When Cos(x) is larger than 1

My son stumbled on an amazing graph completely by accident the other day. He’s doing some work reviewing trig functions this week and I asked him to just play around with some graphs in Mathematica to get a feel for how Sin[x] and Cos[x] behave. One of the graphs he drew was:

y = \cos( \sqrt{x}) from x = -100 to 100:

Screen Shot 2018-07-01 at 6.17.42 PM

I certainly wasn’t expecting him to make a graph like this one, but was happy that he did. Yesterday we talked through what was going on.

We started by discussing why the graph seemed so strange:

Now we dove into some of the details – which involve complex numbers and the definition:

e^{i \theta} = \cos(\theta) + i \sin(\theta)

as well as the definition of even and odd functions. So, there’s a lot of math to that we need to bring to the table to understand what’s going on in our graph.

Finally, we calculated the exact value of \cos(7i). Again, there’s a lot of advanced math that comes in to the calculation here – but even if some of the math ideas took a bit to sink in, I’d say that all in all it was a good conversation:

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.

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

Yesterday I saw this incredible tweet from John Baez:

We did one project with some of the shapes this morning:

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

The tweet links to a couple of blog posts which I’ll link to directly here for ease:

John Baez’s “Juggling Roots” Google+ post

Curiosa Mathematica’s ‘Animation by Two Cubes” post on Tumblr

The Original set of animations by twocubes on Tumblr

Reading a bit in the comment on Baez’s google+ post I saw a reference to the 3d shapes you could make by considering the frames in the various animations to be slices of a 3d shape. I thought it would be fun to show some of those shapes to the boys tonight and see if they could identify which animated gif generated the 3d shape.

This was an incredibly fun project – it is amazing to hear what kids have to say about these complicated (and beautiful) shapes. It is also very fun to hear them reason their way to figuring out which 3d shape corresponds to each gif.

Here are the conversations:

(1)

(2)

(3)

(4)

(5)

(6) As a lucky bonus, the 3d print finished up just as we finished the last video. I thought it would be fun for them to see and talk about that print even though (i) it broke a little bit while it was printing, and (ii) it was fresh out of the printer and still dripping plastic 🙂

The conversations that we’ve had around Baez’s post has been some of the most enjoyable conversations that I’ve had sharing really advanced math – math that is interesting to research mathematicians – with kids. o

A great complex number question from Art of Problem Solving

My son picked a great problem from Art of Problem Solving’s Introduction to Algebra to talk through today:

A lot of people on Twitter thought the problem was fun and many people commented on the difference between a geometric and an algebraic approach.

A geometric approach is beyond my son’s understanding right now, but the algebraic approach has so many great lessons going for it – many more than you think if you aren’t generally around kids working through math problems:

 

After we were done I talked through some of the geometry for just a little bit – I don’t think introducing the geometry solely through this problem is such a great idea, but I wanted to try anyway. Here’s how it went:

 

Finally, as an extra shout out to AoPS – here’s one of my favorite math videos of all time – Richard Rusczyk illustrating how powerful a geometric approach to complex numbers can be with a stunning solution to an old math contest problem:

Frank Farris’s patterns

A couple of weeks ago Evelyn Lamb’s article Impossible Wallpaper and Mystery Curves introduced me to Frank Farris’s work. On Saturday I stumbled on his book at Barnes and Noble:

  

I was excited to try out some of his ideas with the boys even though they use complex numbers and exponentials which are over their heads. We did the whole exploration this morning using Mathematica.

To start, we just explored the exponential function.

Next we moved to looking at sums of two exponential functions. The boys were surprised by the graphs and we played around with a few more examples. They had some interesting ideas about what the pictures looked like, and I’m glad that the pictures also reminded them a little of Anna Weltman’s loop-de-loops.

Next we moved on to sums of three exponential functions motivated by the idea of trying to produce another kink in the loop. There was a little discussion at the beginning of this part of the talk about complex variables. I thought going down this path was going to be too difficult to explain, so I tried to bring the conversation back to the sums. I love the ideas they had about symmetry here.

Next we looked at Farris’s “mystery” shape and played around a bit more with the ideas. These shapes also led to fun conversations about symmetry:

Finally, I let the kids just play around. As I was writing up this project I got a “hey dad, come here and look at this cool shape” call:

  

So, despite the math underlying these shapes being a little over their heads, the kids seemed to really enjoy seeing these shapes. I loved hearing their ideas and I loved seeing them play around with the ideas for a long time after we turned off the camera.

Also, Farris’s book is absolutely amazing – you’ll love the ideas and the presentation, and probably most of all the incredible pictures he creates from the ideas!

Extending Anna Weltman’s loop-de-loops with Frank Farris’s “Creating Symmetry”

We’ve had a ton of fun in the last couple of weeks with Ann Weltman’s loop-de-loop ideas:

  

Here are two of the projects that we did:

Anna Weltman’s loop-de-loops

Anna Weltman’s loop-de-loops part 2

Last night we stopped at a Barnes & Noble after dinner and I found a book that Evelyn Lamb had written about last month:

  

Here’s Evelyn Lamb’s piece:

Impossible Wallpaper and Mystery Curves by Evelyn Lamb

The book is absolutely wonderful and has so many cool examples. I’d hoped it would be easy to make some of the graphs in the book using Mathematica, and after a little documentation reading to kick off some rust, it wasn’t too hard:

 

At least visually the curves you can make from the idea in Farris’s book remind me of the loop-de-loops. I don’t really think it will be that productive to talk in detail with the boys about exponentials and graphing in the complex plane, but I do think they will like seeing the pictures and talking about them. I’m excited to show them some of the ideas from the book tomorrow morning.

Dan Anderson’s complex mappings part 2

Yesterday we did a fun project about a mapping in the complex plane that I saw from Dan Anderson on Twitter:

That project is here:

Dan Anderson Project part 1

When I asked the boys what they wanted to talk about this morning, they said that they wanted to talk more about Dan’s shapes. Instead of the whiteboard again, today we used Mathematica and had a really fun follow up project.

I stared out by showing them some simple code for the project. That code uses the Sin() and Cos() functions in Mathematica. I did not explain why I used these functions in any detail, but just jumped in to talking about the function Dan was studying:

 

After talking about a few of the simple cases in the first part, we moved on to talk about some of the more complicated cases.

Dan actually made a gif of how the map of the circle changes as you increase the number of terms in the series:

It was fun to hear the boys’ thoughts about the shapes in this part – including a couple of “whoa”‘s!

 

Next we explored another of Dan’s ideas – what about the images of circles having a radius other than 1. We explored a few smaller circles and a few larger circles. Lots of “whoa”‘s here. Seems like the ideas here are a great way to get kids to talk about geometry.

 

Finally, I thought it would be fun to look at a few contour plots of the map. The ideas here are a little more advanced and I’m not sure that the boys fully understood what they were looking at – which is fine. I just wanted to show a few alternate ways to view maps of complex functions:

 

So, a fun morning studying yesterday’s project in a little more depth. It sure was nice to year that they wanted to learn more about Dan’s shapes 🙂

After we finished, my younger son asked if he could play around a little more with the shapes on the computer – an hour later he just asked me again if me can play more. Awesome!