Extending our project on Kelsey Houston-Edwards’s Infinity video

Yesterday we did a project inspired by Kelsey Houston-Edwards’s latest math video:

Here’s a link to our project:

Sharing Kelsey Houston-Edward’s Infinity video with kids

Last night my younger son and I were talking a little bit more about the project and he asked me why Cantor’s diagonal argument for why the set of real numbers is larger than the set of Natural numbers doesn’t work for rational numbers!! Yes!!

We explored that question today. First we did a quick review of the diagonal argument (which was the last part of yesterday’s project) and then we began talking about the rational numbers:

Next we looked at what would happen if you applied the diagonal argument to the rational numbers:

After getting our arms around the diagonal argument when applied to rational numbers, we backed up and looked at the argument why rational numbers are countable.

Unfortunately I made an easy concept hard in this part of the project. I was trying to explain the “easy for me” idea that if a set that is larger than the rationals was the same size as the natural numbers, that meant the rationals must also be the same size as the natural numbers. My explanation started off terribly and went down hill . . . .

Finally we looked at one of the strangest consequences of all of this infinity stuff. In math language – the rational numbers have measure zero.

The idea here always blows my mind and is a really fun idea about infinity to share with kids.

Sharing Kelsey Houston-Edwards’s Infinity video with kids

The latest PBS Infinite Series video came out this week:

This is the 4th video in an incredible series from Kelsey Houston-Edwards. Our projects on the first 3 are here:

Sharing Kelsey Houston-Edward’s [higher dimensional spheres] video with kids

Sharing Kelsey Houston-Edward’s Philosophy of Math video with kids

Sharing Kelsey Houston-Edward’s Pigeonhole Principle with kids

I had the boys watch the new video together and started today’s project by asking them what they thought was interesting.

After hearing what the kids found interesting, we dove into the idea of bijections. We talked a bit about how a bijection has to work both ways using the bus idea from the video.

After the bus example we moved on to the example of the bijection between the points in an interval and points on the real line.

We finished up by talking about the bijection between the national numbers and the positive even integers.

Since we’ve done several prior projects where infinity played some role, the next thing I asked the kids was for some thing that they already knew about infinity – both things that they thought made sense and things that they thought didn’t make sense. The discussion and examples here were amazing – “no one knows what infinity divided by infinity is” 🙂

Finally, we wrapped up the project talking about why the infinity associated with the real numbers is larger than the infinity associated with the natural numbers.

I thought this would be a fun way to end the project since it was one of the key ideas in Houston-Edwards’s video:

So, another really fun project from the new set of math videos from PBS Infinite Series. I love this new series – can’t wait for the next one!

Revisiting 1/3 in binary

Last night we talked about writing pi in base 3.   A long long long time ago we talked about writing 1/3 in binary.  Here are those two projects:

Pi in base 3

Writing 1/3 in Binary

I suspected that the boys wouldn’t remember the project about writing 1/3 in binary, so I thought it would make a good follow up to last night’s project.

I started by just posing the question and seeing where things went. They boys had lots of ideas and we eventually got most of the way there:

At the end of the last video they boys figured out that if our number was indeed 1/3, if we multiplied it by 3 we should get 1. That reminded them of the proof that 0.9999…. (repeating forever) = 1.

We reviewed that proof and applied it to the situation we had now.

Just one little problem . . . what if we apply the idea in this proof to a different series, say 1 + 2 + 4 + 8 + 16 + . . . . ?

We’ve looked at the idea in this video before:

Jordan Ellenberg’s “Algebraic Intimidation”

We felt pretty comfortable believing that 0.9999…. = 1 and that we’d found the correct series for 1/3 in binary, but do we believe the results when we apply the exact same ideas to a new series?

