Multiplying Polynomials

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!