The Washington Post did a nice article last week on measuring the number of deaths related to the corona virus in the US. I learned about it from this tweet from Keith Devlin:

Today I had the boys read the article and we talked through several of the ideas they thought were interesting. Here are their initial thoughts and also their thoughts about how you would count the excess deaths from the graph shown in the cover pic from the article:

My younger son mentioned two ideas that caught his eye in the article – the difference between Republican / Democrat states and the difference in outcomes with large and medium lockdowns. We talked about those ideas here:

My older son had two things that he thought were interesting – the reporting delays and how the article counted the excess deaths vs the corona virus deaths:

Following those discussions we downloaded some data from the CDC’s website to see if we could match the Washington Post’s numbers. We could for Massachusetts, but were off by a bit for Indiana. Not sure why – the trouble of filming this stuff live – but the main ideas was just to show the boys how to check the numbers in articles like these (and why checking is important):

This was a fun project – I think the analysis of excess deaths is a helpful way to understand how bad the pandemic is. I’m glad the Washington Post published this article.

I don’t know why, but this zometool shape we made a few years ago based on Bathsheba Grossman’s Hypercube B migrated back down to the living room this week:

Seeing how that zome creation seems to change as you walk by it once again this week made me want to do a project revisiting 4d shapes with the boys.

We started by looking at a few shapes that we’ve played with before:

Next we looked carefully at Hypercube B by Bathsheba Grossman:

Now I had the boys watch the video about the Zometool version of Hypercube B and react to it:

Next we went to the Wikipedia page for the hypercube and look at some of the 2d representations. The boys reacted to some of the pictures and I asked them to pick one and draw it.

Here are their drawings and explanations. One fun surprise is that after they finished their drawings they noticed that they chose the same shape!

This was a fun project and not meant to dive into great detail. I’m happy that the boys are getting comfortable thinking about higher dimensions – it has been really fun to explore ideas from higher dimensional geometry with them.

The boys got the hang of a few relatively simple examples but also noticed that going to numbers with 4 prime factors would get pretty hard to draw.

After we finished the project I saw a post on twitter about a 5d cube and was reminded that we had a 2d projection of a 5d cube hanging on our living room wall:

We have the projection of the 5d cube you can make with zometool hanging on our living room wall đź™‚ pic.twitter.com/yTVJoJFhjc

I’ve just started the book An Illustrated Theory of Numbers by Martin Weissman with my son:

I've been debating a couple of different math paths to go down with my younger son. I offered up a few ideas to him tonight and he said that he was interested in studying Martin Weissman's An Illustrated Theory of Numbers. Excited to see how it goes! https://t.co/e8lcuXgSzLpic.twitter.com/52xhjneZyw

We are going slowly and are just a few pages in, but I wanted to so a project with Hasse diagrams today because he told me last week that seeing those diagrams in the beginning of the book is what made him want to study the book a bit more.

We started today by looking at the book and exploring a bit about factoring integers:

After that introduction I had the boys read the section on the book on Hasse diagrams (roughly 1 page long) to be sure they understood how they worked. Here’s what they had to say and then a bit of practice:

It turned out that the final exercise in the last video – writing the Hasse diagram for 36 – proved to be a little tricky for my younger son. Because the last video was running long we broke things into two parts. Here we finish the diagram for 36:

We finished up by looking at one of the Hasse diagram exercises in the book. Here the boys wrote the diagrams for 7, 15, 18, and 105.

This project was a nice light touch one. It gave the boys an opportunity to review a bit of arithmetic and introductory number theory. It was also fun to explore this interesting connection between number theory and geometry.

One of the most interesting ideas I’ve seen about the spread of the corona virus this week is discussion about the role that superspreaders play. I thought the topic could be made accessible to kids following a plan similar to what we did last week with Christopher Wolfram’s virus spread model:

I also want to be clear that the code we are playing around with here is from Wolfrman’s project which you can find in the link below. Other than really minor modifications for this project, none of the code is mine and this project wouldn’t have been possible without Wolfram’s work:

Today’s project didn’t go nearly as well as I hoped, though, But even with things no going so well I wanted to share the project.

My idea was to show the boys the distributions of outcomes when a virus spreads through a network. So, unlike last week when we just looked at one simulation for each network, today we looked at 1,000 simulations per network. Then, as a really simplified way to look at the idea of superspreaders, we’d look at how the infection spread through the network when the starting point had different numbers of initial infections.

So, we started by looking at one of the networks from last week and talking about the ideas we learned from that project:

Since simulating 1,000 different runs through a network takes a long time I prepared several graphs ahead of times so that we could just talk about the results. Fortunately I prepared two different visuals for each simulation because the first graph I made ended up being extremely confusing for the boys:

We spent a lot of time in the last video making sure that we understood the visualization of the simulations I was running. It turned out that the histogram was the easiest one for the boys to understand.

