Math that made you go whoa!

Saw this tweet from Dan Anderson a few days ago:

I had a 7 hour round trip drive yesterday and spent a little time thinking about the math ideas that really grabbed me in high school. Three really stuck out in my mind:

(A) The Extended law of sines:

We learned in our trigonometry class that for a triangle with sides A, B, and C, and corresponding angles a, b, and c that:

$\frac{A}{Sin(a)}$ = $\frac{B}{Sin(b)}$ = $\frac{C}{Sin(c)}$

But it turns out that these ratios are equal to 2R where R is the radius of the circumscribed circle. I learned this idea from the wonderful book Geometry Revisited by Coxeter:

This identity made me think that there was a lot more going on in geometry that met the eye. One neat particularly neat thing that the identity shows is that the area of a triangle with side lengths A, B, and C is equal $\frac{ABC}{4R}$. Beautiful!

(B) 1 + 1/4 + 1/9 + . . . . = $\frac{\pi^2}{6}$

Mr. Waterman used the idea that the coefficients of a polynomial were symmetric functions of the roots to prove this sum. It blew me away. (yes, this is a non-rigorous proof, but it is what captured my attention)

In general, for a polynomial of degree n, $x^n + c_{n-1}x^{n-1} + \ldots + c_1 x + c_0$, sum of the reciprocals of the roots is given by $-c_1 / c_0.$

We know that Sin(x) = $x - x^3 / 3! + x^5 / 5! + \ldots.$ Factoring out an $x$ we are left with a polynomial whose roots are $\pm \pi, \pm 2\pi, \pm 3\pi, \ldots,$ namely:

$\frac{Sin(x)}{x} = 1 - x^2 / 3! + x^4 / 5! + \ldots$

making the substitution u = x^2, we see that the polynomial

$1 - u/3! + u^2 / 5! + \ldots$

are $\pi^2, 4\pi^2, 9\pi^2 \ldots$

By the “sum of the reciprocals of the roots” formula above, we see that

$\frac{1}{\pi^2} + \frac{1}{4\pi^2} + \frac{1}{9\pi^2} + \ldots = \frac{1}{6}.$

Multiplying both sides by $\pi^2$ gives us the result.

This result showed me that there was more going on with the integers than I realized! How could they be connected to $\pi$? A few years later I’d see this identity in a complex analysis class and see that $\pi$ and $e$ were connected in a strange way, too!

(3) A formula for the Fibonacci numbers

I think it was my sophomore year in high school when a former student, Anita Barnes, came back to lecture to Mr. Waterman’s Enrichment Math class. Her talk showed a way to find closed form solutions for simple recurrence relations like the one for the Fibonacci numbers:

$F_{n+1} = F_n + F_{n - 1}$

The idea seemed incredibly simple – for the Fibonacci numbers just assume the solution took the form $F_n = x^n$ and solve for x. Solving the recurrence relation for the Fibonacci numbers was reduced to solving the quadratic equation $x^2 = x + 1.$ From there it was not hard at all to show that the Fibonacci numbers were connected to the Golden ratio. If we let $\phi = \frac{1 + \sqrt{5}}{2}$, then

$F_n = ( \phi^n - (-\phi)^{-n}) / \sqrt{5}$

That just blew me away – there was a simple formula for the Fibonacci numbers (and any simple recurrence relation). You could calculate the 100th Fibonacci number by just knowing the first 2 plus the recurrence relation. I think this was the first idea from advanced math that totally blew my mind.

Using Kate Nowak’s rotated parabola with kids

Saw a Kate Nowak exercise via a Dan Anderson tweet earlier in the week:

and wrote about an estimation problem coming from that picture that I thought was fun:

A fun estimate question inspired by Kate Nowak

I thought it would be fun to see what the boys thought about the rotated parabola, so this morning I showed them a few rotated parabolas and asked them what they thought:

My younger son was interested some of the pieces of area that were cut out by the rotated parabola. Funny enough, my older son was interested by a similar question a few years ago:

The area inside of a parabola

It was fun to explore their ideas about the different areas of the graph. We had a neat detour when my older son wondered what the graph of $y = x^2$ would look like of we were really zoomed out.

Finally, my older son was interested in what the parabola would look like under a variety of different rotations. The discussion here ended up being a neat surprise as what grabbed the boys’ attention was how to change the x- and y- coordinates following a rotation so that the graph would display correctly. I wouldn’t have thought to talk about that, but they were pretty interested in understanding how the coordinates changed. The funny thing is that walking down this path gets you really close to talking about trig functions.

So, a fun morning project even if math need to compute or calculate the rotation of the graph is obviously way, way over their heads. But, since the picture is actually pretty simple, there’s still plenty of interesting things to talk about with kids, AND plenty of stuff that kids might be interested in!

Talking about Steven Strogatz’s “N dogs” question with kids

Earlier this week Steven Strogatz posted this question on twitter:

The problem received lots of attention (for a math problem on Twitter!) and I thought it would be fun to talk through the problem with the boys today. This type of problem is obviously beyond their ability to solve, but I thought they would have some interesting ideas about it anyway . . . and they did!

We started by just talking through the problem and getting a little clarity on what was going on with the dogs. One of the challenges in talking about a problem like this with kids is that elementary school kids don’t usually see problems involving motion, so it takes them a while to find the language and the ideas to describe what’s going on. They got there eventually, though, and by the end of this video the kids think that the dogs will move inside of the original polygon and eventually collide in the center.

With the introductions and initial thoughts out of the way, we tried to talk in more detail about how you would even approach solving this problem.

One idea my younger son had was to look at a triangle rather than trying to solve the problem about the n-gon. The interesting reason for this is that he thought maybe the dogs would run the same distance in all of the n-gon’s so why not look at a simple polygon first.

We tried to draw the shape that the dogs ran in the triangle. Their approach involved approximating the path with straight lines. Also, their drawing showed the idea of the dogs spiraling towards the center of the triangle.

At the end of this video my younger son has the idea to look at infinitely many dogs on a circle!

So the idea of looking at infinitely many dogs basically stopped me in my tracks. It seemed so fun to try to see what was going on in this situation. Here’s what they came up with:

Finally we moved to looking at one of the programs that Dan Anderson made for this problem. The program we looked at is from this tweet:

Sorry for the technical difficulties on my computer while we looked at Dan’s program. Despite Firefox crashing, the kids thought the curves were really cool:

So, despite being a pretty advanced problem, it is still a fun idea for kids to think through. From a “watching kids learn math” perspective, it is neat to hear the ideas that they have about how to approach this complicated problem, too.

Dan Anderson’s “Lights Out” game

Saw this tweet from Dan Anderson yesterday:

He’d been inspired by an article that Tim Chartier post to make an interactive Light’s out game.

There’s no school today so my wife and kids are heading up to New Hampshire for a hike, so I used Dan’s program for a light little morning project with the boys.

I just showed them the game and had them play around. It gave them an opportunity to talk about how the game works and talk a little about the strategy for winning the game.

Two things were interesting to me:

(1) My older son sort of got stuck on, or maybe “anchored to” is better, one of the patterns, thinking that figuring out how to solve this particular pattern was the key to solving the game. It was interesting for me to see how his thoughts about that pattern sort of stopped him from exploring the game more.

(2) My younger son noticed a connection to the “checker stacks” game we explored a little last week. That was neat because that connection does provide an interesting way to analyze the game.

Our check stack blog posts are here:

Walking down the path the surreal numbers Part 1

Walking down the path to the surreal numbers Part 2

So, below are their thoughts on Dan’s game – two (roughly) 5 minute videos from each kid.

Older son’s thoughts:

Younger Son’s thoughts:

A neat problem Dan Anderson shared with us

We’ve spent the last couple of days studying divisors of integers – mainly the number of divisors and the sum of those divisors. This topic came to us via a “things you should know for math contests” list that math team at my older son’s school gave to the kids.

We’ve used Mathematica to help us get a feel for these topics and that computer work (I assume) prompted Dan to share this problem with us:

The problem is: Find the first triangular number with more than 500 divisors.

I asked the boys if they wanted to try to tackle this problem, and they wanted to give it a try. So . . . off we went:

Once the kids understood the problem, I thought it would be useful to spend some time talking about how we could approach the problem. The boys had some pretty good ideas:

The one thing I wanted to spend some extra time on was an alternate way to calculate the triangular numbers. The method that the boys proposed was actually fine, but it seemed like an extra couple of minutes talking about a different approach would be time well spent:

Now we went to the computer to implement our plan. We found that the 12,375th triangular number, 76,576,500, was the first triangular number with more than 500 divisors:

The boys were a little surprised to learn that the first triangular number with more than 500 divisors was smaller than the one with exactly 500 divisors. In fact, we didn’t even find one with 500 divisors yet. In the next part of the project we looked for that number. We did find that number, but it was much larger than we expected – the 1,569,375th triangular number is 1,231,469,730,000, which has exactly 500 divisors!

