I begin this lesson with the graph of f(x) = -x^2+8 projected on my board from my calculator with a standard viewing window. I say that part of this function represents the path of an flowerpot pushed off an 8 meter tall building, with height in meters a function of time, and challenge my students to identify which part I'm referring to. (MP7)
There are usually several who can identify first quadrant as the relevant section, so I add to the challenge by asking them to explain why it has to be so, mathematically. (MP3, MP5) This is a bit tougher, but generally there are at least a few who can describe that the path of the flowerpot as having both a horizontal (x) component of change in time and a vertical (y) component of change in height and that it starts at the point (0,8) on the graph.
I ask my final challenge question by asking where the flowerpot lands. (MP1, MP2) Some students will solve this by trying to factor or using the quadratic formula while others will estimate using their calculators. While they're working I walk around observing and sometimes asking questions like "Why are you trying to factor this problem? What are you looking for?" This helps my students recognize that factors and zeros are related and that both are ways of describing real-world solutions, a focus for this lesson that I discuss in my video.
I tell my students that for the first part of this activity they will practice graphing and factoring functions independently to identify zeros and end behavior if possible, then they will get to challenge each other by creating real functions and trying to identify zeros and end behavior from each others' graph.
I distribute the Zero to Hero Practice handout, ask if there are any questions, then say they have about 15 minutes to complete the assignment. (MP1) While they're working I walk around offering encouragement and redirection as needed. Some students may still struggle with factoring or graphing. Depending on the what problem they're having I might schedule some additional one-to-one work time or suggest they put is some extra time on their own working with their calculator, factoring or graphing.
When everyone is done or after about 15 minutes I tell my students that they now get to challenge each other. I explain that they may use paper or whiteboards, must take turns creating and solving, and that the goal is for each team to share at least ten graphs between them! I add that each graph must represent some possible real-world function and that their explanations must include a description of what the zeros represent. I check for understanding with fist-to-five, answer any additional questions, then tell them they have about 15 minutes for this section. (MP1) I again walk around while they work, this time mostly observing. If any student or team asks me to arbitrate a dispute about who is correct, I instead challenge them to figure out which solution makes sense mathematically.
My students are accustomed to comparing answers and discussing who is correct, but if your students aren't comfortable with this practice, you might have each student create two or three functions, including their zeros, end behavior and explanations, and collect those.
I close this lesson by asking my students to write a brief definition/explanation in their own words of what "zeros" are and how to identify them from a graph. (MP2) This gives them a chance to cement what they've just been studying into their thinking and also to put into words what they've been doing mathematically. I specifically choose not to develop a class definition/explanation because I think it's more important for my students to put this into words they can understand and that have meaning for them, especially since not everyone chose the same method for finding zeros.