# Starting Art of Problem Solving’s Introduction to Counting and Probability

The math project I’m doing with the boys this summer is working through Art of Problem Solving’s Introduction to Counting and Probability

We also spent a little time in this book last summer, which was fun. What I like about this book is that the mathematical ideas do not require much beyond basic arithmetic which means that the boys can work together pretty well. The extra work that my older son has done in algebra and geometry doesn’t put him too far ahead of my older son in this particular topic.

I also think the ideas in counting and probability are great for helping kids begin to understand how to organize ideas when solving math problems. The first problem from today is a good example of how organizing a few ideas helps with problem solving. The problem is a Venn diagram puzzle. In the first part of this project we talk through a problem that gave my younger son a little trouble and hear his initial thoughts on the problem:

In this next part of the talk we dive into the solution. We’ve got a good picture on the board, but my younger son is struggling a little to see how to pull together the information from the problem. My older son has an important idea – the number of people in four overlapping regions must add up to 20. That idea allows them to see the solution to the problem.

In the last part of this project I show them an alternate solution. The idea here is to use the fact that the 20 people are separated into two different groups in two different ways: left vs right handed, and like jazz vs. don’t like jazz. Recognizing either of these subdivisions allows you to see a little more structure than was initially apparent in the problem.

So a nice start to our little summer project. My hope is that we’ll have a nice and stress free walk through the book this summer and the boys will learn a bit about counting and problem solving.