

Unit 1 Algebra
II 


Suggested Unit Length and Description: 45
days Building
on their work with linear, quadratic, and exponential functions in Algebra I,
students extend their repertoire of functions to include polynomial,
rational, and radical functions. In this Algebra II course, rational
functions are limited to those whose numerators are of degree at most 1 and
denominators of degree at most 2.
Radical functions are limited to square roots or cube roots of at most
quadratic polynomials. Certain
standards in this course require students to revisit the topics of linear,
quadratic and/or exponential functions to build conceptual understanding. ·
Reason quantitatively
and use units to solve problems. ·
Perform arithmetic
operations with complex numbers. ·
Use complex numbers
in polynomial identities and equations. ·
Interpret the
structure of expressions. ·
Perform arithmetic
operations on polynomials. ·
Understand the
relationship between zeros and factors of polynomials. ·
Use polynomial
identities to solve problems. ·
Rewrite rational
expressions. ·
Understand solving
equations as a process of reasoning and explain the reasoning. ·
Solve equations and
inequalities in one variable. ·
Solve systems of
equations. ·
Analyze functions
using different representations. ·
Translate between the
geometric description and the equation for a conic section. 

State
Standards: NQ.A.2 Define appropriate quantities for the
purpose of descriptive modeling. NCN.A.1 Know there is a complex number i such
that i^{2} = −1, and every complex number has the form a
+ bi with a and b real. NCN.A.2 Use the relation i^{2} = –1
and the commutative, associative, and distributive properties to add,
subtract, and multiply complex numbers. NCN.C.7 Solve quadratic equations with real
coefficients that have complex solutions. NCN.C.8(+) Extend polynomial identities to the
complex numbers. For example,
rewrite x^{2} + 4 as (x + 2i)(x – 2i). NCN.C.9(+) Know the Fundamental Theorem of Algebra;
show that it is true for quadratic polynomials. ASSE.A.2 Use the structure of an expression to
identify ways to rewrite it. For example, see x^{4} – y^{4}
as (x^{2})^{2} – (y^{2})^{2}, thus
recognizing it as a difference of squares that can be factored as (x^{2}
– y^{2})(x^{2} + y^{2}). AAPR.B.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). AAPR.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. AAPR.C.4 Prove polynomial identities and use them
to describe numerical relationships. For example, the polynomial identity
(x^{2} + y^{2})^{2} = (x^{2} – y^{2})^{2
}+ (2xy)^{2} can be used to generate Pythagorean triples. AAPR.D.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. AAPR.D.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. AREI.A.1 Explain each step in solving a simple
equation as following from the equality of numbers asserted at the previous
step, starting from the assumption that the original equation has a solution.
Construct a viable argument to justify a solution method. AREI.A.2 Solve simple rational and radical
equations in one variable, and give examples showing how extraneous solutions
may arise. AREI.B.4b Solve quadratic equations in one variable. b.
Solve quadratic equations by inspection (e.g., for x^{2} =
49), taking square roots, completing the square, the quadratic formula and
factoring, as appropriate to the initial form of the equation. Recognize when
the quadratic formula gives complex solutions and write them as a ± bi
for real numbers a and b. AREI.C.6 Solve systems of linear equations exactly
and approximately (e.g., with graphs), focusing on pairs of linear equations
in two variables. AREI.C.7 Solve a simple system consisting of a
linear equation and a quadratic equation in two variables algebraically and
graphically. For example, find the points of intersection between the line
y = –3x and the circle x^{2} + y^{2} = 3. FIF.C.7c Graph functions expressed symbolically and
show key features of the graph, by hand in simple cases and using technology
for more complicated cases.★ c.
Graph polynomial functions, identifying zeros when suitable factorizations
are available, and showing end behavior. GGPE.A.2 Derive the equation of a parabola given a
focus and directrix. Focus Standards of
Mathematical Practice: MP.1 Make
sense of problems and persevere in solving them. MP.2 Reason
abstractly and quantitatively. MP.3
Construct viable arguments and critique the reasoning of others. MP.4 Model
with mathematics. MP.5 Use
appropriate tools strategically. MP.6 Attend
to precision. MP.7 Look for
and make use of structure. MP.8 Look for
and express regularity in repeated reasoning. Instructional
Outcomes: Full Development of the Major Clusters, Supporting
Clusters, Additional Clusters and Mathematical Practices for this unit could
include the following instructional outcomes: NQ.A.2 ·
I can define
descriptive modeling ·
I can determine
appropriate quantities for the purpose of descriptive modeling NCN.A.1 ·
I can define i as the square root of −1 or i^{2} =1. ·
I can define complex
numbers. ·
I can write complex
numbers in the form a+ bi with a
and b being real numbers. NCN.A.2 ·
I can recognize that
the commutative, associative, and distributive properties extend to the set
of complex numbers over the operations of addition and multiplication. ·
I can use the
relation i^{2} = 1 to
simplify. NCN.C.7 ·
I can solve quadratic
equations that have complex solutions. NCN.C.8(+) ·
I can define
identity. ·
I can give examples
of polynomial identifies. ·
I can extend
polynomial identities to the complex numbers. NCN.C.9(+) ·
I can state the
Fundamental Theorem of Algebra. ·
I can verify that the
Fundamental Theorem of Algebra is true for second degree quadratic
polynomials. ASSE.A.2 ·
I can identify
patterns of factoring. ·
I can factor a
polynomial or rational expression. ·
I can classify
expressions by method of factoring. AAPR.B.2 ·
I can define the
remainder theorem. ·
I can use the
remainder theorem to show the relationship between a factor and a zero. AAPR.B.3 ·
I can factor
polynomials using any method. ·
I can sketch graphs
of polynomials using zeroes and a sign chart. AAPR.C.4 ·
I can prove
polynomial identities. AAPR.D.6 ·
I can rewrite
rational expressions using inspection or by long or synthetic division. AAPR.D.7(+) ·
I can add, subtract, multiply, and divide rational
expressions. AREI.A.1 ·
I can demonstrate
that solving an equation means that the equation remains balanced during each
step. ·
I can recall the
properties of equality. ·
I can explain why,
when solving equations, it is assumed that the original equation is equal. ·
I can determine if an
equation has a solution. ·
I can choose an
appropriate method for solving the equation. ·
I can justify
solution(s) to equations by explaining each step in solving a simple equation
using the properties of equality, beginning with the assumption that the
original equation is equal. ·
I can construct a
mathematically viable argument justifying a given, or selfgenerated,
solution method. AREI.A.2 ·
I can determine the
domain of a rational function. ·
I can determine the
domain of a radical function. ·
I can solve radical
equations in one variable. ·
I can solve rational
equations in one variable. ·
I can give examples
showing how extraneous solutions may arise when solving rational and radical
equations. AREI.B.4b ·
I can solve quadratic
equations by inspection (e.g., for x^{2} = 49), taking square roots,
completing the square, the quadratic formula and factoring. ·
I can determine
appropriate strategies (see first knowledge target listed) to solve problems
involving quadratic equations, as appropriate to the initial form of the
equation. ·
I can recognize when
the quadratic formula gives complex solutions. AREI.C.6 ·
I can solve systems
of linear equations by any method. ·
I can justify the
method used to solve systems of linear equations exactly and approximately
focusing on pairs of linear equations in two variables. AREI.C.7 ·
I can transform a
simple system consisting of a linear equation and quadratic equation in 2
variables so that a solution can be found algebraically and graphically. ·
I can explain the
correspondence between the algebraic and graphical solutions to a simple
system consisting of a linear equation and a quadratic equation in 2 variables. FIF.C.7c ·
I can graph
polynomial, rational, and radical functions accurately. GGPE.A.2 ·
I can define a
parabola including the relationship of the focus and the equation of the
directrix to the parabolic shape (using y= a(xh)^{2} + k) · I can derive the equation of a parabola given the focus and directrix using y= a(xh)^{2} + k 

