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!
I had the boys work on the problem on their own and then talk through their progress.
My older son went first – his solution is along the same lines as most of the solutions in Catriona’s twitter thread, though is reasoning is pretty interesting to hear:
My younger son went next. He wasn’t able to find the solution on his own, but was able to get there while we talked about his work. I’m sorry that I forgot the camera was zoomed in on the paper here. I do zoom out a little over half way through. Hopefully the words are clear even if some of the work is off screen:
At the end of the last video my son had worked through the main idea of the problem. Here he finishes the solution and talks about what he liked about the problem:
As usual, having the boys work through one of Catriona’s puzzles made for a great project. I really liked the algebra / geometry combo that this problem had as I think that was great practice for my younger son. I also think the more intuitive solution my older son had shows how mathematical intuition develops as kids get older.
For a math project this month I’m having my younger son (in 9th grade) read a chapter of Jordan Ellenberg’s How not to be Wrong each day. The book is terrific if you’ve not read it.
Yesterday my son read chapter 2 and today I asked him to pick three things that he thought were interesting or just caught his eye.
The first was the way the Greeks thought about pi:
The next thing he found interesting was Zeno’s paradox:
Finally, we talked about some of the really neat ideas about infinite series in the chapter:
I doubt that we’ll do a project on every chapter, but there are so many neat ideas in the book so I bet we’ll get to have at least 5 fun discussions. The book isn’t really aimed directly at kids, but I think most of the ideas are accessible. It’ll be fun to see what he thinks the interesting ideas are!
Yesterday my younger son and I explored the perimeter of the Koch snowflake. He was surprised that the perimeter was infinite, but he also thought that the area was finite. Today we calculated the area.
I started off by asking my son for a plan for how we should approach the problem:
In the last video we calculated the additional area that was added in each step of the process of creating the Koch snowflake. Here we look carefully at those amounts and take a guess at the pattern.
Now that we had the pattern, we added up all of the areas. I really enjoyed seeing my son’s approach to adding up the infinite series. I think this calculation is a great one to show to kids to sneak in a little arithmetic practice as well as introducing some simple ideas about sums of infinite series.
This morning I asked my son to flip through Martin Gardner’s The Colossal Book of Mathematics and pick out a chapter he thought would be interesting to talk through. One of the chapters had a discussion of the Koch snowflake that caught his eye.
Here’s that chapter and why he thought it was interesting:
For today we decided to explore the perimeter to see if we could figure out why it was infinite. For starters we tried to calculate the perimeter of the first four iterations. He had a little trouble with the 4th, but I think that trouble shows why this is such a great arithmetic exercise for kids:
Next we went back to look more carefully at the 4th step to make sure that we had the right number. With this review we found the correct perimeter.
For the last step we found the pattern for how the perimeter changed at each step. This was, unfortunately, slightly rushed as we were about to run out of memory in the camera. But still, I thought my son gave a nice explanation of why the perimeter eventually went to infinity.
This week I was doing a fun overview of higher dimensions with my younger son and we finished by playing around with the famous “central sphere” problem. The earliest reference I know to the problem is in Chapter 11 (“Spheres and Hyerspheres”) of Martin Gardner’s Collossal Book of Mathematics. There the problem is attributed to Leo Moser. I learned about the problem from hearing Bjorn Poonen discuss it with a student.
There are some nice youtube videos about the problem – this one by Kelsey Houston-Edwards is the one I watched with my son this week (the whole video is terrific, but I’m starting it around 6:15 when the discussion of the central sphere problem begins):
Grant Sanderson also has a video about the problem that I’m going to have my son watch later today:
I started the project last night by introducing the problem in 2 dimensions – just like both videos do. My younger son is in 9th grade and calculating the radius of the inner circle is a problem he was able to solve:
Next we moved to 3 dimensions and again my son was able to find the radius of the central sphere and also guess at the formula for the radius of the central sphere in any dimension:
Now we tackled two well-known questions:
(i) Is there a dimension where the radius of the central sphere is larger than the radius of the spheres in the smaller boxes?
(ii) Is there a dimension where the diameter of the central sphere is larger than the side length of the large box?
Finally, off camera we investigated the amazing question that as far as I know is due to Bjorn Poonen:
Is there a dimension where the volume of the central sphere is larger than the volume of the box?
We had to work in Mathematica and work with logarithms because some of the numbers in the calculations got so large. This video is a summary of what we found (and also several interruptions from our cat):
Definitely a fun problem – higher dimensions sure are strange 🙂