A talk I’d love to give to Calc students

I saw a neat video from Gary Rubenstein recently:

In the video he presents a neat Theorem about partitions due to Euler.

Simon Gregg, by coincidence, was looking at partitions recently, too, and has written up a nice post which includes some ideas from Rubenstein’s video:

The part that struck me in Rubenstein’s video wasn’t about partitions, though, it was about the manipulation of the infinite product. It all works out just fine, which is pretty neat, but sometimes manipulating infinite quantities produces strange results. See this famous video from Numberphile, for example:

Just as an aside, here’s a longer and more detailed explanation of the same result:

The fascinating thing to me is that Euler’s proof in Rubenstein’s video is easy to believe, but the sum in the Numberphile video is not easy to believe at all. Both are examples, I think, of what Jordan Ellenberg called “algebraic intimidation” in his book How not to be Wrong. I used Ellenberg’s idea when I talked about the -1/12 sum with my kids:

Jordan Ellenberg’s “Algebraic Intimidation”

The talk I’d like to give to calculus students would start with the theorem presented in Rubenstein’s video. Once the students were comfortable with the ideas about the infinite products and the ideas about partitions, I’d move on to the idea in the Numberphile video. It would be a fun way to show students that infinite sums and products can be strange and you can sometimes stumble on really strange results.

From Ed Frenkel to Natalie Wolchover to Amy Hogan

A few years ago this Numberphile of Ed Frenkel inspired me to think more about how to share math with the public – and especially kids:

This morning I was reminded of the line around 5:41 in the video where Frenkel talks about how other fields do a better job sharing their work with the public than mathematics does:

“And I feel as though other scientists are doing a much better job; physicists, biologists. We keep talking about the solar system about the universe, about galaxies, about atoms and molecules, elementary particles and DNA.

Those concepts are no more complicated than things we do in modern mathematics. Why is it that, you know, DNA and stars and elementary particles are part of our cultural discourse but mathematical ideas are not? Well, in part because we are not doing nearly enough. We professional mathematicians are not doing nearly enough.”

What reminded me of Frenkel’s quote here was this incredible article from Natalie Wolchover (which I was listening to on Quanta magazine’s podcast):

Neutrinos Hint of Matter-Antimatter Rift

I won’t put words in Frenkel’s mouth, but the study of neutrino interactions feels like exactly the type of thing he was talking about physicists study that is “no more complicated than things we do in modern mathematics.”

Wolchover’s article does a fantastic job of making both the problem physicists are studying and their experimental results accessible to the public. This paragraph in particular struck me as a great bit to share with high school students:

“If the seesaw is balanced, signifying perfect CP symmetry, then (accounting for differences in the production and detection rates of neutrinos and antineutrinos) the T2K scientists would have expected to detect roughly 23 electron neutrino candidates and seven electron antineutrino candidates in Kamioka, Tanaka said. Meanwhile, if CP symmetry is “maximally” violated — the seesaw tilted fully toward more neutrino oscillations and fewer antineutrino oscillations — then 27 electron neutrinos and six electron antineutrinos should have been detected. The actual numbers were even more skewed. “What we observed are 32 electron neutrino candidates and four electron antineutrino candidates,” Tanaka said.”

I love how the reader gets to see what the scientists expected in two situations (i) 23 / 7, or (ii) 27 / 6, and then what they actually found – 32 / 4. What a great example of the scientific process!

This paragraph is also a great opportunity to talk with kids about statistics. I’m sure that high school students could understand the basic statistical ideas here and have have a great discussion about the data presented in the article. In fact, this short lecture from New York master teacher Amy Hogan discusses a similar statistics problem:

So, I loved Wolchover’s article and think it is really a great model for how to communicate complicated ideas from math and science with the public. I especially love that there’s something that teachers can use in their classrooms right away. I hope that we’ll see more and more articles similar to this one that bring advanced ideas from math to the public.

