Over the last month I’ve seen several great twitter threads on ellipses. Sort of a strange coincidence, I guess. Today I finally got around to sharing them with the boys.

We started with this tweet from Lucas Vieira. I’m having a fight with WordPress on the embedding of this one – the tweet we are looking at is the “ballistic ellipse” one at the bottom.

The highest point of ballistic trajectories at constant speed form what I call the "ballistic ellipse". I discovered this when I was in high school, but apparently this was only published about in 2004! https://t.co/eSmpw9bQu6pic.twitter.com/TwlXsyPwvc

— Lucas Vieira (LucasVB/1ucasvb) (@LucasVB) April 2, 2020

Here’s what the boys thought of the ideas in the animation:

Next we moved on to a post from Greg Egan that was inspired by Lucas’s post:

A satellite has the least kinetic energy when farthest from the central mass.

If it releases mini-satellites moving tangentially to its orbit, but whose KE per kg is less than its own by that minimum, they will all pass through the 2nd focus of its orbit, all at the same speed. pic.twitter.com/j9HljOTXwM

The kids had a tough time explaining what they were seeing here, so we talked about this picture for a little bit longer than usual:

Now we moved on to Jacopo Bertolotti’s Physics Factlet #216 on 1/r^2 orbits. This animation helped me make sense of a point about General Relativity that I’d heard, but never really understood.

#PhysicsFactlet (216) Central forces that decay as 1/rΒ² are special, as they guarantee that all bound orbits are going to be closed (Bertrand's theorem). Small changes in the power will lead to significantly different kind of orbits. pic.twitter.com/AllIFyTwZd

Finally, since Jacopo share his Mathematica code, we took a look at the program. The boys were surprised by how short it was. After looking at the code for a bit we changed some of the parameters and got a fun surprise:

I love that so many people share their amazing work on Twitter. Looking at these animations was a fun way to share a bit of math and physics with the boys this morning!

— γ Berger | Dillon γ (@InertialObservr) March 6, 2020

After seeing it I thought that talking about the orbits of the Earth and Mars would make for a great project with the boys today. We started with some basic ideas and the kids actually had some ideas that related to Kepler’s Laws, which was super fun:

Next we moved to the computer to look at Berger’s tweet and see what he boys had to say about it:

Now we went back to the white board to talk about representing the orbits with polar coordinates. My younger son is just learning about polar coordinates now and I started the conversation here a bit too quickly, I think, but even with the bad start hopefully we had a productive conversation:

Finally, we went back to the computer to look at a simplified program illustrating what the orbit of Mars looks like when viewed from the Earth.

This project was really fun – I’m really grateful that people like Dillon Berger share so many amazing ideas about math and physics on twitter. All of that sharing makes finding fun projects to do with kids so much easier!

‘Today the boys and I talked through three fun to see, but maybe tricky to understand, ways that something need to rotate 720 degrees to get back to where it started.

We started by looking at a circle rotating around a second circle of the same size:

Next we looked at the famous “wine glass” problem. I originally wanted to color the water in the glass with food coloring, but chickened out!

Before going on to the Dirac Belt Trick, I showed the boys this really nice video showing the trick in a pretty unusual – and super fun – way:

After the video demonstration, I had the boys try the trick with a belt. At the end my old son made a connection between the belt trick and the complex numbers which was a nice and totally out of the blue surprise to me:

Anyone interested in physics should listen to Weinstein’s interview of Penrose – it is amazing. I was really happy to be able to pull out a few ideas from the interview to share with the kids today!

A lot of people have been talking about recent observations of the star Betelgeuse this week. Here’s one great thread I happened to see:

Regarding #Betelgeuse's "historic" dimming, here's V-band & photovis magnitude estimates from @AAVSO database over past century. *You* try maintaining constant luminosity w/~20 solar masses spread out within the size of Jupiter's orbit, w/convection & nucleosynthesis is 3D! pic.twitter.com/e2kojEZXAC

After seeing this thread I thought it would be fun to share some of the ideas about the recent observations of Betelgeuse with the boys. Although I’m way out of my league here, there were some great resources I found that I thought would help the boys understand what was going on. Two of those resources were:

I started today’s project by showing the boys the article on Astroblog and then the graph in Eric Mamajek’s tweet:

Next we looked at a graph from the Light Curve generator showing how the brightness of Betelgeuse has varied going back about 6 months. Sorry for the glare on the computer screen π¦

The boys had different ideas about how to interpret the data – which was fun to hear:

Next I had each on my son’s create a new graph. My younger son went first and he wanted to look at the observations from a single astronomer. We did this by using the green dots since there were only to people who collected that data. The astronomer whose data we looked at was Wolfgang Volmann:

My older son went second – he wanted to look at the observations of Betelgeuse going back a long time. We were able to zoom in on a time period in the 70s and 80s in which many observations showed that Betelgeuse was pretty dim.

This was a really fun project to work through with the kids. It really highlights the difficulty of collecting data in astronomy, and in the real world in general! It was fun to hear their ideas about how to think through the

For today’s math project I asked my son to play around with the program and pick three examples that he found interesting. The discussion of those three examples is below.

Here’s the first one, with a short discussion of three body problem at the start:

Next up was an orbit shaped almost like an infinity symbol:

Finally, an orbit that it completely amazing – I almost can’t believe a shape like this is possible!

