Connecting Polynomials to Geometric Series
Lesson 4 of 16
Objective: SWBAT explain the connection between polynomial multiplication and the geometric series formula.
Students will work independently to complete Sequences and Equations Post Assessment as a review of the previous day's learning. This set of exercises is similar to today's pre-assessment. Because the students have practiced representing visual patterns with polynomials and using algebra to rewrite polynomial expressions, my hope is that they will now answer with more confidence [MP2]. I plan to collect this post assessment and count it as a quiz grade.
This discussion and note-taking session is designed to revisit the properties of polynomial operations that we discussed a few lessons back. Most students will need reinforcement of these concepts, so I start by asking Always Sometimes Never questions similar to the ones I asked in that lesson before moving on to more advanced questions.
Then, I plan to use Conceptual Questions About Polynomial Operations as Quick Polls - both to initiate a discussion about the nature of polynomial operations and to see how far students have come in their precise use of vocabulary [MP6] and their conceptual understanding of operations with polynomials [MP8].
Today, I will encourage students to debate a bit at their table before answering each poll. I will also pause after the results are revealed to answer students' questions. In evaluating each statement, I encourage students to supply simple polynomial arithmetic examples to support their claims [MP3].
Throughout the above activities, we have revisited the idea that operations with polynomial expressions have similar properties to operations with integers [CCSS APR.A.1]. I have taken this approach to encourage students to think conceptually about the operations that can be performed on polynomial expressions.
The second part of this lesson is designed to employ MP8 to drill deeper into the mathematics, using the example suggested in the narrative of this practice standard:
Noticing the regularity in the way terms cancel when expanding (x - 1)(x + 1), (x - 1)(x2+x+1), and (x - 1)(x3+x2+x+1) might lead (students) to the general formula for the sum of a geometric series.
The connection between the polynomials and the geometric series formula can feel very abstract for some students, so I use concrete examples of polynomial arithmetic to keep it grounded. I begin by handing out Patterns in Polynomials 1, which asks students to multiply special pairs of polynomials. The goal for students is to observe the process as they are carrying it the multiplication, looking for regularity in repeated calculations [MP8]. Students initially work independently to work out these multiplication problems:
I keep the room very quiet so each student has a chance to experience their own moment of discovery. I remind students not to shout out the pattern and wreck the moment for their peers [MP1].
I encourage my students to generalize the pattern to write an expression for (1-x)(1+x+x2+ ... + xn). I hope each student will discovered that the middle terms cancel and the expression simplifies to 1-x^(n+1). It is not essential that each student come up with the general "nth" form on their own, but each student should understand the pattern.
Once students are comfortable with the idea that (1-x)(1+x+x2+ ... + xn)=1-xn+1, we focus in on the factor (1+x+x2+ ... + xn). With some help, students will recognize this as a geometric series with initial term 1 and common ratio of x. We review a bit of what we learned about geometric series at this time and then show how the identity (1-x)(1+x+x2+ ... + xn)=1-xn+1 can be rearranged to obtain the formula for the sum of a geometric series [MP7].
Exit Ticket and Closure
As a closing activity, students independently complete Exit Ticket Polynomial Concepts, which assesses two things:
- Mastery of the polynomial multiplication pattern we worked on in class
- Understanding of how polynomial arithmetic is like integer arithmetic
Students are more likely to be successful in mastering the multiplication pattern than they are with the second bullet, but this is one of the concepts I need to continue checking in on as the unit progresses. I am trying to determine whether students really understand that how to complete an operation (multiplication, subtraction, etc) on two things depends on what type of things you are performing the operation on. Polynomials work like integers and rational expressions work like fractions. This is a powerful understanding that is much more cognitively demanding than being able to perform polynomial operations.