I begin class with f(x) = x^3 on the board and the challenge "Find the inverse function, if possible". (MP1) Because we worked with finding the inverses of rational functions in the previous lesson, I choose to help my students stretch their thinking and try to apply what they learned to a different function rather than just showing them. I explain this further in my video. As students work I walk around observing, then ask for volunteers to show their work and solution on the board. I ask the class to review these solutions, critique and ask questions as needed, and to reach a consensus about the correct answer. (MP3) If they struggle with this or cannot agree, I ask questions like "Do you think we can use the same methods as yesterday? Why or why not?" and "How can we check the solution by graphing?" I check for understanding with fist-to-five to determine if we can just move on to the next section or I need to review using another example.
I tell my students that today they get to practice finding inverse functions with their left-shoulder partner and that when everyone is finished I will randomly select a problem for each team to post on board and explain. (MP1, MP3) I pass out the worksheet and say they have twenty minutes or whenever everyone is ready, whichever comes first. While they're working I walk around offering encouragement and redirection as necessary. I watch for students who are still not quite getting the process and ask questions like "What do you need to do first?" and "Do you remember the steps you wrote at the end of the day yesterday?" I also suggest that they refer to their own notes about how to find the inverse of a function, so that instead of needing me to show them, they can figure it out independently. I also watch for teams that are not checking their work, either by graphing and using the horizontal line test or by using f(f(^-1)(x) = f(^-1)(f(x)) = x. I remind those students that they will need to be able to prove that their answer is truly an inverse when they present to the class.
After twenty minutes or when everyone is ready I ask for two volunteer teams and randomly select a problem for each. While they're working on the board, I advise the rest of the class to be ready to critique their work. When they're ready, each team presents their problem, including work and check, then answers questions from their classmates. (MP3) We continue until every team has presented at least two problems, then I offer bonus points for any teams that want to present additional problems. If there are no takers, I go over the remaining problems and have them do a self-check and critique my solutions.
To close today's lesson I post a function, an example might be f(x) = (x-1)^3, on the board and ask my students to find the inverse function independently as their ticket out door, and to be sure to include their work and check. (MP1) This let's me determine which students might still be struggling with inverses and gives all students a chance to demonstrate what they know.