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

I’m going through Jim Propp’s piece on base 3/2 with my kids this week.

His essay is here:

Jim Propp’s How do you write one hundred in base 3/2?

And the our first project using that essay is here:

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

Originally I wanted to have the kids read the essay and give some of their thoughts for part 2, but I changed my mind on the approach this morning. Instead I asked each of them to answer the question in the title of Propp’s essay -> How do you write 100 in base 3/2?

Propp points out in his essay that his approach to base 3/2 via chip firing / Engel machines / exploding dots is not what mathematicians would normally consider to be base 3/2. The boys are not aware of that statement, though, since they have not read the essay yet.

Here’s how my younger son approached writing 100 in base 3/2. The first video is an introduction to the problem and, from knowing how to write numbers like 100 (in base 10) in other integer bases.

I think the first 3 minutes of this video are interesting because you get to hear his ideas about why this approach seems like a good idea. The remainder of this video plus the next two videos are a long march down the road to discovering why this approach doesn’t work in the version of base 3/2 we are studying:

So, after finding that the path we were walking down led to a dead end, we started over. This time my son decided to try to write 100 as 10×10. This approach does work!

Next I introduced the problem to my older son. He also started by trying to solve the problem the same way that you would for integer bases, though his technique was slightly different. He realized fairly quickly (by the end of the video, I mean) that this approach didn’t work:

My older son needed to find a new approach, and he ended up finding an idea different from my younger son’s idea to find 100 in base 3/2. His idea was to use chip firing:

I thought that today’s project would be a quick reminder of how base 3/2 works (at least the version we are studying). That thought was way off base and was completely influenced by me knowing the answer! Instead we found – by accident – a great example of how to explore a challenging problem in math. Sometimes the first few things you try don’t work, and you have to keep trying new things.

Definitely a fun morning!


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

Jim Propp’s essay on base 3/2 is fantastic:

Here’s a direct link to his blog post in case the twitter link doesn’t work:

Jim Propp’s How do you write one hundred in base 3/2?

and here are links to our two prior base 3/2 projects:

Fun with James Tanton’s base 1.5

Revisiting James Tanton’s base 3/2 exercise

I’m hoping to have time to spend at least 3 days playing around with Propp’s latest blog post. Today we had 20 min free unexpectedly in the morning and I used that time to introduce two of the ideas. They haven’t read the post, yet, but instead I started by having them watch Propp’s short video about the binary Engel machine:

After watching that video I had the boys recreate the idea with snap cubes on our white board. Here’s that work plus a few of their thoughts on the connection with binary:

Next I challenged the boys to draw the base 3/2 version of the machine. After they did that we counted to 10 in base 3/2 and talked about what we saw:

I was happy that the boys were able to understand the idea behind the base 3/2 Engel machine. With the work from today giving them a nice introduction to some of the ideas in Propp’s essay, I think they are ready to try reading the essay tomorrow. It’ll be interesting to see what ideas catch their eye. Hopefully we can do another short project on whatever those ideas are tomorrow morning.

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

Kelsey Houston-Edwards has a new video out about the Axiom of Choice:

The video is amazing (as usual) and I wanted to be able to share it with the boys. This one is a bit hard than usual – the topic is pretty advanced to begin with and is also pretty far outside of my own knowledge – but we gave it a shot.

Here’s what the boys thought after seeing the video:

Next we reviewed how Houston-Edwards divided the numbers from 0 to 1 into buckets. The boys didn’t quite have the details right, but that actually made talking through the idea pretty easy – I learned from their explanation what points needed to be re-emphasized.

Now we talked through the really challenging part of the video -> creating the set with no size. Given the challenge of explaining this idea to kids, I’m pretty happy with how the conversation went here. Also, I only finally understood the argument myself while I was explaining it to them!

Now we backed away from the complexity of the Axiom of Choice and reviewed two other slightly easier ideas that came up in our discussion. Here we discuss why \sqrt{2} is irrational:

Finally, we wrapped up by discussing why the rational numbers are countable:

Although kids will have a hard time understanding all of the ideas that Kelsey Houston-Edwards brings up in her Axiom of Choice video, I think it is fun to see which ideas grab their attention. The idea that you can have a set that doesn’t have a size is pretty amazing. I was pretty happy with how things went today – exploring the ultra complex idea first and then backing off to discuss slightly easier ideas involving infinity. Definitely a fun set of ideas to plant in the minds of younger kids 🙂

Revisiting Jacob Lurie’s Breakthrough Prize lecture

Last night I asked my older son what he what topics were being covered in his math class at school.  He said that they were talking about different kinds of numbers -> natural numbers, integers, rational numbers, and irrational numbers.   I asked him if he thought it was important to learn about the different kinds of numbers and he said that he thought it was but didn’t know why.

