The first example we looked at were the “hexagonal” numbers. Here my son explains what those numbers are and gives a little introduction into the surprising geometric idea that helps understand these numbers:
My son had some difficulty seeing the argument from the pictures in the book, so we tried out a few examples (a few days before this project) using snap cubes. Here’s his explanation of the surprising geometry:
The next thing we looked at in the book were the “tetrahedral” numbers. The book game an amazing proof showing a formula for these tetrahedral numbers. Here he explains this clever proof:
This was a really fun project, and I’m also really happy that this book is teaching my son a bit about reading math books – sometimes even reading and understanding just a couple of pages can take time.
Today we explored continued fractions a bit more using Mathematica. I started by showing my son the relatively simple commands at taking a closer look at the continued fraction for the square root of 2:
Now we explored a few other continued fractions for other square roots and looked for a few patterns – he did notice that there always seemed to be a repeating pattern:
Next we looked at pi and found a few, fun surprises:
Finally, we looked at e. We only had about 2 min of recording time left, so this last part was, unfortunately, a little rushed:
The last few days exploring continued fractions has been really fun – hoping to do a few more projects over February break studying them.
This weekend I thought it would be fun to explore my favorite proof – the approach using continued fractions.
We’ve talked about continued fractions before, but probably not for a few years, so I started the project today by asking my son what he remembered about them:
Before moving on to the square root of 2, we talked about why rational numbers would always have finite continued fractions:
Now we calculated the continued fraction for the square root of 2 – it has a pretty fun surprise:
Finally, and this part was just for fun, I showed him the neat little mathematical trick for quickly calculating the convergents. We looked at the first few fractions that were good approximations to the square root of 2.
Since I didn’t think I did a great job communicating the main ideas in Apostol’s proof yesterday, I wanted to try again today. First we started with a review of the main ideas:
Next we tried to take a look at the proof through a slightly different lens -> folding. I learned about this idea yesterday thanks to Paul Zeitz. It takes a bit of time for my son to see the idea, but I really like how this approach helped us understand Apostol’s proof a bit better:
Finally, to really drive home the idea, I asked my son to see if he could see how to extend the proof to show that the square root of 3 is irrational. We were down to about 3 min of recording time, unfortunately, so he didn’t finish the proof here, but you can see how a kid thinks about extending the ideas in a proof here:
So, as I was downloading the first three films, my son continued to think about how to use the ideas to prove that the square root of three was irrational. And he figured it out! Here he explains the idea:
I’m definitely happy that we took an extra day to review Apostol’s proof. It feels like something that is right on the edge of my son’s math ability right now, and I think really taking the time to make sure the ideas could sink in helped him understand a new, and really neat idea in math.
I thought it would be fun to share this proof with my younger son since the geometric ideas in it are both surprising and super interesting. Unfortunately this one didn’t go nearly as well as I’d hoped. I missed a good idea that he had and got caught up in a few details that weren’t that important. Oh well, even after 10 years of doing these projects, I don’t have a good feel ahead of time for how they’ll go.
That said, here’s what we did. I started by having him walk through what is probably the most common proof that the square root of 2 is irrational:
Next we looked at Apostol’s proof and talked about some of the geometric ideas, and I just 100% missed that he was absolutely on the right track:
Now we took a look at an algebraic approach to the problem using the Pythagorean theorem. This part also didn’t go as well as I hoped and I might revisit it tomorrow just to make sure that these algebraic ideas made sense:
Finally, we came back to the geometric ideas since I realized that he was on the right track. Unfortunately I spent way too much time at the end of this part on a minor point. But hopefully the main geometric idea that we talk through in the first half of this video came through ok.
It is always disappointing when these projects don’t go quite as planned – I definitely want to push the “try again” button on this one.
