We had a little surprise this week as we saw e show up three different times. Since each kid only saw one of the ideas, I thought reviewing all three this morning would make for a great math project.
The first thing I did was introduce and review the ideas. Those ideas are:
(1) The proof that e is irrational,
(2) This idea from a Nassim Taleb Twitter thread:
(3) The idea in this twitter post from Sonia on Twitter:
Here’s the introduction to the ideas:
First we talked about the proof that e is irrational. My younger son saw this idea as an exercise in the number theory book he’s working through right now. The proof is accessible to kids, though a bit more difficult than some of the other proofs of irrationality the boys seen before:
Next we moved to the idea in Nassim Taleb’s tweet. The idea that and are so close together is a really important idea from calculus and the general idea has many important applications:
Finally, we looked at the tweet from Sonia and discussed the simplified mathematical problem in the tweet and the surprising relationship to e:
I think these three ideas are fun ones for kids to see. The proof that e is irrational is something that I’m pretty sure I didn’t see until college, but is definitely accessible to kids. The other two ideas are really important ideas from calculus and probability and definitely worth exploring many times!
I saw a really neat tweet from Joel David Hamkins this morning:
It seemed like a fun activity to try out with my younger son tonight – and like pretty much everything Hamkins shares, it was even better than I thought.
We started with the first problem and my son walked through a standard proof that the square root of 2 is irrational to make sure that he understood the problem:
Now we returned to the problem and he looked at each step carefully. He misses the error his first time through the proof, but catches it his second time through. This video is a great example of what a kid thinking about advanced ideas in math can look like:
The work on the first problem really helped him on the second problem. Here he was able to understand the false proof and find the error reasonably quickly. He also saw right from the beginning that the conclusion was correct even though the reasoning was wrong – I was happy that he noticed that feature of this problem:
This was a really fun project – it is really neat to see a kid thinking about proofs!
Yesterday Annie Perkins asked about resources to help a student understand mathematical induction:
Looking for some resources I was reminded of this incredible video that Kelsey Houston-Edwards made in 2017 – the proof by induction example starts around 5:25:
Today I thought it would be fun to have the boys go back and work through that problem. I started by introducing the problem after we’d made some L shapes out of snap cubes. The boys were able to understand the problem and solve the 1×1 and 2×2 cases:
Now we moved to the 4×4 case. My older son started by looking at different cases – which was a good way to solve the specific 4×4 case. At first it was hard to see how to generalize this approach, though my younger son had some nice ideas about how to extend it:
The next step was a deep dive into the 4×4 case and looked carefully at the exact problem we were trying to solve. Here the boys did a great job of seeing how to extend the solution of the 2×2 case to the 4×4 case.
Finally, we looked at how to extend the 4×4 case to the 8×8 case, and then how to extend to all cases of the problem. We ended the project by discussing how / why mathematical induction works on problems like this one:
I saw a really neat unsolved problem shared on twitter this morning:
I thought it would be a really fun problem to talk through with the boys – especially since kids can definitely say something about the n = 2 and n = 3 case.
Here’s the introduction to the problem:
After the introduction we talked about the n = 2 case.
Now we moved on to the n = 3 case. The boys had an interesting idea on this one that caught me a little off guard. The neat thing is that they were able to come up with a pretty good hand waving argument that in the general case the three runners would eventually form an equilateral triangle.
After that neat hand waving argument from the last video, we tried to find a more precise argument for solving the n = 3 case. Solving it in general was just a bit out of reach, but they did find an argument for why at least one runner would become lonely.
Since we ended up pretty close to the proof of the general case for n = 3, I explained the last step after we finished with the last video. I think this is a really nice problem for kids to play around with and I think that lots of young kids will find the ideas in this problem to be really fascinating.
My son is finishing up a chapter on exponentials and logs in the book he was working through this summer. The book had a big focus on e in this chapter, so I thought it would be fun to show him the proof that e is irrational.
I started by introducing the problem and then with a proof by contradiction example that he already knows -> the square root of 2 is irrational:
Now we started down the path of proving that e is irrational. We again assumed that it was rational and then looked to find a contradiction.
The general idea in the proof is to find an expression that is an integer if e is irrational, but can’t be an integer due to the definition of e.
In this part we find the expression that is forced to be an integer if e is irrational.
Now we looked at the same expression that we studied in the previous video and showed that it cannot be an integer.
I think my favorite part of this video is my son not remembering the formula for the sum of an infinite geometric series, but then saying that he thinks he can derive it.
This is a really challenging proof for a kid, I think, but I’m glad that my son was able to struggle through it. After we finished I showed him that some rational expressions approximating e did indeed satisfy the inequality that we derived in the proof.
I just got back from a wo7rk trip to Sydney and I’m going to blame jet lag for goofing up the videos. Because I forgot to zoom out after zooming in during the first video, this is really more of an audio project than a video one!
Today we returned to Martin Weissman’s An Illustrated Theory of Numbers. Flipping through the chapter on prime numbers (which is incredible!) I ran across a problem dealing with the set of numbers and thought it would be a great one to talk through with the boys.
It was really fun as you will
see hear . . .
I stared by introducing the problem and also making it impossible to see what we were doing:
Next we started playing with the first part of the problem. What we talk through here is this idea from number theory: If two numbers A and B are in our set, and A = B*C, then C is also in the set.
The boys looked at a few examples initially and noticed that lots of numbers in the set didn’t factor in the set. Then they noticed that the problem was really a problem about modular arithmetic.
The next part of the problem we played with was going through an exercise similar to the “Sieve of Eratosthenes” procedure to find the “primes” in our set:
Finally, we took at look at the part of the problem that caught my attention -> find elements of our set that factor into irreducible elements in non-unique ways.
My older son found one example -> 100 = 10*10 = 25*4.
The property of our set shows that the integers factoring into primes in a unique way is actually a pretty special property.
Sorry for the filming screw up – fortunately the visuals for this project were quite a bit less important than average. I’m excited to play around in the project chapter this week – I really love this book!
Yesterday we did a project inspired by Mathologer’s “triangle squares” video:
Here’s the project:
Using Mathologer’s triangular squares video with kids
Today we took a closer look at one of the proofs in the Mathologer’s video -> the infinite descent proof using pentagons that is irrational:
Here are some thoughts from the boys on the figure and the proof. You can see from their comments that they understand some of the ideas, but not quite all of them.
Watching Mathologer’s video, I thought that the triangle proof about the irrationality of and the proof of the irrationality of using squares were something kids could grasp, but thought that the pentagon proof presented here was a bit more subtle. We may have to explore this one more carefully over the summer.
After discussing the proof a bit, I switched to something that I hoped was easier to understand. Here we talk about the different pairs of numbers that create fractions close to .
The boys were able to explain how to manipulate the pentagon diagram to produce the fraction 38/17 from the fraction 9/4 that we started with. From there the were able to also show that 161/72 was also a good approximation to :
Next we went to the computer to explore the numbers, and also to see how the same numbers appear in the continued fraction for .
In the last video we tried to do some of the continued fraction approximations in our head, but that wasn’t such a great idea. Here we finished the project by computing some of the fractions we found in the last video by hand.
I love Mathologer’s videos. It is amazing how many ways there are to use his videos with kids. Can’t wait to explore these “triangular squares” a bit more!
Last month Mathologer published an incredible video on what he calls “Triangular Squares”:
I’ve been meaning to use this video for a project for the boys ever since I saw it. Today I finally got around to watching it with the boys.
Here are their initial thoughts after watching the video:
Now we went through some of the ideas. First I asked the boys to try to sketch Mathologer’s argument that is irrational. Then I asked what proof they would have given for that fact without seeing the video:
Next we explored the irrationality proof for :
Finally, we did a bit of exploration of the seeming paradox mentioned at the end of the video. That paradox is essentially -> the argument used to show that is irrational seems to also show that 3 times a triangular number can never be a triangular number. BUT, there are lots of examples showing that 3x a triangular number is a triangular number. What’s going on?
So, another terrific video from Mathologer. His ability to shed light on advanced math topics for the general public is incredible. I love using his videos to help my kids see amazing math ides from new and beautiful angles!
Saw a neat tweet from Joel David Hamkins at the end of last week:
I thought it would be fun to talk through the knight’s tour problems with the boys today and end by showing them the infinite problem. I ran into trouble almost immediately when we began to talk about the tours on the 3×3 and 4×4 boards. The difficulty they had explaining was a big surprise to me. We ended up talking about the 4×4 problem for almost 30 min.
Tonight I sat down with each of them and asked them to talk me through the problem and explain why the knight’s tour on the 4×4 board was impossible. You can see that my older son (in 8th grade) was able to explain the problem pretty well, but my younger son (in 6th grade) still really struggled.
Here’s what my older son had to say:
Here’s what my younger son had to say:
Definitely a much harder problem for kids than I thought. Hopefully will have some time during the week to explore this and maybe a few other tour problems with them.
Earlier in the week my younger son was struggling with a problem about the geometric mean of two numbers. I thought looking at the difference between how my older son saw the problem and how my younger son saw the problem would be interesting.
I asked each kid to go through the problem on his own. My older son had not seen the problem ahead of the project and my younger son had gone through the problem with me two days ago.
Here’s what my older son had to say:
Here’s the first part of what my younger son had to say. You’ll see that the he’s still a little unsure about the problem even though we had talked through it previously:
Finally, I wrapped up with my younger son by talking through the way he was intending to solve the problem originally. This approach is really nice, too – it uses the Pythagorean theorem and a little algebra:
This was a nice reminder to me of what a kid struggling with a math idea can look like.
Sorry for the brief post – needed to get this one out the door before running out for work.