Inundated with Inverses: Algebraic Inverse and Composition to Verify (Day 2 of 2)
Lesson 12 of 16
Objective: SWBAT use "switch and solve" to find the inverse of simple rational functions algebraically and verify inverse functions by using composition.
Warm-up: Homework Quiz
Students will take their homework quiz over last week’s homework (worksheets #3-5). The homework quiz my students took is programmed into the last page of the flipchart.
Old note on homework quizzes: In my class, we do a Homework Quiz every Monday. So for my calendar this lesson is landing on a Monday, so we will take about 10 minutes out of class to do that. I am not posting my homework quizzes at this time because I like to be selective of the problems I put on the quiz based on what is happening in the classroom. If students are really struggling on a particular topic as a whole then I won’t put that question on the quiz. Or if we have gone over a problem in class, I won’t put that on the quiz but would probably choose a similar problem. So for minimal prep work, I would recommend having students do a written homework quiz. Just make a list of problems that you want to check in more detail and give students an allotment of time to copy down their work and answer for this problem. I also sometimes have students do their homework quizzes in their clickers. I will share an example of that, once one is made.
Help student to first solidify their understandings of inverses. Before moving on, it is important that students understand the structure of functions and their inverses (remember this was the goal from the fun with functions lesson). They should recall the questions we have already answered that are presented again here on pages 15 and 16 of the flipchart and have a solid understanding that a function and its inverse have domains and ranges that are flipped and that the inputs of one function become the outputs of the inverse function. In this section, students are going to be asked to explore a composition of functions and relate this to inverses. They will only be successful at this if they have this prior knowledge. A good problem to review would be problem 2 from homework #5 if you think your students need more practice, you can just make up a table of values and ask them to fill in the composition of some of these. Page 17 of the flipchart summarizes these ideas. I am going to just have students read it silently to themselves and take a moment to make sure they understand what it means!
Next, I plan to just put up page 18 of the flipchart and have students do a Think-Pair-Share over each of these questions posed. So first, they will start with the question “What does it mean to compose two functions?” Take a minute to think of their response, then take a minute to pair up at their teams (or triple up) and talk about it, then share out to the class. Then move onto the next question. I am going to set that I want each team to share for each question. The responses will probably be repetitive so depending on time constraints, I may cut it short (before every team has a chance to share) and move onto the next question.
I would like to end this section with having students take a moment to see if they can determine the connection between composition of functions and inverses. Hopefully we will have a few students that can state the connection in student friendly terms. Something like “when we plug in one input, get that output and plug it into the next function, we will get back our original input.”
To wrap up this section, I will present page 18 of the flipchart and have students add that fact to their notes. Then have them practice the two problems on page 19 while working with their teams. Finally, I will have students answer the last clicker question independently which is found on page 20.