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!

Talking kids through the Washington Post’s article on counting corona virus deaths

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.

Talking 4d shapes with my kids

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.

Having kids play with 4 and 5 dimensonal Hasse diagrams

Yesterday we did a fun project on Hasse diagrams:

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:

So for a challenge project this morning I had the boys try to figure out how a Hasse diagram would work in 4 dimensions and in 5 dimensions.

Here’s how the 4d case went:

The 5d case was significantly more challenging – but they got there! Here’s the explanation of their work:

Who would have ever thought that a 5d cube appearing in your twitter feed would be exactly the thing you needed to see for a new math project!

Exploring Hasse diagrams with kids thanks to Martin Weissman’s An Illustrated Theory of Numbers

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

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.