## 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.

## Playing with the nonagon tiles

Two of our recent project have involved studying a tiling of a nonagon from the book “Ernest Irving Freese’s Geometric Transformations”

Those two projects are linked here:

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

nonagon tiles

After school yesterday I had each of the boys make a pattern with the nonagon tiles and then build the two patterns that were in the book. The videos below show there work. My younger son went first:

Here’s what my older son had to say:

This project was super fun from start to finish. Hearing the thoughts from the boys after seeing the pattern initially was really fun. Building and printing the blocks was a nice geometry / trig lesson. Then having the boys play around with them made for a really satisfying end to the project. I hope to do more like this in the near future.

## 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.

## The puzzles (and everything else!) from Nervous System will blow your mind

Yesterday Nervous System in Somerville, MA had an open house and I was lucky to have a few hours free while the boys were at their karate black belt tests. Visiting their shop was absolutely incredible:

Definitely check out their website and their twitter feeds. I follow Jessica Rosenkrantz – @nervous_jessica. Here’s the link to their website:

The Nervous System website

At the open house I bought two new puzzles. The boys had seen one previously at Christmas, too. For our project today I’d already wanted something on the easy to talk about / less heavy math side because of the black belt tests yesterday, so talking about the new puzzles was perfect.

We started with the geode puzzle – one of the fun things we talked about was how the boys thought the computer generated the geode shape:

After the introduction to the puzzles, we moved on to talking about the challenge of putting the puzzle together. Favorite line – “once you get started, it gets pretty hard.” Yep!

Next I showed them the latest creation from Nervous Systems – an “infinity puzzle” inspired by the Mobius strip!

I was incredibly lucky to be able to buy one of the infinity puzzles yesterday. So, for the last part of today’s project we did an unboxing:

If you know kids who like puzzles – or you like puzzles! – all I can say is the Nervous System puzzles are absolutely incredible.

3 hours after we finished the project this morning, my younger son had returned to the Geode puzzle:

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

A few weeks ago we got this book in the mail:

We used the book for one fun project already:

Playing with geometric transformations

Today we were really short on time in the morning, but I still wanted to do a project before we ran out the door. I’d been hoping print some of the shapes from the book, so today for a quick project we looked at one of the transformations.

Here’s the introduction to the shape and some of the thoughts that the boys had:

Next we tried to understand some of the details about the shapes – could we understand anything about the lengths of the sides or the angles?

I think that we now have enough information to make the tiles. It was nice that a little bit of trig came up since that’s what my older son is studying right now. Not sure if I’ll have time this weekend or not, but we’ll hopefully be able to do a project with the 3d printed tiles in the next week.

My younger son has been studying Art of Problem Solving’s Introduction to Geometry book this year. He’s been doing most of the work on his own, but I check in every now and then.

Today he was working in the section about medians, so I thought I’d ask him what he’s learned.

Here were his initial thoughts.

In the last video he was struggling to remember how to prove that the 6 small triangles formed when you draw in the medians of a triangle all have equal area. Here I gave him a hint and he was able to finish off the proof:

We wrapped up with a short discussion about the lengths of the medians.

Overall a fun discussion. I’m a big fan of the Art of Problem Solving books and am happy that my son is enjoying working through their geometry book.

## Ed Southall’s geometry problem

A problem that Ed Southall posed on twitter caught fire on the internet last week:

I thought it would be fun to share with the boys for our project today. Here are their initial thoughts on the problem. My younger son’s initial guess at the amount of area shaded was “a little bit bigger than 1/4” and my older son’s guess is was 1/3.

In the last video the boys decided to use coordinate geometry to solve the problem. Here’s that work:

Next I wanted to have them try to solve the problem by folding paper. Studying eometry through folding is an approach that I always want to do more of, but almost never do more of. This tweet from Patrick Honner last week reminded me to fold more:

Here’s how I introduced the idea of approaching this problem through folding to the kids:

The boys folded the patty paper off camera (and without me – I was publishing the previous videos). Here are the ideas that came from the time they spent folding. A bit of a surprise to me was one of the ideas really was a folding idea, and one was really a similar triangles idea that my younger son noticed when he was playing with the paper folding.