With the boys hopefully understanding what the histograms meant now, we looked at how the spread of a virus through a network changes as the interaction between nodes of the network changes. What we looked at specifically was how the spread changes from almost nothing to spreading through the entire network quite suddenly.

Having looked at the change in spread based on the average number of interactions in the last two videos, here we changed to looking at how the spread changes based on the number of initial infections. By changing the number of initial infections from 5 to 10 to 15 to 25 to 100 (out of 1,000 nodes) we saw very different spreading patterns in the network.

This way of looking at spread through a network was my guess for an easy way for kids to see / understand the role of superspreaders.

Definitely not my best executed idea ever, but still hopefully something that helped the boys get a bit more understanding of some of the important ideas in virus models.

Yesterday I was trying to understand why the corona virus hit Massachusets so differently than it hit Georgia and Diego Zviovich shared a really nice bit of Mathematica code with me:

In case the graphs don’t so up show well from Twitter, here are the graphs of new positive cases in Massachusetts and Georgia since March (per 100,000 population)

Zviovich’s code was so easy to use that I made a gif of the charts for all states and territories. It wasn’t working well with WordPress, but you can see it on twitter here:

I used some Mathematica code that @dzviovich shared yesterday to make a state by state (and territory) graph of corona virus cases by day (per 100,000 population). It is wild how different the outbreak has been in different states in the US. pic.twitter.com/eEF3Nut21Z

Tonight I asked me kids to look at the graphs from the different states and territories and pick out 4 that caught their eye.

My older son picked out Washington D.C., Louisiana, Nebraska, and South Dakota

My younger son picked out Kansas, Nebraska, New Jersey, and South Dakota

I thought this was a nice exercise for kids. Both to see how you can use computer programs to sift through lots of data, and also to see how to read and interpret graphs.

My younger son had thought it would be neat to see how the map looked scaled by population, so I spent a little bit of time trying to figure out how to do that. I’m a total novice when it comes to using Mathematica, but I was able to figure out how to make the new presentation for my son last night.

We started our project today by looking at the original map that Bahrami made:

Next we took a look at Bahrami’s code. The goal here wasn’t to understand the details, but rather to see that making a visualization like this in Mathematica is actually not nearly as hard as it seems . . . if you know what you are doing!

Finally, we took a look at Bahrami’s map scaled by county population. I didn’t do as good a job with the colors as I should have – the darkest colors are 3 cases per 1,000 people in the county. Still, it was interesting to hear what the boys thought of this map vs the original one.

Even though fully understanding the underlying code is a but much to ask for kids in a 30 min project, I think Mads Bahrami’s project is a great one for kids to see. It give kids a chance to see how data visualizations are done, and also gives them an opportunity to understand and talk about the data. I really like sharing this project with my kids.

I’d played around with it a bit over the last two days and decided to share some of the ideas with my kids this morning.

We started with the basic idea of networks and graphs:

Now we stepped away from Wolfram’s mode for a second to look at several of the different kinds of graph structures he was studying. The boys had some pretty interesting things to say about the different types of graphs:

Next we looked at one of the results in Wolfram’s project that I thought was particularly fascinating – how a seemingly small change in assumptions can cause a virus to change from hardly spreading at all to spreading across the entire network:

Finally, my older son (in 10th grade) had looked through Wolfram’s presentation yesterday and I asked him to show some of the ideas that had caught his eye:

I love Wolfram’s post – both for showing how mathematical modeling can showing you interesting ideas about the spread of a virus and for showing the power of Mathematica to make these models accessible to everyone. This was a really fun project to share with the boys.

My plan is to look at one more of Catriona’s puzzles tomorrow and then return to his Precalculus studies next week. I’ve really enjoyed this project and think that the combination of trying to find a solution on his own and then explaining one of the solutions from the original twitter thread has been a great way for my son to study geometry.

In the video he’s reference some of his earlier work that was really eye-opening to me. Specifically the discussion around 8:50 in this video:

Tonight I had the boys watch the new video and then we discussed the property of heavy tail distributions that Nassim talked about -> especially that the sample mean for heavy tail distributions is likely going to be below the true mean.

We started with some basic ideas from Nassim’s video and then looked at the alpha = 2 case:

Next we looked at the alpha = 1.2 case. Here we began to see clearly how the sample mean underestimates the true mean of the distribution:

Finally, we looked at the alpha = 1.03 case that Nassim mentions in his video and found that almost always the sample mean underestimated the true mean. We also saw that even with 100,000 samples, our sample mean was not even close to the true mean of the distribution.

This was a fun project. A little on the advanced side for kids, but my main hope is that they start to appreciate that an important part of any statistical analysis is understanding the kinds of distributions that you are likely to be dealing with.