Introduction to infinite series

We’ve been studying series and sequences for a few weeks (with about a 1 week interruption for Thanksgiving) and we finally had time to do a short project tonight.

I started off tonight’s project by asking my son to talk about the Harmonic series and give two different “proofs” for why it diverges:

Next I pulled one of the exercises out of his book. This series is a slightly difficult integral – and we’ve skipped the techniques of integration section for now – so this part was a nice integration review, too:

Following the approach via the integral test, I wanted to show how the comparison test works. The comparison test is actually the topic for tomorrow, so this was new material. He did a nice job thinking through how the test would work:

Finally, I picked a random problem from the book and asked him to work through it. It turned out to be a pretty neat problem asking why the integral test wouldn’t work on a particular series:

It is fun going through sequences and series with my son. It is going to take a bit longer than I initially thought, so we’ll probably be working through the ideas here through mid December. We’ll come back to techniques of integration after that.

Having kids play with the Binary Black Hole explorer made by Vijay Varma, Leo Stein, and Davide Gerosa

Last night I saw an incredible tweet thanks:

Here’s the link in the tweet just in case WordPress doesn’t display everything properly:

The binary black hole explorer

The link goes to the “Binary black hole explorer” made by Vijay Varma, Leo Stein, and Davide Gerosa, and it is one of the most amazing computer visualizations I’ve ever seen!

Even though the math and physics going in in the background is way way way too advanced for kids, I thought it would be fun to hear them describe what they were seeing. Imagine having the opportunity to see simulations of rotating black holes when you were in 7th and 9th grade!!

My younger son went first:

My older son went next – we’ve just finished up the section in his calculus book on parametric equations and polar coordinates, so I thought he’d find these simulations to be especially interesting:

I loved showing these simulations to the boys. Even if they can’t totally understand what’s going on, it sure is a nice peek at what can come down the road if they find physics to be interesting.

And, as always, it is so fantastic to see scientists sharing amazing work like this on twitter!

A fun experiment sharing Grant Sanderson’s Topology video with a kid

I saw a really neat new video from Grant Sanderson this morning:

We’ve actually looked at the ideas Grant is sharing here before, but my son didn’t remember:

Grant Sanderson’s “Fair Division” video shows a great math project for kids

For today I asked my younger son (in 7th grade) to watch the video and take some notes. After he finished we started taking about what he saw. He was interested in the Borsuk–Ulam theorem and also he thought the “stolen necklace problem” was pretty neat:

Next we talked about the proof of the Borsuk-Ulam theorem. I was really happy that most of the main ideas that Grant shared in his video stuck in my son’s mind.

We wrapped up by talking about the “stolen necklace” problem. We did a few examples about that problem and then had a fun discussion about the equation for a sphere. My son was curious about the difference between the boundary of the sphere and the all of the points inside the sphere. In particular, he was wondering why the equation for a sphere Grant used was x^2 + y^2 + z^2 = 1 and not x^2 + y^2 + z^2 \leq 1

From there we had an interesting discussion about dimension. I didn’t expect the conversation to go in that direction, but I guess you never know what a kid is going to take away from a video about some pretty advanced math ideas 🙂

Exploring Wilson’s theorem with kids inspired by Martin Weissman’s An Illustrated Theory of Numbers

I love Martin Weissman’s An Illustrated Theory of Numbers:

Flipping through it last night I ran into an easy to state theorem that seemed like something the boys would enjoy exploring:

Wilson’s Theorem -> If p is a prime number, then

(p-1)! \equiv -1 \mod p.

The proof is a bit advanced for the kids, but I thought it would still be fun to play around with the ideas. There is an absolutely delightful proof involving Fermat’s Little Theorem on the Wikipedia page about Wilson’s Theorem if you are familiar with polynomials and modular arithmetic:

The Wikipedia page for Wilson’s Theorem

Here’s how I introduced the theorem. We work through a few examples and the boys have some nice initial thoughts. One happy breakthrough that the boys made here is that they were able to see two ideas:

(i) For a prime p, (p-1)! is never a multiple of p, and

(ii) There were ways to pair the numbers in (p-1)! (at least in simple cases) to see that the product was -1 \mod p

Next we tried to extend the pairing idea the boys had found in the first part of the project. Extending that idea was initially pretty difficult for my younger son, but by the end of this video we’d found how to do it for the 7! case:

Now we moved on to study 11! and 13! At this point the boys were beginning to be able to find the pairings fairly quickly.

To wrap up this morning, I asked the boys for ideas on why we always seemed to be able to pair the numbers in a way that made Wilson’s theorem work. They had some really nice ideas:

(i) For odd primes, we always have an even number of integers – so pairing is possible,

(ii) We always have a product 1 * (p-1) which gives us a -1.

Then we chatted a bit more about why you could always find the other pairs to produce products that were 1. The main focus of our conversation here was why it wouldn’t work for non-primes:

Definitely a fun project! There’s some great arithmetic practice for the boys and also a many opportunities to explore and experience fun introductory ideas about number theory.

What a kid learning algebra can look like

Section 9.2 of Art of Problem Solving’s Introduction to Algebra is one of my favorite sections in any book that my kids have gone through. The section has the simple title – “Which is Greater?”

One question from that section that was giving my younger son some trouble today was this one:

Which is greater 2^{845} or 5^{362}

I decided our conversation about the problem would make a great Family Math talk, so we dove in – his first few strategies to try to solve the problem resulted in dead ends, unfortunately. By the end of the video, though, we had a strategy.

Now that we’d found that 2^7 and 5^3 are close together, we tried to use that idea to find out more information about the original numbers.

I found his idea of approximating at the end to be fascinating even if it wasn’t quite right. It was also interesting to me how difficult it was for him to see that the two numbers on the left hand side of the white board were each bigger than the two corresponding numbers on the right hand side of the board. It is such a natural argument for someone experienced in math, but, as always, it is nice to be reminded that arguments like that are not obvious to kids.

A really neat way for kids to experience prime and composite numbers

Yesterday I saw an incredible program shared on twitter:

I thought it would be really fun for kids to watch this program play out and describe what they saw, so I shared it with my own kids. Here’s what my 7th grader thought:

Here’s what my 9th grader thought:

I really love this program and especially love how kids can use it to experience some simple ideas in number theory.

Using an idea from one of Katherine Johnson’s NASA technical papers to introduce polar coordinates

Yesterday we did a fun project on parametric equations and touched on the motion of planets at the end:

Playing with parametric equations in Desmos

Even though the discussion of planetary motion in the last project wasn’t even close to a complete picture, the kids seemed pretty interested in it, so I continued with that idea today. The goal for today was to show them why the ideas we talked about yesterday weren’t quite right and how we could use polar coordinates to study the same ideas.

The main idea I drew on for today’s project came from one of Katherine Johnson’s technical notes on NASA website:

Screen Shot 2018-11-11 at 12.01.17 PM

The link to that paper is here:

NASA Technical Note D-233 by T. H. Skopinski and Katherine G. Johnson

and the specific equation I’ll be using from that paper is equation 1a which gives an equation for an ellipse in polar coordinates:

Screen Shot 2018-11-11 at 12.01.37 PM

I’ll also be drawing from some lecture notes on planetary motion I found via a google search:

Richard Fitzpatrick’s lecture notes on planetary motion on the University of Texas website

I started the project by introducing polar coordinates – sort of a high level conceptual introduction for my younger son and a few more details for my older son:

Next I showed the them the polar coordinate description of an ellipse from the Katherine Johnson paper and how if we applied the velocity and acceleration ideas from yesterday that we’d see the acceleration wasn’t always directed at the same point:

Next we looked more carefully at the movement around the ellipse and (after a while) saw that the line from the ellipse back to the origin was moving with a constant angular velocity.

The boys were able to explain that the movement of planets in an elliptical orbit probably wouldn’t have a constant angular speed:

Finally, we used the paper from the University of Texas to explore the ellipses corresponding to the orbits of various planets.

Definitely some neat ideas to share with kids and also a fun way to bring some important pieces of math history to life!

Playing with parametric equations in Desmos

I saw two neat ideas about parametric equations in Desmos during the last week. First from Mr S. on twitter:

And then later from Patrick Honner:

So, I modified the Desmos program that Mr. S. shared to show velocity and acceleration and asked the kids what they saw in the animation. Here we are looking at the parametric curve defined by the equations (4\sin(4t),3\cos(3t)

(When you watch the videos, keep in mind that my older son has been studying parametric equations in calculus but my younger son has essentially never seen them)

Next I asked my older son to pick a new set of equations and we looked at what the velocity and acceleration vectors looked like now:

Now my younger son picked some new equations – this time there was a lot of wiggling around!

Finally, I wrapped up by showing them a fun little surprise – what velocity and acceleration look like for an ellipse. This example shows what’s going on with planetary motion.