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.

Today the second my son was studying in his calculus book was about using first and second derivatives to approximate values of functions.

I had him work through 6 problems in the book – 3 of which had answers in the book so he could check his work. Then we moved on to talking about the problems he did for which he was unable to check the answer.

The first problem involved finding the approximate value of $(1.97)^6$ using a linear approximation:

The next problem was finding the value of $\cos(31.5^o)$ again using a linear approximation:

Instead of working through the 3rd problem, I asked him to use a quadratic approximation to find the approximate value of $\cos(31.5^o)$.

When Cos(x) is larger than 1

My son stumbled on an amazing graph completely by accident the other day. He’s doing some work reviewing trig functions this week and I asked him to just play around with some graphs in Mathematica to get a feel for how Sin[x] and Cos[x] behave. One of the graphs he drew was:

$y = \cos( \sqrt{x})$ from $x = -100$ to $100$:

I certainly wasn’t expecting him to make a graph like this one, but was happy that he did. Yesterday we talked through what was going on.

We started by discussing why the graph seemed so strange:

Now we dove into some of the details – which involve complex numbers and the definition:

$e^{i \theta} = \cos(\theta) + i \sin(\theta)$

as well as the definition of even and odd functions. So, there’s a lot of math to that we need to bring to the table to understand what’s going on in our graph.

Finally, we calculated the exact value of $\cos(7i)$. Again, there’s a lot of advanced math that comes in to the calculation here – but even if some of the math ideas took a bit to sink in, I’d say that all in all it was a good conversation:

3d printing totally changed my approach to talking about trig with my son

For the last two weeks we’ve been playing with this book:

Our most recent project involved one of the pentagon dissections. My son wrote the code to make the shapes on his own. We use the RegionPlot3D[] function in Mathematica. To make the various pieces, he has to write down equations of the lines that define the boundary of the shape. Writing down those equations is a fantastic exercise in algebra, geometry, and trig for kids.

Here’s his description of the shapes and how he made the pentagons:

Next we moved on to talking about one of the complicated shapes where the method he used to define the pentagon doesn’t work so well. I wish I would have filmed his thought process when he was playing with the code for this shape. He was really surprised when things didn’t work the first time, but he did a great job thinking through what he needed to do to make the shape correctly.

Here is his description of the process followed by his attempt to make the original shape (which he’d not seen in two days . . . )

I’m so happy that he’s been interested in making these tiles. I’ve honestly never seen him so engaged in a math project. The original intention of this project was just for trig review, but now I think creating these shapes is a great way to use 3d printing to introduce basic ideas from trig to students.

Nonagon tiles

Last week we did a fun project using a pattern we say in “Ernest Irving Freese’s Geometric Transformations” by Greg N. Frederickson:

Using “Ernest Irving Freese’s Geometric Transformations” with kids

I thought it would be fun to make some of the tiles – especially since my older son is studying trig right now. The tiles finished printing overnight:

Last night my son and I talked about how you could make these tiles, with a focus on the trig and algebra required to define the shapes.

Here’s the introduction to the topic:

Now we talked about how to define the kite shape in the tiling. This involves talking about 40 and 50 degree angles:

Finally, we talked through the last part – finding the final point is pretty challenging. Turns out, though, that we don’t have to find the coordinates of the point because we can write down the equation of the top line pretty easily:

I’ve been happily surprised that 3d printing is a fun way to help kids explore 2d geometry. I’m excited to have my son try to make some other tiles from the book on his own for our next project.

An attempt to share some Katherine Johnson’s math ideas from Hidden Figures with my son

For the last few months I’ve been daydreaming about ways to share some of the math from the movie Hidden Figures with kids. As part of that prep work I found one of Katherine Johnson’s technical papers on NASA’s website:

NASA’s Technical Note D-233 by T. H. Skopinski and Katherine G. Johnson

As you’d expect, there’s a lot of trig, calculus and spherical geometry. I like finding ways to share the work that mathematicians do with kids, but this work is pretty technical and I wasn’t getting any great ideas.

Then my son had a homework problem from his Precalculus book that made me think it was time to stop daydreaming and just try something. Here is that problem, which is a completely standard law of cosines problem:

The problem reminded me of one of the equations for an ellipse used in the Technical Note. One surprising thing is that the equation of an ellipse in polar coordinates is that is is a rational function in $\cos{\theta}$.

So, I drew an ellipse and showed my son that equation.

One of the neat things about the Technical Note is that the solution to some of the complicated trig equations were found by an iteration method. The specific ideas for solving those equations are too advanced for kids, so I decided to show my son a different (and really simple) iteration method that converges to a well known number:

After that introduction to iteration methods, I decided to jump to a second and slightly more complicated example -> solving x = 3*x*(1 – x).