I love projects like this one 🙂

A partial response to Sam Shah

[I apologize at the beginning – this post was written very quickly and probably doesn’t read that well, but I loved Shah’s post and wanted to show a few projects for kids on about 8 out of 10 of the topics he mentioned]

Saw this post from Sam Shah today:

Sam Shah’s Inspiration and Mathematics blog post

It is a great post – honestly, I can’t recommend it enough. He talks about being inspired by Ed Frenkel’s book Love and Math. I loved that book and even turned part of it into a short talk about \sqrt{2} and i with the boys:

Ed Frenkel, the square root of 2 and i

Frenkel’s interview with Numberphile is excellent as well (and probably the thing I’ve linked to the most on this blog!):

My post today is a response to this paragraph from Shah’s post:

But what do my kids learn about modern mathematics — from school or popular culture? Are there any weirdnesses or strangenesses that can capture their imagination? Yes! Godel’s incompleteness theorem. Space filling curves. Chaos theory. The fact that quintic and higher degree polynomials don’t have a general “simple” formula always works like the quadratic formula. Fractals. Higher dimensions. Non-euclidean space. Fermat’s Last Theorem. Levels of infinity. Heck, infinity itself! Mobius strips. The four color theorem. The Banach-Tarski paradox. Collatz conjecture (or any simply stated but unproven thing). Anything to do with number theory! Anything to do with the distribution of primes! But do they capture students’ imaginations? No… because they aren’t exposed to these things.

I only have about an hour to write today, but let me see what I can do . . .

In order:

(1) Godel . . . I wish I had an project, but I don’t. Not sure how I would explain this one to kids. My own introduction to the ideas was reading Godel, Esher, Bach in high school.

(2) Space Filling Curves

Our first project involving space filling curves is here (and hits several of Shah’s poinst):

Banach Tarski, Hilbert Curves, and Infinite Sets

Our second project with the boys involving space filling curves was inspired by a 3D Printing blog post from Laura Taalman:

Laura Taalman’s Peano Curve post

That post led to this conversation:

One other project that could lead to a discussion of space filling curves is studying the Gosper Curve:

Dan Anderson’s Gosper Curves

Oh, and I almost forgot, Evelyn Lamb wrote a wonderful piece on space filling curves just last week:

Evelyn Lamb’s piece on Space Filling Curves

(3) Chaos Theory

I know this isn’t exactly “chaos theory” but it remains one of my all time favorite projects with the boys:

Computer Math and the Chaos Game

The 30 seconds starting around 2:46 shows why:

I also believe that the basics of dynamical systems are accessible to kids. I have fond memories of playing around with the logistic equation in high school.

Steven Strogatz’s video lectures and dynamical systems for kids

Finally, James Gleick’s Chaos is a wonderful book and any high school kid interested in learning about chaos theory will love it.

(4) The idea that quintic equations cannot be solved in general

Explaining this idea to kids is my secret dream. But now that the secret is out, here are a few thoughts that I’ve had.

First, I’ve been reading several books to try and try and try to figure out a way to make the idea accessible. Here’s one of the books I’ve been reading:

I have a few others, too. The problem is that the subject is pretty advanced. However, some of the ideas from group theory involved in the proof that you can’t solve quintics in general are accessible to kids. For example, we did this project about cubes inscribed in a dodecahedron:

A 3D Geometry project for kids and adults inspired by Kip Thorne

The group theory idea hiding in this project is that the group A_5 has an element of order 5, and that’s one of the key ideas in the proof of why quintics can’t be solved in general. The action of element or order 5 shows that there are 5 cubes inside of a dodecahedron.

There’s actually another way – and it is incredible – to see that’s there’s a cube inside a dodecahedron:

Can you believe that a dodecahedron folds into a cube

One other bit of 5 fold symmetry shows up with inscribe Tetrahedrons:

Five Tetrahedrons in a Dodecahedron

So, not perfectly getting at Galois theory, but at least a start down the path . . .

(5) Fractals

I’ve already mentioned a few. Our Gosper curve projects have been extremely fun:

A Fun Fractal project – exploring the Gosper curve

Dan Anderson’s Gosper Curves

The Koch snowflake is always fun:

Using the Koch Snowflake to introduce fractals

The idea that the area is finite and the perimeter is infinite really bothered my younger son:

and just last week we used Matt Parker’s latest video to talk about the Menger sponge and a strange relationship it has with \pi

Using Matt Parker’s Menger Sponge video to talk Fractions with kids

(6) Higher Dimensions

Oh gosh . . . this is such an exciting topic for kids thatI’m not even sure where to start!

