# Revisiting the angle sum arctan(1/2) + arctan(1/3)

Today we did a 3d printing project revisiting an angle sum that we’d looked at last week -> arctan(1/2) + arctan(1/3).

We started by reviewing how to approach the sum using complex numbers:

Next my older son explained a geometric way to approach the problem:

Now we went to Mathematica to create the 4 triangles using the RegionPlot3D function. It is a nice geometry exercise to have kids describe the boundary of a simple 2d object:

At the end of the day I had my younger son use the shapes to assemble the 3×2 rectangle and describe how this arrangement showed that the original angles added up to 45 degrees:

I like using 3d printing to help kids see math in a different way. The problem today was originally inspired from a section on complex numbers in Art of Problem Solving’s Precalculus book. It was nice to be able to use it to explore a little bit of 2d geometry, too.

# What a kid learning trig can look like

My younger son is studying in Art of Problem Solving’s Precalculus book this year. Right now he’s looking at some of the trig problems in

(1) The first problem asks you to prove that the area of a triangle is A*B*C / 4R, where A, B, and C are the side lengths and R is the radius of the circumscribed circle:

(2) The second problem asks you to prove that in an acute angled triangle that:

b = c Cos(A) + a Cos(C), where a, b, and c are the side lengths of the triangle and A and C are the angles opposite sides a and c.

(3) The third problem is Tan(A/2) = r / (S – A), where A is the angle opposite side A, r is the radius of the inscribed circle, and S is half the perimeter of the triangle.

(4) The final problem is pretty difficult -> you are asked to prove this identity:

It takes 10 min for my son to work through this problem, including a couple of false starts. But he gets to the end, which made me really happy:

# Sharing Jez Swanson’s amazing Fourier transformation program with kids

I saw an incredible tweet from Jez Swanson yesterday:

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.

# A fun introductory Taylor series exercise with Sin(x) I saw on Twitter last week

Last week I saw a neat “fun math fact” via a Matt Enlow tweet:

By happy coincidence my older son is studying Taylor Series this week, so I thought it would be fun to talk through the problem.

Here’s the introduction:

My son had some nice ideas about how to approach the problem in the last video, so next we went to the white board to work out the details:

Finally, I asked my son to finish up the details and then asked him for a sort of number theory proof of why 180 multiplied by an integer with all digits equal to 5 was always close to a power of 10:

Definitely a fun little problem – definitely accessible to students learning some introductory calculus.

# 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:

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:

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!

# Struggling through a related rates problem

We’ve been talking about ideas in the related rates chapter in our calculus book for the last few days. My son has struggled with the ideas in this chapter much more than I expected he would. Fortunately many folks on twitter who have much more experience teaching calculus than I do have told me that this section often gives students a lot of trouble.

Over the last few days my son had worked through maybe 15 of the problems in our book – so I just picked one that he hadn’t done yet. Here’s the problem:

Two sides of a triangle have lengths 12 m and 15 m. The angle between them is increasing at a rate of $2^o$ per minute. How fast is the length of the third side increasing when the angle between the sides of fixed length is $60^o$

Here’s the introduction to the problem and his initial thoughts:

(oh, and I should say at the outset, we’d not looked at this problem before and I didn’t realize that we needed a calculator, so I just banged out a few numbers on my computer just to speed things along. Most of my calculations were right, but one at the end of the 2nd video was wrong. Whatever I typed in produced the answer of 0.2 m / min, but if I’d actually typed in what we had on the board I would have found 0.4 m / min. That typo by me led to a little confusion in the 3rd video when we went to check our answer. Sorry about that error.)

Here’s the next part of his work.

There’s one part of this problem that we’d not really talked about carefully yet -> the angle numbers in the problem are given in degrees, but in calculus you need to be using radians. Since the main focus of this problem / session is related rates, I just explained that fact quickly in the video.

Again, sorry for the typo by me in evaluating the value of the answer.  Fortunately we decided to check the answer – which was also an important conceptual calculus exercise.

Here we checked the answer from the last video and found the change was double what we were expecting. This was unexpected!

We checked the prior math off camera and found the typo. Luckily it was easy to find since we’d filmed the project. Hopefully this was a nice (and accidental) way to show that checking your answers is important 🙂