A fun project from Art Benjamin and the Museum of Math

Yesterday Art Benjamin gave a talk at the Museum of Math. One neat tweet from from the talk was this one:

It is a pretty neat problem and I thought it would make a fun project for the boys today. I didn’t show them the tweet, though, because I wanted to start by exploring the numbers with increasing digits:


Next we tried to figure out what was going on. My older son wanted to try to study the problem in general, but then my younger son noticed a few things that at least helped us understand why the sum should be divisible by 9.


For the third video we started looking at the problem in general. The computations here tripped up the boys a bit at first, but these computations are really important not just for this problem but for getting a full understanding of arithmetic in general.


For the last part of the project we looked at two things. First was returning to a specific example to make sure that we understood how borrowing and carrying worked. Next we applied what we learned to the slightly different way of multiplying by 9 -> multiplying by 10 first and then subtracting the number.


After the project I quickly explored Dave Radcliffe’s response to MoMath’s tweet:

It took a bit of thinking for the boys to see what “works in any base” meant, but they did figure it out.

I love this Benjamin’s problem – it makes a great project for kids!

Tadashi Tokieda’s “World from a sheet of paper” lecture

Yesterday we saw an incredible public lecture from Tadashi Tokieda. He showed an hours worth of amazing mathematical ideas that come from paper folding.

This is actually our second project inspired by Tokieda. The first came from his “freaky dot patterns” video with Numberphile:

Numberphile’s Freaky Dot Pattern Video

We’ve also done a project on some of the paper cutting activities in this other Tokieda video, though we saw the idea in a different place:

Here’s that project:

Cutting a double Mobius strip

If you are interested in seeing a longer presentation from Tokieda, he has also given a public lecture at the Museum of Mathematics:

Tokieda’s public presentations are absolutely incredible!

We started our project today by exploring an idea at the beginning of the MoMath lecture – I wanted to show the boys something that they’d not seen previously. The exploration here is the noise that a coffee mug makes when you strike it with a spoon at various different locations (and sorry that my hand was blocking a lot of the shot here):

Next we looked at one of the surprising paper folding patterns you can see without doing any careful folding. It is fascinating to see folding patterns arising as naturally as this one does:

For the last project today we looked at the demonstration that Tokieda used to start his lecture yesterday – passing a large circle through a small hole. It seems as though the task is impossible, but some clever folding makes it work. They boys had a bit of a hard time explaining how this one worked – but that’s fine – it is hard to believe that it works at all!

So, a fun project today following yesterday’s fascinating lecture. It is so great to see lectures like Tokieda’s that bring amazing math to everyone – from kids to tenured MIT math professors!

Amazing math from mathematicians to share with kids

About two years ago I saw this Numberphile interview with Ed Frenkel:

One of the ideas that Frenkel mentions in the interview is that professional mathematicians haven’t done a good job sharing math with the general public. Although I’m not really the kind of professional mathematician Frenkel was talking about, I took his words to heart and have been on the lookout for math to share – especially with kids.

It turns out that there are some fantastic ideas that are out there for kids to see. Some surprising fun I had sharing Larry Guth’s “no rectangles” problem with kids earlier this week (see below) made me want to share some of the ideas I’ve found in the last couple of years, so here are a few examples:

(1) One of the most incredible lectures that you’ll ever see is Terry Tao’s “Cosmic Distance Ladder” lecture at the Museum of Mathematics in New York City:

I used Tao’s video for three projects with my kids – but there are probably 20 math projects for kids you could get out of it.

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

Terry Tao’s MoMath lecture part 3 – the speed of light and paralax

(2) The Museum of Math’s public lectures are a great source beyond Tao’s lecture.

Here’s a project based on Bryna Kra’s lecture:

Angry Birds and Snap Cubes – Using Bryna Kra’s MoMath public lecture to talk math with kids

Eric Demaine’s lecture was part of our Fold and Cut theorem project:

Fold and Cut part 3

and I can’t say enough good thinks about Laura Taalman’s work – she’s inspired dozens of our projects.  Just search for her name on the blog:

(3) and Speaking of Fold and Cut . . .