— Dr. Chanda Prescod-Weinstein π π½ββοΈ π§π§π³οΈβπ (@IBJIYONGI) August 9, 2019

After seeing the first tweet from Kayley Brauer I was hoping to find a way to talk about this new result with the boys, but didn’t really know what to do. Thanks to the tweet from Dr. Chanda Prescod-Weinstein, I learned that the LA Times had put together a terrific presentation that was accessible to kids.

First, I had my older son read the article on his own and then we talked through some of the ideas he had after reading it.

I thought that reading the article on his own would be a little too difficult for my younger son (he’s about to start 8th grade) so instead of having him read it on his own, we went through it together:

Obviously I’m not within 1 billion miles of being an expert on anything related to this new image of the Milky Way, but it was still really fun to talk about it with the boys. I’m very happy that advanced science projects like this one are being shared in ways that kids can see and experience.

Today my older son was back from camp and I thought it would be fun to try an experiment that is described in the first part of chapter 3 of the book. The experiment involves a ball rolling down a ramp and is based on an experiment of Galileo’s that Strogatz describes.

I started by having my son read the first part of chapter 3 and then tell me what he learned:

Now we took a shot at measuring the time it takes for the ball to roll down the ramp.

I misspoke in this video – we’ll be taking the measurement of the distance the ball travels after 1 second and then after 2 seconds. I’m not sure what made me think we needed to measure it at 4 seconds.

Anyway, here’s the set up and the 5 rolls we used to measure the distance after 1 second.

Here’s the measurement of the distance the ball rolled after 2 seconds. We were expecting the ratio of the distances to be 4 to 1. Unfortunately we found that the ratio was closer to 2 to 1.

We guessed (or maybe hoped!) that the problem in the last two videos was that the ramp wasn’t steep enough. So, we raised the ramp a bit and this time we did find that the distances traveled after two seconds was roughly 4 times the distance traveled after 1 second.

This is definitely a fun experiment to try out with kids. Also a nice lesson that physics experiments can be pretty hard for math people to get right π

[sorry for mistakes – this one was written up in a big hurry]

I’m a big fan of the Mathematical Objects Podcast hosted by Katie Steckles and Peter Rowlett. Their most recent episode talked about Newton’s law of cooling and I thought it would be fun to try the project at home. Here’s link to the specific podcast:

— Mathematical Objects (@mathsobjects) July 5, 2019

Note that this project does require some adult supervision because it involves boiling water.

The idea in this project is to explore Newton’s law of cooling two different ways. The first way is to talk about the law, observe some water cooling for a bit, and then make a prediction about how that cooling will proceed. The second way is to take two cups of hot water and compare the temperature when you add cold milk to one initially and to the other 10 min later.

Here’s how we got started:

Next we took two glasses of hot water and measured the initial temperature:

5 min later we returned to measure the new temperatures and then use Newton’s law of cooling to predict the temps 5 min later. This part of the project was a little hard to do on camera, but you’ll get the idea of the things you have to keep track of. Hopefully we did all of the calculations right!

Next we moved on to the “tea” experiment. Here we started with two cups of hot water and added milk to one of them. We are going to wait 10 min and then add milk to the other glass and compare the temperatures of the two cups. Both kids mad a prediction about what would happen:

Finally, we returned to the cups and finished the 2nd experiment. Both kids guessed right on the relative temperatures, but I’m not 100% sure that we got the amount of water and milk exactly equal in the two cups. Still a fun experiment, though.

A saw a really neat twitter thread last week thanks to a re-tweet from Chanda Prescod-Weinstein:

Thinking about brightness of a moving object in magnitudes is complicated, so hereβs a brief tutorial with some calculations at the end. Imagine a telescope with one arcsecond pixels, excellent seeing, and some idealization, and a 4th magnitude satellite.

The thread explained why thinking about the (astronomical) magnitude of an object moving through a telescope’s field of view is a little difficult. It was neat to learn that something I didn’t realize was hard is actually pretty hard (though it feels like basically everything in astronomy is like that!), but another thing that jumped off the page for me in that twitter thread was that it was an excellent example to show to kids learning about logarithms.

For reference, here’s the Wikipedia page we used in the project to learn about the concept of magnitude and also get a few examples:

My younger son (in 7th grade) is just learning about logarithms now and my older son (in 9th grade) has a bit more experience with them. We started by talking about the relative magnitude formula and working through a short calculation to show why the number 2.5 shows up in the formula:

Next we looked at the Wikipedia page linked above to get some examples of magnitudes of a few objects we recognize:

Now we talked through Bruce Macintosh’s twitter thread. I wanted to go through the thread carefully to make sure the kids had a basic understanding of the concepts he was discussing (arcseconds, for example). We talked about some of the calculations, but did not do any calculating ourselves in this part. One question for the kids here was why did Macintosh use a + sign in his formula when the Wikipedia page has a – sign in the formula?

Finally, we did the calculations and found the answer to the mystery of the + and – sign from the last video. Happily, we match the answers from Macintosh’s thread:

This project was really fun. It was a really happy accident that just as my younger son was learning about logs a neat (and “new to me”) example of where logs are used showed up in my twitter feed!

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 π π½ββοΈ π§π§π (@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!).