After reading the post I was super excited to go through it with the boys when they got home from school.

So, we read the post after dinner and then made a code out of snap cubes. Here’s what the boys thought of the post:

and here’s our secret message!

We had a lot of fun with this project. It looks like something that could be pretty fun with a group, too, so I’m thinking about using it for 4th and 5th grade Family Math night at my younger son’s school next month.

This Scrooge feels the need to point out that this only proves that the sum of the 3rd & 4th triangular numbers is a square #. #bahnumbughttps://t.co/KR4dALQIaL

Shortly after seeing the tweet my younger son and I were playing Othello. The combination gave me the idea for today’s project.

We started by talking about the triangular numbers and why consecutive triangular numbers might sum up to be a perfect square. My older son’s idea of how to think about triangular numbers was computational rather than geometric.

Now we moved to the Othello board and looked at the geometry. My younger son found two different geometric ideas which was fun.

Finally, I gave the kids a challenge to try to find another geometric version of the identity. This question was a bit more challenging that I intended it to be, but we eventually got there and even saw how our new picture related to the sum formula that my older son used in the first video:

I’ve spent the last couple of days talking about binary with my younger son. We were inspired a bit by Kelsey Houston-Edwards’s latest PBS Infinite Series video on binary. It has been a fun little review.

Tonight we talked about how to write 1/5 in binary. I didn’t really know how the conversation would go, but it ended up being a nice little arithmetic review.

We started talking about the problem and he settled on the idea that we needed to find a number that would equal to 1 when we multiplied by 5. That got us going on the arithmetic review since that idea works in any base.

Now we had to figure out now to divide 1.000000000…. by 101 in binary. This long division problem gave us an opportunity to talk about subtraction (and borrowing) in binary:

The last step was multiplying the number we thought was 1/5 by 101. Once again this was a great opportunity to review some basic ideas about arithmetic and multiplication.

So, an unexpectedly fun project! We learned what 1/5 was in binary and had a nice review of subtraction, division, and multiplication along the way š

Yesterday we did a really fun project inspired by a tweet from Steven Strogatz:

Here’s tweet:

Unusual intuitive argument for why A= pi r^2 for a circle, found by one of the tables in our #math exploration class. I love these surprises pic.twitter.com/dch9PfmynZ

During the 3rd part of our project yesterday the boys wondered how the triangle from Strogatz’s tweet would change if you had more pieces. They had a few ideas, but couldn’t really land on a final answer.

While we punted on the question yesterday, as I sort of daydreamed about it today I realized that it made a great project all by itself. Unlike the case of the pieces converging to the same rectangle, the triangle shape appears to converge to a “line” with an area of , and a lot of the math that describes what’s going on is really neat. Also, since my kids always want to make Fawn Nguyen happy – some visual patterns make a surprise appearance š

So, we started with a quick review of yesterday’s project:

The first thing we did was explore how we could arrange the pieces if we cut the circle into 4 pieces.

After that we looked for patterns. We found a few and my younger son found one (around 4:09) that I totally was not expecting – his pattern completely changed the direction of today’s project:

In this section of the project we explored the pattern that my son found as we move from step to step in our triangles. After understanding that pattern a bit more we found an answer to the question from yesterday about how the shape of the triangle changes as we add more pieces.

Both kids thought it was strange that the shape became very much like a line with a finite area.

The last thing that we did was investigate why the odd integers from 1 to N add up to be $late N^2$. My older son found an algebraic solution (which, just for time purposes I worked through for him) and then we talked about the usual geometric interpretation.

So, a great two day project with lots of fun twists and turns. So glad I saw Strogatz’s tweet on Friday!

It is a pretty neat problem and I thought it would make a fun project for the boys today. I didn’t show them the tweet, though, because I wanted to start by exploring the numbers with increasing digits:

Next we tried to figure out what was going on. My older son wanted to try to study the problem in general, but then my younger son noticed a few things that at least helped us understand why the sum should be divisible by 9.

For the third video we started looking at the problem in general. The computations here tripped up the boys a bit at first, but these computations are really important not just for this problem but for getting a full understanding of arithmetic in general.

For the last part of the project we looked at two things. First was returning to a specific example to make sure that we understood how borrowing and carrying worked. Next we applied what we learned to the slightly different way of multiplying by 9 -> multiplying by 10 first and then subtracting the number.

