Download Algebra 2 Unpacking the Standards

Algebra 2 Unpacking the Standards - Overview
The intent of this document is to unpack each standard in terms of observable and measurable student outcomes (knowledge, skills, and applications). In
should be used in conjunction with the South Carolina College and Career Ready Math Standards Support Document.
SCCCR
A2.AAPR.1* Add, subtract, and multiply
polynomials and understand that
polynomials are closed under these
operations.
A2.AAPR.3 Graph polynomials identifying
zeros when suitable factorizations are
available and indicating end behavior.
Write a polynomial function of least
degree corresponding to a given graph.
(Limit to polynomials with degrees 3 or
less.)
Unpacking
What do these standards mean a child will know and be able to do?
Student will be able to ...
○ Extend content from Algebra 1 (linear times linear or linear times quadratic) that
includes products of polynomials that are degree 3 or higher and that have
rational and integer coefficients.
○ Demonstrate that polynomials are closed under these 3 operations, which means
when adding, subtracting, or multiplying two polynomials the result is a
polynomial (for example: the integers are closed; 2 X 4 = 8 which is also an
integer).
○ Use the distributive property and not mnemonic devices when multiplying
polynomials. They should understand that FOIL is the distributive property.
○ (move to Algebra 1) Understand the mathematical definition of a polynomial– an
expression that includes one variable raised to whole number powers with
coefficients.
○ Show that a polynomial is a base x expression: comparable to a base 10 number
(for example: 243 = 2 X 102 + 4 X 101 + 3 X 100 comparable to 2x2 + 4x + 3).
Student will be able to ...
○ Graph polynomials of 3rd degree (and lower).
○ Explain with examples the interrelationship of these terms: roots, zeros, xintercepts, and solutions of equations. (All the terms were discussed in Algebra
1.)
○ Explain what happens to “y” as “x” approaches infinity or negative infinity to
describe end behavior.
○ Use the information from a graph to write a polynomial function.
A2.ACE.1* Create and solve equations
and inequalities in one variable that
model real-world problems involving
linear, quadratic, simple rational, and
exponential relationships. Interpret the
solutions and determine whether they
are reasonable.
Student will be able to ...
o Formulate equations and inequalities for various contexts and use models from
real world applications that are more complex than those studied in Algebra 1.
o Solve the equations and inequalities (including compound and absolute value)
and determine if the solutions are reasonable for the situation.
o Formulate examples and models that go beyond quadratic and linear data, so the
need for studying polynomials of degree 3 or higher is evident.
A2.ACE.2* Create equations in two or
more variables to represent relationships
between quantities. Graph the equations
on coordinate axes using appropriate
labels, units, and scales.
Student will be able to ...
o Examine patterns in a variety of forms (given a set of numbers, in a table, and
from situations) that include two variables to create linear, quadratic and
exponential equations.
o Generate formulas to demonstrate the linear, quadratic or exponential
relationship among variables in a context.



A2.ACE.3 Use systems of equations and
inequalities to represent constraints
arising in real- world situations. Solve
such systems using graphical and
analytical methods, including linear
programing. Interpret the solution within
the context of the situation. (Limit to
linear programming.)
Linear: The profits from selling a magazine if you knew the cost. Equate the cost
function [10*c - 300 = p]
Quadratic: D = 4 + vt – 10t2 represents the distance given the velocity of a throw and the
time in the air
Exponential: “In two or more variables” refers to formulas, like the compound interest
formula [(A = P(1 + r/n)nt has multiple variables].
o Demonstrate the ability to label and discuss the graph with the correct units and
scales, as well as identify the independent and dependent variables.
Student will be able to ...
o Understand that linear programming involves linear equations and inequalities with 2
variables. Note: A constraint can have just one variable (x > 0).
o Write equations and inequalities that represent the constraints of the problem.
o Write an objective equation that will be maximized or minimized based on the
constraints in the context.
o Graph the constraints, find the critical points, and test the critical points in the objective
equation or inequality to determine the optimal solution.
o Interpret the reasonableness of the solution in light of the context.
A2.ACE.4* Solve literal equations and
formulas for a specified variable including
equations and formulas that arise in a
variety of disciplines.
Student will be able to ...
o Rewrite equations with particular attention to formulas and equations in their other
classes (Chemistry, Economics, Physics, Health Science 1 – 3, etc).
A2.AREI.2* Solve simple rational and
radical equations in one variable and
understand how extraneous solutions
may arise.
Student will be able to ...
o Determine that a solution is extraneous because it does not make the original equation
true.
o Understand that when you square both sides of an equation it is possible to get
extraneous solutions.
o Understand that when you simplify rational equations the solution may not fit in the
original domain.
Student will be able to ...
o Develop fluency in all methods while understanding that one method may be more
appropriate than others for solving quadratic equations from real-world and
mathematical situations.
o Understand the importance of the value of the discriminant, b2 – 4ac, in the quadratic
formula and that it indicates the number and nature of solutions.
o Solve quadratic equations with real and imaginary solutions.
o Understand the difference between real and imaginary solutions and how they are
represented on the graph.
o Know that the result of a negative discriminant is an imaginary solution.
A2.AREI.4* Solve mathematical and realworld problems involving quadratic
equations in one variable.
b. Solve quadratic equations by
inspection, 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.
(Note: A2.AREI.4b is not a Graduation
Standard.)
A2.AREI.7 Solve a simple system consisting
of a linear equation and a quadratic equation
in two variables algebraically and graphically.
Understand that such systems may have zero,
one, two, or infinitely many solutions. (Limit
to linear equations and quadratic functions.)
Student will be able to ...
o Graph the system on a coordinate plane and identify the possible point(s) of
intersection.

Note: Because it is limited to linear and quadratic, the graphs will be limited to lines or parabolas:
1 line and 1 parabola or 2 lines or 2 parabolas.
o Use substitution or elimination to solve algebraically.
o Determine the number of solutions by analyzing the graph of the two functions.
A2.AREI.11* Solve an equation of the
form (𝑥) = 𝑔(𝑥) graphically by identifying
the 𝑥- coordinate(s) of the point(s) of
intersection of the graphs of 𝑦 = 𝑓(𝑥) and
𝑦 = 𝑔(𝑥).
Student will be able to ...
o Find where two functions cross and know the x-coordinate is the solution of the
equation.


NOTE: The ordered pair (x, y) is the point of intersection and not the solution.
Example: Solve 3x = x2. This is a system (y = 3x and y = x2), and this system can be solved
graphically or algebraically. The value of x is the solution.
A2.ASE.1* Interpret the meanings of
Student will be able to ...
coefficients, factors, terms, and
o Decompose the complex expression and identify the coefficients, factors, terms and
expressions based on their real-world
simpler expressions.
 Example: 3x2 + x = x(3x + 1); x is a factor and the quantity (3x + 1) is also a factor.
contexts. Interpret complicated
 Example: A = P(1 + r/n)nt
expressions as being composed of simpler
o Know what the role of the leading coefficient of a polynomial is and what the degree of
expressions.
the polynomial tells us about the polynomial and its end behavior.
o Describe the coefficients, factors, terms and simpler expressions using information from
the real-world context.
o Find the zeros of the polynomial expression using the zeros of its factors

A2.ASE.2* Analyze the structure of
binomials, trinomials, and other
polynomials in order to rewrite
equivalent expressions.
Example: using synthetic division, factoring, or other methods
Student will be able to ...
o Recognize the structure of having a common factor in a polynomial leading to factoring
out a common monomial. For example: x3 + 2x2 – 4x = x(x2 +2x – 4) and x3 + 2x2 – x = x(x2
+2x – 1).
o 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 ).
o Rewrite algebraic expressions in different equivalent forms such as factoring or
combining like terms.
o Use factoring techniques such as common factors, grouping, the difference of two
squares, the sum or difference of two cubes, or a combination of methods to factor
completely.
o Simplify expressions using the distributive property and other operations with
polynomials.
See examples on the next page.
A2.ASE.3* Choose and produce an
equivalent form of an expression to
reveal and explain properties of the
quantity represented by the expression.
(Note: A2.ASE.3b and 3c are not
Graduation Standards.)
b. Determine the maximum or
minimum value of a quadratic
function by completing the
square. Examples:
https://www.illustrativemathematics.org/HSA-SSE
Excellent Illustration for patterns in Seeing Dots!
 A-SSE A Cubic Identity
 A-SSE Animal Populations
 A-SSE Equivalent Expressions
 A-SSE Seeing Dots
These are more advanced.
 A-SSE Sum of Even and Odd
 N-CN, A-SSE Computations with Complex Numbers
Student will be able to ...
o Transform a quadratic expression into an equivalent form that reveals the maximum or
minimum value:


The expression ax2 + bx + c can be transformed to a(x - h)2 + k by completing the square and
the point (h, k) is the maximum value when a< 0 or minimum value when a>0.
In general, h can be found to be b/2a and k to be c – b2/4a but completing the square is a
more appropriate method for transforming the expression
o Solve problems based off of real-world scenarios that can be modeled with quadratic
expressions for example balls being thrown into the air and finding the maximum point
or economic models that might have minimum/maximum points
A2.ASE.3* Choose and produce an
equivalent form of an expression to
reveal and explain properties of the
quantity represented by the expression.
(Note: A2.ASE.3b and 3c are not
Graduation Standards.) c. Use the properties of exponents
to transform expressions for
exponential functions. A2.FBF.1* Write a function that describes
a relationship between two quantities.
(Note: IA.FBF.1a is not a Graduation
Standard.)
a. Write a function that models a
relationship between two
quantities using both explicit
expressions and a recursive
process and by combining
standard forms using addition,
subtraction, multiplication and
division to build new functions.
Student will be able to ...
o Find an appropriate base of an exponential function that reveals information
about the exponential growth or decay


The expression 1.05t, where t is in years, can be rewritten as [1.05(1/12)]12t to reveal the
approximate monthly interest rate if the annual rate is 5% (12 months per year and .05 as
5%).
The exponential property (xa)b = xab provides the means of finding the equivalent expression.
o Recognize that the expression (1/2)x is equivalent to 2-x and know that
exponential functions with bases greater than 1 represent growth and functions
with based between 0 and 1 represent decay.
o Transform an exponential function to the form A(t) = P(1 +r/n)nt to reveal the
Principal (P), interest rate (r), number of compounding periods (n).
o Interpret applications such as the Richter scale for earthquakes, intensity of
sound D -10 log (I/10-16). pH level, and the cooling law T(t) = Te + (T0 –Te) e-kt as
examples for transforming expressions involving exponential functions and its
inverse, logarithmic functions.
Student will be able to ...
o Combine functions using the four operations to write a different function that
represents the new relationship.
o Write explicit expressions (a function that allows the student to calculate any
value) to represent the relationship between two quantities. For example, f(x) =
3x + 2 allows you to calculate any y-value for any x-value and models a linear
relationship.
o Write recursive equations (a function that uses one or more of the previous
terms) to represent the relationship between two quantities. For example, to
model the sequence 2, 5, 8, 11, ... write as the recursive equation a1 = 2, an = an-1
+ 3 for n = 2, 3, 4, ...
o Determine when to use recursive versus an explicit function.
o Transform among explicit and recursive formulas.