I decided share Jacob Lurie’s Breakthrough Prize lecture with the boys this morning since he touches on the study of different kinds of number systems.  The first 12 or so minutes of the lecture are accessible to kids:

Near the beginning of Lurie’s talk he mentions that the equation x^2 + x + 1 = y^3 - y has no integer soltuions. I stopped the video here to what the boys thought about this problem. It took two about 10 minutes for the boys to think through the problem, but eventually they got there. It was fun to watch them think through the problem.

Here’s part 1 of that discussion:

and part 2:

The next problem that we discussed from the video was Lurie’s reference that all primes of the form 4n + 1 can be written as the sum of two squares. I checked that the boys understood the problem and then switched to a problem that would be easier for them to tackle -> No prime of the form 4n + 3 can be written as the sum of two squres.

Finally, to finish up, we began by discussing Lurie’s question about whether or not numbers were real things or things that were made up by mathematicians. Then we wrapped up by looking at why 13 is not prime when you expand the integers to include complex numbers of the form A + Bi where A and B are integers.

There aren’t many accessible public lectures from mathematicians out there. I’m happy that part of Lurie’s lecture is accessible to kids. It is nice to be able to use this lecture to help the boys understand a bit of history and a bit of why these different number systems are interesting to mathematicians.

Thinking about a math appreciation class

Steven Strogatz had great series of tweets about math education earlier in the week. These two have stayed in my head since he posted them:



I know that last year Strogatz taught a college level course similar to the one he is describing in the tweets. We even used a couple of his tweets about the course material for some fun Family Math activities. For example:

Here’s a link to that set of projects:

Steven Strogatz’s circle-area exercise part 2 (with a link to part 1)

So, thinking back to projects like those got me thinking about all sorts of other ideas you could explore in an appreciation course. At first my ideas were confined to subjects that are traditionally part of pre-college math programs and were essentially just different ways to show some of the usual topics. Then I switched tracks and thought about how to share mathematical ideas that might not normally be part of a k-12 curriculum. Eventually I tried to see if I could come up with a (maybe) 3 week long exploration on a specific topic.  I chose folding and thought about what sort of ideas could be shared with students.

Below are 9 ideas that came to mind along with 30 second videos showing the idea.

(1) A surprise book making idea shown to me by the mother of a friend of my older son:


(2) Exploring plane geometry through folding

We’ve done many explorations like this one in the last couple of years – folding is an incredibly fun (and surprisingly easy) way for kids to explore ideas in plane geometry without having to calculate:

Our Patty Paper geometry projects

Here’s one introductory example showing how to find the incenter of a triangle:

(3) The Fold and Cut theorem

Eric Demaine’s “fold and cut” theorem is an fantastic bit of advanced (and fairly recent) math to share with kids. Our projects exploring “fold and cut” ideas are here:

OUr Fold and Cut projects

Here’s one fun fold and cut example:

(4) Exploring platonic solids with Laura Taalman’s 3d printed polyhedra nets

You can find Taalman’s post about these hinged polyhedra here:

Laura Taalman’s hinged polyhedra blog post on her Makerhome blog

And if you like the hinged polyhedra, here’s a gif of a dodecahedron folding into a cube!

dodecahedron fold

Which comes from this amazing blog post:

The Golden Section, The Golden Triangle, The Regular Pentagon and the Pentagram, The Dodecahedron

[space filled in with random words to get the formatting in the blog post right 🙂 ]

(5) An amazing cube dissection made by Paula Beardell Krieg

We’ve also done some fun projects with shapes that I wouldn’t have thought to have explored with folded paper. Paula Beardell Krieg’s work with these shapes has been super fun to play with:

Our projects based on Paula Beardell Krieg’s work

(6) And Paula didn’t just stop with one cube 🙂

(7) Two more of Laura Taalman’s prints

Seemingly simple ideas about folding and bending can lead to pretty fantastic mathematical objects! These objects are another great reminder of how 3d printing can be used to make mathematical ideas accessible.

Here’s Taalman’s blog post about the Peano curve:

Laura Taalman’s peano curve 3d print

(8) Getting to some more advanced work from Erik Demaine and Joseph O’Rourke

As hinted at early with the Fold and Cut theorem, some of the mathematical ideas in folding can be extremely deep:

(9) Current research by Laura DeMarco and Kathryn Lindsey

Finally, the Quanta Magazine article linked below references current research involving folding ideas. The article also provides several ways to share the ideas with students.

