STATE STANDARDS
ALGEBRA II |
| NUMBER AND QUANTITY |
| The Real Number System |
| Extend the properties of exponents to
rational exponents |
|
N.RN.A.1 |
Explore how the meaning of
rational exponents follows from extending the properties of integer
exponents. For example, we define 51/3
to be the cube root of 5 because we want (51/3)3
= 5(1/3)3 to
hold, so (51/3)3
must equal 5. |
|
N.RN.A.2 |
Convert between radical
expressions and expressions with rational exponents using the properties
of exponents. Note: All radical expressions involving variables
assume the variables are representing positive numbers. Includes expressions
with variable factors, such as the cubic root of
27x5y3,
being equivalent to (27x5y3)1/3
which equals 3x5/3y. |
| The Complex Number System |
| Perform arithmetic
operations with complex numbers |
|
N.CN.A.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. |
|
N.CN.A.2 |
Use the relation i2 = 1 and the
commutative, associative, and distributive properties to add, subtract,
and multiply complex numbers. |
| ALGEBRA |
| Seeing Structure in Expressions |
| Interpret the structure of expressions |
|
A.SSE.A.2 |
Recognize and
use the structure of an expression to identify ways to rewrite it.
e.g.,81x4-16y4 is
equivalent to (9x2)2-(4y2)2
or (9x2-4y2)(9x2+4y2)
or (3x+2y)(3x-2y)(9x2+4y2);
(x2+4)/(x2+3) is equivalent to
((x2+3)+1)/(x2+3)=((x2+3)/(x2+3))+(1/(x2+3))=1+(1/(x2+3));
3x3+5x2-48x+80 is
equivalent to 3x(x2-16)-5(x2-16),
which when factored completely is (3x-5)(x+4)(x-4)
Includes factoring by grouping and factoring the sum and difference
of cubes. Tasks are limited to polynomial, rational, or
exponential expressions. Quadratic expressions include leading
coefficients other than 1. This standard is a fluency expectation
for Algebra II. The ability to see structure in expressions and to
use this structure to rewrite expressions is a key skill in everything
from advanced factoring (e.g., grouping) to summing series, to rewriting
of rational expressions, to examining the end behavior of the
corresponding rational function. |
| Write expressions in equivalent forms to
solve problems |
|
A.SSE.B.3 |
Choose and produce an equivalent form of an expression to reveal and explain
properties of the quantity represented by the expression. |
| a |
Factor quadratic expressions including leading coefficients other than 1
to reveal the zeros of the function it defines. See
A.APR.B.3.htm. |
| c |
Use the properties of exponents to rewrite exponential expressions.
Tasks include rewriting exponential expressions with rational
coefficients in the exponent. |
| Arithmetic with Polynomials & Rational
Expressions |
| Understand the relationship between
zeros and factors of polynomials |
|
A.APR.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). |
|
A.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. |
| Use polynomial identities to solve
problems |
|
A.APR.C.4 |
Prove polynomial identities and use them to describe
numerical relationships. For example, the polynomial identity (x2
+ y2)2 = (x2 y2)2
+ (2xy)2 can be used to generate Pythagorean triples. |
| Rewrite rational expressions |
|
A.APR.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. |
| Creating Equations |
| Create equations that describe numbers
or relationships |
|
A.CED.A.1 |
Create equations and inequalities in one variable to represent a
real-world context. This is strictly the development of the
model (equation/inequality). Tasks include linear, quadratic, rational,
and exponential functions. |
| Reasoning with Equations & Inequalities |
| Understand solving equations as a
process of reasoning and explain the reasoning |
|
A.REI.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 (rational or
radical). |
|
A.REI.A.2 |
Solve rational and radical equations in one
variable, identify extraneous solutions, and explain how they arise. |
| Solve equations and inequalities in one
variable |
|
A.REI.B.4 |
Solve quadratic equations in one variable (solutions
may include simplifying radicals). |
| b |
Solve quadratic equations by inspection (e.g., for
x2 = 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. |
| Solve systems of equations |
|
A.REI.C.6 |
Solve systems of linear equations exactly and
approximately (e.g., with graphs), focusing on pairs of linear equations
in three variables. |
|
A.REI.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 x2 + y2 =
3. |
| Represent and solve equations and
inequalities graphically |
|
A.REI.D.11 |
Given the equations y = f(x) and y = g(x):
i) recognize that each x-coordinate of the intersection(s) is the
solution to the equation f(x) = g(x);
ii) find the solutions approximately using technology to graph the
functions or make tables of values;
iii) find the solution of f(x) < g(x) or f(x) ≤ g(x) graphically; and
iv) interpret the solution in context.
