Sharing Appendix 1 to Steven Strogatz’s Infinite Powers with my son

Yesterday Steven Strogatz shared an unpublished appendix to his book Infinite Powers:

I read it and thought it would be terrific to share with my older son who took calculus last year. This year we’ve been working on Linear Algebra – so not a lot of polynomial calculations (yet!) – so I also thought Strogatz’s appendix would be a terrific review.

I had him read the note first and when he was ready to discuss it we began:

At the end of the last video my son had drawn the picture showing Fermat’s approach to calculating the area under the curve y = x^n. Now we began calculating. He was able to write down the expression for the approximate area without too much difficulty:

The next step in working through the problem involved some work with a geometric series. Here my son was a little rusty, but I let him spend some time trying to get unstuck:

I just turned the camera off and on at the end of the last video and he continued to struggle with how to manipulate the geometric series into the form we wanted. After a few more minutes of struggle he found the idea, which was really nice to see.

Once he understood the simplification, the rest of Fermat’s proof was easy!

I’m really happy that Strogatz shared his unpublished note yesterday. It is terrific to share with kids who have already had calculus, and would, I think, also be terrific to share with kids studying Riemann sums.

Sharing Ricky Reusser’s ‘Periodic Planar Three-Body Orbits” program with my son

I saw an really neat idea in a tweet from Nalini Joshi yesterday:

A direct link to Ricky Reusser’s incredible 3-body problem visualization is here:

Ricky Reusser’s amazing 3-body problem visualization

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!

Using Brian Skinner’s terrific math joke for a lesson about logarithms

Yesterday I saw this fantastic post on twitter:

Since my older son leaned calculus last year, I thought it would be fun to run through the 9 equations with him, and then focus on the one about the logarithm of N!

Here are his thoughts on the equations:

Now we explored the one about the log of N! in a bit more depth – I was happy that after a few months off from calculus some of the main ideas still seem to have stuck around:

Finally, we went to Mathematica and explored the formula a bit more to see how good it was. We then wrapped up by looking at the Wikipedia page for Stirling’s approximation.

I’m glad to have gotten 2 days worth of laughs from Skinner’s post. Happy that it was also a fun starting point for a lesson, too πŸ™‚

Using Chapter 6 of Steven Strogatz’s Infinite Powers with a kid

I’m having my older son read a few chapters of Steven Strogatz’s Infinite Powers this summer. We did a calculus course last school year so he has seen some of these calculus concepts before. I’m finding it both fun and fascinating to review some of the ideas with him – there were always lots of ways to review and freshen up the pre-calc ideas, but I still looking for good ways to do that with the ideas from calculus.

Anyway, think of this project as representing with a high school student with a year of calculus under his belt has to say about some of the main ideas from the course.

So, I had him read chapter 6 this morning – here are his initial thoughts:

I asked him to pick two ideas from chapter to talk about. The first idea he wanted to talk about was “instantaneous speed.” Here’s what he took away from the chapter:

The second thing he wanted to talk about was the “Usain Bolt” problem. This part of Strogatz’s book has received a lot of attention – here’s an article in Quanta Magazine, for example:

Quanta Magazine’s article about Infinite Powers

Here’s what my son had to say about the problem:

I’m always really interested to hear kids describe math concepts, but I’m not used to hearing kids talk through Calculus ideas. Hopefully we’ll have some fun over the next few years finding ways to review the main ideas. Probably Infinite Powers will be a great resource!

Trying an experiment described in Steven Strogatz’s Infinite Powers with my son

I got my copy of Steven Strogatz’s new book back in April:

I’ve used it for two projects with my kids already:

Using Steven Strogatz’s Infinite Powers with a 7th grader

Following up on our conversation about Steven Strogatz’s Infinite Powers with some basic calculus ideas

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 πŸ™‚

A project for kids inspired by the Mathematical Objects Podcast

[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:

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.

Following up our conversation about Steven Strogatz’s Infinite Powers with some basic calculus ideas

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:

Using Steven Strogatz’s Infinite Powers with a 7th grader

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:

Using Steven Strogatz’s Infinite Powers with a 7th grader

My copy of Steven Strogatz’s new book arrived a few weeks ago:

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 \pi r^2:

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!

Revisiting a connection between arithmetic and geometry

I saw a really great thread on twitter this week and wanted to share some of the ideas with the boys for our Family Math project today:

We started off looking at the sum 1 + 2 + 3 + . . . .

Next we looked at the sum of squares and searched for a geometric connection:

Now I showed them the fantastic way of looking at the sum of squares in the Jeremy Kun blog post. This method is a terrific way to share an advanced idea in math with kids – it is totally accessible to them and gives them a chance to talk through a fairly complicated idea:

Finally, I showed how the ideas we were just talking about extend to some of the basic ideas is calculus. It was neat to hear my younger son talking through the ideas here, too:

Definitely a neat morning – it is always amazing to see the connections between arithmetic and geometry.

Celebrating Pi day with a fun calculus paper from Susan Jane Colley

About a week ago I saw a great tweet from Dave Richeson about some journal articles that were being made freely available for the month of March:

The paper from the The College Mathematics Journal by Susan Jane Colley caught my attention for being both an really neat result and being explained at a level a student taking calculus could understand.

So, this morning to celebrate Pi day I decided to use the paper to talk a bit of calculus with my son. Pulling all of the different ideas together was challenging for him, so we went slow but still made it through the main points in about 30 min.

We took a quick look at the paper and then started digging into the math by looking at the famous alternating series for \pi / 4.

I should say for clarification that I forgot to look up Susan Jane Colley’s current position before we started the project and wasn’t sure if she was still at Oberlin or had moved to a different university since the paper was published in 2003. But to be clear, she is the Andrew & Pauline Delaney Professor of Mathematics at Oberlin.

Next we dove in to the connection between the alternating series and \pi. I thought I’d try to introduce the connection in a sneaky way, but it was sort of a dead end. Eventually, though, he thought about arctangent.

At the end of the last video the formula for an infinite geometric series came up, but that formula wasn’t quite at the top of his head. So we took a little detour to re-derive that formula. Once we had that formula we could see that the alternating series we were looking at converged to \pi/4:

Now we looked at the main result of the paper – a different series for \pi that converges really fast.

Here we look briefly at the formula for this series (sorry for the reading typos by me – trying to read the paper and stay behind the tripod and not cast a shadow was hard . . . . )

Finally, we went to Mathematica to evaluate the integral and look at the speed of convergence of the two series we’ve been studying:

I think that Colley’s paper is absolutely terrific and a great resource to use to show calculus students some advanced math. It is an extra terrific resource to use on Pi day πŸ™‚