One of my older son’s homework problems asked him to find 3 digit multiples of 7 whose digit sums were also multiples of 7. I was puzzled by this problem had it on my mind most of the day today.

I hoped that talking through it would help me understand what the math idea was behind the problem. Sadly no, but we still had a good talk.

Here’s the problem and the work my son did:

So – still quite puzzled about the problem – I decided to see if there was anything quirky that came up looking at a divisibility rule for 7 with 3 digit numbers. This gave us a nice opportunity to talk about modular arithmetic:

Finally, since I wasn’t making any progress seeing the point of the original problem, I had him talk about other divisibility rules that he knew:

So, a nice conversation, but I’m actually baffled. I’ll have to ask the author of the problem what he was trying to get at – I feel like I’m missing the point.

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.

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.

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 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 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 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 where and 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.

The second half of the podcast was a really interesting discussion of math education. One thing that caught my attention was comparing math education to music education and the idea of having students do “math recitals.”

Another part that caught my attention was a problem used mainly to see the work of the students rather than the specific answers. That problem is roughly as follows:

Find two numbers that multiply to be 1,000,000 but have the property that neither is a multiple of 10.

Here’s how my younger son approached the problem – it was absolutely fascinating to me to see how he thought about it.

Here’s what my older son did. Much more in line with what I was expecting.

Fun little project – definitely check out the Wrong but Useful podcast if you like hearing about math and math education.

This morning I thought it would be fun to look at the “Russian Peasant” multiplication video with the boys. Here’s Rubenstein’s video:

I had the boys watch the video twice and then we talked through an example. My older son went first. He had a fun description of the process: “It is like multiplying, but you aren’t actually multiplying the numbers.”

Next my older son worked through a problem. This problem was the same as the first one but the numbers were reversed. It isn’t at all obvious that the “Russian Peasant” process is commutative when you see it for the first time, so I thought it would be nice to check one example:

Next we moved to discussing why the process produces the correct answer. My older son had a nice idea -> let’s see what happens with powers of 2.

The last video looking at multiplication with a power of 2 gave the kids a glimpse of why the algorithm worked. In this video they looked at an example not involving powers of 2 (24 x 9) and figured out the main idea of the “Russian Peasant” multiplication process:

This was a really great project with the boys. It’ll be fun to work through Rubinstein’s videos over the next few months. I’m grateful that he’s shared the entire collection of ideas.

When I returned from a trip to Scotland with some college friends the game was on the dining room table – yes!! Today we played.

In this blog post I’ll show how the game ships and two rounds of play (and we might not be playing exactly right) to show how fun and accessible this game is for kids.

First, the unboxing. The game comes out of the box nearly ready to play.

Here’s our first round of game play. I think we misunderstood one of the rules here, but you’ll still see that the game is pretty easy to play:

Here’s the 2nd round of play. I think we understood the rules better this time, which is good. You’ll also see how this game gets kids talking about both numbers and geometry:

Finally, here’s what the boys thought about the game:

I’m really happy that I saw Justin Aion’s tweet and now have this game in our collection. It is a great game for kids!

Last week I saw this problem on the IMO and thought that the solution was accessible to kids:

The problem is problem #1 from the 2017 IMO, just to be clear.

My kids were away at camp during the week, but today we had a chance to talk through the problem. We started by reading it and thinking about some simple ideas for approaching it:

The boys thought we should begin by looking at what happens when you start with 2. Turns out to be a good way to get going – here’s what we found:

In the last video we landed on the idea that looking at the starting integer in mod 3 was a good idea. The case we happened to be looking at was the 2 mod 3 case and we found that there would never be any repetition in this case. Now we moved on to the 0 mod 3 case. One neat thing about this problem is that kids can see what is going on in this case even though the precise formulation of the idea is probably just out of reach:

Finally, we looked at the 1 mod 3 case. Unfortunately I got a little careless at the end and my attempt to simply the solution for kids got a little to simple. I corrected the error when I noticed the mistake while writing up the video.

The error was not being clear that when you have a perfect square that is congruent to 1 mod 3, the square root can be either 1 or 2 mod 3. The argument we go through in the video is essentially the correct argument with this clarification.

It is pretty unusual for an IMO problem to be accessible to kids. It was fun to show them that this problem that looks very complicated (and was designed to challenge some of the top math students in the world!) is actually a problem they can understand.

Today I wanted to boys to explore a bit more. The plan was to explore one basic property together and then for them to play a bit on the computer individually.

Here’s the first part -> Looking at what happens when you compute the continued fraction for a rational number:

Next I had the boys go the computer and just play around.

Here’s what my younger son found. One thing that made me very happy was that he stumbled on to the Fibonacci numbers!

Here’s what my older son found. The neat thing for me was that he decided to explore what continued fractions looked like when you looked at multiples of a specific number.

So, a fun project overall. Continued fractions, I think, are a terrific advanced math topic to share with kids.