Katie Steckles and Numberphile put together an incredible video about the Fold and Cut theorem. I used the video this week for project with 2nd and 3rd graders at my younger son’s school earlier this week.  Steckles’s presentation is so incredible – this is the kind of math that really inspires kids:

We used it for three projects (including the Eric Demaine one above):

Our One Cut Project

The Fold and Cut Theorem is Awesome!

In prepping for the grades 2 and 3 projects I also totally coincidentally ran across a “fold and punch” exercise that is a great activity to try with kids before trying out fold and cut:

(4) Another great success with the 2nd and 3rd graders was Larry Guth’s “no rectangles” problem. I had a great time playing around with this problem with my kids, but nothing prepared me for how enthusiastic the kids in the two programs were about this problem.

Larry Guth’s “No Rectangles” problem

After the 3rd grade night, Patrick Honner sent me this picture that I used to wrap things up with the 2nd graders.

(5) The Surreal Numbers

I’d seen John Conway’s surreal numbers previously via an amazing Jim Propp blog post:

The Life of Games.

and I wanted to revisit them after finally reading Donald Knuth’s book:

Revisiting the Surreal Numbers

Infinity + 1 and other Surreal Numbers

Playing with the surreal numbers via checker stacks is an incredibly engaging way for kids to learn about mathematical thinking.

(6) Speaking of John Conway –

In the 2014 edition of the Best Writing in Mathematics Conway had an article about variations on the Collatz conjecture. It was a fascinating article that even gave us the idea to translate some of the math into music.

The Collatz Conjecture and John Conway’s “Amusical” variation

I’ve also talked with the boys about the standard version of the Collatz conjecture:

It is a great way to introduce kids to an unsolved problem in math while also sneaking in a little bit of arithmetic practice!

(7) Occasional contest math problems

I happened to run across another MoMath lecture yesterday – this one by Po-Shen Loh. He was talking about “Massive Numbers.” I thought maybe he’d be talking about the book “Really Big Numbers” by Richard Evan Schwartz:

A few projects for kids from Richard Evan Schwartz’s “Really Big Numbers”

or maybe Graham’s Number:

An attempt to explain Graham’s number to kids

The last 4 digits of Graham’s number

but instead he talked about a neat problem from the 2010 International Mathematics Olympiad:

His presentation is fascinating and I even talked through the first version of the problem with my younger son:

Another math contest-like problem I really enjoyed talking about with the kids was this one:

Show that any positive integer n has a (positive) multiple which has only the digits 1 and 0 when represented in base 10.

A challenging arithmetic / number theory problem

(8) Building off of popular books by mathematicians as well as public lectures

I was surprised at how much great math writing and speaking there has been for the general public in the last couple of years.

Jordan Ellenberg’s “How not to be Wrong” inspired several projects – probably my favorite was using his idea of “algebraic intimidation” to talk about the famous 1 + 2 + 3 + . . . = -1/12 video by Numberphile. :

Jordan Ellenberg’s Algebraic Intimidation

Jacob Lurie’s Breakthrough Prize public lecture inspired two projects about a year apart from each other:


Using Jacob Lurie’s Breakthrough Prize Lecture to Inspire Kids

Using Jacob Lurie’s Breakthrough Prize talk with kids

And, Ed Frenkel, who got me thinking about sharing advanced math with kids in the first place has inspired a few projects, too:

Fine Ed Frenkel – you convinced me

Ed Frenkel, the square root of 2, and i

and one of my all time favorites:

A list Ed Frenkel will love

(9) Finally, it would be impossible to write a post like this one without mentioning the work that Evelyn Lamb is doing writing math articles for the general public. I’ve lost count of how many projects she’s inspired, but it is probably well over 20. I’m especially grateful for her talk about topology which have generated really fun conversations with the boys. For example:

Using Evelyn Lamb’s Infinite Earring with kids

Evelyn Lamb’s fun torus tweet


Henry Segerman’s Flat Torus

which arose after Lamb pointed out this video:


So, I’m really happy that mathematicians are sharing so many amazing ideas. I think this is the sort of math promotion that Frenkel had in mind. Hopefully it continues for many years to come 🙂

The Breakthrough Prize lectures

Lucky moment for me this morning when I happened to be looking at my screen just as Steven Strogatz posted the links to a few Breakthrough Prize lectures:


Michael Harris’s article in Slate a few weeks ago drew my attention to the Breakthrough prize in math:

Michael Harris writes about the Breakthrough Prizes in math

In particular, this statement from the article got me thinking:

“Tao—the only math laureate with any social media presence (29,000-plus followers on Google Plus)—was a guest on The Colbert Report a few days after the ceremony. He is articulate, attractive, and the only one of the five who has done work that can be made accessible to Colbert’s audience in a six-minute segment. But Colbert framed Tao as a genius (which he assuredly is), not as someone who can get them jumping up and down in the aisles.”

