Because of a careless mistake I made last week asking my son to read a calculus-based section in the statistic book he’s studying, I’m doing a few light touch calculus projects with him. Today’s will likely be the most computationally heavy one as my main goal is simply to show him the ideas.
He mentioned to me yesterday that in his class at school they are studying graphing parabolas. I decided to use a quadratic for today’s calculus example. First I asked him to graph it. As the video shows, I probably should have asked him how they were approaching the problem in his class at school first!
Next we spent a few minutes finding the tangent line to our parabola at the point (0,3). Here we talk about what the tangent line is:
Here we finish the tangent line calculation:
Next we moved on to finding the area under the curve. We’d discussed a similar example earlier in the week, so I thought this part of the project would make for a nice review of that prior talk. We ended up diving a little deeper than I intended, but I still think it was a good discussion.
First we talked about the proof that e is irrational. My younger son saw this idea as an exercise in the number theory book he’s working through right now. The proof is accessible to kids, though a bit more difficult than some of the other proofs of irrationality the boys seen before:
Next we moved to the idea in Nassim Taleb’s tweet. The idea that and are so close together is a really important idea from calculus and the general idea has many important applications:
Finally, we looked at the tweet from Sonia and discussed the simplified mathematical problem in the tweet and the surprising relationship to e:
I think these three ideas are fun ones for kids to see. The proof that e is irrational is something that I’m pretty sure I didn’t see until college, but is definitely accessible to kids. The other two ideas are really important ideas from calculus and probability and definitely worth exploring many times!
Today I had the boys read the article and we talked through several of the ideas they thought were interesting. Here are their initial thoughts and also their thoughts about how you would count the excess deaths from the graph shown in the cover pic from the article:
My younger son mentioned two ideas that caught his eye in the article – the difference between Republican / Democrat states and the difference in outcomes with large and medium lockdowns. We talked about those ideas here:
My older son had two things that he thought were interesting – the reporting delays and how the article counted the excess deaths vs the corona virus deaths:
Following those discussions we downloaded some data from the CDC’s website to see if we could match the Washington Post’s numbers. We could for Massachusetts, but were off by a bit for Indiana. Not sure why – the trouble of filming this stuff live – but the main ideas was just to show the boys how to check the numbers in articles like these (and why checking is important):
This was a fun project – I think the analysis of excess deaths is a helpful way to understand how bad the pandemic is. I’m glad the Washington Post published this article.
There’s been a lot of laughing / crying about cubic models in the last few days, but I thought talking about modeling could make a nice lesson for my son who is reviewing calculus. Then I saw a great tweet from Carl Bergstrom that made me want to give it a go.
First we talked about cubic polynomials in general and what we can learn about these curves from calculus:
Next we talked about fitting a cubic polynomial to data. We have talked a bit about fitting curves before – my son mentioned this project on fitting temperature data which used some amazing work from John Shonder on looking at temperature changes in each US county for the last 100 years:
In this video I asked him for his ideas about why fitting data with a cubic curve might lead to problems:
Next we moved on to looking at the cubic model that was published and focused a bit on the fascinating tweet below from Carl Bergstrom. I think Bergstrom’s tweet is a great lesson for calculus students because he’s noticing that the “cubic fit” can’t be a 3rd degree polynomial because the 2nd derivative isn’t right:
So if it's a third degree polynomial fit, how they get a + / – / + pattern of second derivatives?
(Not being sarcastic; the dotted part of the pink line might just be hand-drawn)
Finally, we looked at a few cubic and log-cubic fits to corona virus deaths in the US. I showed him that forcing a cubic fit to the data ended up with some strange results. Finally, I asked him why those strange results might be coming from the cubic fit (which is a pretty hard question for a 10th grader):
This was a nice project – and an especially nice one to show a calculus example directly related to current events.
Our schools have been closed for the last 10 days. During that time I’ve been taking a little break from having my older son work on problems and am having him read Steven Strogatz’s Infinite Powers instead.
For our project tonight I asked him to pick out three chapters that he’s liked so far to discuss. He chose chapters 2, 3, and 10. As a quick note before diving in to his thoughts on these chapters, he studied calculus last year so I was having him read Strogatz’s book for history and context rather than as an introduction to calculus.
Here’s what he had to say about Chapter 2 which is about Archimedes:
Here’s what he had to say about Chapter 3 which is about Galileo and Kepler:
Finally, here’s what he had to say about Chapter 10 which is about Fourier:
I think he’s gotten a lot out of Strogatz’s book, and I was really excited to learn that he thought Fourier’s work was interesting. Maybe the Who is Fourier book really is the next right step for him.
I saw a great Twitter thread on virus spreading models from Nassim Taleb last week. I’d been meaning to share the ideas in the thread with my kids but didn’t get around to it until today.
The original thread is here:
UK Policy is a speculative lunacy.
Playing with the toy standard epidemiological SIR model. We have no idea how model parameters cause a yuuuge variation in ourcomes. We don't even know the central parameters/whether stochastic. Try to add perturbations for "herd immunity". pic.twitter.com/fTJ7pWRlaT
The tweet I wanted to focus on specifically is here:
So, tonight I used the code that Taleb shared and talked through the graphs with the boys. At the end we talked a bit about why Taleb’s conclusion was that these models were unreliable for decision making.
Here’s how my younger son reacted to the graphs:
Here’s how my older son reacted to Taleb’s graphs:
I think talking through some of Nassim Taleb’s ideas is a great way for kids to get some insight into how to think about the virus spread and also to see some of the dangers / limitations of modeling. For today’s project the important lesson is when you don’t know with any certainty how the models work, you really need to proceed with maximal caution.
Today I had my older son study the animation for a few minutes and then we talked about about some of the ideas it is illustrating:
Now we pulled up the tweet and had him talk about what he was seeing. It is always fun to see advanced math ideas through the eyes of kids:
Finally, we played around with the ideas in Mathematica. Although we didn’t replicate the incredible animation that Berger did, we found some fun mathematical ideas to talk through.
Berger’s tweet is great for sharing with students learning (or reviewing) ideas in calculus. It is also really fun to use to hint at some of the amazing ideas that’ll come later when students study complex numbers in more detail.
Today I had my son read through the post and then we discussed the ideas. His initial thoughts are in the video below – he understood most of the post and also had a couple of good questions:
After we talked through the post we went to Mathematica to take a look at some of the example “reflections” which preserve the 1st and 2nd derivatives:
One of my son’s questions in the first video was why the blog post was using functions like f(x/2) and f(x/4) to make reflections. I’d mentioned that these were essentially arbitrary choices. Below we saw what would happen if we used f(x/3) instead:
We finished up by going back to Mathematica to see what these new “reflections” would look like:
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:
I had originally included two appendices in Infinite Powers, but they felt too mathy and got cut. Still, some of you might find them interesting. Here's a rough draft of appendix 1, about a clever way that Fermat integrated x^n with his bare hands [PDF]: https://t.co/gutuj0R38A
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.