# A 2nd try at looking at Tom Apostol’s geometric proof that the square root of 2 is irrational

Yesterday we did a project inspired by this tweet from Lior Patcher:

That project is here, but also it isn’t one of our best:

https://mikesmathpage.wordpress.com/2021/02/06/sharing-tom-apostols-irrationality-of-the-square-root-of-2-with-my-younger-son/

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.

# Part 2 of talking about infinity with my younger son

Yesterday my younger son and I talked about why the set of rational numbers has the same size as the set of positive integers. That project is here:

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!

# Talking about infinity with my younger son

My younger son is reading Bridges to Infinity right now as part of the math / reading project that we are doing in 2021.

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!

# Revisiting Larry Guth’s “no rectangles” problem – one of the best math activities for kids I’ve seen

I was looking for a relatively stress-free project to do with the boys this morning and thought it would be a good day to review one of my all time favorite math projects for kids -> Larry Guth’s “no rectangles” problem. One of my all time favorite moments doing math with kids came when I used this problem for a 2nd and 3rd grade Family Math night at my younger son’s elementary school a few years ago. The problem is accessible to kids of all ages and also of interest to research mathematicians.

We started with a quick review (and lucky clarification!) of the problem and then the boys tackled the 3×3 case:

Next we moved on to the 4×4 case. The thought process the boys went through here I think shows why this is such a great math problem for kids to talk through:

Next I had a film goof up – luckly it was just 30 seconds of introducing the 5×5 case and telling them that answer to the 5×5 case was 12 squares. Don’t know what happened to this piece of the film, but since the plan was for the boys to play with this part off camera anyway I didn’t bother trying to fix it.

In any case, here’s the 12 square covering for the 5×5 they found and then a brief discussion about the surprise that comes when you move to the 6×6 square:

Again, this is one of my favorite math projects for kids – and kids of all ages. It is a really fun problem to play around with.

# Looking at sums of integers and squares via geometry and algebra

Earlier this week I carelessly asked my younger son to read a section in a statistics book that relied on knowledge of calculus:

My son had some interesting questions about ideas in calculus, so instead of statistics we spent a few days this week talking about calculus (mainly finding the area under a curve).

For our project today I wanted to revisit one of the ideas in arithmetic that we’d relied on in the calculus discussion – sums of integers and sums of squares.

We started with sums of integers and my son gave a geometric proof of the rule for sums of consecutive integers:

Now we moved on to sums of squares – here we talked about the sum (and some of the number theory hiding in the sum formula), but didn’t yet try to prove the result:

Now we looked at a geometric way to understand the sum of squares formula:

Finally, we did a lightning fast review of mathematical induction and showed how we could proved that the sum of squares formula was true in general:

So, a fun week and a bunch of great discussions . . . even if it started with me being pretty careless!

# Sharing Dave Richeson’s proof editing exercise with my younger son

I saw a really fun tweet from Dave Richeson last week:

I thought that editing this “proof” would be a terrific exercise for my younger son. We started the project by walking through the “proof” to make sure he understood what he was supposed to do. After this short introduction he worked on the editing off camera for about 10 min:

After working through Dave’s document, here are the changes my son suggested:

I think the above video shows why this editing exercise is such a nice idea. There were a couple of points that I wanted to add to my son’s notes, so we talked through two specific ideas to end the project:

This project was really fun and definitely something I wouldn’t have ever though to try. Thanks to Dave for sharing this terrific idea!

# What a kid learning geometry can look like – studying the power of a point

I’m going to be doing a geometry review with my younger son this year. He’s studied a bit of geometry before so we’ll probably just be bouncing around with various different topics. Today I thought introducing the power of a point would be a fun way to get going with this project.

Before starting the project today we looked at the definition on Wikipedia. Then we started chatting.

The first thing I asked him to prove was that the power of a point was equal to the square of the length of the tangent drawn from the point to the circle. He did a really nice job with this proof:

Next up was a slightly more complicated formula – the power of a point is also equal to the product of two distances – the distance from the point to the closest part of the circle multiplied by the distance from the point to the point on the circle that is the farthest away. This proof gave him a lot more trouble, but I think it is really interesting to see what it looks like when a kid is struggling through a proof.

He hadn’t quite made it through the proof in the last section by 8 min so I just started a new video. He he finished up and we talk about some of the interesting ideas we’ll encounter while we study more about the power of a point:

# Sharing Andrés E. Caicedo’s amazing twitter thread on even / odd irrationality proofs with my younger son

I thought that some of the ideas would be great to share with my younger son and started by asking him if he remembered the usual proof that $\sqrt{2}$ is irrational:

Following the twitter thread, I asked him how he thought the proof that $\sqrt{5}$ is irrational would go. He gave the proof that I think most math people would give:

Next we walked through the “new to me” proof in Caicedo’s twitter thread. The ideas are definitely accessible to kids. In addition to being accessible, the ideas also provide a nice way for kids to get some algebra practice while exploring a new math idea:

Finally, we talked about the surprise that this method of proof doesn’t work for $\sqrt{17}$. My son had an interesting reaction – since this method of proof doesn’t seem to rely on the underlying number, he was surprised that it didn’t work as well as the method he’d used for $\sqrt{5}$

I really loved talking through Caicedo’s thread with my son and am really thankful that he took the time to share this fascinating bit of math on Twitter yesterday!

# Sharing John Urschel’s great video on rational and irrational numbers with my son

This morning I asked my younger son (going into 9th grade) to watch the video so we could talk about it. Here are his initial thoughts and what he thought was interesting:

Now we talked through three of the ideas he thought were interesting. The first was how to find the rational representation of a number like 0.64646464…..

Next he talked about the proof that $\sqrt{2}$ is irrational:

Finally, we talked about a really neat proof in Urschel’s video -> why log base 2 of 3 is irrational:

I love Urschel’s video and think it is an absolutely terrific one to share with kids. It is a great way for kids to see some advanced mathematical proof ideas, but also a great way to review some important ideas in math. We had a really fun morning going through it.

# Having the kids talk through a neat problem shared by Tim Gowers

Last week Tim Gowers shared a great math problem on Twitter – here’s my retweet of it (again to help avoid the temptation to get hints in the original thread:

If you’ve not seen the problem before I’d definitely suggest spending some time thinking about it – it is really a terrific problem. The videos below give the solution, so fair warnng . . .

I’d talked about it a bit with my younger son on a car ride back and for to his (outdoor) karate class earlier this week. My older son hadn’t seen the problem until this morning. A discovery that my younger son had made in the car earlier in the week helped the boys solve the problem today, but even with that prior discovery the discussion was still really great.

Here’s how I introduced the problem – you’ll see that some of the elements in the statement of the problem that are pretty standard for mathematicians are a little confusing to the boys. This introduction clears up a bit of the confusion:

With the definition of a “repetitive” number now clear, we checked if 1/7th was a “repetitive” number – the boys were pretty sure that it was, though explaining exactly why that was true in a 100% precise way was a little challenging:

Now my younger son gave his explanation for why he thought $\pi$ was repetitive:

At the end of the last video the boys were starting to think that all numbers were repetitive. In this last video they finished the solution to the problem:

I really like this problem and think it is a great way for kids to have a fun – and non-computational – mathematical exploration. As the videos show, some of the ideas can be a little difficult for kids to make precise, but I think that’s just another nice reason to explore this problem with them!