Today we followed up on that project by taking a look at why the set of real numbers is larger. First, though, I wanted to tie up one loose end from yesterday. Unfortunately, though, I left things a little too open ended and we didn’t tie up that loose end in one video:
Now that we understood that we’d over counted (in some sense) yesterday, I wanted to show him why that over counting didn’t really change the proof. I also wanted to show him one real curiosity that comes up with infinite sets:
Now we moved on to talking about real numbers. I suspected that he’d seen Cantor’s diagonal argument before (and he had), so I asked him to sketch the proof. He got most of the way there:
Finally, we tied up the loose ends from his summary of Cantor’s diagonal argument and talked about one other surprise with infinite sets.
We had a great time talking about infinity this weekend. It amazes me that kids can get their heads around ideas in math that were absolutely cutting edge just over 100 years ago!
The chapter discussing various types of infinities caught his eye this week, so today we talked about infinity. The specific goal of the project (for me, anyway) was to help him understand why the size of the set of positive integers and the size of the set of positive rational numbers was the same infinity.
We started with a quick introduction about infinity and what it meant for two infinite sets to be the same size:
Next we moved on to talking about the positive rational numbers and how to make a 1 to 1 map between them and the positive integers. This is a longer than usual video, but it turned out my son had an idea that I wanted to pursue to the end.
In the last video we figured out that if we could get a map from the integers to the rational numbers between 0 and 1 we’d be done. Here my son shows his idea for how to create that map:
This project is going to be one I remember for a long time! It is really fun to see a kid get some ideas on a really challenging math problem, and then follow them all the way through to the end!
Yesterday’s discussion helped the boys understand the problem that Ellenberg is discussion in chapter 10 of his book a bit better (hopefully anyway!). Today we took a crack at replicating the calculations in the book relating to the roulette wheel example.
First we revisited the example from the book to make sure we had a good handle on the problem:
Next we talked through the details of the process that we’ll have to follow to replicate the calculations that Ellenberg does. Following the discussion here the boys did the calculations off camera:
Here we talk through the numbers that the boys found off camera – happily we agreed with the numbers in the book.
At the end of this video I introduce a slight variation on the problem – instead of getting R, R, R, R, R in a test of 5 rolls, we get an alternating sequence of R and B for 20 rolls:
Here are their answers – and a discussion of why they think the answers make sense – for the new case I introduced in part 3 of the project:
This two project combination was really fun. My younger son said that he was confused by the roulette wheel example, but I think after these two projects he understands it. I think it is a challenging example for a 9th grader to understand, but with a little discussion it is an accessible example. It certainly makes for a nice way to share some introductory ideas about Bayesian inference.
My younger son is reading Jordan Ellenberg’s How not to be Wrong and the chapter talking about Bayes’ Theorem caught his attention this week. Looking around for something related to talk about in a project, I found this interesting problem on Wikipedia:
Before talking through that problem, though, we talked about the roulette wheel example from Ellenberg’s book:
Next we began to talk through the problem from Wikipedia. This part of the project shows the initial reaction and some thoughts on the problem from the boys:
Finally, with the initial thoughts out of the way we moved on to solving the problem. My older son was seeing these ideas cold, but what was really neat to me in this part is that the ideas from Ellenberg’s book really helped my younger son see how to solve this problem:
I feel like I got a bit lucky with this project. The ideas about updating probabilities looked a bit too difficult to go through in a 15 minute project – especially since my older son was seeing them for the first time. With this introduction, though, I think we can compute / verify the updated probabilities in the roulette wheel example from Ellenberg’s book in a project tomorrow.
Today I thought it would be fun to approach the idea from the (slightly) different perspective of using fractions and binary.
We started with a review / refresher of how to write integers in binary since we haven’t talked about that in a while:
Then we talked about how you write fractions in binary including fun problem of writing 1/3 in binary:
Now I posed the question of how could we write 1 in binary – this part turned out to be the rare discussion that was as fun as I’d hoped it would be 🙂
Finally, having found an interesting way to write 1 in binary, we moved on to the question of how to write 1 in base 10:
This was a enjoyable project. The discussion of infinite series in How not to be Wrong is fascinating and accessible to a wide audience. Talking through the ideas in that chapter with my younger son has been really fun!