Here are all of our projects with the word “dimension”:

All of our projects on dimension

IMG_0555

These include projects from tiling shapes in 2 dimensions as in the picture above:

Zome Tilings

Up to a fun series of projects about 4 dimensional shapes inspired by a Patrick Honner Pi day post:

A link which includes all of our projects inspired by Patrick Honner’s Pi day post

A few others worth mentioning:

Sharing 4D shapes with Kids

Using Hypernom to get kids talking about math

Carl Sagan on the 4th Dimension

Counting Geometric Properties in 4 and 6 dimensions

(7) Non-Euclidean Space

I don’t have a lot here, and what I haven isn’t necessarily right on the money. We did use the Gosper islands to explore a little bit about non-integer dimensions:

Integer and Non-Integer Dimensions

I also think that we’ve done the classic “angles on a sphere” problem, but I just can’t find it.

(8) Fermat’s Last Theorem

I don’t have a specific project for kids on Fermat’s last theorem, but this Numberphile interview with Ken Ribet about a piece of the puzzle used to prove the theorem is a must see. The video is a wonderful illustration of what research mathematics is like:

(9) Infinity!!

There are so many ways to capture the minds of kids talking about infinity. Here are just a few of the projects that we’ve done:

To infinity . . . and to the next infinity

Exploring infinity and other Surreal Numbers

Possibly my favorite bit of math involving infinity to talk about with kids is this Numberphile video:

I’ve also loved talking about this series with my kids – using the idea of “algebraic intimidation” from Jordan Ellenberg’s How not to be Wrong:

One of the projects I’ve done about that video is here:

Jordan Ellenberg’s “Algebraic Intimidation”

(10) Mobius strips

[note: I’m low on time and am just copying this piece about some fun math to do involving Mobius strips directly from a recent post]

This sequence of tweets inspired a really fun set of projects with my kids as well as some other kids from the neighborhood. You just need strips of paper, scissors, and tape.

Here’s what the initial set up looks like – piece of cake!

Screen Shot 2016-04-03 at 6.57.54 PM

Here’s the project:

Cutting a double Mobius strip

This is a wonderful project for kids because the results are so surprising and so hard to see ahead of time even when you’ve already been surprised a few times!

When you finish the project you can watch Wind and Mr. Ug with the kids!

 

(11) The 4 Color theorem –

I actually can’t believe that I’ve never talked about this theorem with my kids . . . but I haven’t. Oh well.

(12) Banach-Tarski

We did a whole project about it 🙂

Banach Tarski, Hilbert Curves, and Infinite Sets

(13) Finally – the Collatz Conjecture

This is another super fun bit of math to share with kids. We’ve talked about it a bunch:

The Collatz Conjecture and John Conway’s Amusical Variation

Having the kids listen to the music at the end of that project is one of my favorite math moments that we’ve ever had:

 

and here’s a more standard approach to the Collatz Conjecture with kids:

 

Exploring infinity and other Surreal Numbers

[post publication update on March 14th, 2016. Lazily doing a google search to get a link for this blog post I learned of this article in Discover Magazine from 1995:

http://discovermagazine.com/1995/dec/infinityplusonea599

I’d not seen this article before – though I should have as it is linked in Jim Propp’s “The Life of Games” blog post which was the seed of my interest in the Surreal numbers. The original title of this post was completely by accident the same as the title of the Discover Magazine article. After learning about the prior article I have revised the title of this post.]

Yesterday we revisited the surreal numbers by looking at the game “checker stacks”

Revisiting the Surreal Numbers

We explored the values of some of the positions in the game and found some simple stacks that had values of 1/2 and 1/4.  Today we studied some of the more unusual ideas in the game, looking at positions that seem to have infinite and infinitesimal values.

Just to be super clear from the start -I’m not trying to be even remotely formal about the surreal numbers in this project.  Rather, I’m stating a few rules and ideas and we are exploring some simple consequences for fun.  The ideas here are something that I think that many many kids will find fascinating.

So, on to the game following the ideas and terminology in Jim Propp’s Life of Games.

The first thing we looked at was the “deep blue” checker.

The boys seemed to catch on to the idea that the deep blue checker had a value of infinity fairly quickly, though the idea that it was strange that a piece could have an infinite value didn’t become clear until the next part of the talk.  I was pretty happy to see that they wanted to explore what happened when a deep blue played against a deep red – that game seems shows that infinity minus infinity equals 0!

 

After the discussion about infinity in the last part of the talk, we explored some of the strange properties of these new numbers. First we looked at infinity + 1, which the boys assumed would be the same as infinity. Surprise 🙂

For the last part of the project this morning we looked at a new stack – a blue with a deep red on top of it. I had to do a little bit of review of the game for my younger son at the beginning of this part of the talk, but once we got the rules straight he understood the strange property of the blue + deep red stack – it has a positive value, but that value seems to be less than any positive number we can think of.

So, a fun project showing the boys some neat, though odd, ideas from math. I love how easy it is to lay out some basic properties of the game “checker stacks” and have kids explore the implications of these properties. To me, this is what learning math looks like.

Talking through Christopher Long’s neat probability problem with kids part 2

Yesterday we talked a bit about this really neat probability problem that Christopher Long posted on Twitter:

Here’s a link to that conversation:

Talking through Christopher Long’s probability problem with kids (part 1)

We finished up the conversation today.

Yesterday we ended after talking about why the probability that two randomly chosen integers will share no common divisors is 6 / \pi^2. Today we revisited that conversation and also discussed why picking two random positive integers is actually a little bit of a hard thing to do:

 

Next I wanted to move on to discuss the last part of the solution to the original probability problem, but a question about fractions came up and we had a short conversation about which fraction was larger 36 / 81 (which arose in the problem from the approximation that \pi = 3, or the fraction 36 / \pi^4. So, this strange probability problem gave us a surprising way to talk about fractions 🙂

 

After the short conversation about fractions we returned to the problem. We now know that the probability that the two randomly selected pairs of integers will both be relatively prime is 36 / \pi^4. But what about the probability that both pairs will have a common divisor of 2?

The answer to this question is a little subtle, but it is the key to solving the original problem. Thinking about this question led to a great conversation about primes and what GCD means.

 

Now that we had a little bit better of an understanding about primes and GCD, we dove into how to think about the problem where the two pairs of integers share a common divisor of 2. In this video we talk about why the probability that two randomly selected integers will have a the greatest common divisor of 2 is 1/4 of the probability that they are relatively prime.

 

Finally, we arrive at the solution to the original problem: the probability is 2/5. Surprise!! I bunch of powers of \pi all cancel at the end. Super fun problem.

 

The original problem is obviously way too difficult for kids, but walking them through the solution was really fun. Along the way we got to talk about fun concepts like infinity, fractions, primes, divisibility, and even some really advanced topics like how does \pi show up in this problem?

I definitely wouldn’t do a project like this one too often, but once in a while an advanced problem actually has some pretty neat stuff for kids hiding inside of it.

Walking down the path to the surreal numbers part 2

I’ve got less time to write today because of a family trip, but the videos below show part 2 of our Family Math project about “checker stacks” and the surreal numbers.

The first part of the project is here:

Walking down the path to the surreal numbers

and we are following Jim Propp’s blog post about the surreal numbers which is here:

Jim Propp’s “The Life of Games”

The first thing we looked at today was “deep blue” stack. The surprise about this piece in the game of checker stacks is that its value appears to be positive infinity.

Next we quickly looked at the “deep red” piece and then looked at a blue + deep red stack whose value is pretty surprising. It was great to hear the ideas that the kids had about this stack.

Next we moved on to study the “deep purple” piece. This piece is pretty mysterious. I thought pretty hard about how to explain the value of this piece to the boys, but didn’t really come up with any good ideas. Instead we spent about 10 minutes exploring its value. That was a great conversation, but we never did quite get to the value of 2/3 that Propp gives in his blog. I’m ok with that outcome, though – I felt the conversation about the possible values was really great.

So, sorry for the quick write up of this 2nd project about the surreal numbers. I’m really happy to have seen Jim Propp’s blog and think there’s got to be a great way to use checker stacks for a neat math project for kids.

The joy of x^x^x^. . . .

