The Lesson

Date: November 5, 2013

Teacher: Sarah Dobo

Course Name: College Algebra

Lesson: The Remainder Theorem and Theorem for Bounds on Zeros

Total Estimated Time: 90 minutes (60 minutes for the lesson and 30 minutes for the formative assessment/worksheet)

Overview:

The remainder theorem is a theorem that helps students find the remainder of polynomial division without going through the process of long division or synthetic division, which can be time consuming and intricate. The theorem is as follows: “Let f be a polynomial function. If f(x) is divided by x-c, then the remainder is f(c).” The Theorem for Bounds on Zeros involves finding a domain in which all of the real zeros of a function exist. The theorem is as follows: “Let f denote a polynomial function whose leading coefficient is 1. A bound M on the zeros of f is the smaller of the two numbers Max {1, |a₀ |+|a₁ | +…+ |a_n-1 |} and 1+ Max{|a₀|, |a₁|,…, |a_n-1|}.

The purpose of the students learning these two theorems is that they allow students to see how to solve problems in different (and easier) ways. Both of these theorems make a process that students already know a lot quicker and easier. By students knowing and understanding these theorems, they can use the ideas when it would be nearly impossible for them to solve the equations using the theorems and processes they already know. While the proofs of these theorems may be a little out of reach for some of the students to comprehend, all of the students are capable of the processes involved in the actual theorems.

Students should know this information because it helps students understand that there are different approaches to answering problems. Students are learning how to solve polynomial functions for their zeros, and these theorems assist students in doing this. It is important that students fully comprehend the ideas associated with solving polynomial functions, and these two theorems give students another way to look at the problems.

Standards:

· CCSS.Math.Content.HSA-APR.B.2 Know and apply the Remainder Theorem: For a polynomial p(x) and a number a, the remainder on division by x – a is p(a), so p(a) = 0 if and only if (x – a) is a factor of p(x).

· CCSS.Math.Content.HSA-APR.B.3 Identify zeros of polynomials when suitable factorizations are available, and use the zeros to construct a rough graph of the function defined by the polynomial.

21^st Century Skills:

-Communication and Collaboration:

· Students will work together to answer questions and explain their reasoning.

· Students will listen to other’s opinions and have a thoughtful discussion on finding the correct answer.

-Critical Thinking and Problem Solving:

· Students will use the tools and ideas they have learned in the past to solve the current problem.

· Students will interpret their solutions and analyze whether their solution makes sense.

Daily Objectives:

Students will know:

· How to use and understand the Remainder Theorem

· The ideas the make up the Bounds on Zeros Theorem

Students will be able to:

· Use the Remainder Theorem to find the Remainder of Polynomial Division

· Use the Bounds on Zeros Theorem to find the bounds and create a graph of the polynomial

Preparation:

Before class, I need to work through all of the example problems as well as the problems on the assessment. I need to make copies of the handouts and have the calculator ready on the Smart Board for when we work on the Bounds on Zeros Theorem.

Materials:

· White board with markers

· Class set of handouts/assessments

· Smart Board Calculator

Connection to Previous Learning:

The students have learned how to evaluate functions at specific values in all of their previous algebra courses. They are very familiar with this process. Students have also learned how to perform synthetic division and polynomial long division in this course, yet it has been a few months, so some students may have forgotten parts of the procedure. Finally, students have been graphing functions on their calculators in all of their math courses since middle school. They are very familiar with how to type in functions and change the window.

Pre-Assessment:

As a pre-assessment I gave students a 4 question “pre-test” involving the ideas associated with finding a remainder of polynomial division as well as how to choose the window for a cubic function. The pre-assessment was given on Thursday, October 31^st to give adequate time to use the information from the analysis in the lesson for Tuesday, November 5^th.

Agenda for the Board:

I will write the Daily Objectives on the board so the students know what we are doing that day. I will write that we will start with the Remainder Theorem, then the Bounds on Zeros Theorem, and then work on the Worksheet/Assessment.

Instruction:

Anticipatory Set (15-20 minutes):

· The teacher will ask students to recall the pre-test they took last Thursday. She will write the first question on the board: “Find when .”

· The teacher will walk through how to arrive at the correct answer (-14).

· The teacher will then ask students about the second question: “What is the remainder when is divided by . (Hint: Long Division or Synthetic Division)”

· The teacher will ask the students, “What is a remainder? Is it a number or an algebraic expression?”

· The teacher will then walk through the long division and show the remainder (-14).

· Think-Pair-Share: Think about what you notice between the 1^st question and the 2^nd question. Talk about it with a partner, and then we will talk about it as a class.

Student Practice (5 minutes):

· The teacher will ask the class to test their “theory” from the Think-Pair-Share. They will perform both the long or synthetic division and evaluate the function at the value for the polynomial equation: divided by x+2

· The teacher will ask which method they prefer and why.

Direct Instruction: The Remainder Theorem (10 minutes):

· The teacher will write the remainder theorem on the board and describe how this is exactly what they just showed. The Remainder Theorem: “Let f be a polynomial function. If f(x) is divided by x-c, then the remainder is f(c).”

· The teacher will ask, “What happens when the remainder is 0? What does that represent?” This will relate to finding the zeros of the polynomial function!

Direct Instruction: Bounds on Zeros Theorem (20 minutes):

· The teacher will ask the students how they decide on a good window for a function.

· The teacher will explain how it can sometimes be difficult to find the zeros (x-intercepts) of the function.

· The teacher will write the Bounds on Zeros Theorem on the board: Let f denote a polynomial function whose leading coefficient is 1. A bound M on the zeros of f is the smaller of the two numbers Max {1, |a₀ |+|a₁ | +…+ |a_n-1 |} and 1+ Max{|a₀|, |a₁|,…, |a_n-1|}.

· The teacher will go through Example 7(a) and 7(b) from the book.

Group Practice/Check for Understanding (10 minutes):

· In groups of 2-3, have the students work on the problem to find the bounds of zeros and graph the equation:

· The teacher will walk around the class to ask them how they are doing.

· As a class, the teacher will ask what the groups got as their answer.

Closure/Post-Assessment (30 minutes):

· The teacher will pass out a handout to all of the students. They are to complete their own sheet but they may work with a partner.

· The teacher will walk around and answer any clarifying questions for the students.

· As students finish they can work on any assignments they have from this class.

Accommodations/Modifications:

· There may be groups who have difficulty understanding the problems on the handout/assessment. I may spend time with these groups and try to do a simpler version of some of the problems to show them how the basic ideas work.

· There may be students who do not feel challenged. I might ask them why these two theorems work. I will bring in a proof of the Remainder Theorem and ask them if they understand it!