We wrapped up by looking at 5 of the triangular numbers with exactly 500 factors. They all shared a common factor of 16. We decided to look to see if there was an odd triangular number with exactly 500 factors. As of now (3 hours) after finishing up the project, the computer has not found one.

So, a really fun computer project with the boys. Thanks to Dan Anderson for providing this challenging problem!

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

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!

A neat complex number program from Dan Anderson

I drove back to Boston from NYC today and was pretty tired when I got home. BUT, upon arrival I saw this incredible tweet from Dan Anderson:

I had to ask Dan (and click through a few of the related tweets) to figure out what was going on, but the final shape in the Gif is the image of the unit circle under the map:

$z -> z + z^4 / 4 + z^9 / 9 + \ldots + z^{n^2} / n^2 + \ldots$

Amazing!

It was clear that the boys would find Dan’s tweet interesting, so I thought up a short project over dinner. Before diving in to that project, though, I just showed them the gif that Dan made and asked for their thoughts. The pings that you hear from my phone during the four videos are a barrage of more neat gifs from Dan!

After hearing what the kids thought about the shape, we moved to the white board to talk a little bit about complex numbers. The kids have heard a bit about complex numbers in the past – just not recently. Once we finished a quick review of some of the basics of complex numbers, we talked about what the image of the unit circle looks like when you take powers.

The talk here was obviously not intended to be a comprehensive talk about complex numbers. The ideas here are just what came to mind when I saw Dan’s picture plus my attempt to explain those ideas to kids on the fly.

Now I explained the picture Dan was making. When you first write down the map (as above) it looks really complicated. We tried to simplify as much as possible by walking through the images of some easy numbers. I was happy that we were actually able to make some good progress here. (Also happy that my older son thinks every infinite series adds up to -1/12. ha ha – thanks Numberphile!)

After we finished this part, Dan actually published a gif showing the first 20 steps – that picture helps you see the images that we talked about in this video:

Not having Dan’s latest gif handy during our project, we went back to his original picture to see if we had understood how 1, -1, i, and -i behave. It is really neat that you can explain the behavior of these special points to young kids 🙂

We end this video by looking at another new gif that Dan made while we were working out our project – it shows the images of circles with radius ranging from 0 to 1 under the map. Great work by Dan!

So, a really fun project for kids thanks to Dan’s incredible programming work. Such a great way to introduce kids to the cool behavior of complex numbers.

Dan Anderson’s Dragon Curve program

The best way to start off on this project is to say that I wish we’d done better with this one. Not that it went terribly or anything, I just think that Dan’s program is a great thing to use to talk with kids about Dragon curves, fractals, programming, and probably lots of other stuff, too. We just scratched the surface.

With that off my chest . . . .

Earlier this week Dan Anderson posted a really cool program you can use to explore the Dragon Curve (you may have to click on the gif to start the animation):

The code and an interactive version of the program is here:

Dan Anderson’s cool Dragon Curve program

One of the reasons that I was excited to see Dan’s program is that I’d done a Dragon Curve project with the boys a little over 3 years ago. In fact, it was Family Math 8 – this project was Family Math 292!!

Today we started by just playing around with Dan’s program. The main idea here was to see if they remembered what Dragon Curves were and to let them see the basics of how Dan’s program worked:

Next I let the kids just play around with the program. They enjoyed varying the angles in the curve and seeing the resulting shape change from sort of simple looking, to crazy looking, and then *surprise* back to amazingly simple looking:

In the last part we decided to take a closer look at the situation when the angles in the curve produced equilateral triangles. Unfortunately, it proved to be a little more difficult than I thought to talk about the angles here because the clockwise vs counter-clockwise turns were a little confusing for the boys to see. Although they seemed to get to a pretty good understanding by the end of the talk, I wish I would have anticipated the difficulty with the angles here. I’m sure I could have explained what was going on much better if I would have understood ahead of time that understanding the angles would be confusing.

So, as I said in the beginning, there’s way more you can do with Dan’s program that we did in this project. Even just scratching the surface was fun, though. I hope I get to see someone use this program for a “hit it out of the park” math project with kids.

A fun fractal project: exploring the Gosper curve

Over the last few days I’ve been preparing a project with the boys based a great fractal geometry example I found in this wonderful book:

We finally got going with that project this morning. The starting point was watching this Vi Hart video which gives a “proof” that $\pi = 4:$

After we watched the video we sat down to talk about the strange result and what they thought was going on. They seemed to gravitate to the idea that the jagged edges were causing problems, but the fact that the zigs and zags were getting smaller and smaller – and would eventually have a height of 0 – was still a bit confusing:

After talking about the Vi Hart video I introduced the kids to the Gosper curve by showing them the figures in the book that inspired this example. We also made use of an amazing program that Dave Radcliffe shared when I asked for a little help on Twitter:

Playing around with this program really helped the boys see the first couple of shapes in the sequence that eventually leads to the Gosper curve. I definitely owe Dave a big favor!

The next part of the project was to build the first couple of shapes that lead to the Gosper curve out of our Zometool set. The initial hexagon was easy, obviously, but the shape at step 2 gave them a little difficulty. In the video below they talk about building the shapes and then explore a connection between the hexagon from step 1 and the shape from step 2. The fun part here is that the boys saw some of the important connections that lead you to the Gosper curve.

Next we built a level 3 shape. It was lucky that we had the program from Dave Radcliffe since that allowed the boys to a little more confident that they had the right shape. It is interesting to see the 6 new level 2 shapes surrounding the original level 2 shape. Too bad our living room isn’t big enough to make a level 4 shape!

One interesting comment from my younger son is that he thinks that as we increase the size we’ll get closer and closer to a shape that looks like the original hexagon.

For the final part of the project we used our 3d printer to make 7 of the (approximate) Gosper curves. Here’s the shape we used from Thingiverse (our shape is the 2nd of the three shapes, but I can’t get that one to link properly):

The Gosper Curve on Thingiverse (I printed the middle one)

The punch line for the project is the same punch line that caught my attention in the book – when you increase the linear size of the Gosper curve by 3, the area inside the curve increases by a factor of 7 rather than a by a factor of 9. Everything that the boys have learned about scaling up to this point is that area scales as the square of the linear factors, but fractals have a different property. Pretty amazing!

Also, sorry for not explaining the analogy between the two boundaries right. Felt as if I was wrong as I was explaining it, but didn’t see what I got wrong until just now.

As a fun end to the project, I showed them Dan Anderson’s modification of the Gosper Island shape – sort of a combination of the Sierpenski Triangle / Menger Sponge shapes and the Gosper Island:

So, a fun project giving the kids an introduction to fractal dimension – a concept that I never would have guessed could be made accessible to kids. Really happy to have had the luck of running into this fun idea last week.

10 fun math things from 2014

I’ve been paying attention to math a little more in 2014 than I have in previous years and thought it would be fun to put together a list of fun math-related things I’ll remember from this year:

(10) Dan Anderson’s “My Favorite” post

Dan asks his students to talk about things they would like to learn more about in math class, and the students talked about subjects ranging from topology to diving scoring. I was really happy to see the incredibly wide range of topics that the kids thought would be interesting. Beautiful post by Dan and a fantastic list of topics chosen by his students – this one made a big impression on me:

Dan Anderson’s “My Favorite” post

My initial reaction to Dan’s post is here:

A list Ed Frenkel will love

(9) Laura Taalman’s Makerhome blog:

We bought a 3D printer early in the year and it allowed us to do a bunch of math projects that wouldn’t have occurred to me in a million years. Most of those projects came either directly or indirectly from reading Laura Taalman’s 3D printing blog. As 3D printing becomes cheaper and hopefully more available in schools, Taalman’s blog is going to become the go to resource for math and 3D printing. It is an absolute treasure:

Laura Taalman’s Makerhome blog

An early post of mine about the possibilities of 3D printing in education is here:

Learning from 3D Printing

and one of our later projects is here:

Klein Bottles and Möbius Strips

(8) Numberphile

It has been nearly a year since Numberphile’s fun infinite series video hit the web. I know people had mixed feelings about it, but I loved seeing a math video spark so many discussions:

I’ve used so many of their videos to talk math with my kids, I’m not even sure which of them to pick for examples. Here are two:

Using Numberphile’s “All Triangles are Equilateral” video to talk about constructions

Some fun with Numberphile’s Pythagorean Theorem video

(7) Fields Medals and the Breakthrough Prizes

Erica Klarreich’s coverage of the Fields Medals over at Quanta Magazine was absolutely amazing. Two of her articles are below, but all of them (including the videos) are must reads. Her work her made it possible for anyone to meet the four 2014 Fields medal winners:

Erica Klarreich on Manjul Bhargava

Erica Klarreich on Maryam Mirzakhan

A really cool opportunity to understand the work of one of the Fields Medal winners came when the Mathematical Association of America made an old Manjul Bhargava’s paper available to the public. I had a lot of fun playing around with this paper (that he wrote as an undergraduate, btw). It made me feel sort of connected to math research again:

A fun surprised with Euler’s identity coming from Manjul Bhargava’s generalized factorials

The Breakthrough Prizes in math didn’t seem to get as much attention as the Fields Medals did, which is too bad. The Breakthrough Prize winners each gave a public lecture about math. Jacob Lurie’s lecture was absolutely wonderful and a great opportunity to show kids a little bit of fun math and a little bit about the kinds of problems that mathematicians think about:

Using Jacob Lurie’s Breakthrough Prize talk with kids

I’m glad to see more and more opportunities for the general public to see and appreciate the work of the mathematical community. Speaking of which . . . .

(6) Jordan Ellenberg’s “How Not to be Wrong”

Jordan Ellenberg’s book How not to be Wrong is one of the best books about math for the general public I’ve ever read. I have it on audiobook and have been through it probably 3 times in various trips back and forth to Boston. My kids even enjoy listening to it – “consider the set of all integers plus a pig” always gets a laugh.

One of the more mathy takeaways for me was his discussion of infinite series and what he calls “algebraic intimidation.” Both led to fun (and overlapping) discussions with my kids:

Jordan Ellenberg’s “Algebraic Intimidation”

(5) The Mega Menger Project

The Mega Menger project was a world wide project that involved building a “level 4” Menger sponge out of special business cards. We participated in the project at the Museum of Math in NYC. The kids had such a good time that they asked to go down again the following weekend to help finish the build.

It was nice to see so many kids involved with the build in New York. It also made for another fun opportunity to explore the math behind the project a little more deeply:

The Museum of Math and Mega Menger

(4) People having a little fun with math and math results

For some family fun, check out the new game Prime Climb:

our review is here:

A review of Prime Climb by Math for Love

Also, don’t forget to have a little fun when tweeting about new and important math results. Like Jordan Ellenberg tweeting about the solution of an old Paul Erdos conjecture:

Erica Klarreich’s Quanta Magazine article on the same result was just published yesterday by coincidence:

Erica Klarreich on prime gaps

For me the math laugh of the year was Aperiodical announcing the results of an 8 year search confirming the 44th Mersenne Prime:

(3) Evelyn Lamb’s writing

Evelyn Lamb’s blog is a must read for me. I love the wide range of topics and am pretty jealous of her incredible ability to communicate abstract math ideas with ease. Her coverage of the Heidelberg Laureate Forum was sensational (ahem Breakthrough Prize folks, take note!). This post, in particular, gave me quite a bit to think about:

A Computer Scientist Tells Mathematicians How To Write Proofs

My thoughts on proof in math are here:

Proof in math

Away from her blog, if you want a constant source of fun and interesting math ideas just follow her on Twitter. For instance this tweet:

led to a great little project with the boys:

Irrationality of the Square root of 2

(2) Terry Tao’s public lecture at the Museum of Math

On of the most amazing lectures that I’ve ever seen is Terry Tao’s public lecture at the Museum of Math. I don’t know how it had escaped my attention previously, but I finally ran across it about a month ago. What an incredible – probably unparalleled – opportunity to learn from one of the greatest mathematicians alive today:

Explaining a few bits of his talk in more detail led to three super fun projects with the boys:

Part 1 of using Terry Tao’s MoMath lecture to talk about math with kids – the Moon and the Earth

Part 2 of using Terry Tao’s MoMath lecture to talk about math with kids – Clocks and Mars

Part 3 of using Terry Tao’s MoMath lecture to talk about math with kids – the speed of light and paralax

(1) Fawn Nguyen’s work

When one of the top mathematicians around is tweeting about projects going on in a 6th grade classroom 2000 miles away, the world is working the right way!

Fawn is producing and sharing some of the most interesting math projects for kids that I have ever seen, and I’m super happy that her work is getting recognized. She’s probably inspired more than 20 projects with the boys, and I can’t wait for the next 20 in 2015. Here are two from this year:

Fawn Nguyen’s Geometry Problem

A 3d Geometry proof without words courtesy of Fawn Nguyen

If you have even a passing interest in fun, exciting, and generally kick-ass math projects for kids – you have to follow Fawn.