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.
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!
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.
My younger son is reading The Book of Numbers by Conway and Guy right now:
1/5 The project that I'm doing with my younger son today comes from a really happy coincidence with a few math ideas I've seen online in the last couple of weeks. He's currently reading Conway and Guy's The Book of Numbers and the binomial theorem caught his attention pic.twitter.com/dolyPeNin5
Last night he read about the Gaussian and Eisenstein Integers. Today we talked about the them (well, just Guassian integers).
Here’s the introduction to what they are in his words:
Before diving into the Gaussian integers, I asked him to give some of the ideas he thought were important in the regular integers. The talk here gave us some good things to talk about in the next video:
Now we talked about Gaussian integers – one fun thing he talked about here was that multiplication by i was the same as rotating by 90 degrees:
To wrap up today’s project we talked about why the number 5 is not a prime in the Gaussian integers. It was nice that he was able to show this!
I’m excited to introduce the Eisenstein integers tomorrow – hopefully today’s project helps prepare for the slightly more complicated math we’ll see there.
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.