Course Name: Algebra 2/Math III
Unit #1
Unit Title:Polynomials
Enduring understanding (Big Idea): Students will understand how to factor polynomials in multiple forms, analyze polynomial
functions and their graphs by identifying end behavior and roots, and graph polynomial functions.
Essential Questions:
1. How are factors and roots of a polynomial equation related?
2. How does solving and graphing higher-order polynomials relate to solving and graphing quadratics?
BY THE END OF THIS UNIT:
Students will know…
• How to factor, solve, and graph polynomial functions
• Start and end behavior of polynomials
• How to write polynomials functions from points
Vocabulary:
Distributive Property, Closure, Like Terms, Polynomial, Term,
Root, Solution, X-Intercept, Zero, Factor, Remainder Theorem ,
Rational Root Theorem, Pascal’s Triangle, rational expression,
simplified form, excluded value, equivalent fractions, asymptote,
Complex Fractions, Rational Expressions, Minimum, Maximum,
Symmetry, End Behavior, Interval, Periodicity, Domain, Domain
Restrictions, Integers, Start Behavior, End Behavior, Relative
extremes, zeroes, Regression, Function
Unit Resources:
See attached standard guides for additional resources.
Building a Cardboard Box of Greatest Volume:
http://questgarden.com/101/08/6/100420161846/index.htm
Representing Polynomials MARS Task – Analysis of Graphs of
Quadratics and Cubics Based On Intercepts, Relative Min/Max,
and Transformations:
http://map.mathshell.org/materials/download.php?fileid=1271
Mathematical Practices in Focus:
1. Make sense of problems and persevere in solving.
2. Reason abstractly.
3. Model with Mathematics.
4. Use appropriate tools strategically.
5. Look for and make use of structure.
CCSS-M Included:
A-APR.1, 2, 3, 4, 5, 6, 7
F-IF.4, 5, 7c
A-REI.10
Students will be able to:
• Add, subtract, and multiply polynomials (the concept should
have already been taught – include fraction and variable
coefficients in Math 3)
• Geometric and real-world applications of adding, subtracting,
and multiplying polynomials
• Long Division of Polynomials
• Use Synthetic Division to completely factor polynomials
• Use Remainder Theorem to evaluate functions and prove that
values are zeroes of functions
• Use Rational Root Theorem to determine possible roots of
functions (honors)
• Find zeroes of higher-order polynomial functions graphically and
from factors.
• Solve cubic and quartic functions.
• Write polynomial functions given the roots
• Factor Sum/Difference of Cubes
• Binomial Expansion
•Recognize and explain the domain of the function of the original
and simplified rational expression
• Simplify Rational Expressions
• Multiplication and division of rational expressions
• Addition and Subtraction of rational expressions
• Identify the intercepts, relative minimums and maximums, lines
of symmetry, and start and end behavior of graphs (Polynomial
and Rational)
• Identify the intervals on which a graph increases and decreases
(Polynomial and Rational)
• Graph a function based on its key features (Polynomial and
Rational)
• Determine the domain of a function from its graph, including
analysis of the start and end behavior (Polynomial and Rational)
• Determine the key domain restrictions of functions, including
division by 0 and even roots of negative numbers (Polynomial and
Rational)
• Analyze word problems to determine domain restrictions (i.e.
negative numbers, non-integers).
• Descartes Rule of Signs (HONORS)
• x = y2 Parabolas (HONORS)
• Write polynomial functions from their points in real-world
situations and use them to predict future values (regression)
Suggested Pacing:
11 days (including 1 review, 1 test) – NOT including rationals
Standards are listed in alphabetical /numerical order not suggested teaching order.
PLC’s must order the standards to form a reasonable unit for instructional purposes.
PAGE 1
Course Name: Algebra 2/Math III
Unit #1
Unit Title:Polynomials
CORE CONTENT
Cluster Title: Polynomial Operations
Standard A-APR.1. Understand that polynomials form a system analogous to the integers, namely, they are closed under the
operations of addition, subtraction, and multiplication; add, subtract, and multiply polynomials.
Concepts and Skills to Master:
• Add, subtract, and multiply polynomials (the concept should have already been taught – include fraction and variable coefficients in
Math 3)
• Geometric and real-world applications of adding, subtracting, and multiplying polynomials
SUPPORTS FOR TEACHERS
Critical Background Knowledge:
 Rules of exponents
 Combining like terms
 Formulas for perimeter, area and volume of basic geometric shapes
 Fraction Operations
Academic Vocabulary:
Distributive Property, Closure, Like Terms, Polynomial, Term
Suggested Instructional Strategies:
 Teach adding/subtracting polynomials, through realworld and geometric examples and situations
 Teach multiplying polynomials through real-world and
geometric examples
Resources:
Algebra 2 Textbook Correlation: 1-3
MARS Task on Manipulating Polynomials:
http://map.mathshell.org/materials/lessons.php?
taskid=437&subpage=concept
NCDPI Unpacking:
The Closure Property means that when adding, subtracting or
multiplying polynomials, the sum, difference, or product is also a
polynomial. Polynomials are not closed under division because in
some cases the result is a rational expression.
