STATE STANDARDS
ALGEBRA II |
| NUMBER AND QUANTITY |
| The Real Number System |
| Extend the properties of exponents to
rational exponents |
|
N.RN.A.1 |
Explain how the definition of the meaning of
rational exponents follows from extending the properties of integer
exponents to those values, allowing for a notation for radicals in terms
of rational 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 |
Rewrite expressions involving radicals and rational
exponents using the properties of exponents. Includes expressions
with variable factors, such as the cubic root of 27x5y3. |
| Quantities |
| Reason quantitatively and use units to
solve problems |
| N.Q.A.2 |
Define appropriate quantities for the purpose of
descriptive modeling. |
| 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. |
| Use complex numbers in polynomial
identities and equations |
|
N.CN.C.7 |
Solve quadratic equations with real coefficients
that have complex solutions. |
| ALGEBRA |
| