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.
Jim Propp’s August 2017 blog post is absolutely terrific:
Prof. Engel’s Marvelously Improbably Machines
Even though we are visiting my parents in Omaha, I couldn’t resist having the boys watch the video in the essay and then play with the challenge problem.
Here’s what they thought after watching the video – the nice thing is that they were able to understand the problem Propp was discussing [also, I shot these videos with my phone, so they probably don’t have quote the quality or stability of our usual math videos]:
Next I had them play the game that Propp explained in his video. The idea here was to make sure that they understood how the [amazing!] solution to the problem shown in the video:
Next we tried the challenge problem from the essay. I almost didn’t do this part of the project, but I’m glad I did. It turned out that there were a few ideas in the Propp’s video that the boys thought they understood but there was a bit more explanation required. Once they got past those small stumbling blocks, they were able to solve the problem.
I’m really excited to dive a little deeper into the method of solving probability problems that Propp explains in his essay. What makes me the most excited is that the method came from someone thinking about how to explain probability to kids.
The last video shows that understanding Engel’s method does take a little time. Once kids get the general idea, though, I think they’ll find that applying to a wide variety of problems is pretty easy. It is amazing how such a simple method can made fairly complex probability problems accessible to kids.
Last week I saw a really fun new question from Jim Propp:
Here’s the first project that we did on the game:
Jim Propp’s Swine in a Line game
Today we returned to the game to see if we could make any more progress understanding how it worked.
First we reviewed the rules and decided on an initial approach to studying the game for today:
Their first idea was to try to keep the two ends open since they knew the result when you reached the position with only 1 and 9 open.
Now we tried to study a bit more. The kids were having trouble seeing a path forward, so I just let them play.
At the end of the last video we were studying a position with all of the pens filled except for 7 and 9. In this video we searched for a winning move.
Finally, we took one more crack at solving the game. They boys got very close to the main idea, about an inch away(!), but didn’t quite get over the line.
The boys were really interested in the game and we kept talking for about 30 min after the end of the project. During that talk they did uncover the main idea. After that we played several games where they followed the strategy and they were able win against me every time following that strategy. It was a fun way to end the morning.
Saw this great new video from Jim Propp yesterday:
This morning I had the boys watch the video and then we spent maybe 15 min talking through the game and seeing what we could learn.
First I asked them what they thought after seeing the video:
Now we played the game and the boys made a couple of initial discoveries. You can see quickly why this is a fun game for kids to play around with:
Next we played the game one more time. We aren’t trying to solve the game in this project, just to try to learn a few things about it.
Finally, we wrapped up by talking about a few of the things they learned playing the game. This part didn’t quite go how I wanted, but it was still interesting to hear what they had to say.
I’m excited to play around with this game a bit more later in the week. It’ll be interesting to see if the boys can continue to make project towards the solution.
For our math project today we returned a tiling idea that is a really fun idea for kids to explore. Here are a few of our prior projects on the subject:
A fun counting exercise for kids suggested by Jim Propp
Counting 2xN domino tilings
Today the plan was to look at 2xN tilings first and then move on to tilings of 3xN rectangles with 3×1 dominoes.
We stared by exploring some simple 2xN cases and looked for patterns:
In the first video we counted the number ways that we could tile 2×1, 2×2, 2×3, and 2×4 rectangles with dominoes. Now the boys noticed the connection with the Fibonacci numbers and we tried to find and explanation for why the Fibonacci numbers seemed to be showing up here. The nice thing is that the boys pretty much got the complete explanation all on their own.
Now we moved on to counting the tilings of a 3xN rectangle using 3×1 “dominoes” – what would be different? What would be the same?
One really interesting thing here is that my older son and younger son came up with different ideas for how to count the general arrangement.
So, in the last video my older son had a counting hypothesis that I couldn’t quite understand. In the beginning of this video I have him explain his process more carefully. The surprise was that for the 3×6 case we were looking at next both of their counting procedures predicted the same number of domino tilings.
In this part of the project we tried to follow both procedures to see how they worked.
Having sorted out the counting procedure in the last video, we now looked carefully at the 2xN and 3xN tiling procedures and saw that we could compute the number of tilings for the 2×100 and 3×100 cases if we wanted to.
I’d love to come up with more counting projects for kids. These projects are accessible to young kids and I think shows of some really fun ideas from advanced math that kids probably don’t usually see in school.
Saw this neat tweet from Jim Propp yesterday:
After playing the game for a just a few minutes I knew that my kids would love it.
Here’s each of their reaction to seeing and playing the game.
My younger son first:
My older son next:
So, definitely a fun little game for kids. They need to be fairly fluent with the arrow keys on the keyboard, but that’s really all that’s required. Definitely some fun puzzles to solve!
My younger son has been learning a little bit about square roots over the last couple of weeks and I thought it would be fun to show him some proofs that the square root of 2 is irrational. Because this conversation was going to explore some ideas in math that are both important and pretty neat, I asked my older son to join it.
I wasn’t super happy with how this little project went – it felt a bit rushed while we were going through it. Hopefully a few of the ideas stuck.
We started by talking about the square root of 2 and what basic properties the boys already knew about it:
After that short introduction we moved on to the first proof that the square root of 2 is irrational – I think this is probably the most well-known proof. The proof is by contradiction and starts by assuming that = A / B where A and B are integers with no common factors.
The next proof is a geometric proof that I learned a few years ago from Alexander Bogomolny’s wonderful site Cut The Knot. It is proof 8”’ here:
Proof 8”’ that the square root of 2 is irrational on Cut the Knot’s site
If you like this proof, we have also explored some geometric infinite descent proofs in a slightly different setting previously inspired by a really neat post from Jim Propp:
An infinite descent problem with pentagons
Finally, we looked at a proof that uses continued fractions. It has been a while since I talked about continued fractions with the boys, and will probably actually revisit the topic soon. It is one of my favorite topics and always reminds me of how lucky I was to have Mr. Waterman for my math teacher in high school. He loved exploring fun and non-standard topics like continued fractions.
So, although I don’t go deeply into all of the continued fraction ideas here – hopefully there’s enough here to show you that the continued fraction for the goes on forever.
So, although this one didn’t go quite as well as I was hoping, I still loved showing the boys these ideas. We’ll explore them more deeply as we study some basic ideas in proof over the next year.