Sample Assessment Tasks
Skill-based task:
1. (3x3 – 4x2 + 1) – (2x + 4)
2. (2x2 + 3) + 4(x – 2)2
3. Simplify:
a2  b2
ab
Problem Task:
1) The area of a rectangle can be represented by the expression
(2x2 – 17x + 6), and the height can be represented by the
expression (x – 6). What expression represents the volume?
2) The radius of a sphere can be represented by the binomial (x +
2). What expressions represent the volume and surface area?
3) Explain why (x + y)2 ≠ x2 + y2.
NCDPI Examples:
Ex. If the radius of a circle is (5x – 2) kilometers, write an
expression for the area of the circle.
Ex. Explain why (4𝑥2 + 3)2 does not equal (16𝑥4 + 9).
Standards are listed in alphabetical /numerical order not suggested teaching order.
PLC’s must order the standards to form a reasonable unit for instructional purposes.
PAGE 2
Course Name: Algebra 2/Math III
Unit #1
Unit Title:Polynomials
CORE CONTENT
Cluster Title: Understand the Relationship Between Zeroes and Factors of Polynomials
Standard: A-APR.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).
Concepts and Skills to Master:
• Long Division of Polynomials
• Use Synthetic Division to completely factor polynomials
• Use Remainder Theorem to evaluate functions and prove that values are zeroes of functions
• Use Rational Root Theorem to determine possible roots of functions (honors)
SUPPORTS FOR TEACHERS
Critical Background Knowledge:
-Solutions to polynomial functions occur when the function equals 0.
- Factors of polynomials set equal to 0 determine solutions.
- Evaluation Functions
Academic Vocabulary:
Root, Solution, X-Intercept, Zero, Factor, Polynomial, Remainder Theorem
Suggested Instructional Strategies:
Resources:
Algebra 2 Textbook Correlation: 5-4, 5-5
- Teach long division first (for HONORS), closely followed by
synthetic division, highlighting that synthetic only works for a
Remainder Theorem Discovery (to be uploaded)
polynomial divided by a binomial.
- Use the distributive property/properties of equality to show how
to synthetically divide a polynomial where x has a coefficient > 1.
- Have students use synthetic division to discover the remainder
theorem.
NCDPI Unpacking:
The Remainder Theorem states that if a polynomial, 𝑝(𝑥) is
divided by a monomial, (𝑥 – 𝑐), the remainder is the same as if
you evaluate the polynomial for 𝑐, i.e. calculate 𝑝(𝑐). If the
remainder when dividing by (𝑥 – 𝑐) is 0, or 𝑝(𝑐) = 0, then (𝑥 – 𝑐)
is a factor of the polynomial. If 𝑓(𝑎) = 0, then (𝑥 − 𝑎) is a factor of
𝑓(𝑥), which means that 𝑎 is a root of the function 𝑓(𝑥). This is
known as the Factor Theorem.
Sample Assessment Tasks
Skill-based task:
1. For f(x) = 3x3 + 4x2 – 12x + 3, evaluate:
a) f(0)
b) f(6)
c) f(-2)
d) f(1/2)
using division.
2. Let p(x) = x5 – 3x4 + 8x2 – 9x + 30. Find p(-2). What does
your answer tell you about the factors of p(x)?
Problem Task:
1) The area of a rectangle can be represented by the expression
2x3 – 17x2 + 31x – 6. If the width is represented by x – 6, what
expression represents the length?
NCDPI Examples:
Ex. Given 𝑓 (𝑥) = 2𝑥2 + 6𝑥 − 20, determine whether −5 is a root
of the function, then write the function in factored form.
Ex. Compare the process of synthetic division to the process of
long division for dividing polynomials.
Ex. Assume that (𝑥 − 𝑐) is a factor of 𝑓, which means that 𝑓 is
divisible by 𝑥 − 𝑐 . Explain why it must be true that 𝑓 (𝑐) = 0.
Ex. Assume we know that 𝑓 (𝑐) = 0. Explain why it must be true
that (𝑥 − 𝑐) is a factor of 𝑓.
Standards are listed in alphabetical /numerical order not suggested teaching order.
PLC’s must order the standards to form a reasonable unit for instructional purposes.
PAGE 3
Course Name: Algebra 2/Math III
Unit #1
Unit Title:Polynomials
CORE CONTENT
Cluster Title: Interpreting Functions
Standard A-APR.3.Identify zeroes of polynomials when suitable factorizations are available, and use the zeroes to construct a
rough graph of the function defined by the polynomial.