The ideas in the iteration method we use here can be explored purely geometrically:

Next we went upstairs to the computer to see some of the ideas we just talked about. The first idea was the polar coordinate equation for an ellipse:

Now we played with the second dynamical system -> solving x = 3*x(1-x).

By the way, the ideas here are incredibly fun to explore (especially seeing when this method converges and when it doesn’t), but the details of this method wasn’t really the idea here. I just wanted to show him what an iterative method looks like.

Finally, I showed him the actual paper and pointed out some of the parts we explored. Sorry that this film didn’t come out as well as I’d hoped, but you can view the paper from the first link in this post:

This was a fun project – even if it wasn’t planned really well. Showing some of the math behind Hidden Figures I hope helps motivate some of the topics that my son is studying right now. It will be fun to return to a second Hidden Figures project when he is studying calculus.

A fun math surprise with a 72 degree angle.

We’ve been talking a lot about 72 degree angles recently. Yesterday’s project was about a question our friend Paula Beardell Krieg asked:

Paula Beardell Krieg’s 72 degree question

In that project we learned that a right triangle with angles 72 and 18 (pictured below)

Is nearly the same as a right triangle with sides of 1, 3, and $\sqrt{10}$

Today I wanted to show the boys a neat surprise that I stumbled on almost by accident. The continued fraction expansion for the cosine of the two large (~72 degree angles) are remarkable similar and lead to the “discovery” of a 3rd nearly identical triangle.

We got started by reviewing a bit about 72 degree angles:

Now we did a quick review of continued fractions and the “split, flip, and rat” method that my high school teacher, Mr. Waterman, taught me. Then we looked at the continued fraction for $1 / \sqrt{10}$:

Now we looked at the reverse process -> given a continued fraction, how do we figure out what number it represents? Solving this problem for the infinite continued fraction we have here is a challenging problem for kids. One nice thing here was that my kids knew that they could do it if the continued fraction had finite length – that made it easier to show them how to deal with the infinitely long part.

Finally, we went to the computer to see the fun surprise:

Here’s that 3rd triangle:

I love the surprise that the continued fractions for the cosine of the (roughly) 72 degree angles that we were looking at are so similar. It is always really fun to be able to share neat math connections like this with kids.

Paula Beardell Krieg’s 72 degree question

A few weeks ago I got this question from Paula Beardell Krieg on Twitter:

Today I went through this problem with the boys – the difficulty of this exercise surprised me a bit. They really struggled to see how you could tell if an angle was 72 degrees.

Here’s the introduction. The boys noticed a few things about the picture and got some ideas with how to proceed:

Next we drew the two squares on a piece of paper and I let the boys explore the question. Here they struggled to make much progress beyond the things that they noticed in the first part of the project:

The thing giving them trouble was that they didn’t know any relationships between angles in a right triangle with a 72 degree angle. That left them completely stuck. Eventually they decided to measure the squares and found that they had something that looked like a triangle with side $1, 3,$ and $\sqrt{10}$.

Next we explored some of the ideas around $1, 3, \sqrt{10}$ triangles. After a little nudging from me they decided to measure the angles with a protractor.

Now I showed them my solution and let them see where the $1, 3, \sqrt{10}$ triangle comes up:

Finally, I let them play with two sets of triangles that I printed overnight. Two of these triangles are right triangles with 72 and 18 degree angles, and two of them are $1, 3, \sqrt{10}$ triangles. The question is -> are all 4 triangles the same?

Here are pictures (to scale) of the two triangles. You can see how similar they are.

First, the right triangle with a 72 degree angle:

Second, the $1, 3, \sqrt{10}$ triangle:

Tomorrow we’ll explore a second similarity between these two triangles. I found it playing around while I was making the triangles yesterday ðŸ™‚

Playing with Cos(72) -> part 2

Last week we used an old AMC problem to explore Cos(72):

That project is here:

Finding cos(72)

Today we built a decagon with our Zometool set to see if we could approach the problem a different way:

I started by having the kids explore the decagon and having my older son explain where cos(72) and cos(36) were (roughly) on the shape:

Next we used a T-square to try to get good approximations for both Cos(72) and Cos(36). The T-square + Zometool combination was a little harder for the kids than I was expecting, but we got there.

Finally, I wrapped up with a challenge question for my older son. If we know that Cos(36) – Cos(72) = 1/2, find the value of Cos(36). He did a nice job working through this problem:

I’ve enjoyed playing around with properties of angles that arent usually part of the trig curriculum. We might have one more project on 72 degrees this weekend – I’m thinking of playing with the idea that Tan(72) is close to 3, but haven’t quite figured out that project yet.

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.