A2.FBF.1* Write a function that describes Student will be able to ...
a relationship between two quantities.
o Extend A2.FBF.1a to include real-world applications. For example, determining a
(Note: IA.FBF.1a is not a Graduation
function of the monthly cost of owning two vehicles when the cost of owning each is
Standard.)
known.
b. Combine functions using the
operations addition, subtraction,
multiplication, and division to
build new functions that describe
the relationship between two
quantities in mathematical and
real-world situations. A2.FBF.2* Write arithmetic and geometric Student will be able to ...
sequences both recursively and with an
o Connect to Algebra 1 standard A1.FLQE.2* Create symbolic representations of linear and
explicit formula, use them to model
exponential functions, including arithmetic and geometric sequences, given graphs,
situations, and translate between the two
verbal descriptions, and tables. (Limit to linear; exponential.)
forms.
o Know that a sequence is arithmetic when the difference between consecutive terms is
constant.
o Know that a sequence is geometric when the ratio of consecutive terms is constant.
o Write an arithmetic sequence recursively.
𝑎1 = 𝑏
{
, n > 1 ; b, b+d, b+2d, ... (2, 5, 8, 11, ...) where b = 2, d = 3.
𝑎𝑛 = 𝑎𝑛−1 + 𝑑
o Write a geometric sequence recursively.
𝑎1 = 𝑏
{
, n > 1; b, b*r, b*r2, ... (2, 6, 18, 54, ...) where b = 2, r = 3.
𝑎𝑛 = 𝑟𝑎𝑛−1
o Write an explicit formula for the nth term, 𝑎𝑛 , of an arithmetic sequence as 𝑎𝑛 = 𝑎1 +
(𝑛 − 1)𝑑, where 𝑎1 repesents the first term and d represents the constant difference.
o Know that the function 𝑎𝑛 = 𝑎1 + (𝑛 − 1)𝑑 and 𝑦 = 𝑚𝑥 + 𝑏 are related in the following
way: d is the slope, m, of the line connecting the discrete points of the arithmetic
sequence and that 𝑎1 is the y-intercept, b, of that line.
o Write an explicit formula for the nth term, 𝑎𝑛 , of a geometric sequence as 𝑎𝑛 =
𝑎1 (𝑟)𝑛−1, where 𝑎1 is the first term and r is the ratio between consecutive terms.
o Use the explicit or recursive formula of sequences in problem situations.
A2.FBF.3* Describe the effect of the
transformations 𝑘(𝑥), 𝑓(𝑥) + 𝑘, 𝑓(𝑥 + 𝑘),
and combinations of such
transformations on the graph of 𝑦 = 𝑓(𝑥)
for any real number 𝑘. Find the value of 𝑘
given the graphs and write the equation
of a transformed parent function given its
graph.
Student will be able to ...
o Identify and analyze different types of parent functions (graphs and equations).
o Write equations of functions in standard form.
o Write the equation for a transformed parent function from a given graph.
o Identify vertical and horizontal shifts on parent functions (both graphically and
algebraically).
o Describe the effects of a shift on the parent function (write the equation from graphs
and vice-versa).
o Make the connection between the vertical stretch or compression as being a dilation in
geometry.
A2.FIF.3* Define functions recursively and Student will be able to ...
recognize that sequences are functions,
o Explain that a function is recursive when it provides an initial value and each successive
sometimes defined recursively, whose
value is based on the previous value.
domain is a subset of the integers.
o Recognize that sequences are functions, sometimes defined recursively, whose domain
is the number of the term, and the range is the value of the term.

Example, the Fibonacci sequence (1, 1, 2, 3, 5, 8, 13, 21, 34,…) is defined recursively by f(0) =
f( 1) = 1, f(n + 1) = f(n) + f(n - 1) for n > 1.
o Show, by graphing or calculating terms, how the recursive sequence if
a1 = 7 and an = an-1 + 2; then the sequence sn = 2(n-1) + 7; and the function f(x) = 2x + 5
(where x is a natural number) all define the same sequence.
A2.FIF.4* Interpret key features of a function Student will be able to ...
that models the relationship between two
o Identify the key features from a given graph.
quantities when given in graphical or tabular
o Complete a table of key features for a variety of functions.
form. Sketch the graph of a function from a
verbal description showing key features. Key
features include intercepts; intervals where
the function is increasing, decreasing,
constant, positive, or negative; relative
maximums and minimums; symmetries; end
behavior and periodicity.
