I begin this lesson with a simple quadratic on the board: f(x) = -0.2x^2. I ask my students to consider a real-world situation this equation might represent, then give them a few minutes to consider this individually. (MP2, MP4) I ask for volunteers to share their ideas and then have the class decide on one specific circumstance for further discussion. For example, they might choose to let this function represent a rainbow. I select a student to sketch the graph on the board, then challenge all of my students to look over the graph in terms of the real-world situation we had decided it would model. After a moment I ask if anyone has any observations to share. (MP2) I'm hoping that someone will notice that all the points below the x-axis represents values that don't make sense in our real-world scenario, but if they don't I ask leading questions like "Do any of the points represent impossible values for our scenario?" or "Which section of this graph represents the rainbow and which section is not part of the rainbow?" I close this section with a review of the definition of "domain" with a focus on the idea that we can use a restricted domain to help define a real-world function.
The main activity of this lesson is a treasure hunt for domains that best fit different functions. I have my students work together with the partner of their choice to complete the treasure hunt by matching functions with possible domains. I discuss why I chose partners for this activity in my video. Each correct answer gives a letter clue to the "treasure", a phrase that is spelled out if all the answers are correct. I distribute the handout and review the directions with my students, ask if there are any questions, and tell them they have about 30 minutes to find the treasure. (MP1, MP2) As they're working I walk around offering encouragement and redirection as necessary. I anticipate that some students will struggle with determining which domain fits which function. I try to help with guided questions like "What would you expect the heaviest mammal to be? What might its weight be?" As each team completes the handout, I encourage them to review their matches and see if there are any that they think could be better written. When everyone is finished or after about 30 minutes I ask for volunteers to share any improvements they thought might be made to the domains as printed. (MP6) All the domain choices on the handout are written in terms of x to avoid giving clues about which function they fit.
I close this lesson by having each student write a notecard about when and why restrictions on domain are appropriate. (MP6) I give the directions to write using complete sentences and proper grammar, with appropriate mathematical terms.