Talking about Pólya’s Urn with kids – inspired by Jim Propp’s blog post

A few weeks ago I saw another great blog post by Jim Propp – this one was about Pólya’s Urn:

Jim Propp on Pólya’s Urn

I’d meant to talk through it with the boys, but it was one of the topics that slipped through the cracks.

However, I was reminded of Propp’s post yesterday when Steven Strogatz shared the list of new fellows of the American Mathematical Society and Propp’s name was on the list:

So, this morning we talked about Pólya’s Urn. This is a little more risky Family Math talk than usual since I didn’t go back through Propp’s post before having the conversation with the boys. My intention was to highlight some basic ideas just to see what the kids had to say. I think we had a great conversation, and I sure hope that I got the main details right! (and, in all seriousness, if you want to talk about Pólya’s Urn with anyone, study Propp’s post, not this one.)

We started off by talking about some simple ideas about the binomial distribution (without naming it). The idea here was to fill an urn with either black or yellow balls by flipping a coin at each step. The boys had really interesting ideas about how you could fill an urn this way.

Instead of flipping a coin, we decided to use a 6-sided die since (as I learned the hard way) flipping a coin on a video doesn’t work so well.


After the short talk (and one example) about the binomial distribution in the last video, we spent the next part of the talk exploring the distribution in a bit more detail. The boys were surprised to find Pascal’s Triangle make an appearance in the discussion here.

For me the most interesting piece of this part of the project was listening to the kids try to find the right language to describe probability. There are a lot of moving parts even in this relatively simple example, and just finding the words to describe what’s going on stretches them a bit.

But they get there, which was really nice 🙂


With the coin flip example out of the way, we moved on to talking about Pólya’s Urn. The first part of this talk explains the new process and why the process begins with the Urn already having a single black and two yellow pieces.

One interesting piece of math that comes up in this part of the project is this – how can we do a random simulation of this process? The kids find a couple of different ways to use the various dice we have handy in the simulation. I was happy to see a few different ideas from them here.

After figuring out how to use our dice, we ran one simulation and discussed what we saw (which included the phrase “unless a miracle happens”- ha ha).


Next, we tried to replicate the process for the binomial distribution that led to the surprise appearance of Pascal’s triangle. The process is a bit more complicated here, but the boys were able to find a great way to describe it. Working through the binomial distribution earlier in the project was a great help here.


At the end of the last video we’d found the first 3 rows describing the process we followed filling up Pólya’s Urn. They boys were starting to wonder what the next row would look like, so I thought working out the next row would be productive.

Once we wrote down the next row I asked them what patterns they saw. Just as in Pascal’s triangle, there are lots of interesting patterns.


So, a fun project about an idea from probability theory that I’d not seen prior to Propp’s post (with super embarrassing apologies to him if he discussed it when I took his combinatorics class in 1992 – ha ha!!) . It was exciting to be able to share this idea with kids and it seems like a really fun way to get younger kids thinking about basic ideas in counting and probability.

Also, I remember this post from Patrick Honner after he read Propp’s piece:

Funny, indeed, how that works 🙂

Fawn Nguyen’s fraction puzzle

Saw this old blog post from Fawn Nguyen making the rounds again this week:

Drawing Rectangles instead of Writing Equations

and decided to give the problem a try with the kids this morning.

Here’s the problem: In a town, 3/7 of the men are married to 2/3 of the women. What fraction of the people in the town are married?

I did the problem with pairs of snap cubes instead of marriage. Here’s now it went:


After they finished their solution, I showed them Fawn’s solution. My younger son was a little confused about adding fractions versus what’s going on in this problem. Hopefully that confusion was straightened out by the end of the video.


So, another nice problem from Fawn. It is fun to be able to talk through a non-standard fraction problem like this one with the boys.

Walking down the path to the surreal numbers with kids

For reasons that are somewhat mysterious to me, I’ve seen a bunch of media coverage about John Conway in the last couple of months.

Of course, there’s Siobhan Roberts’s awesome biography:

Which Jordan Ellenberg reviews in the Wall Street Journal here:

Jordan Ellenberg’s review of Genius at Play

Roberts also has a great Quanta Magazine article:

John Conway: A life in games

One other item that caught my attention last week was a Jim Propp blog post I saw tweeted out by Jordan Ellenberg:

Here’s a direct link to the blog in case the tweet doesn’t embed properly:

Jim Propp’s “The Life of Games”

Propp actually wrote some of my graduate school recommendation letters back in . . . oh why did I bring this up . . . 1992, so it was cool to learn that he had a math blog!

Anyway, Propp’s piece describes the game of checker stacks so well and so simply that I thought it would be fun to try to talk through some of the ideas with the boys. We didn’t get to some of the more unusual features of the game today – and so didn’t really get to the surreal numbers – but even showing the boys how the number 1/2 shows up in the game was really fun. We’ll take a few more steps down the path to the surreal numbers tomorrow.

Here’s how today’s project went (oh, and sorry that the camera angle is so bad in the last two videos, I didn’t notice that the tripod got bumped until I was publishing the videos.):

The first step was a quick introduction the game of checker stacks and a few thoughts from the boys about the game:

Next we studied what happens when we stack checkers on top of each other. First we studied the relatively simple situation – a game with a single red vs. a stack with two blues. Then we moved on to a more complicated situation of a single red vs. a red blue stack. The boys were able to get their arms around these two situations, which was nice.

I realized at the end of the last movie that I wasn’t following Propp’s presentation correctly – he was using blue red stacks rather than red blue stacks. The difference becomes important when you are trying to find a game when it matters which color goes first. So for the third movie we we now studied the blue red stacks.

Also, the boys remembered in between these two videos that we had some 3d-printed red and blue action figures!

One interesting bit of math from the kids in this video was that they assumed that all of the stacks would have values that were represented by integers. Because of that assumption, they think the value of the blue red stack must be 0 or lower. We’ll explore that idea a little more in the last video.

The last game we study is a game with a single red vs two blue red stacks. Propp uses this game to illustrate a surprising property of the blue red stack. The boys were indeed surprised by the result 🙂

So, a really fun start to our journey to the surreal numbers. It was neat to see that even some fairly simple positions from the checker stacks game gave the kids some challenging things to think about – and even a few surprises!

A challenging number pattern problem

My younger son struggled with a number pattern problem from an old MOEMS test today. I enjoyed talking through it with him tonight because it was interesting to see how he approached the pattern in the numbers once he saw it – his approach was quite a bit different that what I was expecting.

Here’s an introduction to the problem and our initial talk that gets us on the path that surprised me:


So, my surprise in the last video is that he wanted to go to the end of a row and subtract a certain amount to get back to the beginning. I thought it would be interesting to see if he could see that you could also add 1 to the square at the end of the last row. This idea was hard for him to see, but eventually we got there.


At the end of the last video we talked about how the odd numbers relate to the perfect squares. The sequence of rows in the original problem hints at the relationship, though for me, at least, the connection doesn’t jump off the page. To get a better sense of that relationship we went to our kitchen table and looked at the relationship using snap cubes:


So, a fun little project starting from an old math contest problem. Ultimately the lesson I’m hoping to convey with my son here is about looking for patterns. The connection between arithmetic and geometry in the last part is also something that I hope he finds interesting. I always find it fun when geometry helps us understand arithmetic a little better.

Adding up perfect squares with snap cubes

We did a really fun project based on this tweet the day before Christmas.

After posting the project (whose link is embedded in the tweets below), James Key pointed out two videos that he thought my kids would like, and he was right!



This morning my older son and I had a great time working through the ideas in the two videos. Here’s how it went:

(1) First we reviewed the three pyramids that inspired the first project and discussed a little bit about how the project related to adding up squares:


(2) Next we moved on to a slightly bigger example: a geometric interpretation of the sum of the first 5 squares. We show how you the shape we get by combining these three large pyramids is starting to look more like a cube:


(3) The final piece of this project was moving on to the algebra. We come up with a counting formula for the number of snap cubes in our three pyramids and compare it with the formula for the sum of squares that I mentioned in our previous project (well, technically 3x that formula). We don’t work through the algebra – that’s an exercise for the reader! – but we do check that the two formulas give the same answer for the first 5 squares.


So, a nice follow up project, and also a fun little project for the morning after Christmas. Thanks to James Key for pointing out the additional videos!

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!

Complete this sentence: Math is ______!

Yesterday I saw a super cool student project posted on twitter – click the link in the tweet to see the stop motion video:

and a few follow up pictures, too:

The student’s project gave me an idea for a fun activity that would tie together some ideas from arithmetic, algebra, and geometry.  We started by building a replica of the student’s cube out of our snap cube set and then talking through some of the ideas we’d be looking at today:

After the short introduction we moved to the living room to start in on the project.  I like to emphasize the problem solving strategy of looking at easier problems first.  Keeping with that spirit, to start in on the geometry we first looked at a line.

I’m not sure that I did a good job explaining the connection between the algebra and the geometry right away, but hopefully looking at this easier example helped get going with that connection.  Also, because my son mentioned the relationship between Pascal’s triangle and (x + 1)^n in the last video, I decided to count the type of building block we use at each stage so that we return to that connection at the end:

Now we moved on to talking about turning a 3×3 square into a 5×5 square with the snap cube pieces we have handy.  My younger son picked up on a pattern was able to construct the 5×5 square pretty quickly.  He struggled with the algebra (which isn’t surprising, since we’ve not talked about algebra!) so we spent a bit of extra time on the connection to algebra in this movie.

