# Sharing a simple and surprising mathematical idea from “How to gamble if you must” with kids

Yesterday I got a neat book in the mail – I’d seen Nassim Taleb recommend it on Twitter:

One warning – this is not a “popular math” book, it is pretty math heavy. Flipping through the first 1/4 of the book, I really enjoyed the presentation and was once again reminded of the surprising fact that foundational research on basic gambling problems was being done in the late 1950s and early 1960s. An accessible and incredibly interesting account of some of this work can be found in Ed Thorp’s autobiography “A man for all Markets.”

One nice example from the beginning of the book relates to gambling in a 50/50 game called “red and black.” Think of the game as trying to guess the color of a card pulled from a randomly shuffled deck, or just betting on a coin flip. If you want to turn, say, \$100 into \$1,000 by betting on this game, what is your best strategy?

IF you are interested, a shorter account of this problem (with accompanying practice problems) can be found in this nice summary paper by Kyle Seigrist published by the Mathematical Association of America.

Summary of the ideas from “How to gamble if you must” by Kyle Seigrist and the Mathematical Association of America

Because this particular gambling problem is accessible to kids, for today’s project I wanted to introduce the idea of 50/50 gambles and ask them what they thought the optimal gambling strategy would be. The specific question is what is the best strategy to follow if you want to try to turn \$100 into \$1,000?

They had some absolutely terrific ideas. My 6th grade son practically suggested the betting strategy from the Kelly criterion!

Next we turned to the computer to study this game in Mathematica. We looked at some simple betting ideas first. So, if we want to turn \$100 into \$1,000 in this game, what happens if we bet \$100 on each bet? What happens if we bet \$50 on each bet?

After seeing the surprising results from the fist set of trials, we looked at the gambling strategies that the boys proposed. First we looked at a version of the strategy that my old son suggested -> basically bet the maximum amount every time (except when you don’t need to bet the max amount to reach \$1,000).

Are you more or less likely to turn \$100 into \$1,000 with this strategy?

Now we checked the betting strategy that my younger son suggested -> bet 1/2 your money each time (except when you don’t need to bet that much to reach \$1,000).

The boys had some pretty interesting ideas about what would happen here.

So, definitely a fun project and the result is pretty surprising (at least to me!) -> in 50/50 games your betting strategy doesn’t matter.

# Seeing ideas about substitution for the first time

My son had an interesting problem on his enrichment math homework this week, and it gave him a lot of trouble this morning:

Tonight I thought it would be good to talk through the problem since I think the main idea he needed to solve it was new to him.

Here’s the introduction and some of the ideas he tried this morning:

Next we took a look at the equations on the computer and talked about some of the ideas we saw:

After looking at the graphs of the equations on the computer we came back to the whiteboard to talk about substitution.

Finally, having worked through the introductory part of u-substitution in the last video, I let him finish off the project on his own.

I can’t remember talking through this topic previously, but it was fun. It is always neat to be there when a kid is seeing a math topic for the first time.

# What kids learning math can look like -> studying the geometric mean

Earlier in the week my younger son was struggling with a problem about the geometric mean of two numbers. I thought looking at the difference between how my older son saw the problem and how my younger son saw the problem would be interesting.

I asked each kid to go through the problem on his own. My older son had not seen the problem ahead of the project and my younger son had gone through the problem with me two days ago.

Here’s what my older son had to say:

Here’s the first part of what my younger son had to say. You’ll see that the he’s still a little unsure about the problem even though we had talked through it previously:

Finally, I wrapped up with my younger son by talking through the way he was intending to solve the problem originally. This approach is really nice, too – it uses the Pythagorean theorem and a little algebra:

This was a nice reminder to me of what a kid struggling with a math idea can look like.

Sorry for the brief post – needed to get this one out the door before running out for work.

# 15 (+1 bonus) Math ideas for a 6th grade math camp

Saw an interesting tweet last week and I’ve been thinking about pretty much constantly for the last few days:

I had a few thoughts initially – which I’ll repeat in this post – but I’ve had a bunch of others since. Below I’ll share 10 ideas that require very few materials – say scissors, paper, and maybe snap cubes – and then 5 more that require a but more – things like a computer or a Zometool set.

The first 4 are the ones I shared in response to the original tweet:

## (1) Fawn Nguyen’s take on the picture frame problem

This is one of the most absolutely brilliant math projects for kids that I’ve ever seen:

When I got them to beg

Here’s how I went through it with my younger son a few years ago:

## (2) James Tanton’s Mobius strip cutting exerciese

This is a really fun take on this famous scissors and paper cutting exercise:

You will honestly not believe what you are seeing when you go through Tanton’s version:

Here’s the link to our project:

James Tanton’s incredible mobius strop cutting project

## (3) Martin Gardner’s hexapawn “machine learning” exercise

For this exercise the students will play a simple game called “hexapawn” and a machine consisting of beads in boxes will “learn” to beat them. It is a super fun game and somewhat amazing that an introductory machine learning exercise could have been designed so long ago!

