You will want copies of the Polynomial identities for this section. I start by distributing the list of polynomial identities and ask my students to look for patterns and/or any that they recognize. (MP7) After a moment or so I have them pair-share their observations with the front partner, then I randomly select students to summarize what they shared. Generally someone will recognize the quadratic formula and occasionally I have a student who remembers the expansions of (a+b)^2 and/or (a-b)^2. Other than that, the most common observations I hear are that all the identities have exponents, all use at least "a" and "b", and all have parentheses. I'm hoping that today's lesson will give them a chance to see these identities in a different light, as sort of abbreviations for working with polynomials.
You'll need sheets of poster paper and markers for this section. Teamwork 15 minutes: I tell my students that today they get to work with a partner to write "proofs" of at least four of the polynomial identities. (MP1, MP3) I explain my expectation that they may use mathematical symbols, words, or both but that whatever they choose must be "rigorous", ie:their proof must stand up to challenges and must work for all numbers. I also say that they should write their proofs on plain paper first and when they're ready they can get a copy of poster paper from me to write their final proofs on for a "gallery walk". I explain why I chose this method of sharing in my Prove video. Students who have had previous classes with me know what a gallery walk is so rather than going into detail with the whole class, I let partners explain the process to anyone who isn't familiar with it. While the teams are working I walk around offering encouragement and assistance as needed. I also use this time to ensure that students are keeping their work rigorous and are writing down all their work appropriately.
Gallery Walk 10 minutes: When all the teams are ready I have them hang their posters around the perimeter of the room and stand in front of their own poster. When everyone is ready I tell them that they will have be moving with their partner and will have 1 minute at each poster to observe what their classmates have done. I also tell them they will be rotating clockwise, but not to rotate until I tell them to. I tell my students to rotate one poster clockwise and begin timing, calling for them to rotate every minute and encouraging those who want to lag behind. When everyone has had the opportunity to review each poster, I tell them they have three more minutes to go back to any poster(s) they would like to see again. After the gallery walk I tell my students that now they have an opportunity to review their own proofs, make any modifications they choose and turn them in.
You'll want to have some colored stickers for this section - I use four colors, but you can decide what works best for you. I close this lesson with a little fun...and something that gets my students thinking about what a good mathematical argument looks like. I have them use red, blue, green, yellow dots (one of each per student) to vote for which proof is the "most elegant", "most unique", "simplest" and easiest to understand". They have one minute to think about which proofs they want to choose and then one minute to put their dots next to their selected proofs. It gets a bit chaotic, but I like the way it gets my students thinking.