Month: December 2020

Exploring n-dimensional cubes with my younger son

During break I’m going to try to do a fun exploration of n-dimensional cubes and spheres with my younger son. Today we talked a bit about cubes and started with by discussing what a “cube” was in a few different dimensions:

Next we talked about a different way to define a cube in n-dimensions using coordinates. These coordinates will help us do a few calculations in the next video.

We wrapped up today by trying to figure out what the longest distance between two points was in a 4-dimensional cube (with side length 1). My son talking through this problem is a really nice example of how kids can grapple with pretty abstract problems:



Having the kids play with the Lightbot, a $3 intro programming app

This week I learned about Lightbot from one of Michael Pershan’s tweets:



Although he was using the app with his 6 year old, I thought it would be fun to see what my kids (in 9th and 11th grades) would think of it. They really enjoyed it, and both played with it for longer than I asked them to.

My older son went first – he has a decent amount of programming experience and is taking a programming class at his high school this year. Here’s what he had to say:

My younger son went next. He as a tiny bit of programming experience. You can hear that the language he uses isn’t as sophisticated as the language my old son used, but that’s fine. You can also see that the app is very easy to learn how to use as this video shows him solving a level with only about 20 to 30 min of playing around with the game.



I was really happy that the kids liked this app as much as they did. Hopefully they’ll play around with it a bit more – it certainly looks like a neat introductory programming game for kids of all ages!

Revisiting Larry Guth’s “no rectangles” problem – one of the best math activities for kids I’ve seen

I was looking for a relatively stress-free project to do with the boys this morning and thought it would be a good day to review one of my all time favorite math projects for kids -> Larry Guth’s “no rectangles” problem. One of my all time favorite moments doing math with kids came when I used this problem for a 2nd and 3rd grade Family Math night at my younger son’s elementary school a few years ago. The problem is accessible to kids of all ages and also of interest to research mathematicians.

We started with a quick review (and lucky clarification!) of the problem and then the boys tackled the 3×3 case:



Next we moved on to the 4×4 case. The thought process the boys went through here I think shows why this is such a great math problem for kids to talk through:



Next I had a film goof up – luckly it was just 30 seconds of introducing the 5×5 case and telling them that answer to the 5×5 case was 12 squares. Don’t know what happened to this piece of the film, but since the plan was for the boys to play with this part off camera anyway I didn’t bother trying to fix it.

In any case, here’s the 12 square covering for the 5×5 they found and then a brief discussion about the surprise that comes when you move to the 6×6 square:

Again, this is one of my favorite math projects for kids – and kids of all ages. It is a really fun problem to play around with.

Revisiting a neat coin flipping game I learned from Ole Peters

This morning I accidentally stumbled on an old coin flipping game we looked at last year:

I thought it would be fun to take a look at the problem again since the last look was long enough ago that the boys probably wouldn’t remember it. Here are their initial thoughts in the problem. After a bit of discussion, the boys came up with a good argument for why HHHT only would appear more often than HHHH only.



Next we looked through a simple computer program I wrote to model the situation. This isn’t the best or most clever way to write the program, but I thought it was an easy one to explain:



Finally, we looked at how the numbers would change if the sequence had 50 flips instead of 20. It was interesting to hear the boys explain why the numbers had changed – I think this extra discussion helped them understand the original problem a bit better:

Extending our project exploring the Pólya Urn

Yesterday my younger son and I looked at the Pólya Urn problem inspired by this sequence of tweets from Ole Peters and Marcos Carreira:



Today I had my son explore a little further. He was interested to see if different starting positions led to different distributions of endings. He looked at five different starting positions. Here’s the first (with a quick review of the problem) when the urn starts with 5 black and 5 white balls and we play the game 1,000 times:



Next he looked at how the starting position with 1 black ball and 5 white balls evolved. The way the distribution of the number of white balls at the end changes is pretty amazing:



Now for the most surprising one of all – the starting position with 1 white ball and 1 black ball – it seems that ending with 1 white ball or 1001 white balls (or any amount in between!) is equally likely:

Finally he looked at the starting position with 1 black ball and 10 white balls. This one is a little less surprsing having already seen the 1 black ball and 5 white ball game, but still it was neat to see:

This is a fun little game for kids to study. It is also a nice introductory programming exercise, too. Thanks so much to Ole and Marcos for sharing their ideas!


Playing around with Polya’s Urn thanks to Ole Peters and Marcos Carreira

I saw a great twitter thread earlier in the week:



With the results and the code presented so nicely, diving into Polya’s Urn with my younger so this weekend was a no brainer!

I started by explaining the problem and asking for his thoughts. His intuition about how the game and how it would play out was fascinating to hear:



After he explained what he thought would happen, we played the game once:



Next we went to the computer to explore the game in more detail. Before diving in, though, I had my son explain Carreira’s code:



Finally, we played the game a few times and looked at the different outcomes. The results really are amazing. Hopefully we’ll do a follow up project tomorrow:

Talking parabolas and calculus with my younger son

Because of a careless mistake I made last week asking my son to read a calculus-based section in the statistic book he’s studying, I’m doing a few light touch calculus projects with him. Today’s will likely be the most computationally heavy one as my main goal is simply to show him the ideas.

He mentioned to me yesterday that in his class at school they are studying graphing parabolas. I decided to use a quadratic for today’s calculus example. First I asked him to graph it. As the video shows, I probably should have asked him how they were approaching the problem in his class at school first!

Next we spent a few minutes finding the tangent line to our parabola at the point (0,3). Here we talk about what the tangent line is:

Here we finish the tangent line calculation:



Next we moved on to finding the area under the curve. We’d discussed a similar example earlier in the week, so I thought this part of the project would make for a nice review of that prior talk. We ended up diving a little deeper than I intended, but I still think it was a good discussion.

Here’s the introduction to the area problem:



And here’s the final calculation of the area:

Looking at sums of integers and squares via geometry and algebra

Earlier this week I carelessly asked my younger son to read a section in a statistics book that relied on knowledge of calculus:



My son had some interesting questions about ideas in calculus, so instead of statistics we spent a few days this week talking about calculus (mainly finding the area under a curve).

For our project today I wanted to revisit one of the ideas in arithmetic that we’d relied on in the calculus discussion – sums of integers and sums of squares.

We started with sums of integers and my son gave a geometric proof of the rule for sums of consecutive integers:

Now we moved on to sums of squares – here we talked about the sum (and some of the number theory hiding in the sum formula), but didn’t yet try to prove the result:

Now we looked at a geometric way to understand the sum of squares formula:



Finally, we did a lightning fast review of mathematical induction and showed how we could proved that the sum of squares formula was true in general:



So, a fun week and a bunch of great discussions . . . even if it started with me being pretty careless!