James Tanton’s penny and dime exercise

I saw a neat tweet from James Tanton yesterday:

After seeing the tweet, I couldn’t wait to do the project today. It turned out to be much more difficult than I was expecting, though. One of the difficulties that really caught me off guard was the time it took to make accurate measurements of the positions. The difficulty there turned what I mistakenly thought was going to be a really quick part of the project into maybe 90% of the project’s time.

Maybe a nice surprise with this project is that it could be a good introductory project for introducing kids to measurement.

So, although I’ll publish all 6 videos from today’s project, the main ideas are in the first and last videos.

Here’s the introduction to the project:

The boys weren’t totally sure what happened the first time around, so we tried again:

After two tries, they still weren’t sure what was going on, so we tried one more time:

After this third try, they had some interesting observations:

One idea they had in the last video was to see what would happen if the 3 points were on a line.

To wrap up, the boys wondered about a few other set ups. I was happy to hear that they were starting to think about new ideas.

I think Tanton’s problem is an absolutely great exercise for kids. I’m sorry that I misjudged the difficulty coming from the measurements, though.

Revisiting folding a dodecahedron into a cube

Saw this great twitter thread from Vincent Pantaloni a few weeks ago:

It made me want to revisit an old project exploring how a dodecahedron can fold into a cube:

Can you believe that a dodecahedron folds into a cube?

I started by asking the boys to each make a dodecahedron with a cube inside using our Zometool set:

Next I asked them to turn one of their shapes inside out. They did that off camera and we compared the two structures:

Finally, they connected the zome balls that were inside of the “inside out” dodecahedron and found that those balls did indeed form an icosahedron. That’s one of the great surprises in this neat shape!

For me, this project illustrates the fun you can have exploring 3d geometry with a Zometool set. Even though I’ve seen the folding dodecahedron many times now, it never gets old!

My year in sharing math from mathematicians with kids

[this post still needs a little editing, but I’m heading out the door in 5 mint and will publish first and edit second]

Lately I feel like I haven’t been able to do as many math projects with my kids as I would like. Need to make sure that the priorities get back in line next year.

One thing I’m happy about, though, is that in 2017 I was able to share a lot of math from mathematicians with them. The list below isn’t even all of the projects, but I felt like 15 was enough! Here are some of the fun projects we did based on ideas we saw from mathematicians that I follow (mainly on twitter):

(1) Katherine Johnson’s work in Hidden Figures

My older son started looking at some ideas from trigonometry this year – that opened the door for him to be able to peek at some of the work from Hidden Figures.

An Attempt to share some of Katherine Johnson’s math ideas from Hidden Figures with my son

(2) Christopher Long and Nassim Taleb

The most interesting piece of math I learned this year was the “covfefe” problem. At first I thought the problem wasn’t that interesting, and than Christopher Long and Nassim Taleb showed me that there was more to the problem than I realized:

The most interesting piece of math I learned in 2017 -> the “covfefe” problem

(3) Nassim Taleb and Alexander Bogomolny

One of the really enjoyable examples of math in public I saw this year was the ongoing twitter exchange of problems and solutions between Alexander Bogomolny and Nassim Taleb. I assume the exchange will continue in 2018, too, I really enjoyed sharing the occasional problem from them with the boys. For example:

Sharing Nassim Taleb’s dart probability problem with kids

A terrific example for Calculus students from Nassim Taleb and Alexander Bogomolny

(4) Laura DeMarco and Katherine Lindsey

Quanta Magazine published a fantastic article on the work of Laura DeMarco and Kathryn Lindsey. Even though we could only scratch the surface of “3d folded fractals”, it was amazing to explore what we could.

My wife make a version of the “square with cap” from the article:

Here’s some prep work and what I talked about with the boys:

Trying to understand the Demarco and Lindsey 3D Folded Fractals

Sharing Laura DeMarco’s and Kathryn Lindsey’s 3d folded fractals with kids

(5) Kelsey Houston-Edwards

Kelsey Houston-Edwards’s work with PBS Infinite series completely blew me away. We probably have over a dozen projects based on her videos on the blog. Here are two that were especially fun:

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

Sharing Kelsey Houston-Edwards’s Axion of Choice video with kids

(6) Jim Propp

Jim Propp’s blog has been extra fun for me to follow because I learned combinatorics from his course 20+ years ago. I love sharing his blog posts with the boys – it is amazing how many seemingly simple ideas can lead to terrific math conversations:

Jim Propp’s “Swine in a Line” game

Sharing Jim Propp’s base 3/2 essay with kids – Part 1

Sharing Jim Propp’s base 3/2 essay with kids – Part 2

(7) Evelyn Lamb

One of the most enjoyable weeks of the year for me came from playing with a post about pentagons by Evelyn Lamb. Exploring some of the details about this pentagon helped me get a much better understanding of the recent result that there are only 15 types of pentagons which can tile the plane:

Evelyn Lamb’s pentagons are everything

(8)  A new result about the Cantor Set

It is pretty unusual to be able to share current math research with kids, but maybe once per year there’s a result that is within the read of kids. The lucky example from this year was a result showing that any number between 0 and 1 can be written as the product $x^2 * y$ where x and y are in the Cantor Set:

Sharing a new result about the Cantor Set with kids

(9) John Baez and Leo Stein’s posts about Juggling Roots.

A dream of mine for a long time has been to figure out how to explain to kids why 5th degree and higher polynomials cannot be solved in general. This year I got a lot closer to reaching that goal thanks to John Baez and Leo Stein.

Sharing John Baez’s Juggling Roots tweet with kids

Using Leo Stein’s polynomial toy with kids

(10) Swarmalators

“Swarmalators” were another bit of math research from 2017 that I was able to share with the boys. The underlying math itself was much too complicated, but luckily one of the authors shared a computer program which allowed anyone to explore some of the ideas.

Having kids play with “swarmalators”

(11)  Martin Gardner

During our trip to Omaha to see the eclipse this summer I found a copy of Martin Gardner’s Colossal Book of Mathematics in a library book sale. One of the most incredible projects in the book was a “machine learning.”

One fun note on this project is that Alison Hansel was inspired to try out this project with her daughter last week:

Intro Machine Learning for kids via Martin Gardner’s article on “hexapawn”

(12) Joel David-Hamkins

We love the math projects for kids that Joel David-Hamkins shares. This year we did two projects based on his work. The first one had a tie in with one of Kelsey Houston-Edwards’s videos:

Buckets of fish and defeating hydras

The next one – from last week – was a fun logic puzzle:

Sharing Joel David-Hamkins’s fun logic exercise with kids

(13)  Elchanan Mossel’s probability problem

This probability problem from Elchanan Mossel made for a terrific project:

Exploring Elchanan Mossel fantastic probability problem with kids

(14) Steven Wolfram

Wolfram’s talk at MoMath is one of the most incredible examples of “math in public” that I’ve ever seen. You’ll need Mathematica if you want to play along and explore more, but just watching Wolfram’s talk is amazing all by itself.

Sharing Stephen Wolfram’s MoMath talk with kids

Revisiting Stephan Wolfram’s Momath talk

(15) Tim Gowers

Finally, Tim Gowers gave a neat talk about non-transitive dice at Harvard this fall. Some of the results he shared were accessible to kids – and actually really surprising. Luckily we already had a set of non-transitive dice made by James Grime, so the kids had seen some of the basic ideas before.

Sharing Tim Gowers’s non-transitive dice talk with kids

Sharing Joel David Hamkins’s fun logic exercise with kids

Saw a fun tweet from Joel David Hamkins last week:

I was looking for a light project this morning and the problem in this tweet was perfect! All that we really needed to get going was an introduction to what the symbols meant. They got the hang of the symbols quickly and were able to understand one of the statements quickly:

Now I just had the boys pick logic statements and try to match them with the appropriate phrase. My younger son went first:

The statement in the next selection led to a short discussion about what “there exists” means.

We finished up with the last two open options – here the boys got a bit more understanding of why “empty train” was a description in some of the choices:

Definitely a neat exercise, and a fun way to get kids talking about logic!

Using Leo Stein’s “polynomial toy” with kids

[note: sorry for the quick write up -> only had 30 min before having to get out the door. Don’t be fooled by the short writing up – this is an amazing topic for kids to explore!]

I saw this amazing tweet from Leo Stein two weeks ago:

Some work travel prevented me from using it in a project with the boys, but I’m back from the trip now. Today was the day to dive in.