Function
Degree
Turning Points
Intervals of Increase
f(x) = 3x + 5
h(x) = x2 – 2x
k(x) = 6x
n(x) = x3 + 9
o Sketch a graph given its key features.
Intervals of Decrease
Relative Minimum
Relative Maximum
A2.FIF.5* Relate the domain and range of
a function to its graph and, where
applicable, to the quantitative
relationship it describes.
A2.FIF.6* Given a function in graphical,
symbolic, or tabular form, determine the
average rate of change of the function
over a specified interval. Interpret the
meaning of the average rate of change in
a given context.
A2.FIF.7* Graph functions from their
symbolic representations. Indicate key
features including intercepts; intervals
where the function is increasing,
decreasing, positive, or negative; relative
maximums and minimums; symmetries;
end behavior and periodicity. Graph
simple cases by hand and use technology
for complicated cases.
Student will be able to ...
o Determine the appropriate domain (x-values) and range (y-values) for a given function.
o Explain/describe the domain and range given a graph and also given a function.
o Given the graph of a function that models a situation, determine the practical domain as
it relates to the relationship described.
Student will be able to ...
o Understand that the slope of a linear function is the average “rate of change”.
o Find the slope and recognize that it represents the rate of change.
o Identify specific characteristics of a polynomial function in a given context,
understanding that if the function is linear the slope is constant.
o Calculate the average rate of change within a specific interval. For example, piece-wise
or absolute value functions.
o Interpret the meaning of rate of change for the context. For example, if the cost
function is C(x) = 300x + 5000, the rate of change is determined by the slope, meaning
the rate by which the cost is increasing for each item x is 300.
Student will be able to ...
o Graph any given function with and without a calculator.
o Graph quadratic functions showing intercepts, maxima, and minima.
o Graph functions expressed symbolically and show key features of the graph.


Examine graphs describing key characteristics
Graph a function given a set of characteristics
o Graphs piece-wise functions, step functions, absolute value functions showing
intercepts, maxima, and minima.
 Recognize absolute value and step functions are part of piece-wise functions and
have linear pieces
 Recognize piece-wise functions can have quadratic and exponential pieces
o Graph and compare quadratic, piece-wise, step, and absolute value functions expressed
in various forms.
o Explore and discuss periodicity in Pre-Calculus.
A2.FIF.8* Translate between different but
equivalent forms of a function equation
to reveal and explain different properties
of the function.
(Note: A2.FIF.8b is not a Graduation
Standard.)
b. Interpret expressions for exponential
functions by using the properties of
exponents.
Student will be able to ...
o Know and apply the properties of exponents.
o Recognize that a function of the form y = abx can either be describing exponential
growth or exponential decay. If a > 0 and 0 < b < 1, the equation represents decay. If a >
0 and b > 1, the equation represents growth.
Examples:
Identify percent rate of change in functions such as y = (1.02)t, y = (0.97)t, y = (1.01)12t, y =
(1.2)t⁄10 and classify them as representing exponential growth or decay.
Use an equation like y = a(1 + c)x to represent the balance of a savings account
after x years with starting balance a and interest rate c.
If the annual interest rate is 2% and the starting balance is $100, how much money will
be in the account after 10 years?
y = a(1 + c)x = 100(1 + 0.02)10 = $121.90
Other examples include:
Determine population using an exponential model
Determine the age of an object by modeling exponential decay
Find compound interest using an investment model
Compute depreciation
Determine percent rate of change
A2.FIF.9* Compare properties of two
functions given in different
representations such as algebraic,
graphical, tabular, or verbal.
Note: These examples and lesson links were adapted and shared from sites, such as Learn
Zillion, shmoop.com, and the Georgia State Standards.
Student will be able to ...
o Distinguish between polynomial, linear, quadratic, exponential, and rational functions,
and be able to identify them by equation and by graph.
o Represent functions using multiple representations: algebraic, graphical, tabular, and
verbal.
o Identify properties of functions represented algebraically, graphically, in table form, and
verbally.
o Translate between an equation, a graph, a verbal representation, and a table of values,
and understand how certain aspects of one representation impact the rest.
o Transform any graph, table of values, or description into a mathematical equation that
describes the function.
Examples:
1. Given a graph of one quadratic function and an algebraic expression for another, determine
which function has the larger maximum.
2. Given a linear function represented by a table of values and a linear function represented by
an algebraic expression, determine which function has the greater rate of change.
A2.FLQE.1* Distinguish between
Student will be able to ...
o Identify and explain the differences between linear and exponential functions when
situations that can be modeled with
presented in graphs or tables.
linear functions or exponential functions
o Apply both linear and exponential functions to specific situations and explain the
by recognizing situations in which one
appropriateness of each function to each situation.
quantity changes at a constant rate per
o Explain that a linear function has a constant rate of change (slope) over its entire domain.
unit interval as opposed to those in which
o Explain that slope changes over the entire domain for an exponential function.
a quantity changes by a constant percent
o Identify exponential growth functions as having bases greater than 1.
rate per unit interval.
(Note: A2.FLQE.1b
o Identify exponential decay functions as having bases between 0 and 1.
is not a Graduation Standard.)
o Describe the rate of growth or decay of an exponential function in the situational context.
b. Recognize situations in which a
 Include contexts involving interest, radioactive decay, population growth/decay, bacterial growth,
quantity grows or decays by a constant
medicine elimination in the body, etc.
percent rate per unit interval relative to
another.
A2.FLQE.2* Create symbolic
Student will be able to ...
representations of linear and exponential
o Extend the writing of Algebra 1 linear functions when given any of the following:
 a graph by using the slope and intercepts
functions, including arithmetic and
 a description of the relationship
geometric sequences, given graphs,
 data in tabular form
verbal descriptions, and tables.
 an arithmetic sequence
o Extend the writing of Algebra 1 exponential functions when given any of the following:
 the graph by identifying the common ratio
 a description of the relationship
 data in tabular form
 a geometric sequence
A2.FLQE.5* Interpret the parameters in a Student will be able to ...
o Based on the context of the situation, explain the meaning of the coefficients, factors,
linear or exponential function in terms of
exponents, and/or intercepts in a linear (y = mx + b) or exponential function (y = a x+c).
the context.
 Identify the parameters (m and b) and their meaning for linear functions.
 Identify the parameters (a, b, and c) and their meaning for exponential functions
 Examine the behaviors of functions as parameters change
 Compare the rate of change for two functions (a linear to an exponential), looking at the specific
pattern of change that each function demonstrates in order to understand that linear functions
have a constant rate of change while exponential functions have a constant growth rate.
A2.NCNS.1* Know there is a complex
number i such that i2 = −1, and every
complex number has the form 𝑎 + 𝑏i with
𝑎 and 𝑏 real.
A2.NCNS.7* Solve quadratic equations in
one variable that have complex solutions.
Student will be able to ...
o Rewrite radicals with a negative radicand using the complex number i.
o Understand that the commutative, associative, and distributive properties hold true
when adding, subtracting, and multiplying complex numbers.
o Add, subtract, multiply, and divide complex numbers.
o Recognize that the product of complex conjugates, a+bi and a-bi, is always a real
number.
o Rationalize a denominator containing a complex number by multiplying by the
conjugate.
o Realize that some properties of radicals that are true for real numbers are not true for
complex numbers. For example a · b = a · b , but -a · -b ¹ (-a) · (-b)
Student will be able to ...
o Solve quadratic equations that have complex solutions.
o Understand that the existence of i allows every quadratic equation to have two solutions
of the form a + bi, either real when b=0, or complex when b0.
o Recognize that a negative discriminant indicates a complex solution.
o Determine when a quadratic equation has complex roots by looking at a graph or
calculating the discriminant.
o Understand that complex solutions always appear in conjugate pairs.