Concepts and Skills to Master:
• Find zeroes of higher-order polynomial functions graphically and from factors.
• Solve cubic and quartic functions.
• Write polynomial functions given the roots.
SUPPORTS FOR TEACHERS
Critical Background Knowledge:
• Factoring
• Simplifying Radicals
Academic Vocabulary:
Zeroes, Roots, Solutions, Intercepts, Factors
Suggested Instructional Strategies:
 Students learned about solving quadratics in Math 2, so
Math 3 extends to solving higher-order polynomials.
Begin with a brief warm-up review of solving quadratics,
but incorporate solving quadratics into solving the higherorder polynomials for those that need review.
 Relate the x-intercepts to the solutions = 0 for the
equation, highlighting that y = 0 at the x-intercepts.
 Relate synthetic division to the solutions of the factors =
0, and use the process to solve the higher-order
polynomials.
 Relate roots to polynomial functions “in reverse” using a
discovery lesson to extend from quadratics.
 Teach start and end behavior rules and use to graph
polynomials by their zeroes.
Resources:
Algebra 2 Textbook Correlation: 5-2, 5-3
1) Solving Higher-Order Equations Performance Task (see wiki,
use after solving cubics, before solving higher-order, all rational
solutions)
2) Writing Equations from Roots Discovery (it uses the roots of
quadratics to compare to their factors to allow students to see the
application to higher-order polynomials) (to be uploaded)
3)Representing and Solving Real-World Quadratics (should not
be our main lesson, but is a good support):
http://map.mathshell.org/materials/lessons.php?
taskid=432&subpage=problem
For HONORS, extend to include completing the square and
introduce
x  y2
parabolas
NCDPI Unpacking:
Find the zeros of a polynomial when the polynomial is factored.
Then use the zeros to sketch the graph.
Sample Assessment Tasks
Skill-based task:
Solve and sketch rough graphs of the following equations:
(Find 3 cubic/quartic)
Problem Task:
NCDPI Examples:
Ex. For a certain polynomial function, 𝑥 = 3 is a zero with
multiplicity two, 𝑥 = 1 is a zero with multiplicity three, and 𝑥 = −3
is a zero with multiplicity one. Write a possible equation for this
function and sketch its graph.
Standards are listed in alphabetical /numerical order not suggested teaching order.
PLC’s must order the standards to form a reasonable unit for instructional purposes.
PAGE 4
Course Name: Algebra 2/Math III
Unit #1
Unit Title:Polynomials
CORE CONTENT
Cluster Title: Polynomial Identities
Standard A-APR.4. Prove polynomial identities and use them to describe numerical relationships. (For example, the
polynomial identity (x2 + y2)2 = (x2 – y2) + (2xy)2 can be used to generate Pythagorean triples.)
Concepts and Skills to Master:
• Use the Rational Root Theorem to determine possible roots of polynomials (HONORS)
• Factor Sum/Difference of Cubes
• Use Synthetic Division to Completely Factor Higher-Order Polynomials
• Given the roots of a polynomial, write the equation of the polynomial
SUPPORTS FOR TEACHERS
Critical Background Knowledge:
• Factoring
• Synthetic Division
Academic Vocabulary:
Rational Root Theorem, Root, Zeroes
Suggested Instructional Strategies:
• Use synthetic division/factoring higher-orders to review factoring
(maybe include factoring warm-up – they learned how to factor in
Math 1 and 2).
• Expand factoring knowledge to sums and differences of cubes
(relate to difference of squares).
• For HONORS, expand synthetic division to the rational root
theorem to help them find the divisor.
NCDPI Unpacking:
Prove polynomial identities algebraically by showing steps and
providing reasons or explanations.
Sample Assessment Tasks
Skill-based task:
1) Factor x3 – 3x2 + x – 3 if one factor is ____.
2) Factor 8x3 + 125.
Resources:
Algebra 2 Textbook Correlation: 5-2, 5-3,5-4, 5-5
Algebra Tiles Explanation of Sum/Difference of Cubes:
http://www.mathnstuff.com/math/algebra/tt21.htm
Algebra II For Dummies Explanation of Synthetic
Division/Remainder Theorem:
http://books.google.com/books?id=VmPwZHw_DKwC&pg=
PA118&lpg=PA118&dq=synthetic+division+tasks&source=
bl&ots=7L88xbs6Kf&sig=o0eXbIk4N0hIVhhVlLpSB1Rtcs&hl=en&sa=X&
ei=3liQUY2LM4X28wS19ICwAQ&ved=0CDUQ6AEwATgK#v=
onepage&q=synthetic%20division%20tasks&f=false
Problem Task:
1) The volume of a cube can be represented by the expression x3
– 125. What expression represents the width of the cube?
NCDPI Examples:
The following examples are meant to be investigated by students
considering analogous problems, and trying special cases and
simpler forms of the original problem in order to gain insight into
its solution(s).
