Title of Lesson: How to write and graph Polynomial Equations
Subject: Precalculus
Grade Level: 11th and 12th
Teacher: Fallon DeMonte
Objectives:
o Conditions – The students will be able to write and graph polynomial equations.
They must also gain an understanding of how to find complex and real zeros from
a graph.
o Behavioral Verb – Illustrate
o Criteria – Students will demonstrate an understanding of what a polynomial
function is and how to find the roots of a polynomial equation verbally and in
groups. Students should also be able to classify what a degree and a coefficient
are, and be able to identify them. In this lesson, students apply it to prior
knowledge of polynomial functions to create a polynomial equation. Students
will then demonstrate verbally and visually how to graph a polynomial equation
from prior knowledge of graphing equations. They will then distinguish the
differences betweens the graphs of the polynomial equations.
SCSDE Curriculum Standards Addressed:
o Math-PC-3.7 Carry out a procedure to solve polynomial equations graphically.
NCTM National Curriculum Standards Addressed:
o Apply and adapt a variety of appropriate strategies to solve problems.
o Communicate their mathematical thinking coherently and clearly to peers,
teachers, and others.
o Recognize and use connections among mathematical ideas.
o Understand patterns, relations, and functions.
Prerequisites:
o Students must be able to solve algebraic equations.
o Students must be able to articulate that a polynomial function is a function that is
defined by a polynomial in one variable with real coefficients.
o Students must be able to apply prior knowledge of how to write an equation in
order to create polynomial equations.
o Students must be able to read and create a graph.
Materials/Preparation:
o Computer with smart notes
o Smart board
o Vocabulary Worksheet (from previous lesson)
o Paper for students to take notes when doing examples
o Group Worksheet (see Mini-Project/Activity Worksheet)
Procedures:
o Introductory Activity: To start the class, I will give students a problem, that they
will work on in groups of 2 or 3, that involves what they learned about
polynomial functions in the previous lesson. Also, I will ask them questions that
pertain to writing a polynomial equation to lead them into the lesson. The
students will discuss their answers in with their groups, and the groups will
verbally communicate their answers. This will be helpful to see if they learned
the material from the previous lesson, and get them thinking about how to write a
polynomial equation based on this new knowledge. (see notes attached for
example problem)
o Main Activity: I will start out by having one student from each group come up
and write their answer, showing all of their work, to the problem from the
introductory activity. I plan to have the students evaluate each group’s answers
and see where some may have made mistakes. When I get to the questions that
pertain to the new lesson, I plan to have a discussion on how to answer/solve the
problem with the students to see if they can derive a way to answer the problem.
Some questions that I will ask the students are: What is a polynomial equation,
What is a degree in a polynomial equation, What is a coefficient, and Based on
what you said a coefficient was, what is a leading coefficient? After the
discussion, I plan to show them how to solve the problem, if they could not derive
the solution in the discussion. I then plan to give a few more examples using
whole numbers. After going over a few whole number examples, I plan to give
students some examples using complex and imaginary numbers. After some
practice, I then plan to lead into the concept of graphing polynomial equations.
Using the previous examples (the equations the students wrote using the given
roots) the students will try to graph the equations. After some exploration, I plan
to discuss with the students how they graphed the equations. I will ask them
questions about how they went about graphing the equation, why they chose to
graph it that way, and if there are any other ways to graph the equation. I will
then have the students come and graph the example problems. After showing the
graphs for the example problems, I will have the students try to find similarities
between the graphs. Some of these similarities include the direction in which the
graph is going and the general shape of an equation with degree 2 or 3. This will
help when they are given a graph and they have to state the number of complex
zeros and the number of real zeros, which will be the next topic discussed in this
lesson. After the students discuss the similarities I will introduce the idea of
finding zeros from a graph. I will show many examples on how to find the
complex and real zeros on a graph. (see notes attached for example problems)
o Closure: To end the lesson, I plan to randomly divide the class into groups of 2 or
3 where they will work on a mini-project/activity. They will be given a
worksheet with many problems where they will have to use what they learned in
this lesson and the previous lesson to complete the worksheet. The problems will
be challenging because they are problems applying what the students just learned
in the lesson, which is why the students will be in groups. This miniproject/activity will be used as an assessment to see how much the students have
learned in the two lessons. The mini-project/activity will also be used to assess if
the students can connect the two lessons. (see mini-project/activity worksheet)
Assessment:
o I plan to divide the students into groups of 2 or 3 and give them a miniproject/activity worksheet that is meant to challenge their knowledge of the
material they learned in this lesson and the previous lesson. (see miniproject/activity worksheet)
o Each student will be required to complete the mini-project/activity with equal
contribution. (I will be observing students participation)
o I will include a challenge problem that connects to the next lesson, quadratic
equations.
Adaptations:
o If it seems like students are having a difficult time figuring out how to create a
polynomial equation then most likely they did not fully understand the previous
lesson. In this case it would probably be best to review the material from the
previous lesson.
o If students are having a difficult time graphing the polynomial equations then it
may be helpful to review the basic skills for graphing a function, which are
making an x-y chart and plugging in values.
o If students are having a difficult time distinguishing between the different graphs
to find the number of complex and real zeros then it may be helpful to review
graphs for different equations, such as linear, quadratic, and exponential.
Follow-Up Lessons/Activities:
o The next lesson will be solving quadratic equations.
o Based on the mini-project/activity (assessment) that the students will complete at
the end of the lesson, I will adjust the next lesson to include any material that
students still do not quite understand.
o In the next lesson, I plan to give an assessment that will incorporate material from
the previous lesson, this lesson, and the new material they learn in the next lesson.