Yesterday Nervous System in Somerville, MA had an open house and I was lucky to have a few hours free while the boys were at their karate black belt tests. Visiting their shop was absolutely incredible:

At the open house I bought two new puzzles. The boys had seen one previously at Christmas, too. For our project today I’d already wanted something on the easy to talk about / less heavy math side because of the black belt tests yesterday, so talking about the new puzzles was perfect.

We started with the geode puzzle – one of the fun things we talked about was how the boys thought the computer generated the geode shape:

After the introduction to the puzzles, we moved on to talking about the challenge of putting the puzzle together. Favorite line – “once you get started, it gets pretty hard.” Yep!

Next I showed them the latest creation from Nervous Systems – an “infinity puzzle” inspired by the Mobius strip!

I was incredibly lucky to be able to buy one of the infinity puzzles yesterday. So, for the last part of today’s project we did an unboxing:

If you know kids who like puzzles – or you like puzzles! – all I can say is the Nervous System puzzles are absolutely incredible.

3 hours after we finished the project this morning, my younger son had returned to the Geode puzzle:

Solving this problem requires calculus, and trig to even begin to understand how to approach it, but it still seemed like one that would be interesting to talk through with kids. Especially since a Monte Carlo-like approach is going to lead you down a surprising path.

So, I presented this problem to the boys this morning. It took a few minutes for them to get their arms around the problem, but they were able to understand the main ideas behind the question. That made me happy.

Here’s the introduction to the problem:

Next I asked the boys what they thought the answer to this question would be. It was fascinating to hear their reasoning. Both kids had the same guess -> the expected average distance was 1.

Now we went to the computer to see what the average was when we did a few trials. We started by doing 100 trials to estimate the average and then moved up to 10,000 trials.

Next we went to 1 million trials and found a few big surprises including this amazing average:

We wrapped up by discussing how you might get an infinite expected value by looking at the values of Tan(89), Tan(89.9), Tan(89.99), and so on. It was interesting for them to see how individual trials could have large weights, even with large numbers of trials.

Definitely a fun project to show kids, and a nice (though advanced) statistics lessonm too -> What happens when the mean you are looking for is infinite?

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.

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.

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

Interesting problem on my son's enrichment math homework. Solve { x^2 – xy + y^2 = 13 and x – xy + y = -5 }. The section is on solving quadratics. I wonder if the idea they are trying to teach with this problem is u-substitution (u = x + y, v = xy) or something else.

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.

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

Ok #MTBoS and #iteachmath tweeps! If you were asked to plan a 4 day math themed summer camp for rising 6th graders, what would you dream up?? You have 80 mins a day and no more than 20 kids. Go!!

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:

(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!

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:

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 🙂

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:

(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:

(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):

— John Allen Paulos (@JohnAllenPaulos) June 15, 2016

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).

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:

My older son is on a school trip this weekend, so this project is just with my younger son (in 6th grade). I thought he’d had a lot of fun playing around with the program, so I let him explore it (with no instruction or even explanation) for about 10 min and then asked him what he thought was neat:

At the end of the last video he was playing around with the different numbers. I didn’t want to go into what those numbers represented, but I did think it would be great to hear some of his ideas and conjectures.

He found some ideas that seemed to work and a few that didn’t – so that was great to hear. By the end we’d found a shape that we could make from our Zometool set.

To finish this morning’s project we built the shape – here’s are his thoughts about having the shape in front of him vs seeing it on a computer screen:

This was a super fun project. I think it might be a nice challenge to try to dive a little deeper into the general Wythoff constructions that the Matt Zucker’s program is designed to explore. For now though, even with any details, the program is really fantastic for kids to play with.

I didn’t do a very good job managing the time on this project today. The trouble is that there are lots of different directions to go with the ideas and we walked down a lot of different paths.

But – I think this is a great topic to show off the beauty of math and we end with an amazing connection between sums of divisors of integers and .

The topic of sums of divisors of an integer came up in my younger son’s weekend enrichment math program yesterday. I thought it would make for a good topic for a project, so I gave it a go this morning.

The first part of the project was mostly about divisors and the kinds of questions that we could ask about them. A lot of the discussion here is about a question you can ask about the product of a number’s divisors:

Next we began to look at the sum of the divisors of a few different numbers. The boys noticed a few patterns – including a pattern in the powers of 2.

