Last week Grant Sanderson published a fantastic video showing some simple models of how a virus can spread through a population.

All of the common pandemic models are pretty complex and have tremendous uncertainty in their parameters, but Grant’s video does an incredible job of showing their strengths and weaknesses.

Today I watched the video again, but this time with my kids. I asked them to take some notes and then we talked about what they thought was interesting. It is always fascinating to hear what kids take away from math / science content.

Here’s what my younger son (in 8th grade) had to say:

Here’s what my older son (in 10th grade) had to say:

With so much terrible news about the corona virus lately, I thought it would be good to talk through some of the numbers and models with them. One thing I thought would be particularly interesting for them to see is why the virus didn’t show up on some of the flu tracking maps, yet.

We started by looking at some of the flu maps from the CDC so they could see how those maps work. Those maps are here:

Next we moved to this interesting flu tracking map which uses internet connected thermometers. The interesting thing about this map is that it indicates that the flu-like systems are declining rapidly right now:

Here’s what they boys had to say looking at Miami, New York, and Boston on this map:

The next chart we looked at tracked movement in the US using cell phone data. This map allows us to see how the lock downs around the US are working. The map is here:

Our schools have been closed for the last 10 days. During that time I’ve been taking a little break from having my older son work on problems and am having him read Steven Strogatz’s Infinite Powers instead.

For our project tonight I asked him to pick out three chapters that he’s liked so far to discuss. He chose chapters 2, 3, and 10. As a quick note before diving in to his thoughts on these chapters, he studied calculus last year so I was having him read Strogatz’s book for history and context rather than as an introduction to calculus.

Here’s what he had to say about Chapter 2 which is about Archimedes:

Here’s what he had to say about Chapter 3 which is about Galileo and Kepler:

Finally, here’s what he had to say about Chapter 10 which is about Fourier:

I think he’s gotten a lot out of Strogatz’s book, and I was really excited to learn that he thought Fourier’s work was interesting. Maybe the Who is Fourier book really is the next right step for him.

Yesterday we did a project designed to help kids get a better understanding of some of the log-linear plots relating to the spread of the corona virus. That project is here:

I’ve seen an a lot of log-linear plots about the corona virus. My guess is that these plots are a little confusing to kids so I thought I’d spend 20 min tonight talking about them with my kids.

We started by just talking about what exponential graphs were. My younger son had a little misconception, so I was extra glad that we were having the conversation:

Next we talked about how an exponential graph changes when you switch from a regular graph to a log-linear graph:

So, with this very short introduction we took a look at two graphs about the corona virus that I’ve seen in the last week. The first was in a tweet from Steven Strogatz:

A Different Way to Chart the Spread of Coronavirus — Ken Chang @kchangnyt on the clarity and insight offered by log-linear plots https://t.co/fFAsO2An15

This tool allows us to look at the spread of the corona virus in countries all over the world. The plots are presented in log-linear form. You’ll see from this video that the boys seem to have a decent handle on what these plots are saying:

This week I’m going to share math projects based on items you can purchase from small businesses (in the US) who make amazing math-related products. The first project is based on the tiles from Cherry Arbor Design:

Yesterday my friend Suzanne Fields shared Sal Khan’s corona virus video with me:

Khan’s explanation is nice and definitely accessible to kids (and I assume aimed at helping kids understand). I thought it would be nice to have the boys watch the video and then talk through what they learned.

Here’s what my younger son had to say:

Here’s what my older son had to say:

Finally, we talked about some of the limitations of the analysis and a bit about the ideas you need to think about when you are making decisions facing extreme uncertainty:

I saw a great Twitter thread on virus spreading models from Nassim Taleb last week. I’d been meaning to share the ideas in the thread with my kids but didn’t get around to it until today.

The original thread is here:

UK Policy is a speculative lunacy.

Playing with the toy standard epidemiological SIR model. We have no idea how model parameters cause a yuuuge variation in ourcomes. We don't even know the central parameters/whether stochastic. Try to add perturbations for "herd immunity". pic.twitter.com/fTJ7pWRlaT

The tweet I wanted to focus on specifically is here:

So, tonight I used the code that Taleb shared and talked through the graphs with the boys. At the end we talked a bit about why Taleb’s conclusion was that these models were unreliable for decision making.

Here’s how my younger son reacted to the graphs:

Here’s how my older son reacted to Taleb’s graphs:

I think talking through some of Nassim Taleb’s ideas is a great way for kids to get some insight into how to think about the virus spread and also to see some of the dangers / limitations of modeling. For today’s project the important lesson is when you don’t know with any certainty how the models work, you really need to proceed with maximal caution.

Today we did a 3d printing project revisiting an angle sum that we’d looked at last week -> arctan(1/2) + arctan(1/3).

We started by reviewing how to approach the sum using complex numbers:

Next my older son explained a geometric way to approach the problem:

Now we went to Mathematica to create the 4 triangles using the RegionPlot3D function. It is a nice geometry exercise to have kids describe the boundary of a simple 2d object:

At the end of the day I had my younger son use the shapes to assemble the 3×2 rectangle and describe how this arrangement showed that the original angles added up to 45 degrees:

I like using 3d printing to help kids see math in a different way. The problem today was originally inspired from a section on complex numbers in Art of Problem Solving’s Precalculus book. It was nice to be able to use it to explore a little bit of 2d geometry, too.

I saw an amazing resource for looking at the spread of the Corona virus today – Alison Lynn Hill is a researcher specializing in mathematical biology at Harvard:

I’m sharing a simulation tool I put together for studying COVID19 dynamics and generating visualizations without having to do a bunch of coding yourself. Hoping it can help with your coronavirus-related research and teaching. https://t.co/GV5ErDkfE9 (1/7)

I thought that Hill’s program would be a terrific one for kids to use to see how a researcher studies the spread of a virus. We loaded the program on they boy’s computers and they played around with it for about 15 min.

Here’s what my younger son (in 8th grade) had to say – he was particularly surprised by how many variables there were:

Here’s what my older son (in 10th grade) had to say – he was interested in how the curves changed when he played with the transmission rates.

I’m grateful to Hill for sharing her incredible program, and really think it could help kids see (and understand a little) the modeling the modeling involved in studying pandemics.