I begin this class with a quadratic equation on my board like ((-3x^2)/7 -2x-4) that doesn't have real factors and challenge students to solve it in whatever ways they choose. (MP1) By now they should be thinking about complex numbers and imaginary roots, so after a few moments I ask for volunteers to post their answers, along with the work they did. There's room for three or four students at a time to work on the board, so we end up with at least a couple of different examples of how to find the answer, and maybe a few different answers. I encourage my students to reconcile any discrepancies they see by careful review and critique of the posted work as a class. (MP3) It would be faster for me to just point out any errors and tell them which answers were correct, but that keeps the "math magic" in my hands instead of letting my students feel their own power. Once they've agreed on the answer, I tell them that today they get to build their factoring skills.
You will need copies of the complex quadratics handout for this part of the lesson. For this part of the lesson I have my students do some individual practice on factoring quadratics and putting results in proper form. I believe there are lots of interesting and engaging ways to learn new skills and strengthen existing ones, and I also believe that sometimes practice does make perfect, or at least better. While my students are working I walk around offering encouragement and redirection as necessary. (MP1) After about 35 minutes or when everyone is done, I ask them to pass their papers in, then randomly redistribute them to other students for grading. I remind them that they are not only responsible for catching any mistakes, (MP6) they also need to ask questions about answers they're not sure of. I go through all the answers, ask if there are any questions, then have my students sign their name (as corrector) in the upper right corner and return the paper to the proper person. I give about five minutes for them to look over their own papers and ask any questions. I discuss my reasons for having students check each other's papers in my video.
To close this lesson I give my students notecards and ask them to write one question they have about the day's assignment on the front and one thing they believe they understand better on the back. I assure them that I will answer their questions if they take mine seriously and write what they really think. This gives my students a chance to ask questions individually and it gives me some insight into what aspects of complex numbers are still confusing to them. For example, I often have students who express frustration about not being able to just graph the equations using their calculators and find the zeros that way. Notecards allow me to open a dialogue with these students in a less stressful venue than the general classroom might be.