Ex. Is (2x – 3)2 - 64 equivalent to(2x – 11)(2x + 5)? Explain why
or why not.
Ex. Jessie claims that (x + y)2 = x2 + 2xy + y2. Is he correct? Prove
why or why not.
Ex. Prove x3 – y3 = (x – y)(x2 + xy + y2). Justify each step.
Ex. Solve the quadratic ax2 + bx + c = 0 Justifying each step.
What was interesting about the result?
Standards are listed in alphabetical /numerical order not suggested teaching order.
PLC’s must order the standards to form a reasonable unit for instructional purposes.
PAGE 5
Course Name: Algebra 2/Math III
Unit #1
Unit Title:Polynomials
CORE CONTENT
Cluster Title: Interpreting Functions
Standard A-APR.5. Know and apply the Binomial Theorem for the expansion of (x + y)n in powers of x and y for a positive
integer n, where x and y are any numbers, with coefficients determined by Pascal’s Triangle.
Concepts and Skills to Master:
• Binomial Expansion
SUPPORTS FOR TEACHERS
Critical Background Knowledge:
• Multiplying Polynomials
• Identifying Patterns
Academic Vocabulary:
Pascal’s Triangle
Suggested Instructional Strategies:
Have students fill out Pascal’s Triangle for the first 12 rows. Then,
they should multiply (x + y) to the 6th power manually until they
discover the pattern. (Start the standard with the argumentation
task so they discover the pattern.)
NCDPI Unpacking:
The Binomial Theorem describes the algebraic expansion of
powers of a binomial. There are patterns that develop with the
coefficients and the variables when expanding binomials. Pascal’s
triangle is a triangular array that identifies the coefficients of an
expanded binomial. The numbers in Pascal’s triangle are also
evaluations of combinations, nCr. The values of the combinations
correspond with the coefficients of the expanded binomial, which
indicates how many times that term will appear in the completely
expanded form. This is a connection between Probability and
Algebra that should be made explicit. For example, when squaring
the binomial (𝑎 + 𝑏), note that the product 𝑎𝑏 occurs twice: (a +
b)2 = a2 + ab + ab + b2 = a2 + 2ab + b2. Using combinatorics, the
coefficient of the second term would be 2C1 = 2.
Sample Assessment Tasks
Skill-based task:
1) Expand (x + y)5.
2) What is the fourth term in (2x – y)6?
Resources:
Algebra 2 Textbook Correlation: 5-7
Pyramid Power (Pearson Activities/Games/Puzzles 5-7,
accessible through Pearson Success Net or wiki)
Binomial Theorem Enrichment (Pearson Enrichment 5-7,
accessible through Pearson Success Net or wiki)
Problem Task:
1) Textbook pg. 329 #24
Task:
1) Fill out the first 12 rows of Pascal’s Triangle.
2) Distribute and write out the solutions to: (x + y)0, (x + y)1, (x +
y)2, (x + y)3, (x + y)4, (x + y)5, and (x + y)6.
3) Explain the relationship between Pascal’s Triangle and your
solutions, and apply this relationship to compute (x + y)10.
4) How do you think the answer would change if you computed (x
+ 5)10? Justify your answer.
NCDPI Examples:
Ex. Explain how to generate a row of Pascal’s triangle.
Ex. What are the coefficients of the expanded terms of (𝑎 + 𝑏)5?
Ex. Using the binomial theorem, expand (𝑎 + 𝑏)5.
Ex. Why are the coefficients of a binomial expansion equal to
values of nCr?
Standards are listed in alphabetical /numerical order not suggested teaching order.
PLC’s must order the standards to form a reasonable unit for instructional purposes.
PAGE 6
Course Name: Algebra 2/Math III
Unit #1
Unit Title:Polynomials
CORE CONTENT
Cluster Title: Rewrite Rational Expressions
Standard A-APR.6 Rewrite simple rational expressions in different forms; write a(x)/b(x) in the form q(x)+r(x)/b(x), where
a(x), b(x), q(x), and r(x) are polynomials with the degree of r(x) less than the degree of b(x) using inspection, long division or,
for the more complicated examples, a computer algebra system.
Concepts and Skills to Master:
• Recognize and explain the domain of the function of the original and simplified rational expression
• Simplify Rational Expressions
SUPPORTS FOR TEACHERS
Critical Background Knowledge:
• Simplifying numerical fractions
• Factoring (GCF, difference of squares, trinomials)
• Domain and excluded values
Academic Vocabulary:
rational expression, simplified form, excluded value, equivalent fractions, asymptote
Suggested Instructional Strategies:
Resources:
Algebra 2 Textbook Correlation: 8-2, 8-3, 8-4
 Stress that students need to factor first to simplify
rationals, and they can only divide out factors in factored
Jogging Rates (Real-World Application):
form.
http://www.google.com/url?sa=t&rct=j&q=&esrc=s&frm=1&source=
 Relate the domain restrictions of division by 0 to the
web&cd=1&cad=rja&ved=0CC8QFjAA&url=http%3A%2F%2F
graph (no point at that x-coordinate, asymptote).
www.gwinnett.k12.ga.us%2FPhoenixHS%2Fmath%2Fgrade09%
 Students should relate computational skills to real –
2Funit02%2F08-Task-Jogging_Rtnl%2520Expression_.
world applications, see problem tasks.
pdf&ei=6FqQUZX_ FIam8QSz1YCACA&usg=
NCDPI Unpacking:
AFQjCNFhRIckRWHle2yvRnmOw9L5IuKjBA
Rewrite rational expressions,
r ( x)
a( x)
in the form q ( x) 
b( x )
b( x )
using long division, synthetic division or with expressions that
pose difficulty by hand, use a computer algebra system such as
the TI Inspire CAS or Ipad applications. When dividing a
polynomial by a polynomial, the new form is the quotient plus the
remainder divided by the divisor. This process should be
connected to dividing with numbers. The quotient represents the
number of times something will divide, plus the parts or pieces
remaining. Know that the degree of the quotient is less than the
degree of the dividend. Connect division of polynomials to the
remainder theorem when 𝑏(𝑥) is in the form (𝑥 − 𝑐).
Graphing Rationals (Great progression from technology to by
hand, highlights domain restrictions well):
https://www.google.com/url?sa=t&rct=j&q=&esrc=s
&frm=1&source=web&cd=3&ved=0CDkQFjAC&url
=https%3A%2F%2Fwww.georgiastandards.org%
2FFrameworks%2FGSO%2520Frameworks% 2FAcc-Math-IIIUnit-3-SE-RationalFunctions.pdf&ei=T1uQUYyFNZGy9gT3qYDICg
&usg=AFQjCNF4MDd0XdfVJsY3YQezv0lAVohLlA
Standards are listed in alphabetical /numerical order not suggested teaching order.
PLC’s must order the standards to form a reasonable unit for instructional purposes.
PAGE 7
Course Name: Algebra 2/Math III
Unit #1
Sample Assessment Tasks
Skill-based task:
2
Given the rational expression x  7x  10 ; state the domain
x 2  25
and the simplified value.
Unit Title:Polynomials
Problem Task:
2
1) Given the rational expression f (x )  8x  2x  15 ; state
2x 2  x  6
the domain and the simplified value. Explain why the domain of
the simplified function must reflect the domain of the original
function. Justify that a complicated rational function is equivalent
to a simplified functions using graphs and tables.
2) Using a real-world situation discussed in class (STANDARD) or
creating your own real-world situation (HONORS), use a rational
expression to illustrate the situation. (For example, a class with x
students is throwing a party that will cost $50. How much does
each member of the class contribute?) Now, based on your
situation, explain any limits on the domain that would exist based
on the situation or real-world limitations. Is the rational expression
the best mathematical means of describing your situation? Why or
why not?
NCDPI Examples:
Ex. We know from arithmetic, that a fraction like
327
indicates
10
the division of 327 by 10. The result can be expressed 32 R 7 or
7
. Use division of polynomials to show that
10
 x 2  4x  8
can be written with an equivalent expression in
x 1
r ( x)
the form of q ( x ) 
.
x 1
as 32 +
Ex. Divide. Write the answer in the form of quotient plus
remainder/divisor.
x 4  3x
x2  4
Ex. Use a computer algebra system to rewrite the following
rational expression in quotient and remainder form
9x3  9x 2  x  2
x  23
Standards are listed in alphabetical /numerical order not suggested teaching order.
PLC’s must order the standards to form a reasonable unit for instructional purposes.
PAGE 8
Course Name: Algebra 2/Math III
Unit #1
Unit Title:Polynomials
CORE CONTENT
Cluster Title: Rewrite Rational Expressions
Standard A-APR.7 Understand that rational expressions form a system analogous to the rational numbers, closed under
addition, subtraction, multiplication, and division by a nonzero rational expression; add, subtract, multiply and divide rational
expressions.
Concepts and Skills to Master:
• Multiplication and division of rational expressions
• Addition and Subtraction of rational expressions
SUPPORTS FOR TEACHERS
Critical Background Knowledge:
• Factoring
• Simplifying, add, subtract, multiply and divide numerical fractions
• Like and unlike denominators
Academic Vocabulary:
Complex Fractions, Rational Expressions
Suggested Instructional Strategies:
• Teach multiplication and before addition and subtraction
• Students should be encouraged to factor all numerators and
denominators prior applying the operations, and stress that they
cannot divide out factors unless the rationals are in factored form
• Complex fractions should be selected based on difficulty –
HONORS students should have more rigorous problems
• Students should relate computational skills to real – world
applications, see problem tasks.
NCDPI Unpacking:
When performing any operation on a rational expression, the
result is always another rational expression, which is the Closure
Property for rational expressions. Compare this to the Closure
Property for polynomials. Perform operations with rational
expressions, division by nonzero rational expressions only.
Sample Assessment Tasks
Skill-based task:
Given f (x )  3 and g (x )  7
x 5
x 6
Find
f  x   g  x  ;f  x   g  x  ;f  x   g  x  ;f  x   g  x 
State all restrictions on the domain of the functions and the
simplified expression.
NCDPI Examples:
x2  x  2
A rectangle has an area of
sq. ft. and a height of
x3
x2
ft. Express the width of the rectangle as a rational
x 1
expression in terms of 𝑥.
Resources:
Algebra 2 Textbook Correlation: 8-4, 8-5
Rational Equations Paideia (helps students solve and determine if
solutions are realistic in a real-world setting): Linked to wiki
Pearson Enrichment 8-5 – The Superposition Principle
(accessible through Pearson Success Net or wiki)
Problem Task:
8y 2
1) Given x

27 x 2
y
y
x

4 x 9y
, students will explain the techniques and
rationale for simplifying, including the domain and excluded
values.
2) Real world problems - see textbook pg. 540 problems
#37,45,46
3) Write and simplify a complex fraction to find the average rate of
speed for a plane flight from North Carolina to California that
travels 400 mph west and 520 east on the 3,200 mile flight.
a) Why is the average speed NOT 460 mph? [(400 + 520)/2]
Explain the difference between the rational expression and the
basic formula for computing mean.
b) What could account for the differences in speeds between the
trip east and the trip west?
Standards are listed in alphabetical /numerical order not suggested teaching order.
PLC’s must order the standards to form a reasonable unit for instructional purposes.
PAGE 9
Course Name: Algebra 2/Math III
Unit #1
Unit Title:Polynomials
CORE CONTENT
Cluster Title: Interpret Functions That Arise in Applications
Standard F-IF.4. 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.
Concepts and Skills to Master:
• Identify the intercepts, relative minimums and maximums, lines of symmetry, and start and end behavior of graphs.
• Identify the intervals on which a graph increases and decreases.
• Graph a function based on its key features.
SUPPORTS FOR TEACHERS
Critical Background Knowledge:
• Graph and identify points on the coordinate plane
• Solve one-variable equations
Academic Vocabulary:
x-intercept, y-intercept, Minimum, Maximum, Symmetry, End Behavior, Interval, Periodicity
Suggested Instructional Strategies:
Resources:
• Ensure that students can identify intercepts, maximum points,
Algebra 2 Textbook Correlation: Mainly 5-1 and 5-9 for this unit
minimum points, start and end behavior, symmetry, and
(polynomials), but the concept repeats itself throughout several
periodicity early in the unit from actual graphs, then using
units
technology
• Use technology/graph paper to plot key features and functions.
1) Start and End Behavior Discovery (students use calculator to
graph various functions with +/- leading coefficients and even and
odd degrees to see relationships) (to be uploaded)
NCDPI Unpacking:
This standard should be revisited with every function your class is
studying. Students should be able to move fluidly between
graphs, tables, words, and symbols and understand the
connections between the different representations. For example,
when given a table and graph of a function that models a real-life
situation, explain how the table relates to the graph and vise
versa. Also explain the meaning of the characteristics of the graph
and table in the context of the problem as follows:
At the course three level, students should extend the previous
course work with function types to focus on polynomial and
trigonometric functions.
• Polynomials – emphasis should be on the commonalities and
differences between quadratics and power functions, relative
max/mins.
• Trigonometric functions –emphasis should be on periodicity,
amplitude, frequency, and midline. (F-TF.5)
Note – This standard should be seen as related to F-IF.7 with the
key difference being students can interpret from a graph or sketch
graphs from a verbal description of key features.
Standards are listed in alphabetical /numerical order not suggested teaching order.
PLC’s must order the standards to form a reasonable unit for instructional purposes.
PAGE 10
Course Name: Algebra 2/Math III
Sample Assessment Tasks
Skill-based task:
On what interval does the graph of f(x) = x2 increase?
Unit #1
Unit Title:Polynomials
Problem Task:
1) Given the graph of a function, identify the intercepts, minimum
and maximum values, lines of symmetry, and end behavior. Using
the basic functions learned in the previous objective, how is this
function similar/different?
2) Find the y-intercepts of f(x) = 2x, f(x) = 0.5(2)x, and f(x) = 4(2)x?
Based on the value of f(0) in each function, explain the rationale
for the difference in y-intercepts. Use the pattern to form a
hypothesis involving the y-intercept of an exponential function.
NCDPI Examples:
Ex. Insert graph or equation of a polynomial function from a reallife situation.
a. What are the x-intercepts and y-intercepts and explain them in
the context of the problem?
b. Identify any maximums or minimums and explain their meaning
in the context of the problem.
c. Describe the intervals of increase and decrease and explain
them in the context of the problem.
F-IF.4 When given a verbal description of the relationship
between two quantities, sketch a graph of the relationship,
showing key features:
Ex. Jaquan found that for a period of 7 days, the daily high was
85° and the low was 65°. He also noticed that the sunrise and
sunset temperature was 75°. Sketch a graph showing the
fluctuation temperatures during those 7 days.
Standards are listed in alphabetical /numerical order not suggested teaching order.
PLC’s must order the standards to form a reasonable unit for instructional purposes.
PAGE 11
Course Name: Algebra 2/Math III
Unit #1
Unit Title:Polynomials
CORE CONTENT
Cluster Title: Interpret Functions That Arise in Applications
Standard F-IF.5. Relate the domain of a function to its graph and, where applicable, to the quantitative relationship it
describes.
Concepts and Skills to Master:
• Determine the domain of a function from its graph, including analysis of the start and end behavior.
• Determine the key domain restrictions of functions, including division by 0 and even roots of negative numbers.
• Analyze word problems to determine domain restrictions (i.e. negative numbers, non-integers).
SUPPORTS FOR TEACHERS
Critical Background Knowledge:
• Ability to identify x values by looking at a graph
Academic Vocabulary:
Domain, Domain Restrictions, Integers, Start Behavior, End Behavior
Suggested Instructional Strategies:
• Compare the graphs of parent functions to their equations to
determine any restrictions algebraically.
• Students write word problems to satisfy various domain
restrictions (positive numbers, non-negative numbers, integers, all
real numbers).
• Identify the range of functions based on their graphs and
equations.
Resources:
Algebra 2 Textbook Correlation: Introduced in 2-1, but additional
resources can be found in the radical unit (Unit 6) and rationals
unit (Unit 8)
Domain Discovery (discovers domains of parent functions): on
wiki
Domains of Radicals and Rationals Discovery (highlights the
domain restrictions of radicals and rationals) (to be uploaded)
NCDPI Unpacking:
Given a function, determine its domain. Describe the connections
between the domain and the graph of the function. Know that the
domain taken out of context is a theoretical domain and that the
practical domain of a function is found based on a contextual
situation given, and is the input values that make sense to the
constraints of the problem context.
Standards are listed in alphabetical /numerical order not suggested teaching order.
PLC’s must order the standards to form a reasonable unit for instructional purposes.
PAGE 12
Course Name: Algebra 2/Math III
Unit #1
Sample Assessment Tasks
Skill-based task:
What are the domains of f(x) = x, f(x) = x2, f(x) = x3,
f(x) = √x, and f(x) = 1/x? Explain the rationale for any restrictions
(or lack of restrictions).
Unit Title:Polynomials
Problem Task:
1) The summer before going to college, a student earned a
promotion to shift supervisor at her job at Starbucks! The new
position pays $10.20/hour. If the student’s paycheck was modeled
by a function, what would be the domain and range? Explain your
answer. Why is this domain and range different than other similar
functions?
2) Create graphs of functions with the following domains: {All real
numbers}, {All real numbers for x ≠ 0}, {All real numbers for x ≥
0), {All real numbers for x < 0}, {All positive integers}, and {All
integers}.
1. How do your graphs accurately represent the required
domains?
2. What are the similarities and differences between your graphs?
3. For each graph, think of either a math function OR a real-world
situation where the domain would apply. Explain why your math
function OR real-world situation must have the domain you
identified.
NCDPI Examples:
Ex. A rocket is launched from 180 feet above the ground at time t
= 0. The function that models this situation is given by h(t) = – 16t2
+ 96t + 180, where t is measured in seconds and h is height
above the ground measured in feet.
a. What is the theoretical domain for the function? How do you
know this?
b. What is the practical domain for t in this context? Explain.
c. What is the height of the rocket two seconds after it was
launched?
d. What is the maximum value of the function and what does it
mean in context?
e. When is the rocket 100 feet above the ground?
f. When is the rocket 250 feet above the ground?
g. Why are there two answers to part f but only one practical
answer for part e?
h. What are the intercepts of this function? What do they mean in
the context of this problem?
i. What are the intervals of increase and decrease on the practical
domain? What do they mean in the context of the problem?
Standards are listed in alphabetical /numerical order not suggested teaching order.
PLC’s must order the standards to form a reasonable unit for instructional purposes.
PAGE 13
Course Name: Algebra 2/Math III
Unit #1
Unit Title:Polynomials
CORE CONTENT
Cluster Title: Analyze Functions Using Different Representations
Standard F.IF.7. Graph functions expressed symbolically and show key features of the graph, by hand in simple cases and
using technology for more complicated cases. (Specifically c – Graph polynomial functions, identifying zeroes when suitable
factorizations are available and showing end behavior.)
Concepts and Skills to Master:
• Graph higher degree polynomial functions from their zeroes and end behavior.
• Identify relative extremes.
• Descartes Rule of Signs (HONORS)
• x = y2 Parabolas (HONORS)
SUPPORTS FOR TEACHERS
Critical Background Knowledge:
• Zeroes as x-intercepts on a graph
Academic Vocabulary:
Relative extremes, zeroes
Suggested Instructional Strategies:
• Relate everything we’ve done this unit with higher-order
polynomials (solving by graphing/factoring/synthetic division, end
behaviors, key points, etc.) to graphing higher-order polynomials.
• Show how graphs of some polynomials can have minimum and
maximum values for different intervals (use graphing technology
to set left bounds and right bounds).
NCDPI Unpacking:
Part a., b., and c. are learned by students sequentially in Courses
I – III. Part e. is carried through Courses I – III with a focus on
exponential in Course I and moving towards logarithms in
Courses II and III. Part d. is introduced in course III with further
development in a fourth course option.
This standard should be seen as related to F-IF.4 with the key
difference being students can create graphs, by hand and using
technology, from the symbolic function in this standard.
Sample Assessment Tasks
Skill-based task:
1) Find the relative maximum, relative minimum, and zeroes of:
y = 2x3 – 23x2 + 78x – 72
Resources:
Algebra 2 Textbook Correlation: 5-2, 5-5, 5-9
Cubic Graphs MARS Task:
http://map.mathshell.org/materials/tasks.php?
taskid=265&subpage=apprentice
Problem Task:
1) Textbook pg. 294 #46
2) Find the zeroes and relative minimum and maximum values for
y = (x + 1)4, y = (x + 3)4, and y = (x + 1)4 + 2. What are the
similarities and differences between the values? Explain why this
occurs based on what you have learned about transformations.
NCPDI Examples:
Ex. Graph f (x) = 2x+1, identifying its intercepts and asymptotes,
and describe the end behavior of the function.
Ex. Graph f(x) = x2 + 6x + 5 and f(x) = x2 + 6x – 16, identifying and
comparing the key characteristics in these two graphs.
Ex.A roller coaster’s track design can be modeled by the
polynomial f(x)=x4- 8x3+16x2. Analyze the graph of this function
and describe the ride of the roller coaster. Is there a possible error
to using this function to model the roller coaster? Why or why not?
Standards are listed in alphabetical /numerical order not suggested teaching order.
PLC’s must order the standards to form a reasonable unit for instructional purposes.
PAGE 14
Course Name: Algebra 2/Math III
Unit #1
Unit Title:Polynomials
CORE CONTENT
Cluster Title: Represent and Solve Equations and Inequalities Graphically
Standard: A.REI-10: Understand that the graph of an equation in two variables is the set of all its solutions plotted in the
coordinate plane, often forming a curve (which could be a line).
Concepts and Skills to Master:
• Write polynomial functions from their points in real-world situations (regression)
• Use polynomial functions to predict future values and solve for needed values
SUPPORTS FOR TEACHERS
Critical Background Knowledge:
• Finding regression equations in the calculator
• Evaluating Functions
• Solving Polynomial Equations Using Technology and Algebraically
Academic Vocabulary:
Regression, Polynomial, Function
Suggested Instructional Strategies:
- Begin by relating polynomial regression to linear and quadratic,
which students covered in Math 1 and 2
- Expand to cubic and quartic regression to evaluate and predict
future values
- Use technology to solve cubics and quartics for independent
variable when given the dependent variable in a problem
- HONORS: Can expand to using common differences to
determine appropriate regression
NCDPI Unpacking:
The solutions to equations in two variables can be shown in a
coordinate plane where every ordered pair that appears on the
graph of the equation is a solution. Understand that all points on
the graph of a two-variable equation are solutions because when
substituted into the equation, they make the equation true.
Sample Assessment Tasks
Skill-based task:
(Give table): What is the value of the cubic function at x = 8?
Resources:
Algebra 2 Textbook Correlation: 5-8
Pearson Activity 5-8 (uses polynomials to show monetary growth,
good intro to when we will use exponentials later): Accessible
from Pearson Success Net or wiki
Pearson Enrichment 5-8 (building a cube from cardboard):
Accessible from Pearson Success Net or wiki
Problem Task:
(Give table of values for quartic): The amount of liquid y that flows
through a tube of radius x each hour is given by the table. If
_____ gallons of water needs to pass through the tube in one
hour, what radius does it need to be?
Standards are listed in alphabetical /numerical order not suggested teaching order.
PLC’s must order the standards to form a reasonable unit for instructional purposes.
PAGE 15