Reviewing Inverses

The first few weeks of Pre-Calculus are always a little bit overwhelming for me. The course is designed to fill gaps of knowledge before students get in the ring with Calculus in an AP class or when they're off at college. Great, in theory. However, it means we need to make the gaps glaringly visible so we can address them. Before we tackle any of the really cool concepts, we have to tackle the misconceptions. One that I find the kiddos struggle with the most is inverses.

I've taught inverses so many different ways and always feel like I haven't quite hit the mark. I need a way to efficiently review the concept so we can get into some more skill based practice, but I don't want to do a full blown, hour long exploration. I also don't want to just lecture. I was playing around in Desmos and I decided I'd try a different approach this year. Spend less time trying to remind them of what they've learned in Algebra II myself and give them a little more time to think through it with a partner.  Here are my thoughts right now as to why I'm using Desmos instead of my beloved TI-84:

1) We can examine the concept of an inverse without worrying about the notation or the algebra just yet

The kids always are able to tell me "you just switch x and y," but they often have no idea what an inverse actually is or why switching x and y is important. I like we can build on what they do remember  (switch x and y) to tease out the actual concept. We don't need to get into messy algebra ("YOU EXPECT ME TO USE A LOGARITHM??") to illustrate it....yet. 

2)  The graphs are clear and the commands are user-friendly

We have block scheduling, so I have 2nd semester seniors who haven't seen a graphing calculator since they were first semester sophomores. It's definitely a challenge. Sometimes we spend so much time trying to reintroduce them to the steps ("Okay...2nd....calc....5.....enter....enter....enter") that our intentions get muddled. Yes, I know I need to reteach those, and we will get there. On a conceptual day, I'd rather not. 

We will definitely be examining the fact that the intersections all happen at a point where x=y.....why would that happen? What is demonstrated about the graphs here? This kind of conversations can lead to one to one functions, horizontal line tests, invertibility....all that good stuff. I'm hoping this gives us something concrete to refer back to throughout the year as we repeatedly see inverses in our curriculum. 

This is definitely simple and in the planning stages.  Here's what I have so far. Constructive feedback always welcome!