### A Good Lesson

Based on my experience earlier this semester trying to get the gen-ed students to prove that the square root of a prime number is irrational, I know that a large number of students don't understand the difference between a "factor" and a "multiple." So today in the Education class, I started the section on tests for divisibility by spending the entire class on the difference between those two concepts.

I divided my class into five groups and assigned each group one of the following phrases:

It went pretty well! But I was surprised that it took all class. Friday the groups will reconvene to decide whether

I divided my class into five groups and assigned each group one of the following phrases:

*a*is divisible by*b*,*a*is a multiple of*b*,*b*is a factor of*a*,*b*is a divisor of*a*, or*b*is divides*a*. I had each group come up with a mathematically precise working definition of their term, had them come up with examples that illustrated the idea, and had them use their term to describe the relationship between pairs of quantities that I provided (some were numbers, some were algebraic expressions). Then I had each group select a representative to present their work to the class.It went pretty well! But I was surprised that it took all class. Friday the groups will reconvene to decide whether

*d*divides*a*+*b*given the relationship between*d*,*a*, and*b*. Then finally on Monday we should start on the tests themselves -- and why they work.