Today as sort of a unusual way to play around with fractions I thought it would be fun to try to write some fractions in binary. It has been a while since we talked about binary, though, so I had my son tell me what he knew about binary first:

Next we moved on to writing fractions in binary – we started with some simple cases:

Finally, we tried to write 1/3 in binary. This video shows what a kid thinking through a math problem can look like, and also shows why I thought this exercise would be a nice fraction review:

Yesterday I had my younger son (in 7th grade) read chapter 1 in Steven Strogatz’s new book Infinite Powers and we had a fun time talking about what he learned:

Today I wanted to show him a hand waving overview of two of the more well-known ideas from calculus – finding tangent lines and finding areas under a curve.

I started with the tangent line problem:

He was struggling to remember some of the basic ideas about lines, so I broke the talk about tangent lines into two pieces to let him take his time remembering how to describe lines. Here’s the second part of the discussion:

With the tangent line discussion finished, we moved on to finding the area under a curve. To keep things simple, I stuck with the same function:

The book is terrific and the math explanations are so accessible that I thought it would be fun to ask my younger son to read the first chapter and get his reactions.

Here’s what he thought and a short list if things that he found interesting:

After that quick introduction we walked through the three things that caught his eye – the first was the proof that the area of a circle is :

Next up was the “riddle of the wall”:

Finally, we talked through a few of the Zeno’s Paradox examples discussed in chapter 1:

I think you can see in the video that Strogatz’s writing is both accessible and interesting to kids. I definitely think that many of the ideas in Infinite Powers will be fun for kids to explore!

Saw this problem from the 2019 EGMO last week and though it would be a great one to share with kids:

My older son was working on something else tonight, so I talked through the problem with my younger son (he’s in 7th grade). The aim of the project tonight was not to solve the problem, but just to have him play around with a few simple cases and see if he could take a guess at the answer.

I started by just sharing the problem and making sure he understood the ideas and the constraints:

Having looked at the 2×2 case in the prior video, we moved on to the 4×4 case in this video. He had some pretty interesting ideas about how to check if we’d found the maximum number of dominoes:

Now that we’d satisfied ourselves with the 4×4 case, we moved to the 6×6 case. This case is a little harder, but still accessible to kids. Here’s his first attempt at a solution – the trouble is that we weren’t sure if it used the maximum number of dominoes:

It took a bit more experimenting to see that we’d used the most dominoes we could in the last video, and you can see him starting to understand some of the patterns in the problem as he experiments.

By the end of this video he guessed that the maximum number for the 8×8 grid would be 10.

The final challenge was an 8×8 grid. In the first attempt at filling in the board with dominoes we kept getting stuck – but his thoughts about the problem are very interesting:

With one more try through the 8×8 board we were able to fit 10 – yay!

This is a great problem to share with kids. Again, even if they can’t get all the way to understanding the general solution, there are plenty of things they can play with and understand – and also tons of ways to approach the problem!

The first ever image of a black hole was released last week, and it blew my mind! For our Family Math project this morning I decided to try to share some of the ideas relating to that image with my younger son. He’s in 7th grade and had not heard about the announcement, so it was especially fun to hear his off the cuff reactions.

I started by simply showing him the picture and asking him to describe what he saw and if it looked like what he expected a picture of a black hole to look like:

Next we watched Katie Bouman’s Ted Talk video (published in 2017) about the black hole imaging project. This video is fantastic and great to share with students because the explanation of the project, and especially why the project is so difficult, is done at a level that kids can appreciate even if they can’t understand all of the details. Here’s that video:

After we finished watching, I asked my son to tell me some things from Bouman’s talk that caught his eye:

Next I had my son read through two great twitter threads from when the announcement happened last week. Those threads are from two physics professors – Katie Mack from North Carolina State University and Chanda Prescod-Weinstein from the University of New Hampshire. Those two twitter threads are here:

— Chanda Prescod-Weinstein 🙅🏽♀️ 🇧🇧🌈 (@IBJIYONGI) April 10, 2019

First ever direct image of a black hole! The supermassive black hole in the galaxy M87 — 6.5 billion times as massive as the Sun! #EHT#BlackHole The image is better than I expected! pic.twitter.com/Tv7I36v4xQ

Here are two of the tweets from those threads that caught my son’s eye and his explanation of why he thought those two tweets were interesting:

Now I had my son play with a on online program that Leo Stein made a few years ago that shows the amazingly beautiful paths light can take orbiting a black hole. You can find the program here:

Unfortunately our camera was having a strange fight with the computer screen at the beginning of this video, so I cut that part out. Because of that cut the video starts mid conversation, but you’ll still be able to see that Leo’s program is really great to use with kids. I love his comment at the end: “Black holes seem pretty mysterious and neat and have weird properties.”

Finally, I showed him a picture from one of the papers about the black hole image that was published last week. The link to the paper with the image is here:

(2/2) Take this picture, for example, which shows what the pic would look like leaving out one of the telescope sites. It really does a great job of illustrating the deep level of care that went into the final work. pic.twitter.com/UqHtqPcWZE

Here’s our quick discussion about that image and what my son thought about it. Talking a little bit about this image can help younger students see and understand some of the statistical work that went into producing and checking the image:

It was really incredible to see the announcement of the black hole image last week. It is equally incredible that so many people in the physics community take time to share their ideas about discoveries like this with the public. I’m super grateful for the public-facing work those people do because it makes sharing new discoveries with kids possible (and fun!).

For today’s project I tried a little challenge – I had my younger son read the first chapter in the wonderful book Count like and Egyptian and then we talked about what he learned. Here’s the book and and link to Evelyn Lamb’s review of it:

The goal for today was more for him to talk through what he learned as opposed to getting the math details right. We definitely had a few stumbles, but it was still fun and the multiplication and division ideas are really neat.

Here’s the introduction to the book and the arithmetic ideas:

Next I asked my son to talk through a few multiplication problems:

I was re-reading How to Gamble if you Must by Dubins and Savage and though it would be fun to talk through the gambling problem in the beginning of the book with my younger son.

The book isn’t exactly light reading, but definitely interesting if you want to understand a bit of the math behind gambling

The problem is fairly straightforward to understand -> you start with $1,000 and you need to get to $10,000 by making bets. How should you bet if the game is unfavorable, fair, and favorable?

I started the project today by explaining the game to my son and asking how he thought you should bet in the various games:

Next we wrote a short program in Mathematica (off camera) and then played the game. Here’s a discussion of the program and a few times through the unfavorable game:

Now we played the fair game and looked to see if the strategy was the same or different than the unfavorable game:

Finally, we played the favorable game – again we looked for what might be different in the betting strategy for this game: