Following my algebraic vocabulary and identifying components of polynomials lesson, my students should be ready to find all the terms, factors, coefficients, and exponents. Today's focus is on grouping and regrouping the components so that students become comfortable manipulating complicated polynomials (MP7). This is actually the first day of a two-day lesson with tomorrow's focus on rewriting the polynomials.
I begin with an example on my board of a complicated polynomial like the one listed below. My Puzzle it Out video explains the Common Core Algebra II connection.
(4x³ - 5)²
I ask for volunteers to help identify all the expressions component, then ask if there are any ways to make an equivalent expression by just rearranging the components. Some students will want to cube the 4 or square the entire expression but at this point I'm asking for changes to the arrangement of the components only. I focus on this because I've found that many students jump to the exponents or try to factor before they really look at the structure of the expression (MP7). This Educreations video that I created details how I walk my students through possible re-groupings for the expression. I discuss the delivery of this opening activity in my Asking myself why? reflection.
Independent work (15 minutes): I begin this section by telling my students that they will be working independently to create as many different equivalent expressions/equations as they can in the time given. As I pass out the Puzzles directions and cardstock I ask my students to read through the directions carefully and ask questions if needed. I remind them that the principal goal is to rearrange the terms while keeping the new expression/equation equivalent to the original (MP7). When all questions have been answered I allow my students to get their scissors and begin the activity. I wait until after the discussion to have them get their scissors to help ensure that they actually read the directions before beginning to cut! As the students are working, I walk around offering encouragement and assistance as needed. I try to offer as little help as is necessary to continue the development of personal responsibility -- I always try to encourage Student Ownership of mathematical ideas.
Whole Class work (10 minutes): Toward the end of the twenty minutes I divide my board into four sections and write each Polynomial_Expression in one section. Then I randomly select students to post one of their created problems on the board in the appropriate section. (I have multiple students at the board at one time to speed this process up.) When everyone has posted a problem, I ask the class to review the board and see if all the expressions/equations are truly equivalent to the "parent" problem that I wrote at the top of each section. This gives students a chance to critique the work of others without putting anyone on the spot (MP3).
Obviously, the success of this conversation depends on establishing and sustaining Trust and Respect in my classroom. I work hard at this all year because it supports student learning. By reviewing the work on the board as a class, students can see whether or not their on work is on target. I usually identify those who are struggling during my walkabouts, but this gives my students a chance to assume responsibility of their learning by letting them identify if they have a problem.
Independent work (15 minutes): To close out this section of the lesson I challenge my students to regroup each of the expressions/equations on the polynomials handout into at least two new expressions/equations (MP1). I again walk around while they work, offering encouragement and assistance. For students who are having particular difficulty, I try to schedule some one-on-one time to work with them outside of class. When they complete this work, I have them exchange papers with their left-shoulder partner and correct that paper as I go over possible correct answers. This is not as easy as it sounds, since there are several ways to regroup each problem, which means my students have to be reviewing their partner's answers critically to determine if each answer meets the criteria I'm giving the class.
Teacher's Note: I require the person making the corrections to write what he/she thinks is wrong, why it's incorrect, and how they would fix it, then sign the upper right corner of the paper. We use MLA heading throughout our secondary school, so name, class, etc are in the upper left corner.
To close out this lesson students get their "corrected" polynomial worksheets back and have a chance to respond to the corrections. They can either agree that the correction is necessary or they can explain why the correction is not needed. If a student has no corrections on their paper they can earn credit by creating additional expressions/equations OR by helping classmates who are struggling with this lesson. If they're helping another student respond to corrections, they need to sign the bottom of that student's paper.