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!

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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 🙂

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!

What do kids see when they see ideas from advanced math?

Saw a really neat tweet from Steven Strogatz tonight:

I thought it would be fun to share it with the boys and just listen to how the described what they saw.

Here’s what my older (8th grade) son said:

Here’s what my younger son said when he saw the video:

It is fun to see ideas from advanced mathematics through the eyes of kids 🙂

Playing with 3d printed versions of shapes theorized by Hermann Schwarz

Saw a neat tweet earlier today about 3d printing, math, and engineering:

I recognized some of the shapes in the article as ones that we’d played with before:

The grey shape displayed in the article is a “made thicker for 3d printing” version of the surface \cos(x) + \cos(y) + \cos(z) = 0. I thought it would be fun to print that shape today and use it for a little project with the kids tonight. Here’s the Mathematica code and what the print looks like in the Preform software:

8 hours later the print finished and I asked the boys to describe that shape plus the gyroid. It is always fascinating to hear what kids see in unusual shapes. My younger son went first:

Here’s what my older son had to say (and he’s starting to study trig, so we could go a tiny bit deeper into the math behind the shape I printed today):

Next we watched the video about the shapes made by Rice University:

After watching the video I asked the boys to talk about some of the things they learned:

Of course, mostly they didn’t want to talk about the shapes – they wanted to stand on them! So much for an 8 hour print and 45 min of trying to clean out the supports . . .

Here’s how the standing went:

Definitely a fun project and a fun way to show kids a current application of both theoretical math and 3d printing!

Having kids play with ” swarmalators”

Saw a couple of neat tweets on a new paper in Nature by Kevin P. O’Keeffe, Hyunsuk Hong, and Steven Strogatz:

It looked like playing with the “swarmalator” program would be a really fun way for kids to experience ideas from current math research even though the math underneath these results is a bit out of reach.

So, this morning we just played. Here’s how I introduced the ideas of the program – the two most important ones are (i) the strength of attraction of similar colors, and (ii) the strength of the desire for neighbors to have the same color (and both of these “strengths” can be negative):

After that short introduction I had my younger son (in 6th grade) play with the program to see what he found:

Next I let my older son (in 8th grade) play:

Finally, to talk about the ideas a bit more we went through 4 of the 5 examples at the bottom of the web page of the program we were using. I had the kids try to guess what was going to happen before we set the coordinates. Here are the first two examples:

Here are the last two examples – in this video the boys are getting the hang of how the program works and have several pretty neat things to say about what they are seeing (and what they expect to see):

We played with the program for about 20 min more after we turned off the camera. This program is definitely fun to play with and it was really fun to hear what the boys were guessing the various different states of the program would look like. Even with just two parameters, the kids really had to think hard to talk about the expected behavior. I think that lots of kids will really love playing around with this program.

Thinking about a math appreciation class

Steven Strogatz had great series of tweets about math education earlier in the week. These two have stayed in my head since he posted them:

Tweet9

Tweet10

I know that last year Strogatz taught a college level course similar to the one he is describing in the tweets. We even used a couple of his tweets about the course material for some fun Family Math activities. For example:

Here’s a link to that set of projects:

Steven Strogatz’s circle-area exercise part 2 (with a link to part 1)

So, thinking back to projects like those got me thinking about all sorts of other ideas you could explore in an appreciation course. At first my ideas were confined to subjects that are traditionally part of pre-college math programs and were essentially just different ways to show some of the usual topics. Then I switched tracks and thought about how to share mathematical ideas that might not normally be part of a k-12 curriculum. Eventually I tried to see if I could come up with a (maybe) 3 week long exploration on a specific topic.  I chose folding and thought about what sort of ideas could be shared with students.

Below are 9 ideas that came to mind along with 30 second videos showing the idea.

(1) A surprise book making idea shown to me by the mother of a friend of my older son:

 

(2) Exploring plane geometry through folding

We’ve done many explorations like this one in the last couple of years – folding is an incredibly fun (and surprisingly easy) way for kids to explore ideas in plane geometry without having to calculate:

Our Patty Paper geometry projects

Here’s one introductory example showing how to find the incenter of a triangle:

(3) The Fold and Cut theorem

Eric Demaine’s “fold and cut” theorem is an fantastic bit of advanced (and fairly recent) math to share with kids. Our projects exploring “fold and cut” ideas are here:

OUr Fold and Cut projects

Here’s one fun fold and cut example:

(4) Exploring platonic solids with Laura Taalman’s 3d printed polyhedra nets

You can find Taalman’s post about these hinged polyhedra here:

Laura Taalman’s hinged polyhedra blog post on her Makerhome blog

And if you like the hinged polyhedra, here’s a gif of a dodecahedron folding into a cube!

dodecahedron fold

Which comes from this amazing blog post:

The Golden Section, The Golden Triangle, The Regular Pentagon and the Pentagram, The Dodecahedron

[space filled in with random words to get the formatting in the blog post right 🙂 ]

(5) An amazing cube dissection made by Paula Beardell Krieg

We’ve also done some fun projects with shapes that I wouldn’t have thought to have explored with folded paper. Paula Beardell Krieg’s work with these shapes has been super fun to play with:

Our projects based on Paula Beardell Krieg’s work

(6) And Paula didn’t just stop with one cube 🙂

(7) Two more of Laura Taalman’s prints

Seemingly simple ideas about folding and bending can lead to pretty fantastic mathematical objects! These objects are another great reminder of how 3d printing can be used to make mathematical ideas accessible.

Here’s Taalman’s blog post about the Peano curve:

Laura Taalman’s peano curve 3d print

(8) Getting to some more advanced work from Erik Demaine and Joseph O’Rourke

As hinted at early with the Fold and Cut theorem, some of the mathematical ideas in folding can be extremely deep:

(9) Current research by Laura DeMarco and Kathryn Lindsey

Finally, the Quanta Magazine article linked below references current research involving folding ideas. The article also provides several ways to share the ideas with students.

Quanta Magazine’s article on DeMarco and Lindsey’s work

The two blog posts below show my attempt to understand some of the ideas in the article and share them with kids. The video shows some of the shapes we made while studying the article.

Trying to understand the DeMarco and Lindsey 3d folded fractals

Sharing Laura DeMarco’s and Kathryn Lindsey’s 3d Folded Fractals with kids

So, these are just sort of ideas that popped into my head thinking about one part of a math explorations class. Feels like you could spend three weeks on folding and expose kids to lots of fun ideas that they’d (likely) never seen before.