Looking at the complex map z -> z^2 with kids

Yesterday we did a fun project using Kelsey Houston-Edwards’s compex number video:

Sharing Kelsey Houston-Edwards’s Complex Number video with kids

The boys wanted to do a bit more work with complex numbers today, so I thought it would be fun to explore the map $Z \rightarrow Z^2.$ The computations for this mapping aren’t too difficult, so the kids can begin to see what’s going on with complex maps.

We started by looking at some of the simple properties. The kids had some good questions right from the start.

By the end of this video we’ve understood a bit about what happens to the real line.

After looking at the real line in the last video, we moved on to the imaginary axis in this video. The arithmetic was a little tricky for my younger son, so we worked slowly. By the end of this video we had a pretty good understand of what happens to the imaginary axis under the map $Z \rightarrow Z^2.$

At the end of this video my younger son noted that we hadn’t found anything that goes to the imaginary axis. My older son had a neat idea after that!

Next we looked at $(1 + i)^2$. We found that it did go to the imaginary axis and then we found two nice generalizations that should a bunch of numbers that map to the imaginary axis.

Finally, we went to Mathematica to look at what happens to other lines. I fear that my attempts to make this part look better on camera may have actually made it look worse! But, at least the graphs show up reasonably well.

It was fun to hear what the boys thought they’d see here versus their surprise at what the actually saw 🙂

I think this is a pretty fun project for kids. There are lots of different directions we could go. They also get some good algebra / arithmetic practice working through the ideas.

Sharing Kelsey Houston-Edward’s complex number video with kids

I didn’t have anything planned for our math project today, but both kids asked if there was a new video from Kelsey Houston-Edwards! Why didn’t I think of that 🙂

The latest video is about the pantograph and complex numbers:

Here’s what the boys thought about the video:

They boys were interested in the pantograph and also complex numbers. We started off by talking about how the pantograph works. With a bit more time to prepare (and probably a bit more engineering skill than I have), building a simple pantograph would make a really fun introductory geometry project.

Next we talked about complex numbers. We’ve talked about complex number several times before, so the idea wasn’t a new one for the boys. I started from the beginning, though, and tried to echo some of the introductory ideas that Kelsey Houston-Edwards brought up in her video.

To finish up today’s project we looked at some basic geometry of complex numbers. The specific property that we looked at today was multiplying by i. At the end of this short talk I think that the boys had a pretty good understanding of the idea that multiplying by i was the same as rotating by 90 degrees.

Complex numbers are a topic that I think kids will find absolutely fascinating. I don’t know where (if at all) they come into a traditional middle school / high school curriculum, but once you understand the distributive property you can certainly begin to look at complex numbers. It is such a fun topic with many interesting applications and important ideas – many of which are accessible to kids. Just playing around with complex numbers seems like a great way to expose kids to some amazing math.

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 🙂

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.

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

One of the math mountains that I’ve always wanted to try to climb is to find a way to explain to kids why 5th degree polynomials can’t be solved in general.

The “one step closer” came from a comment by Allen Knutson on one of our projects on John Baez’s “juggling roots” tweet. Here’s the tweet:

Here are the two recent projects that we’ve done after seeing that tweet. Knutson’s comment is at the end of the first post:

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

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

The comment pointed me to a video that shows how the “juggling roots” approached can be used to show that there is no general formula for finding the roots of a 5th degree equation:

The neat thing about the combination of this video and Baez’s post is that you can see some of the ideas from the video in the “juggling roots” gifs in the post.

Tonight I used some of the 3d prints of the juggling roots that I’ve made in the last few days to talk about the ideas a bit more and then we watched just a few minutes of the video.

We started with with a print that I accidentally made twice – but luckily the two prints give us a way to view the juggling roots through two cycles:

Next we looked at a different print to see a different juggling roots pattern. Here I was trying to set up the idea that the roots can move around in different ways. The way those different movements interact is the key idea in the video that Allen Knutson shared.

Finally, we went upstairs to watch a little bit of the video. Sorry for the sound issues, I don’t know why I left the sound on in the video. I mainly wanted the boys to see a different view of the juggling roots and I told them that the video gave the explanation for why 5th degree polynomials can’t be solved in general:

So, although I don’t quite have a full explanation of 5th degree polynomials for kids – I feel like I took a giant step towards getting to that explanation today. It is an extra happy surprise that 3d printing is going to come into play for that explanation!

Evelyn Lamb’s pentagons are everything

Last week Evelyn Lamb published a fantastic article:

Math Under My Feet

In a way – a super serious way – I don’t want you to read this blog post. I want you read her article and just think about some of the properties that the tiling pentagons in article probably have.

The question that same to my mind was this one -> Why are the pentagons in her article Type 1 pentagons?

The resources I used initially to help with this question were:

(i) the pictures of the different tiling patterns in the article:

(ii) Laura Taalman’s Tiling Pentagon resource on Thingiverse:

(iii) and then when I was stumped and wrote to Evelyn she pointed me to the Wikipedia page for tiling pentagons – which is really good!

Wikipedia’s page on pentagon tilings

So, honestly, stop here and play around. You don’t have to have the nearly week long adventure with these pentagons that I did, but I promise that you will enjoy trying to figure out the amazing properties of this damn shape!

If that adventure is interesting to you, I think you’ll also find that sharing that adventure with students learning algebra and geometry would be pretty fun, too!

Here are some of our previous pentagon tiling projects:

Using Laura Taalman’s 3d Printed Pentagons to talk math with kids

Learning about tiling pentagons from Laura Taalman and Evelyn Lamb

Also, here are the first two projects that I did with the boys after reading Evelyn Lamb’s latest article:

Evelyn Lamb’s Tiling Pentagons

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

The problem with those last two projects is that they aren’t actually right. I hadn’t properly understood the shape . . . dang 😦

With a bit more study, though, I did *finally* understand this damn shape!!

So, I printed 16 of them and set off on one more project with the boys tonight. The goal was to show them the 3 completely different tilings of the plane that you can make with Evelyn Lamb’s pentagon.

I won’t say much about the videos except that watching them I hope that you will see that (i) this is a great way to talk about geometry with kids (building the shapes is a great way to talk about algebra), and (ii) that understanding these tiling patterns is much harder than you think it is going to be. As an example of the 2nd point, it takes the boys nearly 10 minutes to make the tiling pattern in Lamb’s article.

So, here’s how things went:

(1) An introduction to the problem:

(2) Using the pentagons to make the “standard” Type I tiling pattern

This tiling pattern is in the upper left hand corner of the picture above that shows the collection of pentagon tiling patterns.

(3) Using the pentagons to make the “pgg (22x)” tiling pattern from the Wikipedia article:

(4) Part I of trying to make the tiling pattern in Evelyn Lamb’s article:

(5) Part 2 of Evelyn Lamb’s tiling pattern:

Don’t really know what else to add. I think playing around with the math required to make these pentagons AND playing with the pentagons themselves is one of the most exciting algebra / geometry projects for kids that I’ve ever come across.

I’m so grateful for Evelyn Lamb’s article. It is really cool to see how a mathematician views the world and it is so fun to take her thoughts and ideas and turn them into projects for kids

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 🙂

Evelyn Lamb’s tiling pentagons

Since the 15th tiling pentagon was discovered in 2015 we’ve done some fun projects with tiling pentagons. A key component in all of our project was Laura Taalman’s incredible work that made all 15 pentagon tilings accessible to everyone:

Here are a few of those projects:

Using Laura Taalman’s 3d Printed Pentagons to talk math with kids

Learning about tiling pentagons from Laura Taalman and Evelyn Lamb

and, of course, pentagon cookies 🙂

Evelyn Lamb has also written some absolutely fantastic articles on tiling pentagons. Here original article on the subject was critical in helping me understand what was going on in the different tilings:

There’s Something about Pentagons by Evelyn Lamb

And her amazing article from last week (April 2017) inspired today’s project:

Math Under My Feet

The prep work for this project was probably 100x more than I usually do because the tiling described in Lamb’s article turned out to be very hard for me to understand. It didn’t look like the “type I” tiling pictured in the article and I spent days trying to see if it was somehow a sneaky form of one of the other tilings.

Finally I wrote to Lamb and asked her about it and she pointed me to the Wikipedia page here which showed that the type 1 tilings have two different forms. One form has a repeating pattern with 2 pentagons and the other has a repeating pattern with 4 pentagons. Ahhhhhh – at last I saw what I was missing and why this “new to me” type 1 tiling was so elusive:

Wikipedia’s page on pentagon tilings

So, having finally understood what was going on with this octagon / pentagon tiling, I got to work making some of the pentagons. I didn’t quite match the pentagons in Lamb’s article, but the ones I made still have the property that they can produce two different tilings.

I got started this morning by having the kids read Lamb’s new article. Here’s what they thought:

Next I had the boys try to make a tiling from the pentagons I made last night. They made the first type of tiling (the one that has two repeating pentagons) and we talked about whether or not that was the tiling in Lamb’s article.

I include the whole process of finding the tiling here to show that even a tiling with two repeating pentagons isn’t so easy to find as you might think.

Now we went to the both Lamb’s article and to the Wikipedia pentagon tiling page to study the various different types of Type I tilings. I’m still a little confused as to what makes tilings different, but however the classification works, here’s our discussion of the various Type I tilings.

Off camera I had the boys try to make the new type of tiling. It took a while (though not super long – from the time they started reading the article until the time we finished the project was roughly 30 min).

Once they had the tilings I turned on the camera to talk about the shapes:

This was such a fun project! Tomorrow I hope to do a second project to show how making these pentagons is a great way to help kids learn about / review basic properties of lines.

Sharing Grant Sanderson’s Calculus Ideas video with kids

Yesterday I saw an incredible new video from Grant Sanderson:

As is the case with all of his videos, this one totally blew me away. I also thought that it has some fantastic ideas to share with kids. So, this morning we tried it out!

I started by asking the boys about the area of a circle – how do you find the area?

We have studied the idea before. Here’s the previous idea (that we got from a Steven Strogatz tweet):

and here are the projects inspired by Strogatz’s tweet:

Steven Strogatz’s circle-area exercise

Steven Strogatz’s circle-area project part 2

Fortunately, the boys were able to remember that idea and explain it pretty well:

After this short discussion I had the boys watch the new “Essence of Calculus” video. I actually left the room so that I wouldn’t interfere. The video below shows the ideas that they found interesting. One thing – luckily! – was the idea of making lots of slices and getting a better and better approximation to a shape. We were able to connect that idea to our prior way of finding the area of a circle, which was nice 🙂

Next we talked about the new (to them) way of finding the area of a circle that Sanderson explains in his video. What made me really happy here is that my younger son was able to understand and explain most of the ideas. It think that a 5th grader being able to grasp these ideas really shows the tremendous quality of Sanderson’s explanation in his video. I also think that it shows that many important ideas from advanced math are both accessible and interesting(!) to kids

Finally, I showed the boys some 3d prints that I made overnight.

These prints were pretty easy to make and I hoped that they would make some of the approximation ideas seem more real. In the middle of the video I remembered that I’d actually tried this idea before (ha!):

3d printing and calculus concepts for kids

After remembering the old project, I ran and got the old shapes, too:

I’m really excited for the rest of this new calculus series. Some of the more advanced ideas might not be so great for kids, but I hope to share one or two more with the boys just to show them a few ideas that they’ve probably never seen before. Plus – I’ve got no doubt at all that this whole series is going to be amazing!