# Using 3D printing to help kids learn about algebra and 2d geometry

In the last couple of weeks I’ve started to get a better understanding of how to use our 3D printer to help teach the boys about both algebra and 2d geometry. The project that got me thinking about this particular application of 3d printing was based on a problem that Patrick Honner shared on his blog back in November:

Inequalities and Mr. Honner’s Triangles

Then, just last week Tina Cardone shared a neat problem on Twitter:

A Cool Geometry Problem Shared by Tina Cardone

Then, by lucky coincidence, my son struggled with a challenge problem in his Geometry book just a couple of days later:

A Follow Up to Our Tina Cardone Geometry Project

All of that background left me with 3D printing on my mind when my older son stumbled on this problem at the end of last week:

Problem 12 from the 2005 AMC 10a

I wanted to try something a little different for this problem, though, so when my wife and older son were out at an art class, I tried using this problem as a 3d printing project with my younger son. He’s not had much geometry or algebra, and certainly nothing about lines. The goal for this project was to see how 3D printing would work out as an introduction to those topics.

We started by talking through the problem:

Next we drew the two shapes in Mathematica in three steps. The first of the three steps was to make a 3D version of an equilateral triangle:

The next step was making the little circle parts. Unfortunately we drew the wrong shape:

The last step was correcting the mistake in step two so that we draw only the little circular arc. This time we got it right:

It took about two hours to print all of the pieces. I hadn’t really thought about how big they were going to be, but probably 4 cm sides would have been plenty big instead of the 6 cm sides I made for the triangles. Once they were all printed I revisited the challenge problem from “A follow up to our Tina Cardone Geometry Project” and the new geometry problem from the AMC10 with my younger son. Without the printed puzzle pieces, I wouldn’t even consider giving him these problems. With the manipulatives, though, he is able to talk through the problems and test out some ideas. I especially liked his ideas to try to make circles out of the shapes and seeing that the first couple of attempts were not circles:

Finally, I showed the shapes to my older son. We spent probably 15 minutes talking about the problem yesterday since there are several different ways to solve it. Having the shapes in your hand helps you see one of the geometric solutions pretty quickly:

Although it may seem like a strange application of 3D printing, getting Mathematica to draw some 2D shapes and then printing them is a fun exercise in both algebra and geometry. I think it is also an educational exercise, too, as getting the shapes from the board to the computer and then to the printer involves some math that might be interesting and challenging for kids. I’m excited to try out a few more projects like this one.

# A short talk about probability and game theory

At least that’s what I thought – it ended up being another good example in the “what learning math sometimes looks like” series.

Last night I saw this book out of the bookshelf.

I don’t think anyone was reading it, we just had a few different people staying with us for the holidays and had to move a few things around. Seeing the book made me think of using something from it for our Family Math project this morning. I ended up finding two good ones.

The first is a problem about the archers shooting at a target. You are told the probability for each one of the archers hitting the target on a given shot and then asked to calculate the probability that at least one person hits the target if each of them takes one shot.

A fairly standard probability problem but we haven’t talked about probability in a long time, so I didn’t quite know what to expect. During the first part of the talk the kids were pretty confused. We don’t make a lot of progress towards the solution and approaching the 5 minute mark my younger son has the idea of adding up the individual probabilities. We’ll explore doing that in the next video:

I decided to stop the last video after 5 minutes just to split the conversation into two pieces. All I did was turn the camera off and on and we picked up where the last movie left off: talking about adding up the probabilities. Eventually our little winding path takes us to the idea of looking at the probability that 0 people hit the target in a round. The idea of looking at the times when 0 people hit the target – a concept called complementary counting – is an important (and not obvious) idea in counting and probability. Our conversation about complimentary counting here is far from complete!

The next problem is involves a little more probability and a little game theory. I think this is the first time that we’ve ever talked about game theory, so I was really excited to see how the kids would react to this problem. The problem goes like this:

Three people play a dodge ball-like game. Each person takes a turn throwing a ball at an opponent. If you hit an opponent that person is out, and the goal of the game is to be the last person standing. Two of the three people get hits 100% of the time and the third person gets a hit only 50% of the time. Who is most likely to win this game?

Fun little problem with a solution that isn’t at all obvious. I find problems like this to be more difficult to think through than the first problem, but the kids seemed to find this problem to be much easier to understand. It was fun to hear them think through the different ideas and get to the surprising result:

So, a fun morning with counting and probability (and a little game theory!). It has been a while since we talked about these topics and the first problem gave the kids quite a bit of trouble. The lesson there for me is that it probably makes sense to return to old topics occasionally rather than just leaving them for good. The game theory problem turned out to be more fun than I was thinking it would be. Hopefully I can find some more fun game theory problems are accessible to kids.