Currently, my group of Algebra I students is in the midst of a chapter on polynomials. We’ve just finished our major unit on linear functions, which ranged from graphing and solving linear equations (individually and as parts of systems) to graphing and solving linear inequalities (also individually and as part of systems). This chapter is part of our dive into higher-order polynomials, the primary piece of which will be on quadratic functions and their solutions.
The chapter we’re working on now (Chapter 7, a little over half-way through the Glencoe Texas Algebra I text) is partially a review of what students finished last year looking at: the classification of polynomials (by name and degree) and operations on polynomials. Most students remember the names, but we’ve needed to spend some time reviewing degree of a monomial and then degree of a polynomial. I introduce and review degree as it applies to graphs in the coordinate plane, so we do a little preview of the fundamental theorem of algebra graphically.
I teach at a private classical school, so we don’t necessarily follow Common Core standards. But I make sure our lessons are loosely aligned with standard curriculum, so that as students graduate and move on to different high schools (my school only goes through 8th grade) there’s not a drastic difference in what students encounter. My goals are for students to meet the Common Core standard shown below:
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.
Specifically, I want students to be able to multiply monomials, binomials, and trinomials by the end of this week, and to start to gain familiarity with some special products like perfect squares and the difference of squares.
Warm-up: Visual Pattern #51 from http://www.visualpatterns.org.
Since I’ll be using the area model to demonstrate both how to multiply polynomials and what their products represent, I chose a visual pattern that reduced to a simple rectangle with dimensions n by n + 1 if you rotate the top of the right-most column counter-clockwise. Students who don’t rotate that extra bit of column mostly see the pattern as an n by n square with an n by 1 “antenna,” so we are able to discuss the equivalence between n(n + 1) and n² + n. I relate this equivalence to the distributive property, which will be one of three “methods” we use to multiply monomials and polynomials in today’s lesson.
We work primarily out of the textbook most days. Students have a composition notebook meant to be structured like a $1 textbook. They keep a table of contents for each lesson and all their warm-ups in the notebook. Each lesson contains between three and five examples we work together, interspersed with practice problems assigned either for group work and discussion, individual work, or modeling if a particularly ambitious student wants to run through an example. Today, the lesson we’re covering begins with an area model:
We used algebra tiles last week to model adding and subtracting polynomials, so students are familiar with the representations. I demonstrate three methods for our first example problem: –2x²(3x² – 7x + 10). The first method is the area model. I have students tell me what kind of box I’ll need based on the number of terms (1 by 3), then we quickly review exponent rules as we multiply:
There’s some good class discussion about how the powers of x go in order within the boxes. I ask the students to think through why that might be, and if it will always be true, depending on how we initially write the polynomials. My hope is that they retain this pattern once we branch into binomial or trinomial products, so that they can begin to anticipate the like terms along diagonals. Although it doesn’t come up today, I know when we get to, for example, binomial multiplication, students will trip over why certain boxes get combined while others don’t, and whether to combine those boxes through addition/subtraction or through multiplication. Hopefully, the time we’ve spent today on the difference between the area of the boxes and the lengths that produce those areas will help alleviate those bumps.
Using the same problem, we then do simple distribution, showing the intermediate steps for those students who weren’t quite sure where the “areas” were coming from in the previous model. Then we finish up with a stacking method, demonstrating the similarity between a problem like 372 × 2 and our polynomial problem.
Some students ask why we don’t “carry” like we do with stack multiplication. I put the question back on them and relate stack multiplication back to the distributive property.
No homework for tonight. Tomorrow, we’ll finish 7-5 lesson and students will do selected homework problems from the textbook.
Combine what we did today with multiplication of monomials and binomials with what we did last week in adding and subtracting polynomials. So, they’ll be multiplying multiple monomial/binomial or trinomial pairs and then combining like terms. Then move on to equations in which any squared/cubed terms cancel out, and students solve what remains.