Inundated with Inverses: Restricting the Range of an Inverse (Day 1 of 2)
Lesson 11 of 16
Objective: SWBAT generate the graph of an inverse by reflecting across y=x and restrict the range of an inverse to make it a function.
As a warm-up to today’s lesson, students should answer the question on page 2 of the flipchart. This question just requires students to determine if a given relation is a function. Students should text in their answer on their personal response systems.
Environment: For the warm-up question, I put in the option of I don’t know. Sometimes I think it is important to include this option to eliminate guessing. For this warm-up, I want to know who knows this and who doesn’t. I don’t want students magically guessing the correct answer since they have a 50/50 shot at it.
Differentiation: For struggling students, I would encourage them to think about a function machine or the analogy set in the function video we watched. When you input something (say a chicken) how many outputs can we have for that specific input? Sure a chicken can output nuggets and legs, but only on two different machines, two different functions. If this silly analogy isn’t working to help students understand functions better, then I would prompt students graph the points and to think about the vertical line test. They should be familiar with the vertical line test from other courses.
Present page 3 and 4 from the flipchart and encourage students to copy the new vocabulary to their personal dictionaries. On pages 5-7 of the flipchart there are built in clicker questions so that students can apply these new definitions. To be able to successfully answer the question on page 5, students will need to be able to write the equation of a vertical line. After students text in their answers on page 6, I plan to ask for a volunteer students to share their explanation.
Differentiation: If students are struggling on the question on page 5, I plan to have them draw a sketch and show me where there could be a vertical line drawn to show this is not a function. If students can successfully do this, then I know that they understand the vertical line test and a little about functions, but are just struggling to write the equation of that line. Then I will ask them to make a table of values for a few points on that line. Then I will guide them to the equation by asking “What is common in this table? Or what rule can we use about x that will always be true?”
Before students get into teams for group work today, I wanted to talk with students about the last mathematical practice that we have not yet talked about, Mathematical Practice 3: Construct Viable Arguments and Critique Reasoning of Others. While students are working in teams I want them to practice the 4 “I will” tasks outlined on page 8 of the flipchart.
Also, just to note this lesson is designed to continue over two class periods. So if you have a longer class period, continue on with tomorrow’s lesson.
Preparation: Today’s activity calls for students to have one colored pencil. The color does not matter.
Narrative: Present page 9 of the flipchart to set the stage for students to discover a few functions that have inverses that are not functions. Then give every student the An Inverse that is Not a Function handout. Guide students through finding the inverse of the quadratic function by using the carbon copy method. Have students check their work by revealing the graphed inverse on the flipchart. Then set students free to work through part a, b and c of this activity in their teams. This activity is designed to be completed in teams as it is important that students have people to share their ideas with as we want them talking about the math and practicing mathematical practice 3! For part c, we really just want students to brainstorm at least one other function. I do not expect them to come up with one for every graph on the back side of this page. This goal of this activity is mainly to have students practice with using the line to find the inverse of a graphed function. Even if students don’t find a single function that has an inverse that is not a function, but have spent time exploring different inverses, then I would consider this successful completion of the activity. Take about 5 minutes at the end of this section to have any teams present an inverse they found that is not a function. Try to guide the class discussion into what the original function needs to look like to insure that its inverse will be a function. (Needs to pass the ‘horizontal line test’)
Differentiation: I am guessing that students are going to have difficulty writing the equation for the inverse. To help remediate this for teams that need it, I am going to have students make an input output table for the original function. Then ask them to find the table of values for the inverse. Hopefully, students quickly realize that we can just flip the inputs and outputs as they should have learned this 2 lesson ago. To help students who don’t recall this, I will relate it back to the function machine concept. And that the inverse is like taking the output back up into the machine and getting the input back. Now, students should be able to come up with the equation from this table of values. If not, I will ask questions like “What rule (ok to tell me in words) would help us change every input to it’s output?”