We have played with these ideas before – see the collection of project here:

My week with “juggling roots”

One of the most enjoyable parts of that week was printing the paths the the roots of 5th degree polynomials took as we played:

Those pretty prints convinced me that playing with roots of polynomials can be a really fun project for kids.

I started the project today by simply talking about polynomials. My kids are in 6th and 8th grade and have heard about polynomials, but haven’t heard enough for us to just dive right in to Stein’s program. So, here’s the 5 min introduction to the project:

Next we dove in to Stein’s program. I used a 2nd degree polynomial as an example of how to play with the program. From the beginning, the movement of the roots as you move the coefficients was really interesting to the kids.

I had the boys play with the program off camera for about 10 minutes. Here are some of the things that they thought were interesting:

My younger son went first – simply describing what you are seeing is a good exercise in using mathematical language for a kid:

My older son went next – the movement of the roots reminds him of gravity and planets:

A long term goal of mine is to find a way to explain to kids (maybe high school students) why a 5th degree polynomial cannot be solved in general. The ideas from these “jugging roots” programs feel like the right path. The movement of the roots as the coefficients change is simply mesmerizing!

Geometry Snacks

I returned from a trip to London to find that the new book from Ed Southall and Vincent Pantaloni had arrived:

I’d just flown back from Cologne, and was really out of gas, but wanted to take a quick look at the book. My younger son spent a bit of time flipping through it and picked a problem.

Here’s his reaction to the book and the problem he picked:

and here is his solution to that problem:

I think these two short videos give a nice little peek at the book. Looks like it’ll be a fun way to explore geometry. Thanks to Ed and Vincent for sending me a copy – can’t wait to play with it a bit more.

The most interesting piece of math I learned in 2017 -> the COVFEFE problem

This question from a math exam in India was flying around math twitter last week:

I mistakenly thought the question was just sort of a fun joke and not all that interesting, but then I saw a series of tweets beginning with:

and ending with:

That last tweet was definitely a “wait . . . what??” moment for me.

Thinking about ABRACADABRA right off the bat was too hard, so I simplified the problem drastically too see what was going on. Suppose you are flipping a fair coin, what is the expected number of flips until you see the sequence H H? What about H T? These two problems are also, I think, great ways to introduce ideas about stopping time problems to kids. (the answers are 5 flips and 4 flips respectively).

Playing around with the easier problems showed me why the ABRACADABRA problem could have a longer stopping time than I would have guessed, but I couldn’t solve the problem exactly. Then I found this paper (written as part of an undergraduate research program at the University of Chicago!) which gave a wonderful explanation of the ABRACADABRA problem and (almost incredibly) a way to think about the problem that allows you to solve it in your head!

Martingale’s and the ABRACADABRA problem by Di Ai

After going through that paper I was happy to have learned some new ideas about stopping time problems and more or less moved on. But then one more nice surprise came from the COVFEFE problem when Nassim Taleb shared his Markov chain solution:

I’d never played with Markov chains in Mathematica (or, basically anywhere really) so I thought it would be fun to use what I learned from Taleb’s code to explore the ABRACADABRA problem. Working through that code gave me a much better understanding of Long’s “which states lead to which states” comment above. It took me a bit of time to realize, for example, that the state ABRA can move to the state AB, for example.

Again, copying Taleb’s code, here’s the transition matrix:

and the graph of the states plus the stopping time which matches 26^11 + 26^4 + 26:

It was fun to learn that the original COVFEFE problem was part of a class of problems that are much more subtle than they might seem at first glance. Learning about the connections to martingales and learning how to implement Markov chains in Mathematica was a really nice surprise, too.

Revisiting non-transitive dice

We’ve done a few projects on non-transitive dice in the past:

Sharing Tim Gower’s non-transitive dice talk with kids

Non-Transitive Grime Dice

Today we’ve got some snow to shovel, so I was looking for a fairly light project this morning so we could get out the door to shovel. I grabbed our Grime dice off of the shelf and asked the kids to talk about them:

I asked the boys to pick two pairs of dice and test them to see which color would win. They worked independently and here’s how they explained what they found:

Finally, for a bit of a challenge, I had them work together to put the dice in a circular arrangement so that every color beat the one coming after it and lost to the one before it. This arrangement illustrates the seemingly odd non-transtive nature of these dice:

Although short, this was a fun exercise. These “Grime” dice are really fun for kids to play with!

