Primary Type: Lesson Plan
Status: Published
This is a resource from CPALMS (www.cpalms.org) where all educators go for bright ideas!
Resource ID#: 47823
Dancing Polynomials/Graph Me Baby
Dancing Polynomials is designed to lead students from the understanding that the equation of a line produces a linear pattern to the realization that
using an exponent greater than one will produce curvature in a graph and that further patterns emerge allowing students to predict what happens
at the end of the graph. Using graphing calculators, students will examine the patterns that emerge to predict the end behavior of polynomial
functions. They will experiment by manipulating equations superimposed onto landmarks in the shape of parabolas and polynomial functions. An end
behavior song and dance, called "Graph Me Baby" will allow students to become graphs in order to physically understand the end behavior of the
graph.
Subject(s): Mathematics
Grade Level(s): 9, 10, 11, 12
Intended Audience: Educators
Suggested Technology: Document Camera, Graphing
Calculators, Computer for Presenter, LCD Projector
Instructional Time: 30 Minute(s)
Resource supports reading in content area: Yes
Freely Available: Yes
Keywords: Polynomial Leading Co-efficient Power of Polynomial
Resource Collection: CPALMS Lesson Plan Development Initiative
ATTACHMENTS
CPALMS St. Louis Arch.docx
CPALMS Equations for the Lesson.docx
graph me baby.docx
CPALMS Summative Assessment Graph Me Baby.docx
LESSON CONTENT
Lesson Plan Template: General Lesson Plan
Learning Objectives: What should students know and be able to do as a result of this lesson?
Students will identify the end behavior of rational functions by examining only the equation.
Students will create equations that will have a given end behavior.
Prior Knowledge: What prior knowledge should students have for this lesson?
The prior knowledge students should have for this lesson is the basic understanding that an equation produces a set of points which follows some pattern that can be
predicted by its equation.
Guiding Questions: What are the guiding questions for this lesson?
The following will act as guiding questions/prompts for this lesson:
In planning a roller coaster ride, playground equipment, the orbit of the shuttle and re-entry into space, what must the designer do to communicate the idea to the
builders (create equations to model the ride, equipment or path of the shuttle)?
Using your graphing calculator, what happens to a graph of an equation such as y = 4x when we change the power of x from 1 to 2? (The line becomes a curve.)
How does the graph change from the previous question if the leading co-efficient is changed to -4? (The graph is reflected across the x-axis.)
page 1 of 4 Examine the picture of the St. Louis Arch that has just been sent to you. Collaborating with a partner, you have 4 minutes to come as close as possible to
superimposing the equation of the parabola to the picture.
Using your graphing calculator and the following set of equations, what happens to the new graph when we the highest power increases by one? (The graph
changes direction.) See "Equations" attachment for the equations.
Reference the directions in the GRAPH ME BABY attachment for the following questions:
Use your graphing calculator to describe what the highest power of the graph tells you about the appearance of the graph? (The maximum number of direction
changes.)
Do graphs that have an even degree (largest exponent) have something in common when we look at BOTH ends of the graph? (The left and right side of the graph
will be the same. They will either both go toward positive infinity or both go toward negative infinity.)
Do graphs that have an even degree have something in common when we look at the end of the graph? (The left and right side of the graph will go in opposite
directions.)
Do graphs that have a negative leading co-efficient have anything in common? (The right side of the graph goes toward positive infinity.)
Do graphs that have a positive leading co-efficient have anything in common? (The right side of the graph goes toward negative infinity.)
Does the combined effect of positive/negative leading co-efficient and even/odd powers impact the direction in which the graph goes at its ends? (The combined
effect of these factors create end behavior.)
What impact does a positive or negative leading co-efficient have on the end of the graph? (It controls what happens on the right side of the graph. A positive
leading co-efficient means that as x approaches infinity, f(x) approaches positive infinity.)
What impact does an even power have on the end of the graph? (The power controls whether the left side of the graph is going in the same direction as the right
side or the opposite direction. If the power is even, the left side will go in the same as the right side. If the power is odd, the left side will go in the opposite
direction of the right side.)
Teaching Phase: How will the teacher present the concept or skill to students?
The teacher will use the guiding questions/prompts to help students make the connections between the power and the appearance of the graph in terms of its
curvature.
1. A picture will be shown (sent via WiFi to student calculators if available) of the St. Louis Arch (or any other picture of a landmark with a parabolic shape), and the
students will be asked to pick a partner and try to create an equation which best models the actual path of the arch.
2. After a short period of time, students graphs are shared, superimposed on the picture (if using WiFi, displayed at the same time on the overhead projector, without
the name of the student).
3. Students will vote on which equation best modeled the picture. If using WiFi, the teacher will activate the name feature of the WiFi to show which students came
close to modeling the equation. Teachers who do not have a WiFi unit can use a doc cam and a calculator that can receive pictures and have the students call out
equations to model the St. Louis Arch. If a doc cam is not available, teachers can copy the picture of the St. Louis Arch, ensuring that the photo is approximately the
screen size of the graphing calculator, and provide patty paper for the students to trace the arch and place the traced image on the calculator screen.
4. The Guiding Questions that address the impact of increasing the power will lead the students to the understanding that the graph and its end directions will change.
A separate attachment (with several polynomials that have clear directional changes) is included.
5. The Guiding Questions about even and odd exponents will lead to a discussion of end behavior to enable the students to discover what combinations of odd/even
powers and positive/negative leading coefficients create end behavior from the equations.
6. The lesson closure will be the End Behavior dance which has students physically modeling end behavior of equations when the power and leading coefficients are
called aloud. (The students' arms will be used to show whether the graph goes to positive or negative infinity on the right and left of the graph.) A short video clip
can be created by the instructor/students (and can be made available via social media or through the teacher's classroom website as deemed appropriate according
to school district policy.) See End Behavior Dance attachment entitled "Graph Me Baby."
Guided Practice: What activities or exercises will the students complete with teacher guidance?
The activities that the students will complete with teacher guidance are the creation of the equation for the landmark picture and the end behavior dance that the
teacher is calling aloud as the students are modeling the end behavior of the graph with their arms.
Independent Practice: What activities or exercises will students complete to reinforce the concepts and skills developed in the
lesson?
The exercises that the students will complete to reinforce the concepts and skills developed in the lesson will be questions that lead them forward and backward in the
concept of end behavior and curvature of the graph.
One of the ways in which the learning from the lesson will be reinforced over this unit is the use of the end behavior dance either at the beginning of class as a warm
up or in the middle of the lesson as a bridge to a new concept.
Additionally, students will be asked to create equations with a particular end behavior, rather than the traditional process of finding the end behavior from a given
equation.
Closure: How will the teacher assist students in organizing the knowledge gained in the lesson?
The teacher will assist students in organizing the knowledge gained in the lesson concerning end behavior by presenting "The End Behavior Dance." The End Behavior
Dance makes each student act as a human graph, with their arms pointing in the direction that the graph goes (negative or positive infinity.) The lyrics to the song (to
the tune of "Call Me, Maybe") describe what is occurring in the equation. The teacher will bring the lesson to a close by recording The End Behavior Dance, Graph Me
Baby, and uploading it to social media for future viewing. Students could record this on their cell phones for future reference.
Summative Assessment
The teacher will determine if the students have reached the learning targets for this lesson through the use of a practice quiz (administered via the WiFi calculator
technology if available).
The follow up questions will respond to student common misconceptions in connection with the concept addressed.
Formative Assessment
The teacher will gather information about student understanding and prior knowledge throughout the lesson using questioning techniques.
Feedback to Students
1. The students will get feedback about their performance or understanding during the lesson with each question asked (via the WiFi calculator technology if
available).
page 2 of 4 2. After each question, teachers will use the responses (received via the technology if available) to guide the level of detail needed to allow students to have an
opportunity to use this feedback to improve their performance on the very next question posed so that the learning goals and objectives are met.
3. Additional feedback will be given by selecting individual students to be the instructor. This is accomplished by using the technology to allow the individual student's
calculator to be the projected on the screen so that the other students will see the step-by-step procedure used to work the question.
ACCOMMODATIONS & RECOMMENDATIONS
Accommodations:
Students with special needs would be given a special selection of problems which include all of the key concepts provided in the work for the class as a whole. The
special selection would have a more structured sequence so that the key concepts are more evident through the progression of the assignment.
Extensions:
This lesson idea could be expanded to include the trig functions at the end of the Algebra II course.
Suggested Technology: Document Camera, Graphing Calculators, Computer for Presenter, LCD Projector
Special Materials Needed:
Graphing Calculators
Picture of landmark
Graph Me Baby lyrics
Sample Equations
LCD projector
WiFi system OR Document Camera OR Computer with graphing calculator simulator
Optional Patty Paper
This lesson requires a graphing calculator for each student. The lesson is enhanced by the use of a class set of graphing calculators linked by an accompanying Wi-Fi
system. Although any graphing calculator will allow students to perform the actual graphs identified in the lesson, the specific Wi-Fi system allows the teacher to
determine if the group of students understands the concepts, both individually and as a whole. The Wi-Fi system stores the data for each question posed by the
teacher for examination after the lesson has been completed. Additionally, the Wi-Fi system allows teachers to see every calculator in action at the same time to
ensure that students are not one step behind and to allow students to act as the presenter when answering questions. In the absence of this system, patty paper and
a document camera will be used. Please note that the time estimate for the lesson is based on the lesson delivered using the Wi-Fi system.
Further Recommendations:
1. This lesson takes students who have worked exclusively with linear equations and allows them to explore the end behavior of polynomials beginning with parabolic
function in the form
. The students will observe the impact of changing the degree of the polynomial in combination with the sign of the leading
coefficient. Students will be asked to come to a conclusion regarding the effects of the highest exponent and the maximum number of directions of the graph of a
polynomial and its end behavior. This lesson is written with the expectation that a class set of graphing calculators linked to a WiFi system is available for the
instruction and feedback. Modifications to these directions are noted for teachers who do not have this technology available to them.
2. Students can photograph landmarks that appear parabolic and find the closest equation to model the photograph taken. Examples of landmarks could be satellite
dishes, contact lenses, fast food locations whose logos include two parabolas, a portion of a roller coaster, etc. It is important for them to understand that these
products are created by an equation which governs the size, shape, steepness and are built to specifications dictated by the equations (rather than the students'
image that something is built like a sandcastle on the beach.)
Additional Information/Instructions
By Author/Submitter
The following standards of math practice may align with this lesson:
MAFS.K12.MP.5.1 - Use appropriate tools strategically.
MAFS.K12.MP.6.1 - Attend to precision.
SOURCE AND ACCESS INFORMATION
Contributed by: Marie Causey
Name of Author/Source: Marie Causey
District/Organization of Contributor(s): Seminole
Is this Resource freely Available? Yes
Access Privileges: Public
License: CPALMS License - no distribution - non commercial
Related Standards
Name
Description
Graph functions expressed symbolically and show key features of the graph, by hand in simple cases and using
technology for more complicated cases. ★
page 3 of 4 MAFS.912.F-IF.3.7:
a. Graph linear and quadratic functions and show intercepts, maxima, and minima.
b. Graph square root, cube root, and piecewise-defined functions, including step functions and absolute value
functions.
c. Graph polynomial functions, identifying zeros when suitable factorizations are available, and showing end behavior.
d. Graph rational functions, identifying zeros and asymptotes when suitable factorizations are available, and showing
end behavior.
e. Graph exponential and logarithmic functions, showing intercepts and end behavior, and trigonometric functions,
showing period, midline, and amplitude, and using phase shift.
page 4 of 4