3 proofs that the square root of 2 is irrational

My younger son has been learning a little bit about square roots over the last couple of weeks and I thought it would be fun to show him some proofs that the square root of 2 is irrational. Because this conversation was going to explore some ideas in math that are both important and pretty neat, I asked my older son to join it.

I wasn’t super happy with how this little project went – it felt a bit rushed while we were going through it. Hopefully a few of the ideas stuck.

We started by talking about the square root of 2 and what basic properties the boys already knew about it:

After that short introduction we moved on to the first proof that the square root of 2 is irrational – I think this is probably the most well-known proof. The proof is by contradiction and starts by assuming that \sqrt{2} = A / B where A and B are integers with no common factors.

The next proof is a geometric proof that I learned a few years ago from Alexander Bogomolny’s wonderful site Cut The Knot. It is proof 8”’ here:

Proof 8”’ that the square root of 2 is irrational on Cut the Knot’s site

If you like this proof, we have also explored some geometric infinite descent proofs in a slightly different setting previously inspired by a really neat post from Jim Propp:

An infinite descent problem with pentagons

Finally, we looked at a proof that uses continued fractions. It has been a while since I talked about continued fractions with the boys, and will probably actually revisit the topic soon. It is one of my favorite topics and always reminds me of how lucky I was to have Mr. Waterman for my math teacher in high school. He loved exploring fun and non-standard topics like continued fractions.

So, although I don’t go deeply into all of the continued fraction ideas here – hopefully there’s enough here to show you that the continued fraction for the \sqrt{2} goes on forever.

So, although this one didn’t go quite as well as I was hoping, I still loved showing the boys these ideas. We’ll explore them more deeply as we study some basic ideas in proof over the next year.

Some extensions of Matt Enlow’s tweet

Saw a neat tweet from Matt Enlow yesterday:

The pictures on the bottom row reminded me of two fun projects that we’ve done.

This old Numberphile video featuring Harvard’s Barry Mazur discusses shapes similar to the bottom left picture:



Here’s the project that we did with that video:

Using Numberphile’s “Blob Pythagorean Theorem” video in a lesson

The picture on the bottom right reminds me of a really neat geometry problem that I heard MIT’s Bjorn Poonen discussing. The problem starts with the circle inscribed in the center of the picture below and then ask some questions about higher dimensional versions of the same situaiton:



Here are my blog posts about that problem:

A Strange Problem I overheard Bjorn Poonen discussing

Bjorn Poonen’s n-dimensional sphere problem with kids

A fun surprise in Bjorn Poonen’s n-dimensional sphere problem

Finally, Enlow actually provided a fun extension of his tweet, too! This is a great geometry problem!

Exploring some shapes in Henry Segerman’s new book

Henry Segerman’s new book – Visualizing Mathematics with 3D Printing – arrived last week:

I asked the boys to flip through the book and pick out two shapes each that we could then order from Shapeways. Those shapes arrived today – yay!! I’ll do a more extensive project with these shapes later, but for today I just wanted to hear their reaction to seeing the shape and holding it in their hands.

My younger son happened to be home when the package arrived so he went first:

(1) A 120-Cell


The fun thing about this shape is that we’ve played with versions of the 120-Cell before:

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A Stellated 120-cell made from our Zometool Set

We’ve also played around with the game Hypernom and experienced the 120-cell first hand!

Using Hypernom to get kids talking about math

(2) A hyperbolic paraboliod

This shape really caught my son’s attention in Segerman’s book – I’m glad he thought it was as cool in person as it was in pictures!


(3) Double Zarf

When my older son got home we unpacked his two shapes. I’d never heard of either shape before and am really excited to explore both of them. Both shapes come from the mathematical artist Bathsheba Grossman

Here’s his reaction to “Double Zarf”


(4) Tentacon

Finally, here’s his reaction to “Tentacon”


The Koch Snowflake with Squares

I asked the boys what they wanted to talk about today and got a fun response – the Koch Snowflake with squares!

Luckily we still had the Zome set out from the “tribones” project from last week – so making the first couple of iterations wasn’t that hard.

Before we started, though, I asked the boys what they thought the shape would look like:

While they were building I searched for something on line that would let us play with this particular fractal. I found these two Wolfram Demonstration projects:

Creat Alternative Koch Snowflakes by Tammo Jan Dijkema


Square Koch Fractal Curves by Robert Dickau

We’ll explore these to demonstrations below.

Here’s what the boys had to say about the first three iterations of the square Koch Snowflake. A fun thing that happened here was that during the discussion the boys found a small mistake in the construction of their level 3 curve:

Next we moved to the computer to explore the Wolfram Demonstration projects. First up was Robert Dickau’s “Square Koch Fractal Curves.” Sorry that this video (and probably the next one, too) is so fuzzy – it looked ok in the view finder. Oh well.

Finally, we explored Tammo Jan Dijkema’s “Create Alternative Koch Snowflakes” demonstration project. This project allows you to alter the fractal. The boys had a lot of fun playing with this project. Again, sorry for the fuzz in the video.

Using a Natalie Wolchover article to talk about hyperuniform distributions with kids

I’d somehow missed this article from Natalie Wolchover when it was published in July:

A Bird’s-Eye View of Nature’s Hidden Order

Luckily, though, it started playing on Quanta Magazine’s podcast while I was at the gym yesterday. It is such a great article! As soon as I got home I checked out the print version to see the pictures (which are pretty important for this article🙂 ) and then started thinking about how to share this article with kids.

As an aside, Wolchover has won a ton of awards for science and math journalism:

Natalie Wolchover wins Evert Clark/Seth Payne Award

Check out her Quanta magazine articles – they are all incredible.

We started today’s project by looking at different ways to pack pennies. We’ve talked once or twice about sphere packing in the past, but it still felt like the best thing to do was start with the basic problem of packing circles. It was fun seeing the boys explore different ways to pack pennies on the board.

Also, sorry this video ends abruptly – the camera ran out of storage space. Oops!

While I was fixing the storage space problem (!) I had the boys skim through Wolchover’s article. We re-started with a quick review of the penny packing problem and then I also had the boys give a quick summary of what they saw in Wolchover’s article.

The fun thing in this part of the project was seeing the two different ways the boys found for packing the two coins.

Finally, with the camera off I had the boys try to create a hyperuniform distribution of pennies and nickels. I was pretty curious to see what they’d do and what the pattern would look like. We’d already discussed (off camera) the “Finding Hidden Order” graphic in Wolchover’s article, so they knew a tiny bit about what a truly random distribution would look like. What would they think a hyperuniform distribution of pennies and nickels would look like?

One of the great things about Wolchover’s article is that the high level ideas are something that kids can understand. The article is also a really neat way for kids to see how different fields – from biology to math – can work together. Oh how I wish I’d seen articles like this one when I was a kid!

Math year in review part 2 – Fold and Cut

I’m starting to write up some thoughts on the projects we’ve done in the last year.  The difference between the past school year and the ones before that is that this was the first year that we weren’t home schooling.  That change altered  some of what I was doing with the boys, but maybe not as  much as I thought.  Yesterday I wrote up our tiling projects:

Math school year in review part2 – our tiling projects

Today I want to look back on a second idea we came back to a few times – fold and cut.

Our year in math projects got off to a super fun start last September when I saw this incredible video from Numberphile featuring Katie Steckles:

This video led to several projects:

Our one cut project

Fold and cut project #2

Fold and cut part 3

One of my great mathematical thrills of the year was running into Martin Demaine over at MIT and showing him parts of these projects!

Our next interaction with the fold and cut theorem came when I stumbled on a “fold and punch” activity while preparing to run Family Math night at my younger son’s elementary school:

As you might imagine, this activity was one of the most popular ones at the Family Math nights. I tried it out with my younger son before trying it out with the larger groups:

Fold and Punch


A good (and fun!) thing that happened today – half a punch!

After seeing my tweet about the fold and punch activity, Joel David Hamkins created this incredible activity for kids:

Math for nine year olds: fold, punch and cut for symmetry

Later in the year Steven Strogatz shared the “Mathematical Etudes” videos on twitter:

We used one of their videos to look at a slightly more difficult fold and cut problem than we’d tried before:

Screen Shot 2016-08-19 at 5.06.40 PM.png

Using the Mathematical Etudes videos with kids

Finally, in April we saw an absolutely amazing public lecture from Tadashi Tokieda which led to two really fun projects that involve folding and cutting paper – one of the projects involves trying to pass the circle in the picture below through the much smaller square:

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Tadashi Tokieda’s “World From a Sheet of Paper” lecture

So, a fun year playing around with paper folding. I love how these activities engage kids with math. We’ve also explored a little bit in this geometry book that teaches geometry through paper folding and I plan to use it more this year to help the boys see a different side of geometry.


Math school year in review part 1 – our tiling projects

Roughly a year ago the boys started their first year in school after 5 years of home schooling. Both kid’s packets for the next school year arrived last week. That got me daydreaming about the math we’ve done in the last year. Turns out that we’ve done a lot and I wanted to write about some of it before it all slips out of my mind.

One subject that we’ve spent a fair amount of time on this year is tilings. Not by design but I just happened to see a lot of neat tiling ideas in the last year.  Here’s a review of the tiling projects we did in the last 12 months:

(1) The project from Zome Geometry that got us going

This project was one of the few ones that I didn’t film.  The reason was that we had several kids from the neighborhood over working on it and I don’t feel comfortable filming kids that aren’t mine!

Anyway, this was a really fun project from Zome Geometry by George Hart and Henri Picciotto

Zome Tilings


(2) That project led to two 3d versions:

The natural thing to do after this project was to look at ways that you could cover 3 dimensional space with shapes – again we made use of our Zometool set:

Tiling 3-dimensional space with our Zometool set


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(3) A problem from a UK math exam led to a fun tiling project

I saw this neat problem from a UK math test circulating on Twitter back in February.

The UK Intermediate mathematics challenge part 2

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(4) A domino counting exercise form Jim Propp

We’d done a couple of projects based on Jim Propp’s blog, he thought that we might enjoy studying how 2×1 dominoes tile a 2xN square.  The project was so fun that we actually did it twice!

A fun counting exercise for kids suggested by Jim Propp

Counting 2xN domino tilings

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(5) Propp’s suggestion above came after we did these two projects on the Arctic Circle Theorem

I learned about the Arctic Circle theorem from a graduate student at MIT who thought it might be possible to share this fairly advanced mathematical idea with kids:

The Arctic Circle Theorem

A second example from tiling the Aztec diamond

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(6) Dan Anderson’s Gosper Curves

Maybe stretching a little to call this tilining, but we had fun exploring how the Gosper Island’s that Dan Anderson sent us fit together:

Dan Anderson’s Gosper curves

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(7) Inspiration from Eugenia Cheng’s Shapes video

I saw this neat video from Eugenia Cheng over the summer:

Thinking about how to use it with my kids inspired these two projects:

Tiling pentagon cookies

Learning about tiling pentagons from Laura Taalman and Evelyn Lamb

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(8) Richard Stanly’s Tiling presentation

Finally, just last week I stumbled on a presentation that Richard Stanley – a math professor at MIT who specializes in combinatorics – had put together about tilings.  There were a couple of ideas that were accessible to kids:

Talking through some examples from Richard Stanley’s tiling presentation

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Starting Algebra with my younger son

I’ll be working through Art of Problem Solving’s Introduction to Algebra book with my younger son this year. He’s excited, I’m excited . . . we all scream for ice cream, I guess.

For now I”m having him work through the introductory material alone. Here are the problems he wanted to talk about for the first two days of work. They are both basic arithmetic problems:




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