After the project I quickly explored Dave Radcliffe’s response to MoMath’s tweet:

I’d played around with it a bit on Mathematica and the code was still up on my computer screen when we were playing with base 3/2 yesterday, so the kids asked about it.

Radcliffe’s proof is a bit too difficult for kids, I think, but the general idea is still fun to explore. I stumbled through a few explanations throughout this project (forgetting to say the series should be finite, and saying “denominator” rather than “numerator” at one point), but hopefully the videos are still clear.

I started by explaining the problem and looking at a few simple examples:

Next we looked at how it could be possible for a finite sum of distinct numbers of the form 1 / (an integer) could add up to 100, or 1000, or some huge number:

Now that we understood a bit about the Harmonic series, we jumped to Mathematica. I sort of half explained / half skipped over the “greedy algorithm” procedure that Radcliffe uses in his paper. I thought seeing the results would explain the procedure a bit better.

We played around with adding up to 3 and then a couple of numbers that the boys picked.

After playing around with a sum adding up to 3, we tried 4 and the boys got a big surprise. We then tried 5 and couldn’t get to then end!

After we turned off the camera we played around with the sum going up to 5 a bit more sensibly and found that there are (from memory) 102 terms and “n” in the last 1/n term has 142,548 digits!

So, a little on the complicated side, but still a fun math fact (and computer project!) for kids to explore.

When I asked them about their favorite unsolved problem, they both mentioned the Collatz conjecture. Unfortunately they couldn’t remember the details, but that made the choice of topic for today’s Family Math project easy!

I decided to approach the problem using sound much like we did when we looked at John Conway’s version of the Collatz conjecture:

Before diving in to the sound, though, we reviewed the details of the Collatz conjecture:

Next we moved to Mathematica to listen to (a version of) the sound of the Collatz Conjecture. Sadly the camera was way out of focus here. I didn’t notice until the movies were published. So, sorry about that, but at least the sound comes through ok.

Next I asked the boys to change the procedure a little. My older son’s suggestion was to change the procedure from “multiply by 3 and add 1” to “multiply by 3 and add 3.”

My younger son noticed from the sound that the loop didn’t start with 1 – that was really fun to hear! Maybe a good thing, too, since the video is so out of focus š

Finally, we made one more change to the procedure – this time “divide by 2” was replaced by “divide by 2 and then add 4.” We saw some new patterns again.

So, I love playing around with the Collatz conjecture with kids. First, it is always really fun to be able to show kids unsolved math problems. Lior Patchter has an incredible blog post about various different unsolved problems to share with kids at each grade level if you want more than just the Collatz conjecture:

The question is a really deep and really challenging one for kids. Truthfully it is probably a little over the head of my kids, but I thought I’d give it a try anyway. I’ll revisit this one (hopefully!) several times over the course of this school year – although the question confused my kids a little bit, I really like it.

Here’s my older son’s (started 7th grade today!) thoughts on Dave’s question:

Here’s my younger son’s thoughts – he’s in 5th grade. I took a little extra time at the beginning with him to work through some examples with numbers so that the abstract symbols wouldn’t be so confusing:

It is fun to hear the boys struggle to try to explain / reconcile the strange ideas in Grandi’s series. I’m also glad that they are learning to think through what’s going on rather than just believing the algebra.

My younger son has been working through Art of Problem Solving’s Introduction to Number Theory book this summer. The topic for the last few weeks has been arithmetic in different bases. Today he can across a problem that gave him a lot of trouble. He worked on it alone for about 15 minutes and then we talked about it.

Here’s the problem:

In a certain base (12)*(15)*(16) = 3146. What is that base?

Here’s the first part which summarizes his initial thoughts on the problem. The work he does here shows that the base used in this problem must be lower than 10. Once he discovers that fact we talked about a few other ways that we could have seen that the base wasn’t 10.

Next we tried to see how we could identify the base we were looking for using some of the ideas from the last video. We used the last digit idea to eliminate 7 and 8, but the last digit idea told us that base 9 might actually work.

Then we did the arithmetic to show that we were indeed looking for base 9.

So, a really challenging problem, but a fun talk for sure. Working through problems like this one are a great way to review arithmetic and a neat way for kids to learn some basic ideas in number theory.