Polynomial Functions Unit Learning Targets
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Students will be able to identify independent and dependent variables and domain
and range in polynomial situations, as well as model the situations with tables,
graphs, and rules, and describe the pattern in the table (both the pattern and the
number that shows up).
Students will be able to identify the degree of a polynomial in both factored and
simplified form, and use it to identify the potential number of local max/mins, xintercepts, and number of possible solutions, as well as describe the general shape
and end behavior of a polynomial’s growth (in combination with the leading
coefficient).
Students will be able to find the x-intercepts given any polynomial rule in
factored form, as well as distribute rules into simplified form.
Students will be able to write a polynomial rule in factored form given a degree
‘n’ and the x-intercepts.
Students will be able to write out the calculation steps for a given rule, write the
inverse rule or identify when the inverse rule is not possible, and invert and solve
root functions.
Title of Unit
Curriculum Area
Developed By
Polynomial Functions
Algebra
Grade Level
Time Frame
10th Grade
Approximately 10 assignments,
approximately 4-6 weeks
Isaac Frank
Identify Desired Results (Stage 1)
Content Standards
High School: Algebra
SSE.A.1.A: Interpret parts of an expression, such as terms, factors, and coefficients.
SSE.A.2: Use the structure of an expression to identify ways to rewrite it.
SSE.B.3: Choose and produce an equivalent form of an expression to reveal and explain properties of the quantity represented by the expression.
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.
CED.A.1: Create equations and inequalities in one variable and use them to solve problems.
High School: Functions
IF.A.1: Use function notation, evaluate functions for inputs in their domains, and interpret statements that use function notation in terms of context.
IF.B.1: For a function that models a relationship between two quantities, interpret key features of graphs and tables in terms of the quantities, and sketch graphs showing key features given a verbal description of
the relationship.
IF.C.7: Graph functions expressed symbolically and show key features of the graph, by hand in simple cases and using technology for more complicated cases.
IF.C.7.A: Graph linear and quadratic functions and show intercepts, maxima, and minima.
IF.C.7.C: Graph polynomial functions, identifying zeros when suitable factorizations are available, and showing end behavior.
IF.C.8: Write a function defined by an expression in different but equivalent forms to reveal and explain different properties of the function.
BF.A.1: Write a function that describes a relationship between two quantities.
BF.B.4: Find inverse functions.
BF.B.4.A: Solve an equation of the form f(x)=c for a simple function f that has an inverse and write an expression for the inverse.
Standards for Mathematical Practice
CCSS.MATH.PRACTICE.MP1: Make sense of problems and persevere in solving them.
CCSS.MATH.PRACTICE.MP2: Reason abstractly and quantitatively.
CCSS.MATH.PRACTICE.MP3: Construct viable arguments and critique the reasoning of others.
CCSS.MATH.PRACTICE.MP4: Model with mathematics.
CCSS.MATH.PRACTICE.MP6: Attend to precision.
CCSS.MATH.PRACTICE.MP7: Look for and make use of structure.
CCSS.MATH.PRACTICE.MP8: Look for and express regularity in repeated reasoning.
Understandings
Overarching Understanding
Students will understand how the coefficients and constants in a
polynomial rule (in factored form, primarily) affect the graph of the rule
Students will understand when an inverse can be found for a polynomial
and when one cannot be found
Students will understand how to describe what a graph looks like based on
the factored form of the rule
Students will understand how the degree of the polynomial rule affects the
graph and table
Essential Questions
Overarching
What does the graph of this
polynomial look like?
What is causing it to look that
way?
How can I solve this polynomial?
What is the domain/range of this
function?
What is the end behavior of this
function?
Topical
How many local max/mins
are there/can there be?
How many x-intercepts are
there/can there be?
Why can/can’t we find the
invers of this function?
What numbers need to go in
the rest of the rule to fulfill
the given criteria?
Related Misconceptions
Students may just apply the math from quadratics to polynomials with
higher degrees without realizing that different math may apply. Students
may also try to use the process we used to get quadratics into vertex form
and end up turning a different degree polynomial into a quadratic.
