Quiz and Intro to Graphs of Polynomials

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SWBAT demonstrate understanding of polynomial theorems and ask good questions about the graphs of polynomial functions.

Big Idea

The graph of a polynomial can look a variety of ways depending on the degree, lead coefficient, and linear factors.

Quiz on Polynomial Theorems

45 minutes

To be successful on Quiz Polynomial Theorems, students must know the Remainder Theorem and the Fundamental Theorem of Algebra and understand the implications of these theorems for finding roots of polynomials.  Although I do not require my students to memorize a specific wording of these theorems, I do ask them to state them in their own words using polynomial vocabulary terms correctly[MP6]. 

The questions on this quiz probe for understanding of the idea that a consequence of these theorems is the relationship between linear factors and zeros of polynomials.  I hope they can explain that any polynomial, P(x) ca be written as (x-a)*Q(x) + P(a), and that if P(a)=0 then (x-a) is a factor and a is a root of P(x) [MP2].

Post-Quiz Warm-up

15 minutes

As students turn in their quiz, I hand them Polynomial Graph Questions.  The goal of this exercise is for students to graph some polynomials using a graphing calculator and then try to develop a set of questions about why the graphs look the way they do.  They first work independently to generate questions.

I anticipate that students will ask questions like

  • Why do polunomials sometimes open up and sometimes open down?
  • Why do the functions sometime cross the x-axis and sometimes not cross it?
  • Why do some polynomials seem to "wiggle" more than others?
  • What is the relationship between the degree of a polynomial and the graph of the polynomial?

Group Time to Generate Questions

15 minutes

After students have had some time to generate questions individually, they join together with 3 or 4 other students to assemble a group set of questions.  

Each group is given a sheet of post-it paper and markers.  The goal is to find commonalities among their questions and come up with their favorite 5 questions.  These posters will be put on the wall at the end of the group work time.

Structured Discussion

15 minutes

I ask each group to put their poster on the wall.  Then students circulate around the room to read the questions developed by each group.  We have a brief discussion about commonalities between the sets of questions generated by the groups and begin a discussion about how we might go about answering the questions generated.  The posters will remain on the wall for the next class period, when we will attempt to answer the questions [MP3].