# Working through a neat circle / rectangle puzzle from Catriona Agg

Earlier this week I saw a really neat puzzle from Catriona Agg:

We’ve done lots of projects with Catriona’s puzzles in the past, so just search for “Catriona” and you’ll find them.

My younger son spent some time off camera solving the puzzle and then I asked him to walk through his solution. His solution gets the main idea about tangents and circles, and then computes the radius of the semicircles using the Pythagorean theorem:

Typically when we play with one of Catriona’s puzzles I have my son look through the twitter thread afterwards and find a neat solution. I took a different approach today and showed him how to use similar triangles to get to the answer with slightly less computation:

I really like Catriona’s puzzle. I also think that my son’s explanation is a great example of what kids doing math looks like.

# Showing that 1/40 is in the Cantor set is a great arithmetic exercise for kids

Yesterday we did a fun project on the Cantor set inspired by an amazing tweet from Zachary Schutzman:

That project is here:

Today we extended some of the ideas from that project by showing that the number 1/40 is in the Cantor set. Here’s how my son approached the problem – the idea he uses builds on the idea we talked about with the number 1/10 in yesterday’s project. I was happy to see that those ideas had stuck with him!

Now that we knew 1/40 was in the Cantor set, we talked about what other numbers of the same form must be in it. Although we don’t prove it (that’s what the paper in Schutzman’s tweet does), he’s now found all of the numbers with finite decimal expansions that are in the Cantor set

Finally, I wanted to go down a path relating these base 3 expansions to infinite series, but my son’s ideas took this last part in a slightly different direction. Which was fine and also fun. It really shows that kids can have fun exploring – and also have the capacity to have some great ideas about – infinite series.

These two projects have been really fun. I think the ideas about the Cantor set are great for kids to play around with!

# A fun fact about the Cantor set and a great arithmetic exercise for kids!

Yesterday I saw an amazing tweet about the Cantor set:

The amazing paper posted by Zachary Schutzman was in response to this question posed by Jordan Ellenberg:

I thought explaining some of the ideas about the Cantor set to my younger son and then having him play around with some fractions in base 3 would make a pretty fun project. So we tried it out tonight.

First we talked a bit about the Cantor set and he shared some initial thoughts:

Next I asked him to try to compute 1/4 in base 3. I always like projects like these with kids as they sneak in a little extra practice with fractions. Here’s his work:

Finally, I asked him to compute 1/10 in base 3 using an idea I mentioned at the end of the last video. After he did that, I asked him to find a few other fractions of the form k/10 that must also be in the Cantor set.

This was definitely a fun project. The math ideas here are slightly tricky, but hopefully the work here shows that the are accessible (and interesting!) to kids.

# Continuing our exploration of Collatz-like sequences

Yesterday my younger son and I did a fun project inspired by this blog post from Stephen Wolfram:

That project is here:

https://mikesmathpage.wordpress.com/2021/03/06/playing-with-the-collatz-conjecture-inspired-by-a-stephen-wolfram-blog-post/

Today we continued our exploration and started with a quick look at the 5n+1 problem we finished with yesterday:

Next we moved on to the mod 3 version that Stephen Wolfram looks at in his blog post.

Finally, we looked at what happens for the number 469 in this version of the Collatz conjecture, and talked through some ideas about how we would study if it ever repeats.

This was a fun exploration with my younger son. Wolfram’s blog post is incredible, but also accessible to kids. I love being able to explore ideas like these with kids.

# Playing with the Collatz conjecture inspired by a Stephen Wolfram blog post

I saw a really neat blog post from Stephen Wolfram last week:

The blog post has so many different ideas that you could share with kids, but I decided to spend the weekend exploring various versions of Collatz-like sequences with my son.

We started by looking at some simple code in Mathematica to generate Collatz sequences:

Next we looked at how long it took various numbers to get to 1 in the Collatz sequence and looked at a histogram of the numbers. We got a fun surprise:

We wrapped up today’s project by looking at what happens when you replace the 3x + 1 rule in the Collatz conjecture with a 5x + 1 rule. I don’t remember ever seeing this idea before and it was one that really surprised me reading through Wolfram’s blog post this morning:

Tomorrow we’ll extend today’s project by looking at the 4x + 2 and 7x + 1 version that Stephen Wolfram mentioned in his tweet. Hopefully that’ll make for a really fun project, too.

# Continuing our continued fraction exploration in Mathematica

Yesterday we played around with continued fractions and showed that the square root of 2 is irrational:

https://mikesmathpage.wordpress.com/2021/02/13/my-favorite-proof-that-the-square-root-of-2-is-irrational-continued-fractions/

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.

# My favorite proof that the square root of 2 is irrational -> continued fractions

Last week my younger son and I did two fun projects studying the proof in this tweet from Lior Patcher:

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.

# 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.

# Talking through Chapter 2 of Jordan Ellenberg’s How not to be Wrong with my younger son

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!

# Computing the area of a Koch snowflake

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.