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.