanderson’s essence of calculus series. The series made me see that some of the high level ideas are completely accessible to kids and it has been fun to explore some of those concepts.

Today I thought it would be fun to see what they thought the derivative of absolute value would look like – they had some neat ideas:

Next I thought I would turn the problem around – what if absolute value was the derivative! What would the function look like. This problem was much more challenging. In the first video they spent most of the time just getting their head around the problem:

So, now that they had the ideas in place to solve the problem, they started drawing pictures. The process of getting to the correct graph was really interesting to watch:

The more I think about this calculus project, the more fun I think it is going to be. Many of the ideas in Sanderson’s series will be out of their reach, but some of the high level concepts are incredibly fun to share with kids.

## More calculus ideas for kids inspired by Grant Sanderson

I’m enjoying thinking about how to share Grant Sanderon’s latest calculus video series with kids. My goal is not remotely to develop a calculus course, but just to give kids an opportunity to see and explore some of the basic ideas that Sanderson shares in his video series. At a high level, things like slope of the graph of a function are easily accessible to kids even if the calculations required to make the ideas precise might be beyond them. Our projects so far are here:

Sharing Grant Sanderson’s Calculus ideas video with kids

Sharing Grant Sanderson’s “derivative paradox” video with kids is really fun

Sharing Grant Sanderson’s derivative paradox video with kids part 2

Sharing Grant Sanderson’s “derivatives through geometry” video with kids

So, walking the dog tonight I came up with two ideas for discussion:

(i) How does the length of the hypotenuse of a right triangle change as the length of one of the sides changes?

(ii) If a function has the property that the slope of the tangent line is the same as the value of the function, what would that function look like?

We began with a quick review / discussion of slope in the context of a curve. This concept is still new to the boys and I wanted to have one quick review before we dove into the main project:

Next we moved on to the right triangle problem – how would the length of the hypotenuse change when one of the side lengths changed? The boys were able to grasp some basic ideas around when the changing side was short (near zero length) and very long (near infinite length), and we were able to make a sketch of what the derivative might look like just from these basic observations:

The next project was a basic (the most basic?) differential equation -> a function has the property that the derivative at a point is equal to the value of the function at that point. The value of the function at 0 is 1. What does this function look like?

Finally, we went up to the computer to use Mathematica to explore our two questions. For purposes of this higher level conceptual overview, it is nice that Mathematica’s built in functions allow us to study these two questions without having to do the calculations ourselves:

The more of these project I do, the more I’m convinced that this is a useful exercise for kids. For now at least, I can’t think of any reason why learning about these basic ideas at the same time you are learning about functions will cause problems.

## 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!