Title of Lesson: Polynomial Functions and How to Find Their Zeros
Subject: Precalculus
Grade Level: 11th and 12th
Teacher: Fallon DeMonte
Objectives:
o Conditions – The students will be able to find the zeros, or roots, of polynomial
functions. Students will also learn the necessary vocabulary, which will help
when trying to derive the process to find the zeros in a polynomial function.
o Behavioral Verb – Identify
o Criteria – Students will verbally demonstrate an understanding of what a
polynomial function is before they can move on to finding zeros, or roots. A
basic understanding of a function is a prerequisite for learning about polynomial
functions. Individually, Students will apply what they know about variables and
the concept of a degree, or coefficient, to solve polynomial equations. Students
will learn how to find zeros, or roots by applying the new concepts. They will be
given vocabulary and be introduced to imaginary and complex numbers after a
couple of examples on how to find zeros that are whole numbers. Students need
to know how to apply what they know about finding zeros that are whole numbers
before they get introduced to imaginary and complex numbers
SCSDE Curriculum Standards Addressed:
o Math-PC-3.3 Carry out a procedure to calculate the zeros of polynomial functions
when given a set of possible zeros.
o Math-PC-3.6 Carry out a procedure to solve polynomial equations algebraically.
NCTM National Curriculum Standards Addressed:
o Apply and adapt a variety of appropriate strategies to solve problems.
o Monitor and reflect on the process of mathematical problem solving.
o Analyze and evaluate the mathematical thinking and strategies of others.
o Use the language of mathematics to express mathematical ideas precisely.
o Understand patterns, relations, and functions.
Prerequisites:
o Students must be able to articulate that a function is a mathematical relation such
that each element of a given set (the domain of the function) is associated with an
element of another set (the range of the function).
o Students must be able to articulate that a variable is a symbol (like x or y) that is
used in mathematical or logical expressions to represent a variable quantity and
how to substitute given quantities for variables
o Students must be able to articulate that a degree is the sum of the exponents of
each of the variables appearing in the monomial.
o Students must be able to articulate that a coefficient is a constant number that
serves as a measure of some property or characteristic
o Students need to be able to solve algebraic equations.
Materials/Preparation:
o Computer with power point or smart notes
o Smart board
o Vocabulary worksheet
o Paper for students to take notes when doing examples
o Practice problems worksheet (also put in smart notes so the students can use the
smart board to show how they got their answers)
Procedures:
o Introductory Activity: To start the class, the students will be given four
questions that pertain to variables and equations. The students will work
individually on these problems. Once the students have time to answer the
questions, about 5-10 minutes, I will then go over the questions with the class.
The students will explain how they got their answers and get them thinking about
the possible material that they will see in this lesson.
o Main Activity: This is the first lesson of the chapter and students may not know a
large amount of the vocabulary. The students will be given a vocabulary
worksheet with important terms to complete through-out the lesson. To begin, we
will discuss what a polynomial function is. Based on the introductory activity I
hope the students will be able to verbally define what a polynomial function is. I
plan to give them an example of what a polynomial function is, and I also plan to
have them create a polynomial function. The next terms I will go over are degree
and leading coefficient. These two terms kind of go hand-in-hand, and I plan to
use the same example for the students to find both the degree and leading
coefficient. I plan to give the students a problem where the leading coefficient
and degree are not the first term of the function. This will show the students that
they may need to re-write the function to find the leading coefficient and degree.
The next term that I will introduce is zeros in polynomial functions. The students
will be given a function where they will have to find the degree, leading
coefficient, and if a number is a zero of the function. I plan to have the students
solve this individually, and have them verbally communicate their answers to me.
Polynomial functions and Polynomial equations are very similar, so I plan to
compare and contrast the two. I then plan to introduce roots, and compare and
contrast them to zeros. The next concept students will be introduced to are real,
imaginary, complex, and pure imaginary numbers. These four things are all
connected. I plan to start with real numbers first and lead into imaginary. Once
students understand what an imaginary number is I plan to lead into complex
numbers. After complex numbers I plan to teach students about pure imaginary
numbers. Understanding the differences between these four types of numbers will
help students in the next lesson.
o Closure: To end the lesson, I plan to give the students a worksheet with practice
problems that they will either complete in class or for homework. The practice
problems will consist of finding the degree and leading coefficient, determining
whether a number is a root of a polynomial equation, and identifying real,
imaginary, complex, and pure imaginary numbers.
Assessment:
o Each student will be assessed on how well they learned the new material through
practice problems that will either be completed in class or for homework.
o There is a different level of difficulty for each problem.
Adaptations:
o If students are having a difficult time understanding degrees and leading
coefficients then I will go over more examples.
o If students are having a difficult time understanding whether a number is a root,
or a zero, of a polynomial equation, or function, I will go threw more examples.
Follow-up Lessons/Activities:
o The next lesson will be how to write polynomial functions.
o Based on the practice problems worksheet given at the end of the lesson, I will
adjust the next lesson to include any material that students still did not quite
understand.
o In the next lesson, I plan to give a little group project that will incorporate
material from the new lesson and material from this lesson.