So, a fun project on a couple of different levels. The problem is definitely great, and it was also a really nice surprise to see ideas that at first glance seem like similar triangles emerge from playing around with paper folding.

## Geometric tilings inspired by Annie Perkins

Annie Perkins has been sharing some amazing math art on Twitter. Probably as good of an example as any is the work she shared while I was writing up this blog post!

If you go back through her feed you’ll find amazing pic after amazing pic after amazing pic.

I’d been wondering how to share some of these ideas with the boys. We recreated one of the drawings using our Zometool set a few weeks ago:

Making one of Annie Perkins’s drawing from Zometool

After that I saw this tweet and ordered a set of tiles:

As a point of full disclosure, the tiles arrived in a box from the “Post of Iran”. I do not know what the status of sanctions and trade between the US and Iran is right now, so I don’t know if I accidentally tripped over a rule that I shouldn’t have tripped over by ordering these tiles. Hopefully these plastic geometric tiles are not somehow banned in the US right now, but in any case they did arrive.

Today I had the boys make some shapes. The first one we made was from the company’s website:

Next I had the boys create their own shapes. Here is what my older son made and what he had to say about the shape:

Here’s what my younger son made:

My younger son seemed to especially enjoy this project and creating his own pattern. Hopefully there will be more patterns to come!

My younger son is working his way through Art of Problem Solving’s Introduction to Geometry book. He’s been doing almost all of the work on his own – I just check in every now and then.

Today we had a little extra time so we did a project on the section he’s currently studying -> angle bisectors.

We started with some of the basic properties -> why is the intersection point of the angle bisectors the same distance away from every side:

Next we moved on to a slightly harder problem -> what is that distance?

This problem gave him a little trouble. BUT, after a hint to think about how the 1/2 base * height formula for the area of a triangle might help, he made some nice progress:

Finally, I had him work put the ideas we talked about to work in a specific triangle. Here he finds the radius of the inscribed circle in a 3-4-5 triangle:

It was interesting to see him pull some old ideas from geometry in to help understand some of the new ideas he’s learning here. I haven’t looked ahead in the book, but assume that the angle bisector theorem is coming soon. That theorem was really difficult for my older son to grasp, so I’m going to try to work a little more carefully with my younger son in the coming weeks to help him with any difficulty he might have there.

## Sharing some number theory with kids thanks to Jim Propp’s “Who knows two?” blog post

Jim Propp published a terrific essay last week:

Who knows two? by Jim Propp

Yesterday we did a fun project about card shuffling using the ideas from Propp’s post:

Sharing a card shuffling idea from Jim Propp’s “Who knows two?” essay with kids

Today we did a second project for kids based on some ideas from Propp’s post. The topic today was “primitive roots”. Unfortunately this isn’t a topic that I know well and I messed up one explanation in the first video below. Oh well . . . still a really neat idea to share with kids.

So, I started by introducing the concept of primitive roots by reminding them of the 8 card and 52 card shuffles we looked at yesterday (pay no attention to my explanation about powers and mods at the end. It will become clear in the next video that I goofed up that explanation . . . . ):

Now we looked at some examples of primitive roots with small numbers. These simple examples give a nice way for kids to get a little bit of arithmetic practice and also help them see the main ideas in the problem that we are studying.

After working through these smaller examples, we moved to the computer to continue studying the problem. My older son noticed that the examples that seemed take the longest time to work were primes, but not all primes took a long time. That’s exactly the math idea we are looking at here.

Next we made a small change to the program to study all of the odd numbers up to 1,000 all at once. After correcting a little bug we found that the numbers we were looking for were indeed all primes.

We wrapped up be talking about what was known and what wasn’t known about these primitive roots. I was happy that my older son seemed to be particularly interested in this problem.

Definitely a fun project. It is always fun to find unsolved problems that are accessible to kids (and lots of them seem to come from number theory!). We will definitely have to do some follow up projects to explore the ideas here in a bit more detail.