# 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:

# Having my younger son work through Graph Theory for Kids by Joel David Hamkins

I’ve been playing around with a big of graph / network theory with my younger son the last two days. This morning I had him take a look at a graph theory project made for kids. This project is actually aimed at kids who are a little younger than my son, but I thought it would still be a good exercise for him.

You can find the pdf for the project on Joel David Hamkins’ website:

http://jdh.hamkins.org/math-for-eight-year-olds/

My son spent about 20 min working through the project and then we talked through all of the pages. Here are his ideas. If you listen to the conversation we have, you’ll see what a great little exercise Hamkins’ project is for kids:

# Having my younger son play around with different types of graphs in Mathematica

In our last project we played around with a really terrific site shared by Bill Hanage which shows how a virus can spread across a network:

That project is here:

https://mikesmathpage.wordpress.com/2020/09/10/having-my-younger-son-explore-fantastic-website-shared-by-bill-hanage-explaining-how-social-distancing-changes-network-connections/

Following that project, I thought my son would enjoy seeing different types of graphs (much smaller ones) and the different ways those graphs can be represented. I showed him some simple commands in Mathematica that would allow him to play around with these simple graphs and asked him to show 4 that he found interesting.

He started with a plain vanilla triangular graph. I was a little surprised that he wanted to start with a basic example like this one, but it ended up leading to a really nice discussion:

The next graph he thought was interesting is called the Levi Graph. I haven’t looked to see where this graph comes from, but the different ways of representing it were fascinating.

The next graph he picked is called the Gem Graph. This one is easier to understand than the Levi Graph as it has only 2 different representations. We had a good discussion about how to see that those two representations were the same:

Finally, for the last example, he chose the Icosahedral Graph. This is another graph with many different representations – some of which are really cool! It is hard to believe that all of these graphs are the same, and that fact / surprise led to a fun discussion:

Definitely a fun project. It is fun to see how kids react to seeing graphs / networks. This year I think we’ve all learned one critical application of the ideas in graph theory is understanding how a virus spreads, so I think helping kids see some ideas in graph theory is important. Playing around with different types of graphs definitely makes for a fun introduction to the subject.

# Having my younger son explore a fantastic website shared by Bill Hanage explaining how social distancing changes network connections

Yesterday I saw a tweet from Bill Hanage linking to a really interesting website:

From just a little bit of time on the site, I thought having my younger son read and play with some of the ideas would make a great project. So I asked him to spend 20 min reading and exploring, and then we talked.

Here are his initial thoughts:

One of the things he thought was interesting was the idea of 3 and 6 degrees of separation when you have a few connections and how much the network changes when you just add one connection (on average) per person:

Another thing he thought was interesting was the companion site that allowed you to modify connections in the network. Here he looked at the size of the largest group when you made the change from connections with only essential works to again adding 1 connection on average for everyone:

I really like how the ideas of network connections are explained on this site. Their work makes a fairly complex idea accessible to everyone – including kids. Thanks to Bill Hanage for sharing this site!

# 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 Grant Sanderson’s Hamming Code video with my younger son

Yesterday Grant Sanderson published a fantastic set of videos on Hamming codes. I watched the first one with my younger son last night:

Today we talked about some of the ideas in the video – starting with some of the things he thought were interesting:

Next I had him work through one of the examples in Grant’s video – I didn’t realize it was an example of an error since I just pulled it off of a screen shot, but we discovered the error talking through the example:

Finally, we went back to the same example. This might seem like a strange thing to do, but Grant’s example had an error in the parity bit and I wanted to make sure my son understood that the error correcting codes could also detect that kind of error.

I love Grant’s work – it makes for such a fun and easy way to explore ideas that kids wouldn’t normally see in their school math.