I just received this email from a student:
I FINALLY UNDERSTAND YOUR WITCHCRAFT OF WHY 3 TO THE POWER OF 0 IS ONE.
3^0 = 3^(1 + -1) = (3^1)*(3^-1) = 3 * (1/3)
Talk about an accomplished summer.
This group in Algebra 2 took a lot of convincing. I went through about four or five different approaches to proving this. They objected to using laws of exponents since 30 is one of the rules of exponents. They didn't like writing out factors and dividing them out. They didn't like following patterns. While they did accept that they could use the exponent rule as fact, they didn't like doing this. I really liked that they pushed me so far on this, and I don't entirely believe that their disbelief was simply a method of delaying the lesson of the day.
Whatever it was that led this particular student to have such a revelation, it makes me incredibly proud that this student chose to follow that lead, especially given that it is the middle of summer vacation. Despite labeling the content of the course 'witchcraft', I'm marking this down in the 'win' column.