I begin this lesson with a polynomial graph projected on my front board. As my students come in I expect them to discuss what they see because they know from previous experience that I'll be asking questions! After the bell I ask them to identify points of interest, reminding them to use appropriate vocabulary! Because I project directly onto my whiteboard I can mark the points as my students identify them. When they've identified several "points of interest" I ask for volunteers to explain what each point represents on the graph. (MP2) For example I expect someone to mention that the function doesn't go through the origin, that it crosses the x-axis three times and the y-axis once. If one of my volunteers mentions that the points where the function crosses the x-axis are called "zeros", fantastic! If not, I use leading questions like, "Do these points of intersection have any other names? and simpler examples (perhaps quadratic) to help my students make that connection.
Individual work 15 minutes: I tell my students that today they get to work with several quadratic and cubic functions to solve for the zeros. I distribute the ZEROS handout and assure them that they can use whichever factoring method works best for them since some students prefer traditional while others like synthetic division or rectangular factoring. (MP1, MP2, MP5) As they're working I walk around giving encouragement and assistance as needed paying particular attention to those students who are still struggling with factoring.
Graphing 15 minutes: When everyone has finished finding all the zeros, I tell my students that now they get to use graphing calculators to graph only the first five functions and compare the zeros they've calculated to the points of interest on each graph. I ask them to sketch the graphs next to their calculations (or on a separate piece of paper if they need more room) and mark the zeros they've calculated directly on their sketches. I ask for a volunteer to summarize the relationship between zeros, factors and points on a graph then ask for fist-to-five to check for understanding before moving on. After answering any questions I tell my students to put their calculators away and use what they've just discussed to sketch graphs of the remaining five polynomials. (MP1, MP2) This is a challenge for some because they haven't really made the connection between zeros and x-intercepts, but I can usually help those students individually as I walk around the room.
Real-world connection 10 minutes: When everyone is done I tell my students that I have one more challenge for them today. I post the Real-World Polynomial example on the board and ask if they see any connection between the example and the graph at the beginning of class. I tell my students that while some of the math they've done has been simply to gain skills, like running drills for sports, the purpose of all those drills is so that they can make sense of real-world problems like the one I've just posted. I ask them to use their skills to explain what the zeros of the function are AND what they represent in terms of the problem. I tell them the challenge will be to communicate their thinking to their right-shoulder partner so that he/she can understand it. (MP1, MP4, MP6)
To wrap up this lesson I have my students write in their own words the relationship between the x-intercepts of a graph and the zeros/factors of a polynomial. I tell them that I expect complete sentences and that while they can include a sketch in their explanation it must be supplementary to the writing rather than the primary description. Sometimes my students complain that they're in math class not English to which I reply that even Einstein would be an unknown today if he didn't write down his ideas well enough for other people to understand them! ask if they see any connection between the example and the graph at the beginning of class. As I discuss in my video, a shift new to most in the common core is the importance of mathematical written expression and discourse.