This week I’ve been doing a fun 3d printing project with my younger son who is learning trig (from Art of Problem Solving’s Prealgebra book). We have used 3d printing to explore 2d geometry before – see some of the projects here, for example:
3d Printing ideas to explore math with kids
Exploring Annie Perkins’s Cairo Pentagons with kids
Evelyn Lamb’s Pentagons are Everything!
This week I had my son create, code, and then print some simple 2d shapes – the project combines ideas from trig, geometry, and algebra.
Here’s his description of the first shape -> a 3-4-5 triangle:
Here’s the 2nd shape – a 7-6-3 triangle. Creating this shape shows how ideas from introductory trigonometry come into play:
Finally, here’s a regular pentagon that we made yesterday. Unfortunately we made a mistake in the code for the print – mixing up a Sin() and a Cos(), but here is explanation of how to make the shape is correct:
I’d forgotten how useful 3d printing can be as a tool to explore 2d geometry – this week was a happy reminder of how fun those activities can be!
My younger son is working through Art of Problem Solving’s Precalculus book this year. Today we were working through some problems in chapter 4 on right triangle trig. He had been working through this section on his own while I was traveling last week.
One of the problems struck me as one that would make a nice project – the problem gives two lengths of a triangle plus an angle and asks you to calculate the area.
Since the angle is 30 degrees, his solution to the problem did not really use trig:
Now I changed the angle to 40 degrees. My expectation was that this change would not produce a solution that was all that different – wow was I wrong.
This school year my younger son is working through Art of Problem Solving’s Precalculus book. This week we started chapter 4 -> Applications to Geometry. The first section is on right triangle trig.
As we started working through the example problems, I was struck by technique my son was using to work through the problems. Once he finished, we went back through one of the problems on camera to share the technique.
Here’s the problem and how he got started:
Next he dove into the solution. His technique is to first draw a right triangle with a hypotenuse of 1 and then scale up:
Finally – here’s how he solved the equation he found in the last video (with some not so great calculator help from me . . . )
My younger son is studying trig right now and I thought it would be fun for him to play around with some 3d curves made with trig functions.
I showed him how to use Mathematica’s ParametricPlot3D function and then just let him make some shapes on his own. He settled on a curve that looked like this:
Here’s the code just in case it is not legible in the video:
After he made the curve we printed it – it was really fun to see him working on the print when it was finished. I wish the picture was better!
When everything was finished I asked him to tell me about the curve. I’d not seen the code before and didn’t know there was a stray Cos[x] in it. Talking about that piece of the code led to a great conversation about elementary trig functions (totally by accident!):
Finally, I had him talk about the 3d printed shape:
I really enjoyed this project today – it is fun to use 3d printing to explore so many different areas of math.
I’ve been looking forward to sharing two calculus ideas from Patrick Honner with my son for the last week. We were, unfortunately, a little rushed when we sat down and there are a couple of mistakes in the videos below. Even though things didn’t go perfectly, I really enjoyed talking through these ideas.
Here’s the first idea – a twist on integration by parts that Honner learned from the British mathematician Tim Gowers:
Here’s the second idea – a fun surprise when a student made a creative substitution in a integration problem:
So, I stared the project by talking about how to integrate arctangent without using integration by parts:
In the last video we found a possibly surprising connection between arctan(x) and ln(x). Here I introduced the integral from the 2nd Patrick Honner tweet above and showed my son how you solve that integral using partial fractions. The point here wasn’t so much the integral, but rather to show that ln(x) showed up in an integral similar to the one we looked at in the first part of the project:
How I showed the technique that Honner’s student used (though I goofed up the substitution, unfortunately, using u = ix rather than x = iu. By dumb luck, that mistake doesn’t completely derail the problem because it only introduces an incorrect minus sign):
Now that we’ve found two connections between arctan(x) and ln(x), we went to Mathematica to see if the two anti-derivatives were really the same. It turns out the are (!) and we got an even bigger surprise when we found that Mathematica uses the same technique that Patrick Honner’s student used 🙂
Also, in this video I find a new way to introduce a minus sign by reversing the endpoints of an integral . . . . .
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
(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 using a linear approximation:
The next problem was finding the value of 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 .