Revisiting card shuffling after seeing a talk by Persi Diaconis

Last week I took the boys to a talk given by Persi Diaconis at MIT:

Despite most of the talk going over their heads, the boys were really excited after the talk and had lots of different “shuffling” ideas that they wanted to explore.

Since this was going to be a long project, I divided it into two pieces – studying “pick up 52” with my younger son last night and studying shaking the cards in a box with both kids this morning.

The idea of using Shannon Entropy to study how random the shuffles are is something that we explored in these two prior projects:

Chard Shuffling and Shannon Entropy

Chard Shuffling and Shannon Entropy part 2

The original idea for those projects came from on an old Stackexchange post (well, the first comment) here:

See the first comment on this Stackexchange post

So, I kicked off the project last night with my younger son. Here are his thoughts about the Diaconis talk and about his “pick up 52” idea

We did 4 trials without re-sorting the cards in between. Here are some quick thoughts about how the deck was getting mixed up between the 2nd and 3rd throws:

After we finished I had my son do a few minutes of riffle shuffling to completely mix up the deck (starting from where the deck was after the 4th pick up 52). While he was doing that I entered the numbers from our throws into a spreadsheet.

The surprise was that even after the first throw the cards were really mixed up. I was even more surprised by this because he basically threw the deck in the air rather than what (to me anyway) is the normal way of throwing cards for pick up 52.

This morning we continued the project with my older son’s idea of putting the cards in a box and shaking the box. Here the introduction to that idea:

Here are some thoughts from my older son after the first mixing. He didn’t think they were all that mixed up. We did a total of 3 more mixings – the 3rd and 4th were off camera.

Finally, we wrapped up by reviewing the numbers for mixing the cards up in the box. The first mixing had more entropy than we thought, and after the 2nd mixing the cards appear to be pretty close to as mixed up as you can get (equivalent to about 10 riffle shuffles, I think).

This was a really fun project. The math you need to describe what’s going on here is much to advanced for kids (and worthy of a math lecture at MIT!), but kids can still have a lot of fun exploring some of the ideas. The seemingly simple idea of how can you measure how mixed up a shuffle is is a pretty interest idea all by itself.

An attempt to share some Katherine Johnson’s math ideas from Hidden Figures with my son

For the last few months I’ve been daydreaming about ways to share some of the math from the movie Hidden Figures with kids. As part of that prep work I found one of Katherine Johnson’s technical papers on NASA’s website:

NASA’s Technical Note D-233 by T. H. Skopinski and Katherine G. Johnson

As you’d expect, there’s a lot of trig, calculus and spherical geometry. I like finding ways to share the work that mathematicians do with kids, but this work is pretty technical and I wasn’t getting any great ideas.

Then my son had a homework problem from his Precalculus book that made me think it was time to stop daydreaming and just try something. Here is that problem, which is a completely standard law of cosines problem:

The problem reminded me of one of the equations for an ellipse used in the Technical Note. One surprising thing is that the equation of an ellipse in polar coordinates is that is is a rational function in $\cos{\theta}$.

So, I drew an ellipse and showed my son that equation.

One of the neat things about the Technical Note is that the solution to some of the complicated trig equations were found by an iteration method. The specific ideas for solving those equations are too advanced for kids, so I decided to show my son a different (and really simple) iteration method that converges to a well known number:

After that introduction to iteration methods, I decided to jump to a second and slightly more complicated example -> solving x = 3*x*(1 – x).

The ideas in the iteration method we use here can be explored purely geometrically:

Next we went upstairs to the computer to see some of the ideas we just talked about. The first idea was the polar coordinate equation for an ellipse:

Now we played with the second dynamical system -> solving x = 3*x(1-x).

By the way, the ideas here are incredibly fun to explore (especially seeing when this method converges and when it doesn’t), but the details of this method wasn’t really the idea here. I just wanted to show him what an iterative method looks like.

Finally, I showed him the actual paper and pointed out some of the parts we explored. Sorry that this film didn’t come out as well as I’d hoped, but you can view the paper from the first link in this post:

This was a fun project – even if it wasn’t planned really well. Showing some of the math behind Hidden Figures I hope helps motivate some of the topics that my son is studying right now. It will be fun to return to a second Hidden Figures project when he is studying calculus.