Higher Degree Polynomials, Day 2 of 2

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SWBAT compare the properties of two polynomials represented in different ways. SWBAT apply the Remainder Theorem to answer challenging questions about polynomials.

Big Idea

Students test their mettle against polynomial challenges, comparing graphs to tables and doing long division with unknown coefficients.

Getting Started

10 minutes

At the beginning of today's class, I will take some time to briefly review the solution to problem #2 from yesterday's assignment.  I  may begin this lesson with a gallery walk for students to examine the polynomial graphs generated by different groups for problem number 2.  My goal is emphasize the variety of valid graphs!

Next, I will let students know that today's assignment is to tackle the four problems on Higher Degree Polynomials 2.  However, before giving the handout to any student, I need to first see that they have finished part I.  Once a student receives my seal of approval, they can begin working on part II.  I do this to ensure that no students move on the more challenging problems with unresolved misconceptions about the basics.  If a student seems to still be struggling, my first approach is to pair that student with another classmate for a few minutes while I check in with others.  I make sure to return to this student before too long to clear up any lingering problems. (See the video Mastering Fundamentals for more details on this.)

This shouldn't take more than about 10 minutes.

Comparing Two Polynomials

15 minutes

Next, I give students time to work individually on problem 1 of Higher Degree Polynomials 2.  It's important to provide some individual time for students to complete at least 1a and 1b on their own. Many will have incorrect solutions, but that's okay for now (see below).

I expect a number of students to fail to identify the third root in the first function.  When students fall into this trap, I don't give away that they've done something wrong, but innocently suggest that they might find it helpful to plot the points from the table on the given graph.  This should force them to recognize that the graph must cross the x-axis more than twice.

Another potential misconception is to think that since the function has three real roots, it must be a cubic function.  In this case, I might point to student work from problem 2a of the previous lesson (a quintic polynomial with three real roots).  This will be enough for many students to recognize their mistake.  Alternatively, I might look for two confident students with conflicting answers and point out the difference.  Typically, this will start a debate that will draw in enough other students for the truth to win out (MP 3). If not, I'll step in.

By the time all the students have an answer written for 1a and 1b, I will allow them to begin comparing answers and collaborating on the rest of the handout.

After working in groups until about 5 minutes are left in this section, I'll call for a quick summary discussion.  Student volunteers will share out their answers to the various parts of problem 1. Hopefully, there will be universal agreement, but if not, I'll help students to see the reason behind the correct answer.

Three Challenging Problems

20 minutes

Now that everyone has completed problem 1, the rest of the class will be spent trying to solve the final three problems on Higher Degree Polynomials 2. What I really like about these problems is the different approach each one takes.  In many ways, they don't involve any new concepts, but they approach familiar concepts from an unexpected angle.

  • For Problem #4, of course, the biggest challenge will be framing the known zero in a linear binomial, as well as recognizing that one complex zero implies another: its complex conjugate.
  • For Problem #5, the challenge is the presence of the second variable.  I find that many of my students get scared when they see terms with both x and y, but it's important to face your fears!  Courage!
  • For Problem #6, the long division is complicated somewhat by an unknown coefficient.  The greatest challenge in this case (for most students) is attending to precision as they carry out the long division.  Many will get their signs mixed up at some point!

It's worth noting that I don't necessarily require all of my students to complete all of these problems on their own.  These provide a healthy challenge for even my best students, but it's okay if some students get stuck.  Help them to get unstuck, offer hints, and encourage them to learn from the correct solution provided by another student.  With long division problems, it's often useful for small groups to work through the algorithm together step-by-step at the whiteboard so they can all watch out for sign errors.


Wrapping Up

5 minutes

By the end of today's lesson, I expect every one of my students to have mastered the essentials of higher degree polynomials.  The final problems go above and beyond the standards, so I'm not too concerned with getting all of my students to the same level of comprehension.  These problems are not homework. 

Tomorrow we'll discuss the solutions to these problems and review the most important aspects of polynomial functions.