Quanta Magazine’s article on DeMarco and Lindsey’s work

The two blog posts below show my attempt to understand some of the ideas in the article and share them with kids. The video shows some of the shapes we made while studying the article.

Trying to understand the DeMarco and Lindsey 3d folded fractals

Sharing Laura DeMarco’s and Kathryn Lindsey’s 3d Folded Fractals with kids

So, these are just sort of ideas that popped into my head thinking about one part of a math explorations class. Feels like you could spend three weeks on folding and expose kids to lots of fun ideas that they’d (likely) never seen before.

An introductory stars and bars problem

Yesterday a counting problem from my son’s math team homework gave him a little trouble. The problem went something like this:

There are 5 different types of fasteners and you need to buy 10 total. If you need to buy at least one of each, how many different ways can you do it?

First we talked about the problem and got their initial thoughts. Then I introduced the stars and bars counting idea:

Next I tried to go through a few more examples by changing the numbers a little. The main ideas seemed a little confusing to the boys and I’d hoped a few extra examples would help. Unfortunately things weren’t going so well.

The last example in the prior video confused my younger son, so I moved on to the next video to talk about that example in more detail. By the end of this example I hoped that the general idea had sunk in, but there was still a little confusion.

So, we talked through the problem a few more times. Now the ideas seemed to be sinking in. IF you have N groups of objects (in the original problem 5 fasteners) and you have to pick M total objects (in the original problem we were trying to pick 5 fasteners) you can represent the problem with M stars and N – 1 bars. So the total number of different ways to make the selections are (M + N – 1) “choose” M or, alternately, (M + N – 1) choose (N – 1).

Not all of our projects go super well. Here my mistake was thinking that I could introduce an advanced concept and the boys would immediately understand it. I feel like the ideas here are definitely within their grasp and will probably spend a bit more time this weekend covering the concept. Hopefully a few more examples will do the trick. Stars and Bars.jpg

Learning from Mantis’s Silke Delafortrie, Julia Lischka, and Trixi Peterstorfer

This week I’m writing about the gold medal game between Mantis and Box at the Austrian ultimate championships. The game is here:

Livestream Österreichische Staatsmeisterschaften Ultimate Men/Women

and my first blog post about the game is here:

Learning from Lisa-Maria Hanghofer and Paula Haubenwallner

The play I want to focus on today is Mantis’s scoring possession to go up 6-3 and I would like to highlight the great movement and spacing from #11 Silke Delafortrie, #16 Julia Lischka, and #18 Trixi Peterstorfer. When I saw this play for the first time I just about jumped out of my chair and the more I watch it the more I think it is an outstanding example to learn from.

The first thing I’d like to highlight is the downfield movement of Silke Delafortrie who starts back in a handler position at the top of the screen. Watch her begin to move down the field immediately as the disc moves away from her. This quick reaction is what puts her in such great position after the huck.

I’ve written about some dangerous downfield attacking from the handler position previously. Players like Carolyn Finney and Jenny Fey come to mind (not to mention the amazing attacking strategy that the Colombian national team employs!). Delafortrie movement in this possession is a super example to learn from!

The next thing I want to talk about appears on the screen for only about 1 second – it is the line of three Mantis players moving down the field behind the huck. Those players are #11 Silke Delafortrie at the top of the screen, #16 Julia Lischka in the middle and #26 Lisa-Maria Hanghofer at the bottom of the screen.

The way these three players maintain their spacing across the field as the play develops is what makes Mantis’s scoring play look so easy. I don’t want to give away the punch line here – having watched the score already in the first video, as you watch this video think about how their spacing contributed to the score. Also thounk about what each of these players could have done that would have made it more difficult for Mantis to score:

The way the cutting teamwork on this possession contributed to the score is one of the best downfield cutting lessons I’ve ever seen on film.

Now I have to give a shout out to Trixi Peterstorfer for some incredibly smart play after catching the huck.

The quick dish to Delafortrie is a really natural play and equally natural would be to head down the field to score. If you freeze the shot as Delafortrie catches the disc it looks like Peterstorfer has the whole endzone open. But she doesn’t. Her defender has dropped up to stop the deep cut. Peterstorfer stops her cut immediately and turns around to get an easy 10 yard gain.

Even better, she turns in the direction the disc is moving and fakes a backhand (yay lefties!!) which draws her defender away from the forehand side of the field.

This is a fantastic cutting and throwing lesson – there’s not one bit of wasted movement and not one thing that I can think of that Peterstorfer could have done better.

Finally comes the score. After the nice backhand fake, the throw out to space from Peterstorfer is perfect.

The cut to score from Julia Lischka is terrific and is the end result of the great spacing I talked about above. She had the middle of the field, maintained great spacing, and was in the best position to attack the open space in the endzone.

And I have to highlight the non-cut in the endzone from Hanghofer. She maintained her wide position on the field all the way through the play. That positioning (combined with Peterstorfer’s backhand fake) helped draw the deep defenders to the middle of the field and opened the far side of the field for Lischka. This kind of play doesn’t show up directly in the stat sheets (and barely shows up in the film!) but learning from great spacing examples like this is so important as you look to improve your own play. Sometimes (probably always, actually) the right non-cut is just as critical to the team as the right cut. Great work from Hanghofer!

I hope to write one more blog post about this game – thanks again to Michelle Phillips for writing about it originally!

Working through two old contest problems

I’ve been sort of on pause doing new math with the boys for the last couple of weeks. I want them to find their stride with the new school year before seeing what additional enrichment math we can do at home.

So, while on pause they’ve just been working through problems from old AMC tests in the morning. When they finish we talk through some of the problems that gave them trouble. Both problems were pretty interesting lessons (for them and me, I mean) today.

Here’s what my older son had to say about problem 18 from the 2013 AMC 10b. It was fascinating to me how he counted the numbers in this problem.

and here’s what my younger son had to say about problem #17 from the 1985 AMC 8 (which then was the American Junior High School Math Exam). It was fascinating to me to see both how he played with the averages and how he found his arithmetic mistake.

I love using these old AMC problems to keep the kids engaged with math. It is always fun to see what sorts of ideas give them problems and just as fun to see their problem solving strategies.

Exploring Elchanan Mossel’s fantastic probability problem with kids

Saw a really great problem via a Lior Patcher tweet:

Here’s the problem:

You throw a dice until you get 6. What is the expected number of throws (including the throw giving 6) conditioned on the event that
all throws gave even numbers.

Here are direct links to Kalai’s two blog posts on the problem:

Gil Kalai’s “TYI 30: Expected number of dice rolls

Gil Kalai’s follow up post: Elchanan Mossel’s Amazing Dice Paradox (Your Answers to TYI 30)

It is fun to click through to the first Kalai blog post linked above to cast your vote for the answer if you haven’t seen the problem before.

We actually started the project today by doing that:

Next we rolled some 6-sided dice to see how this game worked. I note seeing the video that a few of the rolls went off camera, sorry about that 🙂

At the end we discussed what we saw and why what we found was a little surprising.

The next part of the project was having the boys play the game off camera until they found 5 rolls meeting the criteria.

After this exercise the boys started to gain some confidence that the answer to the problem was 3/2.

Now I walked them through what I think is the easiest solution to understand. It comes from a comment on the first Gil Kalai’s blog post linked above:


Listening to this discussion now, I wish I would have done a better job explaining this particular solution. Still, I hope the discussion is instructive.

Finally, we went to Mathematica to evaluation the sum from the last video and then to explore the problem via a short program I wrote.

At the end of this video the boys some up their thoughts on the problem.

I love this problem. It isn’t that often I run across a clever problem that is interesting for both professional mathematicians and kids. Those problems are absolute

Counting in 4 dimensions

Yesterday we did a neat project based on problem #12 from the 2015 AMC 8:


That project is here:

A great counting problem for kids from the 2015 AMC 8

Then I got a nice comment on the project from Alison Hansel:

So, for today’s project we extended the problem from yesterday to 4 dimensions.

Here’s the introduction and a quick reminder of yesterday’s problem. I had both boys review their solutions and then we began to discuss how to approach the same problem in 4 dimensions.

Next we dove a bit deeper into how to approach the 4 dimensional problem. They boys thought a bit about the symmetry that a 4d cube would have and at the end (after a long and quiet pause) my younger son thought that looking at how a square turns into a cube might help us.

In studying how a square transforms into a cube, the critical idea is how 4 edges turn into 12 edges. This video is a little on the long side, but I think the discussion is really interesting. By the end the boys have found the main idea for how to count edges as you move up in dimension.

Next I brought out Henry Segerman’s 4-d cube model and compared the model to the ideas we’d developed up to this point.

An important idea from earlier in this project was that my older son thought that each edge of the 4d-cube would be part of two 3d cube “faces”. Using the model we were able to see that, in fact, each edge is part of 3 cubes.

Finally – with the 4 pieces of prep work behind us! – we were able to answer the AMC 8 question about a 3d cube in 4 dimensions. So . . . how many pairs of parallel edges does a 4d cube have? The answer is 112 🙂

Thanks to Alison Hansel for the great suggestion for how to extend yesterday’s project. I think her idea makes a great way to introduce kids to some simple ideas in 4d geometry.