I got so caught up in the algebra in the last video, that I forgot about the geometry.  Instead of moving on to the three dimensional case right away we returned to the 2 dimensional case to talk about dimensions and geometry.  The specific geometry that I wanted them to see here is how the number building blocks of each type we have at a given stage relates to the number of those blocks in the next stage.    Understanding how to count the number of building blocks at each stage helps understand the connection to Pascal’s triangle (although, I did not emphasize this connection at the end, unfortunately):

With all of this background we are now able to understand the student project that was posted on Twitter yesterday.  We look at (i) how this cube arises from sliding a square in the third dimension, (ii) how to count the number of different building blocks, and (iiI) how to write down an expression for (x + 2)^3 from our construction.  There is so much fun math hiding in this shape!

. . . and whoops – watching the movie just now it turns out that we did not actually write down the expression for (x + 3)^3  Dang!  Oh well, it comes from the numbers on the screen:  (x + 2)^3 = x^3 + 6x^2 + 12x + 8.

Now that we’ve understood a little bit about how to make 3 dimensional cubes from 2 dimensional cubes, we tried the next step – understanding how to make a 4 dimensional cube!  Everything works the same way as before, though it is now much harder to visualize so we have to let the math guide us..   The multiple connections that we’ve talked about in the prior stages help us understand the 4 dimensional shape even without being able to see it.  Amazing!

To wrap it all up we went back to the white board to see the connection between Pascal’s triangle and the number of pieces in each of our constructions.  I wrote the two triangles on the board side by side and my younger son noticed the connection between the two number triangles.  I was hoping one of them would see it, so I was super happy when he noticed the relationship!  We finished up just by talking about how cool all these connections are:

The connection between the algebra, geometry, and arithmetic demonstrated by this student project are really are amazing.  Seeing the project posted on Twitter yesterday helped me see a great way to explore these connections – that’s why I love looking at all of the math people share online!  Probably no better way to end the blog post than with what my younger son said at the end of the last video:

Me:  Complete this sentence:  Math is . . . . .

him:  crazy!

Studying the difference of squares with snap cubes.

The tour through the “Algebra with Integers” section of our number theory book continues.  My instincts are to just jump ahead, but my younger son seems to really be enjoying the problems so we are plodding through them.  Today we looked at the difference of squares:  A^2 - B^2 = (A - B)*(A + B).

After a few numerical examples I wasn’t really sure what more to do.  On a whim we decided to try playing around with snap cubes and see if there were any geometric connections.  Turned out to be a lot of fun.

To keep the number of snap cubes from getting out of hand we kept things simple by looking at the example 5^2 - 3^2.  I asked my son to make a picture with snap cubes showing what 5^2 - 3^2 should look like and he made the picture below (the white cubes are supposed to be a minus sign and an equals sign).  He described the shape at the bottom of the picture as the yellow square taking a bite out of the blue square.

Diff of Squares 1

The next step was to see if we could find a different geometric way to understand how the bottom shape arose from the original  two squares.  The algebraic formula for the difference of squares is so simple, you would hope a geometric interpretation would be simple, too.  I asked him to see if there was a simple shape that we could make out of the bottom blue cubes.  He made this:

Diff of Squares 2

Oh, ha ha, my attempt to keep the number of cubes simple accidentally produced a Pythagorean triple 🙂  Oops, but what the heck, let’s not pass up the opportunity:

Diff of Squares 3

A few quick numerical examples showed that the difference of squares didn’t always produce a square, so we looked for an alternate way to arrange the 16 blocks and found these two ideas:

Diff of Squares 5

and this:

Diff of Squares 6

All of the pictures above came from probably 15 to 20 minutes of talking about the geometry.  We finished by talking about the relationship between the two last pictures and the algebraic formula back on the board with two quick discussions:

So, what started out just as a whim turned into to something really fun.  I really enjoy finding the ways to find connections between algebra and geometry.  Turns out to be pretty fun even if you’ve not studied either subject yet 🙂

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.

Mixing Problems

One more trip up to Boston today for the last Brute Squad practice weekend before nationals.   Follow along next weekend and maybe watch some great ultimate frisbee on ESPN:

Before heading out the door we went through a quick Family Math project about mixing.  I illustrated the problem using a glass of red liquid and a glass of green liquid.  The problem is this:

(1) Take an amount of liquid from the red glass and put it in the green glass.

(2) Now take the same amount of liquid from the green / red mixture and put it back in the red glass.

(3) The question is – does the green glass have more red in it or does the red glass have more green in it?

It is a pretty cool problem for kids to talk through:

After talking about the problem with liquid, we now try a hopefully easier version of the problem using snap cubes.   One of the lessons I’m hoping for here is learning to find versions of a complicated problem that are a bit easier:

So, a fun problem for kids that we can do with food coloring and with blocks.   Hopefully these two different approaches hope clarify what is a somewhat non-intuitive result.