Back in 2013, Steven Strogatz posed this problem on Twitter:

A month or so ago, Shecky Riemann suggested that I take a look at this book to find a few fun projects with my kids:

Book Pic

Today I finally got around to the book. We opened it up to a random page and found this problem on page 77:

Find the value of x that satisfies the equation x^{x^{x^{.^{.^{.}}}}} = 2

So, we stumbled on a variation of the problem that Strogatz was tweeting about 2 years ago. The fact that he was tweeting about it suggests that there might be more going on with this problem than initially meets the eye.

As an aside, I originally intended to talk about the next problem in the book, too, which was the Monty Hall problem. However, our discussion of the infinite tower of x’s was so interesting that it didn’t really make much sense to move on to the second problem. We’ll talk Monty Hall some other time.

We started the project by talking about each kid’s initial reaction to the problem My older son noticed that x would have to be between 1 and 2 because he thought that an infinitely tall tower of powers of 2’s would have a value equal infinity. My younger son was worried that, actually, any infinitely tall tower of powers of numbers greater than 1 would have a value equal to infinity.

Interesting observations – we decided to check and see what happens with 2’s. Estimating the value of 2^{16} was nice little number sense exercise.

 

Now that we’ve talked a little bit about how a tower of powers works, we returned to the original problem. It was fun to hear that my older son remembered that we played around with some similar towers of powers when we played around with Graham’s Number.

My younger son thought there might be a connection to the series 1 + 1/2 + 1/4 + 1/8 + . . . , which was also nice to hear.

Despite some interesting ideas like these ones, we didn’t seem to be getting closer to the answer in this part of the discussion.

 

At the beginning of this part of the project my younger son identifies the main difficulty in this problem: the tower of powers never ends, so how can you evaluate its value? After a little bit of thinking, my older son notices that one interesting thing about the tower is that it doesn’t matter where you start in the tower – you always have to go infinitely high up to find the value.

That’s the critical observation, but it still takes a little bit of discussion to figure out how to use that idea to solve the problem. But . . . eventually they notice that the equation seems to be equivalent to the much easier equation x^2 = 2!

 

Having found that x = \sqrt{2} seems to solve the equation, we decided to explore that solution on a calculator. I probably should have gone to Wolfram Alpha or Mathematica for this part but I didn’t plan ahead so well. At least the calculator helped the boys see that the tower of powers of \sqrt{2} did seem to converge to 2.

 

For the last part of the project I wanted to show the boys what happened when you try out a few numbers other than 2 – the idea here is related to the Strogatz tweet I mentioned above. The boys were caught off guard by the seeming paradoxes we ran into. After we turned off the camera my older son said:

“Wait, that means that 5 is less than 2 and that 2 equals 4. I’ll have to re-write the number system.”

I love it when math ideas really bother the boys – it shows that the ideas in the project are really making them think!

So, thanks to Shecky Riemann for the book recommendation. Our first project from the book was definitely fun!

To Infinity and . . . to the next infinity

I started in on a geometry course with my older son this week.  On Wednesday we were discussing some basic shapes and he asked a neat question:  If you have a circle in a plane, are there more lines in the plane passing through the circle or more lines in the plane that don’t pass through the circle?  Fun!

I told him that the answer to the question was a little more complicated than it seems, but we’d talk through it over the weekend.  Well, the weekend is here and we talked through it this morning!

We started by introducing the question and talking about some of the non-intuitive properties of infinity.  I thought the easiest place to start would be comparing the set of positive integers with the set of positive even integers since this comparison is a nice way to show that infinite sets are a little strange!  I think that kids can understand some of the basic ideas about infinite sets, even if some of the concepts my be a little over their heads:

Next we moved on to a slightly more difficult question – comparing the set of positive integers with the set of positive integers that are powers of 2. In this case it looks like the second set is much smaller than the first one, and finding a way to see that these two sets have the same size did prove to be a challenge. However, with a little nudge, they were able to find a way to map the two sets to each other and even sort of answer the question “what is the opposite of powers?”

Probably the next natural step would be to show that the rational numbers are also countable, but I decided to skip that proof because I was worried that it would be more of a distraction and wouldn’t help so much with the question about lines and circles. Instead the next thing we talked about was comparing the real numbers to the integers via Cantor’s diagonal argument. This argument shows that there are more real numbers than integers. Although I didn’t necessarily want to focus on the different infinities, I thought it was important to help them understand the idea that just because two sets are infinite, they may not be the same size. In retrospect, I wish I wouldn’t have called this the “next infinity,” I guess we’ll have to correct that little slip the next time we talk about infinity.

