My older son is studying linear algebra out of Gil Strang’s book this year. Currently he’s in the chapter on determinants and we’ve spent the last couple of days talking about Cramer’s Rule.
As we talked about the proof of Cramer’s Rule, I was struck by how similar the ideas were to the ideas used in Fourier analysis. This morning we had a fun discussion showing how the ideas are connected.
I started by asking him to talk about Cramer’s Rule. He did a nice job, especially since his knowledge about this rule is only a few days old:
Next we played around on Mathematica with a 4×4 example and found that the solutions you get from Cramer’s Rule do indeed match the solutions you get from other methods!
Next I gave a really short introduction to a problem that initially seems very different, but has a lot of the same mathematical ideas hiding in the background -> pulling a signal out of noise:
Finally, we went back to Mathematica to play with a few examples of signals hiding in noise. We saw how the ideas from Fourier Analysis could often pull out the signal even though it wasn’t obvious at all that a signal was hiding in our data in the first place!
Last week Steven Strogatz released two previously unpublished appendicies for his book Infinite Powers:
My older son and I did a fun project with Fermat’s idea. He’d taken calculus last year and the ideas Strogatz shared made for a really nice calculus review:
Sharing Appendix 1 to Steven Strogatz’s Infinite powers with my son
My younger son is in 8th grade and has not taken calculus. I thought some of the ideas about finding areas under simple curves would be interesting, so I tried sharing some of those ideas this morning.
We started by taking a look at the first page of Strogatz’s appendix and then talked about finding the area under for small values of
Now we moved on to the case . He had the really neat idea of thinking that this piece of the parabola might be a quarter circle. That idea made for a great little exploration:
I asked for another idea had he decided to chop the parabola up into rectangles. This isn’t an idea that came out of the blue because we have talked about some intro calculus ideas before. I was still happy to have this idea jump to the front of his mind, though:
Finally, I shared the full Riemann sum calculation with him so that he could see how to arrive at the exact answer of 1/3. This part was not as much an exploration for him as it was just me showing him now to do the sum. I was ok with this approach as there is plenty of time after 8th grade to dive into the details of Riemann sums:
I’m very happy that Strogatz shared these unpublished appendixes. They are yet another great way for kids to see some introductory ideas from Calculus.
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 . 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.
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!
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 🙂
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!
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 🙂