Intro “machine learning” for kids via Martin Gardner’s article on hexapawn

## (4) Katie Steckles’ “Fold and Cut” video

This video is a must see and it was a big hit with elementary school kids when I used it for “Family Math” night:

Here are our projects – all you need is scissors and paper.

Our One Cut Project

Fold and cut project #2

Fold and cut part 3

## (5) Along the same lines – Joel David Hamkins’s version of “Fold and Punch”

I found this activity in one of the old “Family Math” night boxes:

Joel David Hamkins saw my tweet and created an incredible activity for kids.  Here’s a link to that project on his blog:

Joel David Hamkins’s fold, punch and cut for symmetry!

## (6) Kelsey Houston-Edwards’s “5 Unusual Proofs” video

Just one of many amazing math outreach videos that Kelsey Houston-Edwards put together during her time at PBS Infinite Series:

Here is how I used the project with my kids:

Kelsey Houston-Edwards’s “Proof” video is incredible

## (7) Sharing the surreal numbers with kids via Jim Propp’s checker stacks game

Jim Propp published a terrific essay on the surreal numbers in 2015:

Jim Propp’s “Life of Games”

In the essay he uses the game “checker stacks” to help explain / illustrate the surreal numbers. That essay got me thinking about how to share the surreal numbers with kids. We explored the surreal numbers in 4 different projects and I used the game for an hour long activity with 4th and 5th graders at Family Math night at my son’s elementary school.

This project takes a little bit of prep work just to make sure you understand the game, but it is all worth it when you see the kids arguing about checker stacks with value “infinity” and “infinity plus 1” 🙂

Here is a summary blog post linking to all of our surreal number projects:

Sharing the Surreal Numbers with kids

## (8) Larry Guth’s “No Rectangle” problem

I learned about this problem when I attended a public lecture Larry Guth gave at MIT.  Here’s my initial introduction of the problem to my kids:

I’ve used this project with a large group of kids a few times (once with 2nd and 3rd graders and it caused us to run 10 min long because they wouldn’t stop arguing about the problem!). It is really fun to watch them learn about the problem on a 3×3 grid and then see if they can prove the result. Then you move to a 4×4 grid, and then a 5×5 and, well, that’s probably enough for 80 min 🙂

Larry Guth’s “No Rectangles” problem

## (9) The “Monty Hall Problem”

This is a famous problem, that equally famously generates incredibly strong opinions from anyone thinking about it. These days I only discuss the problem in larger group settings to try to avoid arguments.

Here’s the problem:

There are prizes behind each of 3 doors. 1 door hides a good prize and 2 of the doors hide consolation prizes. You select a door at random. After that selection one of the doors that you didn’t select will be opened to reveal a consolation prize. At that point you will be given the opportunity to switch your initial selection to the door that was not opened. The question is  -> does switching increase, decrease, or leave your chance of winning unchanged?

One fun idea I tried with the boys was exploring the problem using clear glasses to “hide” the prizes, so that they could see the difference between the switching strategy and the non-switching strategy:

Here’s our full project:

Exploring the Monty Hall problem with kids

## (10) Using the educational material from Moon Duchin’s math and gerrymandering conference with kids

Moon Duchin has spent the last few years working to educate large groups of people – mathematicians, politicians, lawyers, and more – about math and gerrymandering.  . Some of the ideas in the educational materials the math and gerrymandering group has created are accessible to 6th graders.

Here’s our project using these math and gerrymandering educational materials:

Sharing some ideas about math and gerrymandering with kids

## (11) This is a computer activity -> Intro machine learning with Google’s Tensorflow playground.

This might be a nice companion project to go along with the Martin Gardner project above. This is how I introduced the boys to the Tensorflow Playground site (other important ideas came ahead of this video, so it doesn’t stand alone):

Our complete project is here:

Sharing basic machine learning ideas with kids

## (12) Computer math and the Chaos game

The 90 seconds starting at 2:00 is one of my all time favorite moments sharing math with my kids:

The whole project is here, but the essence of it is in the above video:

Computer math and the chaos game

## (13) Another computer project -> Finding e by throwing darts at a chess board

This is a neat introductory probability project for kids. I learned about it from this tweet:

You don’t need a computer to do this project, but you do need a way to pick 64 random numbers. Having a little computer help will make it easier to repeat the project a few times (or have more than one group work with different numbers).

Here’s how I introduced the project to my kids:

Here’s the full project:

Finding e by throwing darts

## (14) Looking at shapes you can make with bubbles

For this project you need bubble solution and some way to make wire frames. We’ve had a lot of success making the frames from our Zometool set, but if you click through the bubble projects we’ve done, you’ll see some wire frames with actual wires.

Here’s an example of how one of these bubble projects goes:

And here’s a listing of a bunch of bubble projects we’ve done:

Our bubble projects

## (15) Our project inspired by Ann-Marie Ison’s math art:

This tweet from Ann-Marie Ison caught my eye:

Then Martin Holtham created a fantastic Desmos activity to help explore the ideas:

It is fun to just play with, but if you want to see how I approached the ideas with my kids, here are our projects:

Using Ann-Marie Ison’s incredible math art with kids

Extending our project with Ann-Marie Ison’s art

## (16) Bonus project!!A dodecahedron folding into a cube

This is a an incredible idea from 3d geometry.

We studied it using our Zometool set – that’s not the only way to go, but it might be the easiest:

Here’s the full project:

Can you believe that a dodecahedron folds into a cube?

# A terrific probability problem for kids shared by Alexander Bogomolny

Saw this tweet from Alexander Bogomolny yesterday and knew immediately what today’s project was going to be 🙂

The problem is, I think, accessible to kids without much need for additional explanation, so I just dove right in this morning to see how things would go.

My first question to them was to come up with a few thoughts about the problem and some possible strategies that you might need to solve it. They had some good intuition:

Next we attempted to use some of the ideas from the last video to begin to study the problem. Pretty quickly they saw that the initial strategy they chose got complicated, and a more direct approach wasn’t actually all that complicated:

I intended to have them solve the 4x4x4 problem with one of our Rubik’s cubes as a prop, but we could only find our 5x5x5 cube. So, we skipped the 4x4x4 case, solved the 5x5x5 case and then jumped to the NxNxN case:

Finally, I wanted the boys to see the “slick” solution to this problem – which is really cool. You’ll hear my younger son say “that’s neat” if you listen carefully 🙂

Definitely a fun problem – would be really neat to share this one with a room foll of kids to see all of the different strategies they might try.

# A terrific volume project I learned from Kathy Henderson

Yesterday afternoon I saw a really neat tweet from Kathy Henderson:

It immediately reminded me of our projects on the volume of a pyramid and a tetrahedron from a few weeks ago:

Studying Tetrahedrons and Pyrmaids

Comparing a tetrahedron and a pyramid and experiment

We had a hard time finding the volume of the pyramid and tetrahedron by filling them with water because, despite our best efforts with tape, our shapes were not even close to water tight. They were definitely “popcorn tight” though, so we *had* to try out this activity.

Kathy was nice enough to share the handout she used, so designing today’s project was a piece of cake:

So, I had the boys make the shape’s prior to filming – we started the project with a quick discussion of the construction of the shapes. Then we talked about their volume.

My older son thought the volumes would be roughly the same. My younger son thought the one with the rectangular base would have the largest volume.

Next we tried to calculate the area of the base of each prism. Rather than using graph paper as the handout suggested, we found the area of each base by measuring. That gave us a chance for a little arithmetic and geometry practice, too.

Next we went to the kitchen scale to measure the change in weight when we filled the shapes with popcorn kernels. We found *very roughly* the relationship we were expecting, which was nice!

Finally, we revisited the pyramid and the tetrahedron project and looked at the two different volumes using popcorn. We found the ratio of the volumes was roughly 1.96 rather than the 1.7 to 1.8 ratio we found using water.

This is such a great project and I’m super happy that Kathy Henderson shared it yesterday. Working through the project you get to play with ideas from arithmetic and geometry. With a larger group you probably also get to discuss why everyone (presumably) found slightly different volumes.

So, a fun project that was relatively easy to implement. What a great start to the weekend 🙂

# What do kids see when they see ideas from advanced math?

Saw a really neat tweet from Steven Strogatz tonight:

I thought it would be fun to share it with the boys and just listen to how the described what they saw.

Here’s what my older (8th grade) son said:

Here’s what my younger son said when he saw the video:

It is fun to see ideas from advanced mathematics through the eyes of kids 🙂

# Sharing Robert Kaplinsky’s pipe stacking problem with my younger son

Yesterday I saw a really neat video collection from Robert Kaplinsky

I found clicking through to his blog post and watching all of the videos to be an absolutely fascinating exercise.

My younger son has been working through Art of Problem Solving’s Introduction to Geometry book this year and I thought he’d find Kaplinsky’s problem to be interesting. So, we watched the video in the tweet and then dove into the problem.

