## Talking about “sums of divisors” with kids and also a pi surprise

I didn’t do a very good job managing the time on this project today. The trouble is that there are lots of different directions to go with the ideas and we walked down a lot of different paths.

But – I think this is a great topic to show off the beauty of math and we end with an amazing connection between sums of divisors of integers and $\pi$.

The topic of sums of divisors of an integer came up in my younger son’s weekend enrichment math program yesterday. I thought it would make for a good topic for a project, so I gave it a go this morning.

The first part of the project was mostly about divisors and the kinds of questions that we could ask about them. A lot of the discussion here is about a question you can ask about the product of a number’s divisors:

Next we began to look at the sum of the divisors of a few different numbers. The boys noticed a few patterns – including a pattern in the powers of 2.

At the end we were looking to see if we could find patterns in the powers of 3.

It was proving to be a little difficult to find the pattern in the powers of 3, but we kept trying. After few ideas that didn’t quite help us write down the pattern, they boys had an idea that got us there.

At the end of this video I showed them that the sum of the divisors of powers of 6 was connected with the sum of the divisors of powers of 2 and powers of 3.

To wrap up I wanted to show some larger patterns in divisor sums, so we moved to Mathematica to play around a bit.

While I was doing the same playing around last night I accidentally stumbled on an amazing fact: As n gets large, the average of the sum of the divisors of the numbers from 1 to n is approximately $(\pi^2 / 12)*n$.

Number theory sure has some fun surprises 🙂

This is definitely a fun topic and also one that could be used in a variety of ways (arithmetic review, intro to number theory, computer math, . . . ). I wish that I’d presented it better. Probably it needs more than one project to really fit in all of the ideas, though.

## Working through a challenging AMC 10 problem

My son was working on a few old AMC 10 problems yesterday and problem 17 from the 2016 AMC 10a gave him some trouble:

I thought this would be a nice problem to go through with him. We started by talking through the problem to make sure that he understood it:

In the last video he had the idea to check the cases with 10 and 15 balls in the bucket, so we went through those cases:

Now we tried to figure out what was happening. He was having some difficulty seeing the pattern, so I spent this video trying to help him see the pattern. The trouble for me was that the pattern was 0, 1, 2, . . ., so it was hard to find a good hint.

Finally he worked through the algebraic expression he found in the last video:

This isn’t one of the “wow, this is a great problem” AMC problems, but I still like it. To solve it you have to bring in a few different ideas, and combining those different ideas is what seemed to give my son some trouble. Hopefully going through this problem was valuable for him.

## A strange homework problem

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.

## Talking through 3 AMC 8 problems with my son

My son was working on the 1993 AMC 8 yesterday and had trouble with a few problems. Today we sat down and talked through those problem.

The first was problem 17:

Here’s what my son had to say:

Next was problem 19:

Next was problem 24:

I like these old contest problems. They lead to really nice conversations!

## 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:

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.

## 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.

## A second project from the Wrong but Useful podcast

Yesterday afternoon I was listening to rest of the latest (as of August 31, 2017) Wrong but Useful podcast. That podcast is here:

The Wrong but Useful podcast on Itunes

A little project we did from a “fun fact” mentioned in the first part of the podcast is here:

Exploring a fun number fact I heard on Wrong but Useful

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.

## Using Gary Rubinstein’s “Russian Peasant” video with kids

Saw a neat tweet from Gary Rubinstein yesterday:

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.

## Playing with Three Sticks

I saw this tweet from Justin Aion at the end of July and immediately ordered the game:

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!