Tasks include cases where f(x) and/or g(x) are linear, polynomial,
absolute value, square root, cube root, trigonometric, exponential, and
logarithmic functions. |
| FUNCTIONS |
| Interpreting Functions |
| Understand the concept of a function and
use function notation |
|
F.IF.A.3 |
Recognize that a sequence is a function whose domain is a subset of the
integers. Sequences will be defined/written recursively and
explicitly in subscript notation. This standard is a fluency
expectation. Fluency in translating between recursive definitions
and closed forms is helpful when dealing with many problems involving
sequences and series, with applications ranging from fitting functions
to tables to problems in finance. |
| Interpret functions that arise in
applications in terms of the context |
|
F.IF.B.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. Key features include: intercepts;
intervals where the function is increasing, decreasing, positive, or
negative; relative maximums and minimums; symmetries; end behavior; and
periodicity (emphasize selection of appropriate models). |
|
F.IF.B.6 |
Calculate and interpret the average rate of change
of a function over a specified interval. Functions may be
presented by function notation, a table of values, or graphically.
Tasks have a real-world context and may involve polynomial, square root,
cube root, exponential, logarithmic, and trigonometric functions. |
| Analyze functions using different
representations |
|
F.IF.C.7 |
Graph functions and show key features of the graph
by hand and using technology when appropriate. |
| c |
Graph polynomial functions, identifying zeros when
suitable factorizations are available, and showing end behavior. |
| e |
Graph cube root, exponential and logarithmic
functions, showing intercepts and end behavior; and trigonometric
functions, showing period, midline, and amplitude.
Trigonometric functions include sin(x), cos(x) and tan(x). |
|
F.IF.C.9 |
Compare properties of two functions each represented
in a different way (algebraically, graphically, numerically in tables,
or by verbal descriptions). For example, given a graph of one
quadratic function and an algebraic expression for another, say which
has the larger maximum. |
| Building Functions |
| Build a function that models a
relationship between two quantities |
|
F.BF.A.1 |
Write a function that describes a relationship between two quantities. |
| a |
Determine a function from context.
Determine an explicit expression, a recursive process, or steps for
calculation from a context.
Tasks may involve linear functions, quadratic functions, and exponential
functions. Sequences will be defined/written recursively and
explicitly in subscript notation. |
| b |
Combine standard function types using
arithmetic operations. For example, build a function that models
the temperature of a cooling body by adding a constant function to a
decaying exponential, and relate these functions to the model. |
|
F.BF.A.2 |
Write arithmetic and geometric sequences both recursively and with an
explicit formula, use them to model situations, and translate between the
two forms. Sequences will be defined/written recursively and
explicitly in subscript notation. |
| Build new functions from existing
functions |
|
F.BF.B.3 |
Using f(x)+k,
kf(x), f(kx) and f(x+k):
i) identify the effect on the graph when replacing f(x) by
f(x)+k, kf(x), f(kx) and f(x+k)
for specific values of k (both positive and negative);
ii) find the value of k given the graphs;
iii) write a new function using the value of k; and
iv) use technology to experiment with cases and explore the effects on the
graph.
Include recognizing even and odd functions from their graphs.
Tasks may involve polynomial, square root, cube root, exponential,
logarithmic, and trigonometric functions. |
|
F.BF.B.4 |
a. Find the inverse of a one-to-one function
both algebraically and graphically. |
|
F.BF.B.5 |
Understand inverse relationships
between exponents and logarithms algebraically and graphically. |
|
F.BF.B.6 |
Represent and evaluate the sum of a finite arithmetic or finite geometric
series, using summation (sigma) notation. |
|
F.BF.B.7 |
Explore the derivation of the formulas for
finite arithmetic and finite geometric series. Use the formulas to solve
problems. |
| Linear, Quadratic, & Exponential Models |
| Construct and compare linear, quadratic,
and exponential models and solve problems |
|
F.LE.A.2 |
Construct linear and exponential functions,
including arithmetic and geometric sequences, given a graph, a
description of a relationship, or two input-output pairs (include
reading these from a table). |
|
F.LE.A.4 |
Use logarithms to solve exponential
equations, such as ab^ct=d (where a, b, c,
and d are real numbers and b>0) and evaluate the logarithm
using technology. |
| Interpret expressions for functions in
terms of the situation they model |
|
F.LE.B.5 |
Interpret the parameters in a linear or exponential function in terms
of a context. Tasks have a real-world context and exponential
functions are not limited to integer domains. |
| Trigonometric Functions |
| Extend the domain of trigonometric
functions using the unit circle |
|
F.TF.A.1 |
Understand radian measure of an angle as the length
of the arc on the unit circle subtended by the angle. |
|
F.TF.A.2 |
Explain how the unit circle in the coordinate plane
enables the extension of trigonometric functions to all real numbers,
interpreted as radian measures of angles traversed counterclockwise
around the unit circle (includes the reciprocal trigonometric
functions). |
|
F.TF.B.5 |
Choose trigonometric functions to model periodic
phenomena with specified amplitude, frequency, and midline. |
|
F.TF.C.8 |
Prove the
Pythagorean identity sin2(θ) + cos2(θ) = 1.