At the end we were looking to see if we could find patterns in the powers of 3.

It was proving to be a little difficult to find the pattern in the powers of 3, but we kept trying. After few ideas that didn’t quite help us write down the pattern, they boys had an idea that got us there.

At the end of this video I showed them that the sum of the divisors of powers of 6 was connected with the sum of the divisors of powers of 2 and powers of 3.

To wrap up I wanted to show some larger patterns in divisor sums, so we moved to Mathematica to play around a bit.

While I was doing the same playing around last night I accidentally stumbled on an amazing fact: As n gets large, the average of the sum of the divisors of the numbers from 1 to n is approximately .

Number theory sure has some fun surprises 🙂

This is definitely a fun topic and also one that could be used in a variety of ways (arithmetic review, intro to number theory, computer math, . . . ). I wish that I’d presented it better. Probably it needs more than one project to really fit in all of the ideas, though.

Two weeks ago I saw an interesting lecture from Gil Strang at MIT about the math behind machine learning. Sharing some of those ideas with kids has been on my mind ever since. Today I finally got around to it!

We’ve done a few previous projects that touched on ways to make machine learning accessible to kids. The Martin Gardner hexapawn project is incredibly fun and also is accessible to really young kids, the other project below uses the same Tensorflow website that we played with today:

Today I began be asking the boys what they knew about machine learning and then I explained a bit about classification problems:

Next I moved on to drawing a clumsy picture of what a neural network might look like and then did a clumsy explanation of how a neural network might work. My older son asked a really great question that gets to the difference between the Hexapawn game and how modern neural networks work – so we chatted about that for a bit.

Then I talked about the so-called “relu” firing function for neurons.

Before moving on to the Tensorflow program, I wanted to spend a few minutes talking about an idea that Gil Strang mentioned in his lecture. That idea is the connection between folding and classification.

This idea, I think, helps make the classification problem accessible to kids.

Next up was playing with the Tensorflow program and exploring some basic classification examples:

Then I let the kids play with the program by themselves for about 15 min – here are a few of the ideas that they found interesting:

Machine learning is an incredibly popular and growing area of math and computer science right now – the Tensorflow website is a great way to share some of the ideas in machine learning with kids.

Saw a really neat tweet from John Carlos Baez last week:

Al Grant has a great interactive page of tilings that move on "hinges". Check it out: https://t.co/GckHyxjGQk And Al is short for "Albert" not "Artificial intelligence". pic.twitter.com/h79GeheA32

Finally got a chance to share this site with my younger tonight. This site is fantastic to share with kids – my son enjoyed playing around with the tiling patterns, and it was also really interesting to hear him try to describe what he was seeing.

Here’s his initial look at the site:

Here’s his reaction and play with the part of the site that allows you to create and manipulate new quadrilaterals:

This is a wonderfully easy site and a really fun idea to play with. I think with older kids it would be nice to see them try to think through why the cyclic quadrilaterals have this hinged tiling property, but I thought that might be a little much for my younger son. We’ll do a follow up exploring those ideas soon, though.

Today we moved on to a really neat surprise, and what makes the math behind this problem incredibly fun -> the “ABRACADABRA” problem.

First, we reviewed the ideas from yesterday:

After that review, we though through a few of the states and the transition probabilities in the new word. The transition probabilities are subtly different than in the “COVFEFE” problem:

Now we went to Mathematica to code in the ideas we discussed in part 2. We did about half of the coding on camera and did the other half off camera:

Finally, having finished the code we discussed what results we expected. I don’t see how anyone could get the right intuition here seeing the problem for the first time, so what do you expect here is almost an unfair question. Still, the boys had some nice ideas and then we checked out the results:

There are other approaches to these problems – the approach via Martingales, for example:

Probably a little bit advanced for your kids, but the martingale approach is definitely a classic. Check it out: https://t.co/NPAw5ZVRI1@jeremyjkun

What that approach is also interesting (and incredible – you can solve the stopping time in your head!) I think the Markov chain approach is a bit more accessible to kidsd. Well . . . maybe because the math is buried in the background.

Anyway – super fun project, and an great piece of math to share with kids.