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.

Exploring trig and 2d geometry with 3d printing

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!

Sharing Laura Taalman’s slices with my younger son

I had an opportunity to visit Laura Taalman at ICERM today who made me a copy of her latest creation. I couldn’t wait to get home to share it with my younger son:

He had some interesting ideas about what the shape was in the last video. Now I shared where the slices came from and had him explain how Taalman’s creation worked:

This is such a fun way for kids to experience shapes from a different point of view. I’m really excited to see if we can create some similar objects to play with!

A morning with the permutohedron

Today we are revisiting an old project on a really neat shape -> the permutohedron:

“A fun shape for kids to explore – the permutohedron

I learned about this shape thanks to Allen Knutson at Cornell – he included a fun pic of a large permutohedron in the comment of the blog post above:

permutohedron

He also pointed me to a 3d print on Thingiverse that we used in the last project and again today:

“Permutahedron” by PFF000 on Thingiverse

So, I started today by having the boys describe the 3d printed shape. We have two versions – a larger one that unfortunately broke a little and a smaller – but in one piece! – version. Here’s what the boys had to say about the shapes:

Next I had the boys read the Wikipedia page on the permutohedron for about 10 min and then we discussed some of the ideas that they thought were interesting:

Finally, we built the 2-D permutohedron and showed how it was embedded in a 3d grid:

Definitely a fun project and it is always great to be able to have kids hold interesting math ideas in their hands!

Exploring introductory trig using 3d printing

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:

OwenPic

Here’s the code just in case it is not legible in the video:

OwenCode

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!

3d Printing

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.

Sharing Vladimir Bulatov’s Tetrahedral Limit Set with kids

Last week I saw a really neat tweet shared by Alex Kontorovich:

I ended up buying Bulatov’s piece from Shapeways and it came today. Here’s a quick video look at it:

When the kids got home from school I asked them to take a look at it and share their thoughts.

Here’s what my younger son had to say about the shape:

Here’s what my older son had to say:

Exploring Calculus with 3d printing

Last week we were exploring volumes of various solids in the calculus course I’m teaching my son. That subject is a great opportunity to use 3d printing to enhance the course.

In fact, this old video from Brooklyn Tech about 3d printing was one of my first exposures to using 3d printing in math education:

Over the last week we printed 3 different shapes. Tonight I used these shapes as props to go back and review some of the volume concepts we’ve been studying. The ideas were new to my son and he hasn’t quite mastered all of them yet, so the review was productive. The videos are a bit longer than usual as we review some of the concepts, though.

Here’s the first shape: y = Sin(x) revolved around the x-axis:

The next set of shapes were not volumes of curves rotated around the x-axis, but rather shapes where the slices were squares:

Finally, we looked at the curve y = \sqrt{x - 1} from x = 1 to x = 5 rotated around the y-axis. Here my son remembered how to calculate this volume by looking at the slices parallel to the x-axis, but struggled a bit with when the slices were cylindrical shells – so we spent a long time on that 2nd part.

I’m happy that we have the opportunity to explore these shapes with our 3d printer – it definitely is incredible to be able to hold shapes like these in your hand!