Solving Polynomial Equations Graphically

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Students will be able to solve polynomial equations graphically.

Big Idea

Any polynomial equations can be solved graphically. Graphs can be your "go-to" method!

Section 1: Warm up and Homework Review

10 minutes

I include Warm ups with a Rubric as part of my daily routine. My goal is to allow students to work on Math Practice 3 each day. Grouping students into homogeneous pairs provides an opportunity for appropriately differentiated math conversations. The Video Narrative explains this lesson’s Warm Up- Solving Polynomial Equations Day 3.docx which asks students to solve a non-factorable cubic equation.  

I also use this time to correct and record the previous day's Homework.  

Section 2: Solving Polynomials Graphically

25 minutes

Calculator Introduction

In this lesson, students will learn how to solve polynomial equations graphically (Math Practice 6) and when it is appropriate to choose (Math Practice 5) a graphical approach.  An emulator or overhead calculator is helpful for the lesson.  The first task  is to ensure that the settings that the calculators are in the right mode and prior data is cleared.   My classroom routine for this is outlined in the PowerPoint. As a note, we use TI-84s in my class.  If you use a separate calculator, you will need to alter this lesson.

Solving Equations Graphically

The introductory problem is important for the whole lesson.  Students are asked to solve x3 -  x2  - 9x + 9 = 0 graphically.  I have them enter the polynomial in the calculator’s Y= menu as Y1=. When they look at the graph, I ask them to identify the answers.  If needed I show the factored form as a “hint”. 

Next the students enter the factored form into Y2=.  When they graph it, I ask the class to make observations, specifically why there aren’t two different graphs?  Don’t be surprised that some students don’t make the connection that the factored form is the SAME function as the expanded polynomial.  I may need to remind (teach) some students how to trace and identify the intercepts.  These skills are extremely important for the rest of this lesson.   Also I remind them of the different methods of changing the calculator’s viewing window.  My students have been doing this all year but reminders are still often needed.

Now they try another practice problem.  Once they have graphed it and written down the zeros, I have them write it as a product of binomials. 

Finding Irrational Solutions Graphically

The next problem has a pair of irrational solutions.  I don’t tell them in advance.  The students graph it and then I suggest that they use the trace or the zero function of the calculator to find roots (if they feel stuck).  It is important to keep the pace fairly brisk. 

Why do the zeros are such long decimals? (Fractional answer, irrational)  Once they have discussed this as pairs and the class has some shared ideas, I bring up the factored form (Obviously the students could factor it but I chose not to have them do this as it isn’t the purpose of this lesson and may bog them down in unimportant tasks). 

At this point, a discussion on accuracy would be good.  Which method is better?(Math Practice 5)  Which gives a more accurate solution? (Math Practice 6) Is there any point where the decimals on the graph would be a preferred method?  If we were asked to find the most accurate solutions to a polynomial equation, how could the graph help us?

Now, I present them with a problem that has two pairs of simple irrational roots.  The students graph this problem and find the 4 roots.  How is this different for the other problems that we have done?  I mention to the students that this seems to be in quadratic form and they should see if it is factorable.  Once they have factored it, I pull up the factored form on the PowerPoint.  I ask the students to find the answer to each binomial.  Again, which is more accurate?  Could I have three irrational answers?  (This is a BIG conceptual piece of polynomial functions.)  This will lead to the fact that irrational or imaginary roots must come in pairs. 

The next problem is the culminating problem in the lesson. The key to this problem is that there are no rational solutions.  I ask them to put their calculators down and solve this problem with division.  Since there are only two possible zeros, this won’t take them long.  Once they have realized that none of them work, I ask them why they think this happened.  The goal of this portion is for the students to realize that there are cases when graphing is the only method they currently have to solve a polynomial equation.  This is an important element to the lesson's BIG IDEA.  We then discuss the strengths and weaknesses solving graphically and make sure students understand that while we have solutions, they are imprecise. 

This section is concluded with one additional practice problem.  

Section 3: Dividing Polynomials Quiz

12 minutes

The lesson concludes a quiz on material form the first three lessons in my polynomials unit. This formative assessment will reveal how well students can divide divide polynomials using both the traditional method and synthetically.  It also indicates their knowledge of the uses of polynomial division.  I have included two versions of the quiz as I alternate quizzes like a checkerboard.  

This quiz was created with Kuta Software, a amazing resource for secondary mathematics teachers.

Section 4: Exit Ticket

3 minutes

I use an exit ticket each day as a quick formative assessment to judge the success of the lesson.  

Today's Exit Ticket asks students to find the solutions to a polynomial equation that only contains irrational solutions.  This will give a snapshot of how many of the students understood the day's lesson.