pi = 3.14159… while 22/7 = 3.14285… so 22/7 is bigger. But here's a cute proof that 22/7 is bigger.

The integral that gives 22/7 – pi is surprisingly elegant, and it's clearly positive since you're integrating a positive function. pic.twitter.com/Dj2HJMYRJD

It was an nicely timed tweet for me because my son is beginning a long review of calculus ideas this year. Tonight I finally got around to sharing the idea with him.

We began by talking about some basic properties of the function:

Next we talked about how you could approach integrating the function and then used Mathematica to help with the polynomial division:

Finally, we went to the whiteboard to work through the integral and talk about the nice surprise:

I really like this integral. It is both a neat “fun fact” and a great example to share with kids learning calculus.

The program makes the ideas behind Fourier transformations accessible to kids and I decided to share the program with the boys this morning. So, I had each of them play around with it on their own for about 10 to 15 min. Here’s what they thought was interesting. (sorry for all of the sniffing – I’ve got a cold that’s been kicking my butt for the last few days):

(1) My older son who is in 9th grade:

(2) My younger son who is in 7th grade – it is really fun to hear how a younger kid describes advanced mathematical ideas:

I think Swanson’s program is a great program to share with kids – feels like at minimum it would be fantastic to share with kids learning trig.

At the end of that project a question about finding the volume of a rhombic dodecahedron came up. Since I was going to be out this morning (and my older son was working on a calculus project) I asked my younger son to play around with the Zometool set and see if he could actually find the volume.

Fortunately he was able to – here’s how he described his work:

I saw an interesting tweet from Alex Kontorovich earlier this week:

Our "Crystallographic Sphere Packings" paper (with Nakamura) appeared in @PNASNews last week https://t.co/qDFN3kAroC Here's a geometrized Cuboctahedron and its dual, the Rhombic Dodecahedron, as well as their stereographically projected clusters. (Basically high school math…) pic.twitter.com/MflpHrseiF

We’ve looked at but the Cuboctahedron and the Rhombic dodecahedron before, but I thought it would be fun to revisit the shapes. I also hoped that we’d be able to recreate the shape in the picture with our Zometool set.

So, first we built a cuboctahedron and the boys talked about what they saw in the shape:

At the end of the last video the boys thought that the dual of the cuboctahedron would possibly also be another cuboctahedron. Off camera we built the dual, and happily were able to recreate the shape from Kontorovich’s shape!

They were a little worried that we didn’t have the “true” dual, but I think they came around to believing that these two shapes were indeed duals:

Definitely a fun project – it is always fun to see what you can make with a Zometool set. Maybe tomorrow we’ll revisit an old project of finding the volume of a rhombic dodecahedron. That’s another project which Zometool really brings a lot to the table.

Yesterday I saw an absolutely incredible talk by Cédric Villani on youtube:

Although the talk is a public lecture and fairly accessible to anyone interested in math, it really isn’t aimed at kids. That said, Villani gives a beautiful description of the flat torus starting around 28:00 that I thought my younger son would find interesting. So, I had him watch that part of the video, then play a few rounds of Pac Man, and then we talked about the ideas. As always, it is really fun to hear a kid thinking through and describing ideas from advanced math:

I’ve been teaching my son calculus this school year with the goal of having him take the BC Calculus exam in May of 2019. We’ve finished most of the course material (the main gap is techniques of integration which I’ve put off a bit) and I had him take the publicly available 2012 BC Calculus exam this past weekend.

He missed 7 questions on the multiple choice part of the exam, and we went over those questions this morning.

Our discussion of the 7 multiple choice problems he missed is below . . . and, dang, the actual questions are a little out of focus in the videos. I’ll upload pics before each question to clear up that little technical glitch:

Question #4:

Here’s our talk through this problem – I think the error here was simply being a little careless during the exam:

Question #14:

Again, I think the error was a little careless during the exam.

Question #20

This question is asking about integration using the technique of partial fractions, so a topic we have not yet covered. I showed him the basic idea of partial fractions and also how he could have estimated the value of this definite integral.

Question #24

I like this question. It touches on integration by parts and also the fundamental theorem of calculus. Even though he left it blank on the exam, he was able to find the solution here:

Question #85

This question is a “related rates” question. It still gave him a little trouble today.

Question #90

I like this question – it hits on several important properties of convergent / non-convergent series. Although he got this problem wrong on the exam, he does a nice job talking through the ideas in the problem here:

Question #92

This question might be my personal favorite on the exam. The underlying concepts are basically even / odd functions and the definition of the derivative.

My son is able to recognize why two of the properties listed in the question do not have to be true, but recognizing why the 2nd property has to be true was still a little out of reach for him. Definitely a fun problem to talk through, though (and listening to my explanation for the solution as I publish this blog post . . . I definitely wish my explanation was a little less careless):

This week learned about the book Experiencing Geometry by David Henderson and Diana Taimina. Unfortunately I learned about the book through people sharing news about David Henderson’s death. But despite the terrible circumstances, the book was captivating.

When I learned that Cornell professor David Henderson had died I bought a copy of the geometry book he wrote with @DainaTaimina . The book came today and I’ve spent the last few hours enjoying it. What an absolute treasure – can’t wait to use some of these ideas with kids. pic.twitter.com/aZXRY9K8QC

This morning I picked an idea from the book to share with the boys. The idea is from chapter 16 and is about drawing a circle through three points in a plane chosen at random.

Here’s the introduction to the problem. My younger son struggled a bit in the beginning to remember the ideas, but they did come to him eventually. That little struggle made me happy that we were looking at these geometric ideas today:

After we’d talked through some of the introductory ideas, I had the boys talk about their thoughts on the geometry in a bit more detail. I was especially happy that my younger son was able to sketch a proof that the perpendicular bisector was equidistant from the two endpoints of a line segment:

I had the boys work through the constructions off camera and then explain what they did. My older son approached the problem through folding:

My younger son worked for about 15 min on his construction – he works in a way that is so much more detailed than me! Here’s his work and his explanation which includes a nice discussion of why the center of the circle is outside of the triangle he drew: