I begin this lesson by referring to a previous lesson where my students worked with polynomial identities. I ask my students if they can recall any of the identities we worked with in our last unit and give a few hints like "Remember the Pythagorean Triples" or "Think about those polynomial shortcuts we studied". When they've suggested a few I ask them to reflect silently for a moment (actually about 30 seconds, which is a long time for teenagers to be silent) about whether or not they think the complex numbers will also fit the polynomial identities. After the reflection time I give my students each a notecard and tell them to write their opinion and also their supporting reasoning. (MP3) I collect these and tell them we'll look at them again later. I discuss why I choose to include this plus standard in my video.
For this section you will want copies of the Polynomial identities handout. I give my students a copy of the polynomial identities handout that we used in our algebraic arithmetic unit and tell them they will be working independently to check their conjecture about the identities and complex numbers. I tell them they should try at least five different complex numbers with each of the identities to confirm or refute their conjecture. (MP1, MP3) As they're working I walk around offering encouragement and assistance as needed. My students are familiar with identities from earlier work, so they should be fairly comfortable expanding them to include complex numbers. I've included a Resource of possible numbers to "test" which might be used for students who need extra support for this unit.
After about 30 minutes or when everyone has finished I randomly select different students to share their opinions about one of the identities and support it mathematically. (MP3) When we've covered all the identities I ask if there are any questions about whether or not the polynomial identities should include complex numbers. Rather than answering any questions directly I first offer the other students the opportunity to give answers (including explanations, if needed). I clarify and answer those that are still stumping the class as a whole.
To close this lesson I return the notecards written during the opener to my students. I ask them to consider the work they've just completed and decide whether they have supported or refuted their original opinion, then write out what they've learned. I clarify that what I'm interest in is whether or not they think their original reasoning which they based their conjecture on was mathematically sound. This gives them a chance to critique their own thinking and reasoning, a skill I believe is important to cultivate. (MP1, MP3)