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:
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:
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.
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!
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 is irrational:
Following the twitter thread, I asked him how he thought the proof that 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 . 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
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!
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.
I thought that talking through these problems would make a nice project for the boys today, so we started in on Alex’s problem. The nice thing right from the start is that the boys had different guesses at what the expected value of one die would be when the combined roll was 8:
Now that we had a good discussion of the case where the sum was 8, we looked at a few other cases to get a sense of whether or not the intuition we developed from that discussion was correct:
Next I introduced the problem on Gil Kalai’s blog – again the boys had different guesses for the answer:
I had the boys write computer programs off screen to see if we could find the answer to the problem on Kalai’s blog via simulation. The interesting thing was that the boys approached the problem in two different ways.
First, my younger son started looking at dice roll sequences and he stopped when he found a 6 and always started over when he saw an odd number. He found the expected length of the sequence of rolls was roughly 1.5:
My older son looked at dice roll sequences and he stopped when he found a 6 but instead of starting over when he found an odd number, he just ignored the odd number. He found the expected length of the sequences looking at them this way was 3:
This turned out to be a great project. I’m glad that the boys had different ideas that we got to talk through. These conditional expectation puzzles can be tricky and subtle, but they are always fun!
The game is easy to explain. You roll a fair 6-side die N times, where N is any number you pick. You also choose an amount of money to bet – say X. If you never roll a 6 in your N rolls, you win 2^N times your money back. If any of your rolls are a 6, you get $0 back. Salmon’s questions are -> (1) How much money would you bet if you could play this game once, and (2) how many rolls would you select?
I thought this game would be fun to talk through with my older son. Here I explain the game and he talks about a few of the ideas he thinks will be important for answering Salmon’s questions. He has some interesting ideas about “high risk” and “low risk” strategies. We also talk through a few simple cases:
In the last video my son was calculating the probability of winning the game in N rolls by calculating the probability of not losing. That’s, unfortunately, a fairly complicated way to approach the problem so I wanted to talk a little more so he could see that a direct calculation of the probability of winning wasn’t actually too hard. We talked through that calculation here. We also find that if you roll 4 times you have roughly a 50/50 chance of winning the game.
Before we played the game he wanted to calculate the expected value for your winning in this game. Here we do that calculation and find the surprising answer. We then play the game. He decided to bet $100 and roll three times, and . . .
This was a fun problem to talk through with my son, and I’m excited to talk through it with my younger son tomorrow to see if he reaches a different conclusion. It had never occurred to me to talk through this or any version of the St. Petersburg Paradox with the boys before, so thanks to Felix Salmon for sharing this problem.
20 or so projects is only scratching the surface, though, since she comes out with fantastic geometry puzzles all the time! The one from yesterday is fantastic and I thought it would be great for another project for my son:
His solution to the problem was computational. He explains the main ideas here without going into all of the computational details:
In all of our projects we return to the problem’s twitter thread and my son picks out a solution that he thinks is interesting. Today he picked the solution from @lucythepoet
Consider red-yellow as a unit, we have 3 units on top and 1 unit on the bottom. Everything's similar, so scaling the length of a unit by 3 also scales the height by 3. So green:blue is a 1:3 ratio Similarly, each red-yellow block is also a 1:3 ratio
Here’s his explanation of this solution and a bit about why he liked it:
I think – and have thought for a long time! – that Catriona’s puzzles are great to use with kids. The process of attempting to solve the puzzle (sometimes getting it, sometimes not) and then going to the twitter thread to see all of the neat solutions has been a great way for my younger son to review geometry.
Yesterday we looked at a very simple model of how a virus spreads through a network – the assumption was that everyone infects everyone they are connected to. In that (obviously simplified model) the structure of the network affects the structure of the spread:
In Christopher Wolfram’s model, we use Mathematica to make a network and then study how the virus spread through the network by varying the average number of connections per day that people in the network have. The surprise (that we discuss mostly in the last video) is here the different network structures seem to behave in nearly identical ways. So the result today is very different than yesterday’s result.
I introduced today’s idea by asking the boys to think about how to build a more realistic model of how a virus spreads. The first network we looked at was a simple 2d grid:
Now we looked at a 3d grid:
Next up was a Delaunay triangulation:
Now we looked at a pure random graph network:
For the last two we looked at two graph networks that look a lot like connections in the “real world.” First up was a Watts-Strogatz graph:
Finally we looked at a Barabasi Albert Graph. This graph looks like the pure random graph we looked at, but you can see in the video that the degree distribution is really different. At the end of this video the boys talk about some of the surprises in this project and what they learned.
I think Christopher Wolfram’s program is one of the best I’ve seen for helping students understand some of the difficulties in modeling how a virus spreads. It seems like a big surprise that all of these networks seem to behave the same way, but understanding why it maybe isn’t a huge surprise helps kids see some of the key ideas in these simple models.
This week I watch an interesting live coding video from Stephen Wolfram:
Right at the beginning of this video Wolfram shows how to use some simple Mathematica commands to make a simple model of how a virus spreads through a network. I thought it would be fun to share this idea with the boys for several common networks.
I introduced the idea on a 2d grid:
Then we moved to a 3d grid:
Then we moved to a type of network called a Delaunay triangulation:
Now we moved away from these relatively simple graph networks and looked at a completely random one:
With these examples out of the way, we moved to two types of networks that more more commonly used to model a network of human interactions. The first was a Watts-Strogatz network:
Finally we looked at a Barabasi-Albert graph:
This was a really fun project and I was really excited to hear how the boys thought about the different types of networks. The math to properly describe what’s going on in these networks is over my head but I am really happy that Mathematica makes it so easy to explore.
Finally, the idea for looking at these 6 different graphs comes from Christopher Wolfram’s fantastic agent based modeling example. In that program he dives into these different networks much more deeply than we do here – this program is definitely worth checking out if you’ve not see it already: