Finding Cos(72)

My older son is learning trig out of Art of Problem Solving’s Precalculus book this year. Yesterday he was working on the “sum to product” section, which derives rules for expressions like Cos(x) + Cos(y). It reminded me of one of my all time favorite math contest problems:

Today I thought I would show him my solution to that problem. What we go through probably isn’t the best or easiest solution, but I think it is an instructive solution for someone learning trig.

We started by talking about the problem and how some of the ideas he was currently learning could help solve it:

At the end of the last video we’d found a nice equation that we derived from the original problem:

\cos(36^o) - \cos(72^o) = 2 \cos(36^o) * \cos(72^o)

Now we used the double angle formula to simplify even more and find a cubic equation satisfied by Cos(36):

Now we tried to find the solutions to the cubic equation we found in the last video. This part gave my son a bit of trouble, but he eventually got there.

Now we were almost home! We just had to compute the value of Cos(72) and we’d be able to solve the problem. That involved one last application of the double angle formula:

I think solving this problem from scratch would be far too difficult for just about any kid just learning trig. But, the fun thing about this problem is that the ideas needed to solve the problem are all within reach using elementary trig identities. So, I think that working through the solution to this problem is a nice exercise for kids.

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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.

Finding e by throwing darts

[had to write this a little more quickly than usual, sorry for typos / bad writing]

A few weeks ago Nassim Taleb posted a neat probability problem that let do several neat followups:

That post led to a fun project:

Sharing Nassim Taleb’s Dart Probability Problem with Kids

Alexander Bogomolny followed up with a problem about throwing 16 darts onto a grid with 16 boxes:

and yesterday Steve Phelps showed how one of his students approached Bogomolny’s problem:

Phelps’s post got me thinking about the expected number of empty squares. So I started playing around and remembered (about an hour later!) that we had done a project for kids on that exact topic before based on this John Allen Paulos tweet:

A fun way to estimate e with kids

So, today we revisited that old project. We started by writing a computer program to pick 64 random integers from 1 to 64 and I had the boys plays snap cubes on a chess board on the positions given by the numbers.

Here’s what the boys had to say about this particular random placement of snap cubes:

Next we returned to the code and talked through it. We then looked at what the maximum number of snap cubes on a square was in a few (about 10) different runs.

Now we looked at the number of squares with zero cubes and the number with 1. My older son guessed that the number of squares with zero cubes would be roughly equal to 1/3 of the squares.

Finally, overnight I’d run a program to find the expected number of squares covered. Well, technically something slightly different -> the number of squares that has a 50% probability of being covered.

I then compared that number to (1 – 1/e) times the number of squares and found that the two numbers were nearly the same. That means the number of empty squares is roughly 1/e times the number of squares!

Definitely a fun project for kids. It is a great “hands on” way – and a non-calculus way – to see the number e.

The cube root of 1

After a week of doing a little bit of practice for the AMC 8, my older son has returned to Art of Problem Solving’s Precalculus book. The chapter he’s on know is about trig identities.

Unrelated to his work in that book, the cube root of 1 came up tonight and he said “that’s just 1, right?” So, we chatted . . .

First we talked about the equation x^3 - 1 = 0:

For the second part of the talk, we discussed the numbers \frac{1}{2} \pm \frac{\sqrt{3}}{2} and their relation to the equation e^{i\theta} = \cos(\theta) + i\sin(\theta)

Finally, I connected the discussion with the double angle (and then the triple angle) formulas that he was learning today. You can use the same idea in this video with \cos(5\theta) to find that \cos(75^{o}) = (\sqrt{5} - 1)/4:

So, a lucky comment from my son led to a fun discussion about some ideas from trig that he happened to be studying today 🙂

Playing with Cos(z)

​We did a neat project with Newton’s method yesterday:

Exploring Newton’s Method with kids

That project plus the fact that my son is starting to learn a bit of trigonometry got me thinking about what the transformation z -> Cos(z) “looks like.”

During the day today I printed the real and imaginary parts of this transformation:

After printing those shapes I had a different idea – I’d look at how the function z -> Cos(z) mapped circles centered at the origin. To see the images of different circles, I put the circle with radius R at a height R in the map. That ended up being a bit too squished, though, so I changed the height in the image to 3R. Here’s what it looked like in Mathematica:

[ I’m having a little trouble with the videos below. Maybebecause I took them on my phone – not sure – but hopefully they at least show the shape and a few of the ideas my kids had.]

Here’s what the 3d print looked like:

and here’s how the boys described the shape. My younger son went first:

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My older son went second

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Fawn Nguyen’s Olive problem

Last night Fawn Nguyen posted a neat problem for kids:

I thought it would be fun to try out the problem with the boys this morning. My younger son went first (while my older son was practicing viola in the background). He described his approach as “guess and check”:

My older son went next. I think we wasn’t super focused at first because Fawn’s problem about 75 olives became a problem about 40 apples, but once he got back on track he found a nice solution. His approach looked at the number of non-green olives since that number stayed constant:

It was fun to see the two different approaches, and also interesting to see that my two kids approach percentages very differently. This was a nice problem to start the day.

Exploring Newton’s method with kids

Yesterday we had about a 30 min drive and I had the boys open up to a random page in this book for a few short discussions in the car:

There were some fun topics that were accessible for kids, but then Newton’s method came up. Ha ha – not really drive time talk 🙂

It did seem like it could be a fun project, though, so I took a crack at it today. The goal was not computation, but mainly just the geometric ideas. Here’s how we got started:

Next I asked the boys if they could find situations in which Newton’s method wouldn’t work as nicely as it did in the first video. They were able to identify a few potential problems:

Now I had both kids draw their own picture to play out what would happen when you used Newton’s method to find roots. I think there’s a lot of ways to used the exercise here to help older kids understand ideas about tangent lines and function generally. I mostly let the kids play around here, though, and the results were actually pretty fun:

Finally, we went to Mathematica to see some situations in which Newton’s method produces some amazing pictures. Here we switch from real-valued functions to complex valued functions. Since I wasn’t going into the details of now Newton’s method works, rather than using some easier to understand code, I just borrowed some existing code from here:

The page from A. Peter Young at U.C. Santa Cruz that gave me the Newton’s method code for Mathematica

The boys were amazed by the pictures. For example, (and this is one we looked at with the camera off) here’s a picture showing which root Newton’s method converges to depending on where you start for the function f(z) = z^3 - 2z + z - 1:

Cubic

Definitely a fun project. Even if the computational details are a bit out of reach, it is fun to share ideas like this with kids every now and then.

Counting paths

This week both kids had a homework problem in their enrichment math program that involved counting different paths on the edges of a cube.

I thought it would be fun to use those problems as a way to visit the ideas of counting paths in a lattice.

I started the project with a pretty standard path counting problem -> counting the number of paths that go from corner to corner in a rectangular lattice:

Nice I changed the shape of the lattice and ask the boys how they thought the number of paths would change:

Now we moved on to a problem that was similar to the problem they had for homework -> count the paths going from one corner to the opposite corner on a cube:

Now for a challenge -> count the paths on a 2x2x2 cube going from one corner to the other (in this case each step will have length 1):

This is a fun introductory counting exercise. I was a little surprised how difficult it was to keep track of the numbers on the final 2x2x2 cube, but it was nice to see that they boys could see how to count those paths directly with choosing numbers.

Struggling through an AMC 8 problem

My younger son has been practicing for the AMC 8. This week we’ll be going over a few problems here and there that give him trouble. The problem from the practice test today was #16 from the 2016 AMC 8:

Problem 16

This problem really gave him some trouble – as you’ll see from his 5 min struggle below:

I was caught a bit by surprise over the difficulty he was having. It wouldn’t surprise me if the mistakes he was making were quite common mistakes for a problem like this, but I was stuck on what to do. So, I decided to show him one path that leads to the solution to the problem:

So, having shown him one way to solve the problem, I challenged him to find a different solution. Initially he struggled, but then he did something pretty clever:

I definitely struggle to see a good way forward when a problem is giving one of my kids as much trouble as this one was. Hopefully my son was able to see some of the important ideas in the problem after we talked through it in the second video. I really do like the solution he came up with in the third video, though, especially since it is more geometric and less reliant on calculation.

Sharing Sam Hansen’s “Knotty Helix” podcast with kids

The latest Relatively Prime podcast is fantastic:

The short description from the podcast’s website is:

“Sure DNA is important, some might even claim it is absolutely integral to life itself, but does it contain any interesting math? Samuel is joined by UC-Davis Professor of Mathematics, Microbiology, and Molecular Genetics Mariel Vazquez for a discussion proves conclusively that mathematically DNA is fascinating. They talk about the topology of DNA, how knot theory can help us understand the problems which occur during DNA replication, and how some antibiotics are really pills of weaponized mathematics.”

Since it is only 20 min long, I thought it would be fun to share with the boys. We listened to it in the car when we went out to breakfast this morning. Upon returning home I asked the kids what they thought and what were somethings they learned. Here’s what they had to say:

Next we looked at an interesting process described in the podcast. That process was an example by Mariel Vazquez of how you can go from a link with 6 crossings to two unlinked circles.

The process in the podcast seems simple – maybe even obvious – but I think that the process is actually much more subtle than it seems listening to it.

Here we followed the steps to go from the trefoil knot to the two unlinked circles. I think the ideas we followed here are a great way for kids to explore the process described in the podcast:

The next thing we looked at was the idea that when you cut a loop with an even number of twists in half, the halves would be linked. We took a long strip of paper and gave it 6 twists, taped the ends together, and then cut it down the middle.

I fast forwarded through the taping and cutting part, but forgot to remove the sound. Sorry for the “Alvin and the Chipmunks” middle part of this video.

I think both the podcast and the follow up projects are a great way for kids to explore some math ideas that wouldn’t normally be part of a school curriculum.

We’ve done a few other projects with knots and with paper cutting. Here’s a link to those collections:

Our knot projects

A collection of some of our paper cutting (and folding) projects

It was really neat to hear about how knot theory applies to biology.