Though I’m sure that Harris is correct, it seemed to be quite a shame that the work of these mathematicians wasn’t really accessible to the general public. One of the reasons that I thought this was a shame is selfish – I want to learn from these great mathematicians. I want other people to be able to learn from them, too.

Terry’s Tao’s public lecture at the Museum of Math is a great example of how you can learn from these world class mathematicians. That lecture is filled with beautiful math that the general public (and kids especially) can understand. I’ve done three projects with my kids already from that lecture:

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

I’d love to see more public lectures like Terry Tao’s lecture at MoMath and use them to show kids more about what mathematicians do. Well . . . what I didn’t know until I saw Strogatz’s post this morning was that each of the Breakthrough Prize winners gave a public lecture as part of the prize. Yes!!

I haven’t had time to view all of them, but the lectures by Terry Tao and Jacob Lurie are absolutely tremendous. I’m not sure that they prove Harris wrong, since they aren’t really going into detail about their work, but . . . . If you want to get a better understanding of what math is, watch these lectures. If you want to help other people, including kids, understand what mathematicians do, these lectures are a great starting point:

Jacob Lurie’s Breakthrough Prize talk:


Terry Tao’s Breakthrough Prize talk:


For good measure, here are the other talks. I have not viewed these ones yet, but I am excited to watch them, too. It is so great to see lectures like these ones online. I’m really happy that the Breakthrough Prizes are helping to connect these amazing research mathematicians to the general public. That connection has to be a great step forward for math.

The remaining three lectures are below:

Maxim Kontsevich’s Breakthrough Prize talk:


Richard Taylor’s Breakthrough Prize talk:


Simon Donaldson’s Breakthrough Prize talk:

Angry Birds and Snap Cubes: Using Bryna Kra’s MoMath public lecture to talk math with kids

Last night I watched Bryna Kra’s public lecture at the Museum of Math:


I’m not quite sure how to talk through some of the simple dynamical system ideas in the lecture, but the earlier material about patterns and the pigeonhole principle are definitely fun topics to talk about with kids. We used our collection of snap cubes and Angry Bird stuffed animals as props 🙂

In the first part of the talk we introduce the pigeonhole principle and talk through a simple pattern with only single blocks based on one of the elementary patterns Kra uses in her talk. This simple pattern allows us to get a little bit of practice identifying the “pigeons” and the “pigeonholes” in a problem:


In the second talk we look at a slightly more complicated pattern – patterns you get with two blocks rather than one. For this pattern we consider the order of the birds to be important – so a (red, blue) group is different than a (blue, red) group. The example we look at in the last part of today’s talk will consider those two groups to be the same.

The boys were able to see the four different patters that we could make with the two birds / blocks. My older son even noticed a connection with Pascal’s triangle which was fun to see. We then talked about how to count the different types of pairs by looking at the number of choices we had for the first bird and for the second bird. That led my younger son to wonder if there would be a total of 9 groups of two birds if we allowed three different birds in the pattern. Pretty fun discussion:


At the end of the last talk my younger son wondered what would happen if we used three different colors of blocks rather than two. I hadn’t planned on discussing that problem, but what the heck! It was interesting to see the kids figure out how to group the blocks to make the 9 pairs. They were also now able to see how the patterns would continue if we varied the colors and/or number of blocks in the pattern. Fun little exercise. Watching this again I wish I would have spent a little time responding to my older son’s comment that there was no connection to Pascal’s triangle pattern anymore – oh well, next time!


Our last project was a slightly different twist on the Pigeonhole principle. We looked at a tournament involving 4 birds in which each game involves 2 birds. The question I had the boys look at was this: If there are 7 total games played in this tournament, show that at least two of the games must involve the same two players.

I liked their approach to solving this problem. Their instinct was to solve the problem by listing out all of the types of games that could happen. If we were at our whiteboard I would have drawn a square with its sides and diagonals, but their list of all of the types of games was good enough for this project. They had a little difficulty identifying the pigeons and pigeonholes here, but that’s ok, it isn’t always so obvious how to make that identification.


So, a fun project based on another MoMath talk. See here for our last project based on a MoMath lecture:

Part 3 of using Terry Tao’s MoMath lecture to talk about math with kids

I think the public lectures at the Museum of Math are a great way for kids to see some amazing math. There will surely be some lectures that are too advanced for young kids, but many of these lectures have ideas in them that are not hard at all for kids to understand. With Bryna Kra’s lecture, the ideas about patterns and the pigeonhole principle are topics that kids can play with and really enjoy. I’m super glad that MoMath is making these lectures available to the public. It is really fun to show kids some ideas that professional mathematicians use in their research, and hopefully also a great way to inspire a new generation of mathematicians!

Terry Tao’s MoMath Talk Part 2: Clocks and Mars

Last week I wrote about finding Terry Tao’s incredible public lecture delivered at the  Museum of Math and how that lecture provides many great examples you can use to talk about math with kids:

Terry Tao’s MoMath Lecture Part 1: The Earth and the Moon

for ease, the direct link to the Terry Tao lecture  is here:

Today I wanted to use a second example from that lecture for a little math talk with the boys.  This topic comes from approximately 42:30 into the video when Tao discusses Copernicus’s calculation of how long it took Mars to orbit the sun.   This calculation is an incredible scientific achievement, especially when you consider that telescopes hadn’t even been invented yet!

In the lecture Tao describes the remarkable story behind the calculation, but does not go into the details of the calculation itself.  To be clear, that’s not a criticism – the point of his lecture was to tell the story not to dive into the details.  Exploring the details of this particular calculation is a great topic to discuss with kids, though.  The only background material required is some basic knowledge about fractions.

We began this morning by watching the (approximately) 5 minute portion of the talk in which Tao describes how Copernicus calculated the time it took for Mars to Orbit the sun.  Following that we went to the whiteboard to talk about what we learned, and to head down the path of understanding the calculation in detail.   The starting point I chose for understanding the calculation is asking questions about the angles formed by the hands of clocks.

I will say at the start that it was a little harder for my kids than I was expecting.  The discussion and the explanations below are not at all flawless and have several false starts.  As I’ve said many times, that’s what learning math (and, in this case, a little physics) looks like.  Watching the films of this discussion prior to publishing this post has reinforced my feeling that Tao’s lecture  is a great spring board to talking math with kids.

Having looked at a few examples of when the angles between the hour hand and minute hand of a clock would be zero, in the next part of the talk we began to try to drill down on the math.  The starting point for the discussion here was the observation by my older son that the minute hand moves 12x faster than the hour hand.    In this video we try to write down some expressions that describe how fast the two hands of the clocks are moving:

The next step was writing down an equation that told us how far the hour and minute hands would move in “t” minutes.  In retrospect I wish I would have made a different choice in the approach here since jumping directly to the algebra made a simple idea a little harder than it needed to be.   If I could do it again I’d probably cover the ideas in this video nearly in reverse (and I’m annoyed with myself for getting frequency and period reversed, too.  Can’t get everything right . . . .)

However, even with the little bit of extra time that introducing the algebra at the wrong moment led to, the discussion here did get us to an equation that looked a lot like the equation Terry Tao had written down in his presentation slides.

At the end of the last video we got to an equation that helps us understand when the hands of a clock are exactly on top of each other – now we solve it!  Solving this equation is a great exercise for kids who have a little familiarity with fractions.  We sort of stumble out of the gates with the solution, but once we get on the right track we actually get to the end in sort of a neat way.

With all of this background out of the way we can return to the equation that Terry Tao had in his presentation.  We being this part by briefly talking about difference between our clock equation and the equation that Copernicus solved..  After that introduction we solve the equation and determine how long it takes for Mars to orbit the Sun!

I’m really excited about using more examples from Terry Tao’s lecture to talk math with kids.  There are so many great things about this lecture – for instance the incredible historical information and the great opportunity to see Terry Tao speak on an accessible topic – but for me the new examples the talk contains for talking  about some basic school math with kids is the best thing about this public lecture.    Who would have thought that calculating the orbit of Mars just boiled down to simple fractions?!?

MoMath and MegaMenger

Yesterday we visited the Museum of Mathematics in NYC to help out with their MegaMenger build.   The boys had a blast!

Menger Boys
This was our 3rd visit to the Museum and I’m sure there will be many more.  One of the fun attractions is this tricycle with square wheels (sorry for the poor quality of this video):

The MegaMenger project is an incredible project in which people from all over the world are working together to build a giant Menger Sponge out of business cards.  The website for the project is here:


The boys were actually so excited about participating in this project yesterday that we started the day today building a level 2 Menger sponge out of snap cubes.  Although I enjoyed the project, too, I wouldn’t have described my excitement as “build a new level 2 Menger Sponge at 5:30 am the next day excited,” but hey, I’ll take it:

Snap Cube Menger

With that new morning build, there was really  no doubt at all what our Family Math project for the day would be 🙂  We began by simply reviewing our trip to MoMath and some basics about the Menger Sponge.  The specific topics for the day are going to be volume and surface area.  For all but the last movie the questions will revolve around Menger Sponges of ever increasing sizes, like the one being build in the Mega Menger project:

Having touched on the volume of the Menger Sponge in the last movie, we now dive into the volume calculation in more detail.  What I liked here is that each kid had a different way of calculating the volume.  So fun to see the different approaches to counting here!  I showed a third way, too, that has  a sort of surprising twist.  The Level 2 figure we talk about at the end is the shape that we constructed out of the snap cubes that is pictured above:

The surface area calculation is only slightly more tricky.  As with the approach to volume, both boys had different approaches to counting the surface area of the Level 1 Menger Sponge.  It turned out that my younger son’s method was actually the same as mine, so I didn’t add a third counting method here.    Taking through my older son’s direct counting method and my younger son’s method of counting the overlaps was really enjoyable.    We finished by wondering which of these two methods was easier to generalize to the higher level sponges.

Next we attempted to calculate the surface area of the level 2 sponge.  The level 2 sponge is the one that we made out of snap cubes this morning.  Our contribution to the MoMath Mega Menger build amounted to the construction of two of these sponges out of business cards.  The construction from folding business cards took a bit longer than the one from snap cubes, though the business card construction was at 2:00 in the afternoon and was followed by BBQ at Blue Smoke in Manhattan, so maybe I should call it a draw 🙂

The math in this video is the most difficult to follow in this project, but hopefully we work through it slowly enough.  To calculate the surface area of the Level 2 sponge we use the method my younger son suggested in the last video.  We first assume that the surface area is 20 times the surface area of one of the Level 1 sponges (since it takes 20 level 1’s to make a level 2) and then subtract out the surface area that vanishes when two sides touch.  We break down this calculation into two pieces.  The first part is for the middle pieces that touch two other Level 1 sponges, and the second part is for the corner pieces that touch three.  After a 3 minute calculation, we arrive at the surface area of the Level 2 sponge:

Finally – the punch line!  I thought that ending the project with the lengthy calculation above would kill the excitement we had going this morning, so I went in a different direction for the last movie.  Instead of building ever larger sponges, what happens if we start with a sponge of a fixed size and make a Menger Sponge by cutting holes of ever decreasing size in it?    Even thinking about this question may seem strange, but the result is both fun and a little bit perplexing.  Luckily to answer it we can use the numbers we’ve already calculated in the previous videos – we just need to adjust the scale of the sponges.  Adjusting that scale is an interesting lesson all by itself, btw!    After spending a minute or two talking about what tripling the side length of a cube does to a cube’s volume and surface area, we look at the volume and surface area of a the different levels of Menger Sponges with a fixed edge length.  The result is a neat surprise.

So, despite the super early start (!!), we had a really fun morning.  I’m happy that we had a chance to help out the MoMath team with their Mega Menger build.  Hopefully many other kids around the world will get to help out with this project – it is such a great opportunity to hold an amazing math project in your hand.  Exploring the math behind the Menger Sponge seems like a project that lots of kids would love.

Also, if you’ve made it this far and happen to be in the NYC area, head over to MoMath today (Sunday October 26, 2014) to help them finish the build!  And now having finished this morning’s project and written up this blog post by 8:30 am, it is time to take a nap!! ha ha.