Find the value of any of the six trigonometric functions given any other
trigonometric function value and when necessary find the quadrant of the
angle. |
| GEOMETRY |
| Expressing Geometric Properties with
Equations |
|
G.GPE.A.2 |
Derive the equation of a parabola given a focus and
directrix. |
| Statistics & Probability |
| Interpreting Categorical & Quantitative
Data |
| Summarize, represent, and interpret data
on a single count or measurement variable |
|
S.ID.A.4 |
Use the mean and standard deviation of a data set to
fit it to a normal distribution and to estimate population percentages.
Recognize that there are data sets for which such a procedure is not
appropriate. Use calculators, spreadsheets, and tables to estimate
areas under the normal curve. |
| Summarize, represent, and interpret data
on two categorical and quantitative variables |
|
S.ID.B.6 |
Represent data on two quantitative variables on a
scatter plot, and describe how the variables are related (linear focus,
discuss general principle). |
| a |
Fit a function to the data; use functions fitted to
data to solve problems in the context of the data. Use given
functions or choose a function suggested by the context. Emphasize
linear, quadratic, and exponential models. Includes the use of the
regression capabilities of the calculator. |
| Making Inferences & Justifying
Conclusions |
| Understand and evaluate random processes
underlying statistical experiments |
| S.IC.A.1 |
Understand statistics as a process for making
inferences about population parameters based on a random sample from
that population. |
|
S.IC.A.2 |
Determine if a value for a sample proportion
or sample mean is likely to occur based on a given simulation.
If the statistic falls within two standard deviations of the mean (95%
interval centered on the population parameter), then the statistic is
considered likely (plausible, usual). |
| Make inferences and justify conclusions
from sample surveys, experiments, and observational studies |
|
S.IC.B.3 |
Recognize the purposes of and differences among
sample surveys, experiments, and observational studies; explain how
randomization relates to each. |
|
S.IC.B.4 |
Use data from a sample survey to estimate a
population mean or proportion; develop a margin of error through the use
of simulation models for random sampling. |
|
S.IC.B.5 |
Use data from a randomized experiment to compare two
treatments; use simulations to decide if differences between parameters
are significant. |
|
S.IC.B.6 |
Evaluate reports based on data. |
| Conditional Probability & the Rules of
Probability |
| Understand independence and conditional
probability and use them to interpret data. Link to data from
simulations or experiments. |
|
S.CP.A.1 |
Describe events as subsets of a sample space (the
set of outcomes) using characteristics (or categories) of the outcomes,
or as unions, intersections, or complements of other events (or,
and, not). |
|
S.CP.A.2 |
Understand that two events A and B
are independent if the probability of A and B
occurring together is the product of their probabilities, and use this
characterization to determine if they are independent. |
|
S.CP.A.3 |
Understand the conditional probability of A
given B as P(A and B)/P(B),
and interpret independence of A and B as saying that
the conditional probability of A given B is the same
as the probability of A, and the conditional probability of
B given A is the same as the probability of B. |
|
S.CP.A.4 |
Interpret two-way frequency tables of data
when two categories are associated with each object being classified.
Use the two-way table as a sample space to decide if events are
independent and calculate conditional probabilities. |
| S.CP.A.5 |
Recognize and explain the concepts of conditional
probability and independence in everyday language and everyday
situations. For example, compare the chance of having lung
cancer if you are a smoker with the chance of being a smoker if you have
lung cancer. |
| Use the rules of probability to compute
probabilities of compound events in a uniform probability model |
|
S.CP.B.6 |
Find the conditional probability of A given
B as the fraction of Bs outcomes that also belong to
A, and interpret the answer in terms of the model. |
|
S.CP.B.7 |
Apply the Addition Rule, P(A or
B) = P(A) + P(B) P(A
and B), and interpret the answer in terms of the model. |