Not long ago I had this simple idea for introducing the idea of an antiderivative. Turns out it wasn’t long before the idea would pay off again. In College Algebra, we were looking at radical equations. Solving stuff with square roots wasn’t new, but solving stuff with cube roots and fourth roots was. I place a huge emphasis on connecting algebra to graphs in College Algebra. If we’re solving a cube root equation, I want them to be able to graph it.
I cheat a little bit and tell them Desmos “can’t” accept a cube root as written with a radical symbol. It very much can, if you go digging around in the function buttons. However, I use it as an opportunity to introduce root and fractional exponent equivalence. It was a bit of a struggle to get kids to remember, so I thought we’d play a game with some of the extra index cards I had lying around.
As before, groups of students were given a random pile of cards. Each card had 1 match in the deck. I did not tell them how they were supposed to make matches.
Thrown into the mix were some rational expressions and their equivalent form with negative exponents. Students had no exposure to this other than from previous classes. I made the coefficients unique enough to where they were able to put things together using that as a context clue. As groups concluded their pairs, I went around and had a discussion with each group about what rules they had developed to make their matches.
Within 10 minutes, we had decoded fractional exponents and negative exponents without any boring lecture about exponent laws.