Objectives
Skills
Knowledge
Students will know…
Students will be able to…
How the coefficients/constants in a rule affect the graphs and tables of
polynomial
How to identify the degree of a polynomial
How to simplify polynomial rules
When and how to inverse polynomial rules
When it is and is not possible to invert polynomial rules
Model a cubic situation
Describe patterns, relationships, and characteristics of tables and
graphs of polynomials
Simplify polynomial rules
Create polynomial rules to fulfill given criteria
Identify the domain and range of polynomial functions
Solve polynomial equations using the inverse (when possible)
Recognize when it is not possible to invert a polynomial equation
and explain why
Assessment Evidence (Stage 2)
Performance Task Description
Goal
Role
Audience
Situation
Product/Performance
Standards
Assess the students’: ability to identify the x- and y-intercepts based on a rule in factored form;
understanding of the degree of polynomials; comprehension of end behavior of polynomials; ability
to sketch a graph based on a rule; understanding of when and it is possible/impossible to find an
inverse of a polynomial; understanding of the coefficients in a polynomial rule, and how they affect
the graph and table of the rule; ability to write a rule that satisfies given criteria; ability to solve a
polynomial using the inverse; ability to simplify a polynomial; understanding of domain and range
of a polynomial
End of unit test
Myself, Craig Huhn (mentor teacher)
Summative unit exam (taken during regular class period)
Test
SSE.A.1.A: Interpret parts of an expression, such as terms, factors, and coefficients.
SSE.A.2: Use the structure of an expression to identify ways to rewrite it.
SSE.B.3: Choose and produce an equivalent form of an expression to reveal and explain properties of the quantity represented by the expression.
CED.A.1: Create equations and inequalities in one variable and use them to solve problems.
IF.B.1: For a function that models a relationship between two quantities, interpret key features of graphs and tables in terms of the quantities, and
sketch graphs showing key features given a verbal description of the relationship.
IF.C.8: Write a function defined by an expression in different but equivalent forms to reveal and explain different properties of the function.
BF.B.4: Find inverse functions.
BF.B.4.A: Solve an equation of the form f(x)=c for a simple function f that has an inverse and write an expression for the inverse.
Other Evidence
Learning Plan (Stage 3)
Concept in Unit
Properties of
Cubing
Lesson Topic
Lesson Learning
Objective
Introducing
cubing using
volume
SWBAT model a cubic
situation (volume)
Properties of
cubic
functions
SWBAT describe
patterns, relationships,
and characteristics of the
tables and graphs of
(two) cubic functions.
Properties of
cubic
functions
SWBAT describe
patterns, relationships,
and characteristics cubic
functions
Simplifying/creating Simplifying
rules
polynomial
rules
Creating
polynomial
rules
SWBAT simplify
polynomial rules
SWBAT create
polynomial rules that
fulfill given criteria
Description of how lesson
contributes to unit-level
objectives
This lesson has students modeling
a cubic situation by making boxes
with different dimensions using
regular sheets of copier paper.
This lesson has students evaluating
three linear rules and their
combined cubic product for
multiple x values, making tables
and graphs of the equations,
simplifying the cubic rule, and
comparing significant features of
linears and cubics.
This lesson has students recapping
the properties and characteristics of
cubic functions, and building the
understanding of important
features.
This lesson has simplifying
polynomial rules of degree 3, 4, 5,
and 6.
This lesson has students create
rules with given characteristics (or
explain why it is not possible), as
well as complete rules in factored
form so that they fulfill given
requirements.
Assessment activities
Assignment (Making Rectangular
Containers), small group work, class
discussion on assignment
Table, graph, Writing a rule
Assignment (Product of Three Lines),
small group work, class discussion on
assignment
Table, graph, descriptions of patterns
in both
Simplified rules
Assignment (Cubic Functions Recap),
small group work, class discussion on
assignment
Descriptions of properties and
functions
Assignment (Simplifying Rules), small
group work, class discussion on
assignment
Simplified rules
Assignment (Making Polynomial
Rules), small group work, class
discussion on assignment
Created rules
Characteristics of
polynomials
X-intercepts
of
polynomials
SWBAT describe the
tables and graphs of
polynomial functions
based on rules in factored
form
This lesson has students describe
the characteristics of tables of
graphs of polynomials of degree 3
4, and 5.