With all of this background behind us, we moved on to answering the original question about lines and circles. We began by looking at a problem that is a little easier – what happens if we look only at vertical lines? Restricting our attention to this slightly easier problem allows us to see a surprising result – the number of points between 0 and 1 is the same as the number of points between 1 and infinity!

Now with the discussion of the vertical lines out of the way we can solve the general problem if we can figure out how to deal with lines that aren’t vertical. As luck would have it my older son thought looking at horizontal lines would be a good way to start. That idea got the boys thinking about rotational symmetry and led them to the solution to the original problem! Unfortunately I got confused on one of the pictures, but hopefully that 30s of confusion didn’t cause too much confusion – the perils of illustrating some of these ideas early on Saturday morning!

This was a really enjoyable project and the boys seemed to have a lot of fun and stayed engaged all the way through. I’m extra happy that this project came from a question that my son asked earlier this week. It is nice to talk about some of these ideas from pure math every now and then. It helps show younger kids that math isn’t just about playing around with numbers.

Banach-Tarski, Hilbert curves, and infinite sets

I have my kids write short reports every day on chapters they select in Cifford Pickover’s amazing “Math Book.”  (sorry I don’t know Latex well enough to format the title correctly).
Image

These reports give them a chance to see fun math outside of the standard stuff covered in their school books.   Last week my younger son stumbled across the section on the Banach-Tarski theorem and it really intrigued him.  I finally got around to talking a little more about that theorem with the boys today, though it obviously isn’t the easiest subject to cover with younger kids!

The first thing we talked through was the two different statements of the theorem.  A short, and excellent as usual, summary of the two theorems can be found on the Cut the Knot website:

http://www.cut-the-knot.org/do_you_know/banach.shtml

I covered the the two different statements of the theorem and moved on to a much easier to understand example of infinite sets – why there are the “same number” of positive integers and positive even integers.

With the example with integers and even integers showing us how to compare infinite sets, I moved on to showing them that a line segment of length 1 has the “same number” of points as a line segment of length 2.    The ideas in this proof at least let you see how one object could somehow be the “same size” as something that seems to be twice as large.

 

The next thing we talked about was how we could see that a line segment of finite length could have the “same number” of points as an infinitely long line.    We approach this idea using stereographic projection:

Next we moved on to 3D and I showed them that the sphere has the same number of points as the plane.  The idea here was also to look at stereographic projection, though luckily for this example we have a special prop designed by Henry Segerman that we found on Laura Taalman’s 3D printing blog:

http://makerhome.blogspot.com/2014/01/day-139-stereographic-spheres.html

Goes without saying that holding the model in your hand is quite an improvement over a sketch on the board!

So, by this time we’d seen that a line segment has the “same number” of points as the whole line, and a sphere has the “same number” of points as the plane.  Now we show something really amazing – a line can fill up a square, and hence the plane.  That means that a line segment has the “same number” of points as the whole plane.  Wow.

The approach here took much longer than what is on camera.  We found this great website that gave a tutorial on Hilbert Curves:
https://www.fractalus.com/kerry/tutorials/hilbert/hilbert-tutorial.html

We also found some space filling curves on Laura Taalman’s blog:
http://makerhome.blogspot.com/2013/10/day-60-peano-curve.html

and

http://makerhome.blogspot.com/2014/04/day-225-hilbert-cubes.html

So, the 5 minutes on camera was actually preceded by a couple of hours of printing and drawing Hilbert curves on our own.  It made for a really fun morning:

Lots of people to thank for this one – Clifford Pickover,  Kerry Mitchell, Alexander Bogomolny, Laura Taalman, and Henry Segerman.  So glad to have resources like theirs online to help kids learn about this kind of fun math.

** Addendum **
After finishing up this post we were playing around with our 3D printed Hilbert curve and took it off the base.  After we did that, we found that we could stretch it out into almost a line.  Cool!!  I think it helps kids get a better feel for the fact that it is all one long line segment twisted up into a curve:

**