What learning math sometimes looks like: the triangle inequality

My younger son and I just started looking at some basic geometry in Art of Problem Solving’s Prealgebra book. Today I noticed a neat challenge problem at the end of the section about perimeter and it looked like it would be a fun problem to talk through. It turned out to be a great illustration of ideas that kids have when they are learning math.

In the first part I introduce the problem:

An isosceles triangle has integer side lengths and a perimeter equal to 25. What are the possible lengths of the sides?

We spend the first part making sure that my son understands the problem. He then dives in.

He doesn’t know what to do, so he decides to try some examples. The first example he tries is 1, 1, 23. When he draws a picture he notices that this combination of side lengths doesn’t work. From that he forms a conjecture and proceeds to write down some of the solutions.


We left off the last section with a list of possible solutions. I started off this part by asking him to describe the difference between the 1, 1, 23 case and the 12, 12, 1 case. They seem sort of similar when you look at the numbers, but one works and the other doesn’t. His thoughts on this point are really nice.

Next I ask him to go through some of the other cases he thought would not work based on his original conjecture. The first case is 8, 8, 9. Talking through this case leads him to a new idea, and this new idea is basically the triangle inequality! Again, his reasoning here is really great – kids have such interesting ideas about math 🙂

The rest of the video is just checking the other cases. It takes a little bit of time because of a little arithmetic mistake, but we make it to the end.


So, hopefully a nice example of what learning math can look like. Again, not a straight line to the end, but learning math seldom is. Although kids it would be silly to expect kids to formulate their ideas in super precise mathematical terms, their ideas and instincts about math are the key to helping them learn. It was so fun to hear my son essentially formulate the triangle inequality here:


Pursuing ideas like this one is what makes learning math fun.

Going through Joel David Hamkins’s Graph Theory for Kids

Denise Gaskins published the latest issue of Math Teachers at Play today. As usual it is a wonderful compilation of math activities and one, in particular, caught my eye. Here’s the latest issue:

Math Teachers at Play issue 85

And here’s the activity from Joel David Hamkins that caught my eye:

Graph Theory for Kids

After downloading and flipping through Joel’s activity this afternoon, I thought that it would be really fun to run through it with the boys tonight. I’ll present the videos without much comment since they really speak for themselves. The short summary is that I absolutely love this activity and think that many kids will love it, too. My kids were totally engaged for the full hour that it took to work through it.

The only warning I’d give is that in the part where the kids draw their own graphs – part D in our set of videos – make sure they are careful when counting. My kids drew some pretty complicated graphs and we had to slow down and count really carefully.

So, here’s our tour through Joel David Hamkins’s Graph Theory for Kids:

Part A: getting into the activity and explaining a few of the terms that we’ll be using


Part B: Going through a few more examples and introducing the definition of Euler Characteristic

Part C: There were lots of examples to do in the last section, so I had the boys work through all of them off camera and then had them each explain their work on one of them. This section also introduces the idea of a “connected” graph:

Part D: At the end of last section the kids were asked to draw their own graphs. My kids drew some complicated graphs and had a hard time counting the various pieces. Here we are just going through and counting carefully.

Part E: The next section of the activity introduces the idea of a “planar” graph

Part F: Now we move on to the end of the activity and look at some calculations for 3-Dimensional shapes

Part G: I thought it would be fun to end with a shape that didn’t have an Euler Characteristic of 2, so we built a torus out of our Zometool set. We also looked up the Euler characteristic of a few other shapes off camera and my son mentions a Klein bottle at the end.

So, a really fantastic activity. As I said, my kids were really engaged for over an hour and seemed to find all parts of the activity to be really interesting. There’s a ton of fun math here and lots of ways for kids to get creative. Also lots of great opportunities to hear kids talking about their own mathematical ideas.

Thanks to Denise Gaskins for sharing this one and to Joel David Hamkins for putting this awesome activity together.