I begin this lesson by challenging my students to find the roots of a simple cubic polynomial like (x^3+x^2+x+1) I encourage them to find the roots using whatever methods and tools work best for them and walk around observing while they're working. (MP1, MP5) When everyone has found the roots I ask for volunteers to put their work on the board. I sometimes have students who still struggle with roots, but letting the rest of the class lose focus while waiting for one or two people to finish, I move ahead and give those students individual help later. I have my students review the work and ask questions, then put a fourth degree polynomial on the board like (x^4+x^3+x^2+x+1). I have students who will be trying to factor it even as I'm writing, so I tell them that I don't want them factoring this polynomial yet. Instead I ask them to predict how many roots it will have and to justify their projection. Some of my kids have trouble adjusting to the idea that math is not just about plugging numbers into some equation or memorizing an algorithm. Asking for predictions and justifications in this fairly non-threatening setting (no grade or score directly attached!) helps my students build both confidence and skill. I facilitate a discussion about roots, then tell my students that today they will be working to determine if there is a clear pattern that will allow them to predict the number of roots for any degree polynomial. (MP8)
I tell my students that today they may work with the partner of their choice. I say that the directions for today's activity are simple; they need to factor enough different degree polynomials to come up with a generalization about how many roots a polynomial of degree "n" will have. I suggest trying several 1, 2, and 3 degree polynomials and maybe a 4 or 5 degree polynomial if they're feeling tough. They can find polynomials to factor in their textbook or create their own, but either way should record the polynomial, the roots, and their work for each polynomial. While my students are working I walk around offering encouragement and redirection as needed. (MP1, MP2)
When everyone has had time to factor multiple polynomials, I remind them that the goal is to come up with a generalization about the number of roots of any degree polynomial and that the number can include real and imaginary roots. This comment is a hint for those who've been struggling a bit to see the pattern. (MP8)
After all the teams have finished I ask for volunteers to share their generalization with the class. I serve as scribe to translate the generalizations into mathematical symbols on the board while the rest of the students are talking. This allows me to give some direction to the discussion without appearing to. For example, if my students are having a hard time generalizing and a team suggests that they found two factors for all the second degree polynomials they tried, I might make a table with degree as one column and number of factors as another. I don't give them the values to fill in, but this kind of visualization usually helps students see the pattern more clearly. When everyone has had a chance to share, I ask my students to review the board, what they've just heard, and their own work today. I give them time to talk as a class about the patterns and then congratulate them on finding the fundamental theorem of algebra!
To close this lesson I ask my students to write a brief letter (text!) to themselves summarizing the fundamental theorem and explaining how they know it's true. I specify that even if they're writing a text, they must use proper grammar and punctuation - or else their poor, old teacher might not be able to figure out what they're trying to say. I explain in my video why I chose this method for closing the lesson.