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:

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.

Today I had my son read through the post and then we discussed the ideas. His initial thoughts are in the video below – he understood most of the post and also had a couple of good questions:

After we talked through the post we went to Mathematica to take a look at some of the example “reflections” which preserve the 1st and 2nd derivatives:

One of my son’s questions in the first video was why the blog post was using functions like f(x/2) and f(x/4) to make reflections. I’d mentioned that these were essentially arbitrary choices. Below we saw what would happen if we used f(x/3) instead:

We finished up by going back to Mathematica to see what these new “reflections” would look like:

A few days ago we did a project using Mathologer’s amazing video on Fermat’s “two squares” theorem. At the end of the project the boys were wondering about why so many of he numbers we found that could be written as the sum of two squares in several different ways were multiples of 5. I was wondering the same thing and spent two days playing around and trying to learn more these sorts of numbers. Even after searching the positive integers up to 3,000,000, all of the numbers I found that could be written as the sum of two positive squares in exactly 7 ways were multiples of 5. What was so special about 5?

Overnight I got some great twitter advice on the subject form Stephen Morris and Alex Kontorovich. Their ideas helped me understand a bit more about what was going on. Tonight I explored some of the basic ideas with the boys. I know next to nothing about the number theory here, but am completly amazed by the never ending patterns that are hiding inside of the integers!

We started today’s project by looking at all of the positive integers less than 1,000,000 that can be written as the sum of 2 positive squares in exactly 7 ways. Here’s what they noticed:

At the end of the last video my younger son thought that it might be useful to factor all of the numbers on our list. We did that off camera and then the boys looked for patterns in the numbers and factors. Finding patterns in the factored numbers was more challenging than I expected, but they were able to make some progress.

Based on what we noticed we took some guesses at numbers that were not multiples of 5 that could be written as a sum of two positive squares in exactly 7 ways.

Finally, we used the Wolfram Alpha code that Stephen Morris showed us to check if the numbers we guessed really could be written as the sum of two positive squares in exactly 7 ways.

This project was incredibly fun. It shows how computers (and Twitter!) can really help kids explore some pretty advanced ideas. I’m really interested to see how we might be able to explore a few more related ideas in the next week.

Select three points uniformly at random inside of a unit square. What is the expected area of the circle passing through those three points?

This question turns out to have a lot of nice surprises. The first is that exploring the idea of how to find the circle is a great project for kids. The second is that the distribution of circle areas is fascinating.

I started the project today by having the kids explore how to find the inscribed and circumscribed circles of a triangle using paper folding techniques.

My younger son went first showing how to find the incircle:

My olde son went next showing how to find the circumcircle:

With that introduction we went to the whiteboard to talk through the problem that Steve Phelps shared yesterday. I asked the boys to give me their guess about the average area of the circle passing through three random points in the unit square. Their guesses – and reasoning – were really interesting:

Now that we’d talked through some of the introductory ideas in the problem, we talked about how to find the area of a circle passing through three specific points. The fun surprise here is that finding this circle isn’t as hard as it seems initially:

Following the sketch of how to find the circle in the last video, I thought I’d show them a way to find the area of this circle using ideas from coordinate geometry and linear algebra – topics that my younger son and older son have been studying recently. Not everything came to mind right way for the boys, but that’s fine – I wasn’t trying to put them on the spot, but just show them how ideas they are learning about now come into play on this problem:

Finally, we went to the computer to look at the some simulations. The kids noticed almost immediately that the mean of the results was heavily influenced by the maximum area – that’s exactly the idea of “extremistan” that Nassim Taleb talks about!

This project is a great way for kids to explore a statistical sampling problem that doesn’t obey the central limit theorem!

I really love the problem that Phelps posted! It is such a great way to combine fascinating and fundamental ideas from geometry and statistics

It looked like it could make a neat project both on the computer and with our Zometool set.

First I had my younger son look at Phelps’s visualization – one really interesting observation he had was that the intersecting lines inside the icosahedron dodecahedron:

Nest I had my older son look at a similar program in Wolfram’s Demonstration Project. The thing that caught his attention was all of the underlying structure:

We also created a zometool version of the icosahedron with all of the diagonals. We tried to see if we could see the same interesting things that we saw in the computer programs using the Zome shape:

Later in the day we did build a slightly larger icosahedron in which the diagonals did intersect on a zome ball. This allows you to see the dodecahedron that my younger son thought was there: