In the middle of the book I found a really nice Simpson’s Paradox example and tried it out today with the kids. For more on Simpson’s Paradox, the Wikipedia page is actually great:

So, here’s the first part of today’s project – we have 4 boxes that have fixed amounts of red and blue cubes inside of them. First we divide them into two groups of 2 and ask which one in each group gives you a better chance of selecting a red block. It turns out that this is also a good introductory fraction exercise for kids, too!

Next we see the “paradox.” We combine the two winning boxes into one box and combine the two losing boxes into one box. Now which of the two remaining boxes gives you a better chance of selecting a red block?

So a fun and strange example for kids to see. Again, the Wikipedia page linked above gives a few more fun (and famous) examples. Really happy to have found this example in Moscovich’s book last night!

The 3rd video in Kelsey Houston-Edwards’s amazing new series was published last week. I’ve already used the first two videos for projects with the boys – I love this series so much!

Before diving into the video I asked the boys what they thought about the hair question – fortunately I got two different answers!

Next we watched Houston-Edwards’s new video:

Here’s how the boys reacted to the video:

(1) They were excited about the hair result and were also able to understand and explain it.

(2) They gave a nice summary of the Pigeonhole Principle.

(3) They really liked the example about 5 points on a sphere, so we took a really close look at that example. One of the tricky parts of that problem is understanding *why* you can draw an equator through any two points – both kids gave nice explanations of that idea.

Now I moved on to a couple of fun Pigeonhole Principle examples that weren’t covered in the video. I wanted to show the boys that the idea comes up in lots of different situations, including some that are not at all obvious Pigeonhole Principle situations!

The first example comes from my college combinatorics textbook – Applied Combinatorics with Problem Solving by Jackson and Thoro:

Small twitter math world fun fact – the professor for this class (~25 years ago!) was Jim Propp!

Here’s the problem (which is example 5 on page 35 of the book):

Suppose that we are given a set X of 10 positive integers, now of which is greater than 100. Show that there are two disjoint nonempty subsets of this set whose elements have the same sum.

I had to do a little bit of work on the fly to translate the problem into something that the boys could understand (and also explain quickly why there are 1024 subsets), but it seemed like they enjoyed this example:

The last problem is one I remembered when reading through some of the other examples in Jackson and Thoro’s book and is one that I talked about with the boys last year:

Show that every positive integer has a multiple whose base 10 representation consists of only 1’s and 0’s.

It certainly isn’t obvious at all at the start why this is a Pigeonhole Principle problem!

As I said at the beginning – I love this new series from Kelsey Houston-Edwards. I’m so happy to be able to use these videos to explore fun mathematical ideas with my kids!

This problem gave my son some difficulty yesterday – it is problem #19 from the 2011 AMC 10a

Last night we talked through the problem. The talk took a while, but I was happy to have him slowly see the path to the solution. Here’s his initial look at the problem:

Next we looked at the equation . Solving this equation in integers is a nice lesson in factoring. Unfortunately by working a bit too quickly he goes down a wrong fork for a little bit.

In the last video we found that the original population of the town might have been 484, and it might have changed to 634 and then once more to 784. We had to check if 784 was a perfect square.

Finally, we needed to compute the approximate value (as a percent) of 300 / 484. The final step in this problem is a great exercise in estimating.

So, a really challenging problem, but also a great problem to learn from. We went through it one more time this morning just to make sure that some of the lessons had sunk in.

This week’s video is about philosophy and math. A deep subject, for sure, but one which the kids thought was interesting. Here’s the video (and the twitter link so you know when the new videos appear!):

Need a Thanksgiving dinner topic? Argue about the philosophy of mathematics with the help of our new episode: https://t.co/BjKNgV4RGu

Today we revisited that old snowman and had the boys talk about each of the Archimedean solids in the shape. This is a fun project – not just because the shapes themselves are cool – but you get a nice opportunity to talk about counting and symmetry. You’ll see in the videos that my older son is a bit more comfortable with the idea, but my younger son seems to catch on by the 3rd video.

Here’s a link to all of the Archimedean solids on Wikipedia:

After playing the game for a just a few minutes I knew that my kids would love it.

Here’s each of their reaction to seeing and playing the game.

My younger son first:

My older son next:

So, definitely a fun little game for kids. They need to be fairly fluent with the arrow keys on the keyboard, but that’s really all that’s required. Definitely some fun puzzles to solve!

I happened to see the Raspberry Pi set below at a store earlier in the week:

Â Today I showed it to the kids and we played with it for a bit. Here’s their initial reaction:

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There wasn’t much to assemble which was nice. We did have some trouble connecting it to my laptop, so we switched to connecting it to our TV. Here’s what the boys had to say after we got it running:

Finally, the kids discovered a few of the games that came loaded on the computer. This squirrel one made them laugh. From start to finish was about an hour – and at least 20 minutes of that was trying and failing to get connected to the laptop.

We’ve only scratched the surface of what was in the kit. I’m excited to have the boys play around with the computer a bit more. It has Mathematica (yay!) and some software for introductory programming like Scratch. Hopefully there will be many more projects to come.

Had a great night with the boys tonight. My older son was working on some old AMC 10 problems and we talked through one that stumped him for his movie:

It was #15 from the 2013 AMC 10a:

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Next I spent some time with my younger son. He’s been studying the basics of lines using Art of Problem Solving’s Introduction to Algebra book plus a little bit of Khan Academy (when I’ve been traveling for work). I asked him what he’d learned so far and loved his response. It was a great reminder of the joy of learning new math ideas for the first time:

Everyone always talks about thinking through problem carefully – here’s a great problem and a great opportunity to give some examples of that kind of deep thinking.

Anyway, I ran out to home depot and got some wire and we made some knots. I had each of the boys make a trefoil knot and then make a random knot of their own choosing. In retrospect I wish I’d spent maybe just 5 minutes explaining some of the ideas in Richeson’s blog post – oh well, the excitement got the better of meðŸ™‚

Here’s my older son playing with his trefoil knot and making a Mobius strip bubble. I love the “hey, I actually think I got it” moment:

Here’s him playing with the knot me made – in retrospect I’d argue for a knot that was slightly less complicated:

Next up was my younger son. First up was the trefoil knot and we got another great moment “I think this might be a Mobius strip” !!

Finally we made his own knot and explored. Again, I’d probably ask for a less complicated knot if I was doing this again:

So, that so much to Dave Richeson for posting his old project – this is an incredible project, and an especially great one for kids. The appearance of the Mobius strip is really quite an amazing little math miracle!