Here are his initial thoughts. You’ll see that the problem has confused him a bit and that he doesn’t quite know how to get started:

His first idea was to simplify the problem at look at just two layers of each of the stacks. He was able to solve the simpler problem (!) and then formed a conjecture about the solution to the 20 layer problem.

Next we tested the conjecture.

I would have liked to have gone about 5 min more, but we were already over the time allotted -> he needed to get ready to go to school. I’m pretty happy with his approach to the problem, though. It was really nice to see him work all the way to the complete solution. Thanks to Robert Kaplinsky for this nice problem.

# A simplified version of the Banach-Tarski paradox for kids

Yesterday we were listening to Patrick Honner’s appearance on the My Favorite Theorem podcast. Honner was discussing Varignon’s Theorem. We actually have discussed this appearance before, but the kids hadn’t listened to the podcast, yet:

After listening to the podcast I asked my older son what his favorite theorem was:

However, after giving up on the idea initially (!) I looked at the Wikipedia page for the Banach-Tarski paradox and found an idea that I thought might work. Here’s the page:

The idea was to share the first step in the proof – exploring the Cayley graph of $F_2$ – with kids. Here’s the picture from Wikipedia:

So, here’s what I did.

First I introduced the boys to some basic ideas about a free group on two generators. I used a Rubik’s cube to both demonstrate the ideas and to show why a Rubik’s cube didn’t quite work for a perfect demonstration (I know that part of the video drags on a bit, but stay with it – there is a nice surprise):

Next we talked about the free group with two generators in more detail. My younger son accidentally came up with a fantastic example that helped clarify how this free group worked.

Then there was a bit of a surprise misconception that I only uncovered by accident. That led to another important clarification.

So, completely by accident, we had a great conversation here.

In the last video they boys thought you could use the “letters” $x, x^{-1}, y, y^{-1}$ only once. In the beginning of this video I clarified the rules.

Next we began to talk about the representation of our free group by the Cayley graph from Wikipedia pictured above. I was really fun to hear how the boys described what they saw in this graph.

Finally, we looked at two different ways to break the Cayley graph into pieces. This video is a little long, but it has a simplified version of the main idea in the Banach-Tarski paradox.

The first decomposition of the Cayley graph is into 5 pieces -> the identity element, words that start with $a$, words that start with $a^{-1}$, words that start with $b$, and words that start with $b^{-1}$. This decomposition is pretty easy to see in the picture.

The second – and very surprising decomposition is as follows:

The combination of (i) the words that start with $a$ and (ii) $a$ multiplying (on the left) all the words that start with $a^{-1}$ gives the entire set. The same is true for the combination of (i) the words that start with $b$ and (ii) $b$ multiplying (on the left) all the words that start with $b^{-1}$

Although the words describing this decomposition might not make sense right away, you’ll see that the boys had a few questions about what was going on and eventually were able to see how this second decomposition worked.

And this second decomposition gives a huge surprise -> we’ve taken 4 subsets, combined them in pairs and created two exact copies of the original set. Ta da 🙂

This project is an incredibly fun one to share with kids. I’m pretty surprised that *any* ideas related to the Banach-Tarski paradox are accessible to kids, but the simple ideas about the Cayley graph of $F_2$ really are. Using those ideas you can show the main idea behind the sphere paradox without having to dive all the way into rotation groups which I think are a little more abstract and harder to understand.

Anyway, this one was a blast!

# Another great problem from Matt Enlow’s collection

Today we talked about another problem from the amazing list of problems that Matt Enlow’s published a few weeks ago:

This is our second project from that collection. The first is here:

Sharing a neat problem from Matt Enlow with kids

For today’s problem I introduced the problem and asked the boys for their initial thoughts. My older son noticed an important property about the sum of 9 and consecutive integers. He explained the property that the sum of 11 consecutive numbers would have and then my younger son explained the similar property that the sum of 9 consecutive numbers would have:

Next we had to see if there were any special properties that the sum of 10 consecutive integers would have.

Once we had that property, my younger son was able to explain how you could use them to find a number that would work (though not necessarily the smallest one):

At the end of the last video we though that 495 would satisfy the conditions of the problem. Here we checked that it did and wondered if it was the smallest.

Finally, we checked to see if 495 was indeed the smallest positive number with the properties in the problem.

My older son thought that 0 would have worked, but working it out he saw that it didn’t.

After that, we saw that 495 was indeed the smallest.

Definitely a great problem – it is nice to hear the boys explain some basic ideas in number theory. It is also a nice problem because kids – well, at least my kids – often struggle to see the difference between “find the smallest” and “find an example” and this problem helps show that “find the smallest” requires a bit more work.