Assignment (Focus on x-intercepts),
small group work, class discussion on
assignment
Descriptions of characteristics
Domain and
Range of
polynomials
SWBAT identify the
domain and range of
polynomial functions
This lesson has students both
giving examples of functions with
a given domain or range, as well as
giving the domain or range of
given functions.
Assignment (Domain and Range),
small group work, class discussion on
assignment
Solving
Polynomials
Inverses of
Polynomials
Practice Test
These lessons have students find
the inverses of polynomial rules (if
possible) and use them to solve, as
well as recognize when it is not
possible to invert and explain why.
This lesson gives students the
opportunity to practice taking the
unit test, with a day to review it
with their peers the next day, with
the real test the day after the
review.
Assignments (Using Inverses to Solve
Polynomials; Solving Practice), small
group work, class discussion on
assignment
Practice Test
SWBAT solve
polynomial equations
using the inverse, or
recognize when that is
not possible
SWBAT demonstrate
their knowledge of the
unit
Assignment (Polynomials Practice
Test), small group work (day after)
Test
Both Craig and I will assess much of the evidence of student learning in day-to-day interactions and conversations with students, as well as
class discussions. We will also use warm-ups to quickly check where students are at with their understanding of learning targets, as well as
possible giving one or two quizzes. At the end of the unit, students will also get credit for the day-to-day work they have been doing on the
tasks we have handed out by handing in their unit packet.
Polynomial Functions Unit Test
Name ________________________
Make sure to show all of your work and explain completely. Answer every question, and be sure not to spend
too much time on one problem. Feel free to skip around, but make sure you attempt all problems. If you get
done before the time is up, use your calculator to verify or check your answers.
Answer the following questions about f(x) = -7(x – 5)(x + 2)(x – 1.3)(x – 100)(x + 4).
1)
a.
Identify the x-intercepts, and explain how you knew what they were without searching in the table.
b.
Identify the y-intercept and explain how you got it.
c.
What degree is this polynomial, and how do you know?
d. What does the right end of this graph look/behave like? How do you know based just on the rule?
e. What does the left end of this graph look/behave like? How do you know this based just on the rule?
Answer the following questions still about f(x) = -7(x – 5)(x + 2)(x – 1.3)(x – 100)(x + 4).
f. Sketch a graph with the maximum number of “bumps” for this function (and the correct end behavior).
How do you know each of these properties from the rule?
g. Can you find the inverse rule for this function? If so, how would you do it; if not, why not?
2)
What would have to be true about the leading coefficient (the a value) and the degree of the polynomial
to make a graph like this? For each, say how you know.
3)
Describe the pattern in the table for –5x3 + 4x2 – 8x + 9, without having to make the table and check.
Full credit will be given if the pattern and the number is given.
4)
Write a rule for a polynomial of degree four with the x-intercepts of only (2,0) , (-4,0) and (1.5,0).
5)
Solve the following using inverses. If it cannot be done without searching, say so and why this is the
case.
 ( x  3) 

 = -300
 6 
5
a.
7x – 9 = 0
c.
5( x  4) 3  10  0
d.
e.
4 x 6  7  22
f.
2
b.
 12 x 4  11x 3  10  7.875
11( x  3) 4  9   4
6)
Simplify (-2x + 8)(x – 6)(4x + 1).
7) What is the maximum number of zeroes that could result in a 10th degree polynomial?
8) Define the domain and range of the following function: f(x) = -3x4 – 4x2 + 5x - 4
EXTRA CREDIT:
Given the polynomial f(x) = px3 + 12x2+ q , find values for p and q that give a
polynomial with all positive outputs. Explain why your chosen values of p and q give positive outputs or why
the task is not possible.