As part of our homework for the challenge, we were supposed to try to increase the rigor of an upcoming lesson using the practice standard we had selected as a district to work on implementing more effectively. We had chosen practice standard #1: "Make sense of problems and persevere in solving them."
When they gave us that assignment I said to the people at my table, "I'm multiplying polynomials next week, how am I going to do this?" So I brainstormed the whole way home, and I had a pretty good idea, but I decided to create the UbD template for the standard. (Any feedback on what I have come up with would be welcomed as I am very much still struggling with knowledge vs. understanding and other things necessary for breaking down the standards.) The standard is: Understand that polynomials form a system analogous to the integers, namely, they are closed under the operations of addition, subtraction, and multiplication; add, subtract, and multiply polynomials. Unfortunately, with they way that the common core has ramped up our curriculum, it is unlikely I will be able to dive into the "analogous system" part of the standard as much as I would like :( But maybe next year...
After working on what the standard says, I did an internet search (using websites I know of that have CC aligned assessment) for ideas for this standard, but I came up short, so I decided to run with my idea and see how it went. I used this chart to guide students through a repeated process of individual, small group, class discussions on a variety of multiplication scenarios that slowly increased in difficulty. After going through each pair as a class, I assigned each small group another similar problem. They were to process as a group and present to the class. It took the good part of a week to make it through the chart, and it is still a steep stair between a monomial multiplied by a binomial and a binomial multiplied by a binomial. I had the students complete a survey of several internet sites in order to see three methods of representing the multiplication of polynomials (vertical, distributive, and grid).
Overall, I believe the process was productive for them. I was able to quickly address misconceptions that arose with the entire class (a common one was that (x)(x) = x rather than x^2 which lead to the phrase "ninja one" which was much more successful than I would have ever thought) and give them more opportunities to practice. (I still gave practice problems for students to work on individually, but I post an answer key also so they can check as they go. I have struggled a lot with finding an appropriate balance in beginning mathematics (Algebra) between practicing too little and too much, but I will save all those thoughts for another time.) I also thought that it lead to some of the best mathematical conversations that the students had ever had in my classroom. I heard them discussing exponents, coefficients, and terms! Oh my!