Revisiting Mads Bahrami’s corona virus mapping project with the boys

Back in May, Mads Bahrami made a terrific map of how the corona virus spread in the US:

We did a project based on Bahrami’s work back then:

and today seemed like a good day to revisit it. This project needs Mathematica to do yourself, but I think it is also really interesting to hear what the kids have to say about the maps.

Here’s their first reaction to an animation showing total (population adjusted) cases in the US over time:

I wasn’t happy with the color scheme I chose for the first map, so the main work we did for the project today was making a new map with an improved color scheme. That work required us to look carefully at the data and study the distribution of the population weighted counts by US county. Here’s the new map and how the boys described that work:

This project was a nice way for kids to think about how to present and interpret data. Thanks to Mads Bahrami and to Wolfram for making the original work public.

Revisiting Po-Shen Loh’s approach to the quadratic formula

Last fall Po-Shen Loh shared a really interesting approach to understanding the quadratic formula. His idea got so much attention that they were written up in the New York Times::

We had our first look at these ideas back in December:

I thought it would be fun to revisit some of the ideas today. It turned out to be a really good algebra review for my younger son, and a nice review of ideas about roots of equations for both kids.

We started talking about some general ideas about quadratic equations and a reminder of the sum of roots and product of roots ideas for quandraics:

Now we took a closer look at the sum of roots and product of roots ideas to give the boys a bit more background on the ideas in Loh’s paper. The ideas here were a little confusing for them, so it was good that we took a little time to review them before going to the next step:

With all of the background out of the way we moved on to Loh’s difference of squares idea. This idea wasn’t obvious to the boys, but once they saw it the quadratic formula appears immediately!

Finally, we finished up the project by showing how to derive the quadratic formula for a general equation:

I really like Loh’s approach and think it is a great way for kids to see the quadratic formula. This project showed me that the ideas are a bit more subtle than I thought, though, and we’ll probably have to run through them a few more times for them to really sink in.

Talking through an interesting tweet from Scott Gottlieb with my kids

Last week I saw this fascinating tweet from Scott Gottlieb:

I thought the charts in the tweet would be great for a discussion of the pandemic in the US with kids, so I gave it a shot this morning. We walked through the charts one at a time (my kids just finished 8th and 10th grade). Here’s what they had to say:

Chart 1:

The boys were able to understand why separating out the New York area from the rest of the country made sense. They were concerned about states making decisions about reopening when the cases in the country (ex NY area) were not declinng:

The next chart in Gottlieb’s tweet was one I thought was particularly interesting:

I was really interested to see if the kids could understand why the projections based on the aggregate data would be different than the combined state by state projections.

For the third chart, the boys had some really interesting things to say. I was really happy that they noticed that the color scheme changed chart to chart.

Finally, to try to connect some of the ideas we talked about, we went to the FT’s website to study some of the trends in positive tests in the US. We had a good discussion about a few states and then a really nice discussion about log plots. It was great to hear what kids see in all of these charts:

I’m really happy with how this project went – it is nice to hear what kids have to say about different data sets related to the corona virus. Obviously not all of the information about the corona virus is going to be accessible to kids, talking through a few of the ideas that are accessible will really help them understand the pandemic, and the decisions we have to make around the pandemic, much better.

Talking through one of the Wolfram programming challenges with my kids

One of our projects this summer is working through some of the Wolfram programming challenges. I’m not a particularly good programmer, but am excited to try to help out the kids as best I can.

One of the challenges we worked on this week was really interesting:

A quick summary of the challenge is here:

This challenge was difficult for the kids and took about 3 days working for roughly 30 min each day to complete. I think that part of the difficulty came from having to think about a list of lists, which is a new idea for them (programming or otherwise).

For today’s project I wanted them to talk through their approach to the problem and eventually discuss the solution. We started with looking at the problem statement and talking a bit about what made this challenge a little difficult:

Next we talked about some of our initial ideas about the program and how we thought about the problem with an even number:

Now we discussed what was different (maybe surprisingly different) about the case with odd numbers:

Two wrap up we looked at the program the boys wrote and they talked through the code:

I’m really excited about working through more of these challenges. Some seem absurdly hard and I’m sure won’t be able to solve all of them, but I think we’ve got a fun summer ahead of us!

Talking about some introductory number theory ideas with the boys

My younger son is working his way through Martin Weissman’s An Illustrated Theory of Numbers right now:

He’s in the chapter on greatest common divisor and least common multiple now, and I thought talking through some of the ideas he’s seeing would make for a good project this morning. It gave him a chance to talk about what he’s learning and it gave my older son a chance to review some ideas he’s seen before.

We started by talking about the Euclidean Algorithm:

Next we discussed the interesting identity that the product of the LCM and GCD of two positive integers is equal to the product of those integers:

Now we moved on to discussing how the ideas we talked through in the prior videos could help us solve Diophantine equations. Here my younger son introduces the main ideas:

To finish, I had my older son explain why the general solution my younger son introduced in the first video was

I can’t say enough good things about Martin Weissman’s book – it has really gotten my son interested in number theory. Can’t wait to explore more of the ideas in the book with him!

Talking through the Rubik’s Cube episode of the Mathematical Objects podcast with kids

Yesterday I listened to the new episode of the Mathematical Objects podcast:

I love listening to the discussions that Katie Steckles and Peter Rowlett have on this podcast. This episode made for an especially great project for kids, I thought. So, I had the boys listen to it after breakfast and then we talked about some of the ideas that they thought were interesting:

Now we dove into a tiny bit more detail about groups and modular arithmetic. Here I wanted to show the boys that the idea of an identity element was pretty important even though it seems like a pretty simple requirement when you see it for the first time:

Finally, we moved on to talking about some of the group theory ideas that relate to Rubik’s cubes. The specific idea we talked about from the podcast episode was “commutators”. We tried out three examples that – honestly by accident – turned out to be nice illustrations of the idea:

I really enjoyed the podcast and also learning about what the boys found interesting listening to it. It definitely is fun illustrating some basic ideas from group theory with Rubik’s cubes!