For Pi day today we explored the amazing near integer

I started by showing the boys the numbers as well as just how close it was to being an integer. I measured the closeness both in terms of the decimal expansion and in terms of the continued fraction expansion of the number:

Next I asked the boys to each take a turn finding another number relating to that was either nearly an integer or nearly a rational number. It turned out – especially with my younger son – to be a really nice way to discuss properties of powers of numbers.

The number my younger son found was

The number my older son found was

So – obviously just for fun – but still a neat way to talk about numbers and continued fractions. And a pretty fun number at the start, too ðŸ™‚

We had a snow day today and I finally got around to sharing a neat problem from the 2017 Putnam Exam with the boys.

When I first saw the problem I thought it would be absolutely terrific to share with kids:

Problem A1 from the 2017 Putnam is a great problem to walk through with kids. There's a bit of number theory and a bit of neat mathematical reasoning. Would be fun to talk through in a group. pic.twitter.com/YsIc01STLI

I started off the project today by having them read the problem and spending a little bit of time playing around:

After the initial conversation the boys, I triehd to start getting a bit more precise. The first sequence of numbers they knew was in the set was 2, 7, 12, 17, . . . .

They were not sure if 4 was in the set or not. My first challenge problem to them was to show that if 4 was in the set, then 3 would be in the set.

My next challenge question was whether or not 1 would be in the set.

Now we moved on to one of the number theory aspects of the problem – is 5 in the set?

During this conversation my younger son noticed that we had found a number that was 1 mod 5.

Finally, we talked through how you could find 6 from the number my son noticed in the last video.

I’m really happy with how this project went. This problem is not one (obviously) that I would expect the kids to be able to solve on their own, but most of the steps necessary to solve the problem are accessible to kids. It was really neat to hear their ideas along the way.

As I said in that blog post, I’d seen a few teachers discussing the idea, but I don’t remember who originally shared the project. So, to be clear again, the idea for this line of study isn’t mine, but I’m happy to have some fun with it.

Instead of revisiting the prior project today, I tried something slightly differnt -> comparing double stuffed oreos with thin ones. The prep work for this project proved to be a little harder than I was expecting because the double stuffed oreo shells were really fragile. So, if you want to repeat this project, be prepared for lots of broken oreo shells!

To start I introduced my son to the problem we were going to try to solve today and asked for his thoughts. The problem was to try to find the ratio of the volume of stuffing in the double stuffed oreos to the volume of stuffing in the thin ones.

Our original intention was to weigh 10 of the crackers from each of the 2 types of cookies. We were able to get only 8, though. The 8 thin crackers weighted 22 grams and the 8 double stuffed crackers weighed 33 grams.

Sorry the writing was off screen.

Next we moved to weighing the full cookies. I didn’t communicate really well at the start of this video, and confused my son a bit. Eventually we got back on the same page weighed 4 cookies of each type.

The 4 double stuffed cookies together weighed 60 grams. That led us to conclude that the stuffing weight was approximately 27 grams for 4 cookies.

Finally, we repeated the process in step 3 with 4 thin oreos. We found that the 4 cookies together weighed roughly 30 grams, meaning the total weight of the filling was 8 grams.

So, my son’s guess of 4 to 1 for the ratio of the filling weight was pretty close. Turned out be 27 grams to 8 grams for 4 cookies, or about 3.5 to 1.

Definitely a fun project. I haven’t done much in the way of introductory statistics for kids – this project definitely gets kids engaged!

I asked the boys what they waned to talk about for a project today and got a bit of a surprise when my 6th grader suggested polynomials. It seems that the topic has just come up in his math class at school and he’s interested in learning a bit more.

To start the project I asked him what he knew about polynomials:

Next I asked my younger son to explain adding and subtracting polynomials, and then to try to see how to multiply them since he said that he didn’t know how to multiply in the last video:

Now I wanted to show an unusual property of polynomials that was relatively easy to understand. My hope was to show my older son something that he’d not seen before but also something that was still accessible to my younger son. I chose to show them a short exploration of a difference table for a quadratic

Finally, I showed how you could use the difference table to reconstruct a quadratic function if you knew the values of the function at three consecutive integers:

So, despite the surprise topic request, this was a fun little project. It was nice to be able to find a topic that you could explore if polynomials were “new to you” and still get something out of if polynomials were a familiar topic.

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 .

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 .

Number theory sure has some fun surprises ðŸ™‚

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.

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.

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.