Enduring Understandings: ·
Odd functions begin
and end in opposite directions. ·
Even functions begin
and end in the same direction. ·
Zeros are the x
intercepts of an equation and can be used to find factors and write a
polynomial equation. ·
Odd and even
functions have graphs that are symmetric with respect to the origin or
yaxis. ·
Zeros of a polynomial
can be found by factoring or by graphing the polynomial. ·
Rational functions can
be represented as fractional exponents and follow the same rules as regular
exponents. ·
Solve rational
expressions. ·
Computational skills
applicable to numerical fractions also apply to rational expressions
involving variables. ·
Radical expressions
can be written and simplified using rational exponents. ·
Graphs of radical
functions look like curves, are not symmetric and can be transformed using
predictive indicators. ·
Radicals are the
opposite of exponents. ·
Only radicals with a
common radicand and index can be added or subtracted. ·
Radical equations can
be solved by graphing or inverses. 
Essential Questions: ·
How do exponent
value, zeros, and factors affect the appearance of a graph? ·
How can a polynomial
inequality be solved? ·
How can real world
data be used to generate a polynomial model? ·
What does it mean to
be an odd or even function? ·
How do the elements
of a polynomial equation determine its general shape? ·
How can the solutions
or zeros of a polynomial be found? ·
How do the zeros of a
polynomial relate to its graph? ·
What is a rational
function and what does its graph look like? ·
How are operations
extended to rational functions? ·
How are rational
expressions simplified? ·
How are rational
equations solved? ·
How are rational
equations graphed? ·
What are radicals and
how can they be simplified? ·
What do graphs of
radical functions look like? ·
How can a radical
function be identified from its graph? ·
How can a radical
expression be simplified and combined? ·
How can a radical equation
be solved? ·
How can operations be
extended to radical expressions and equations? 
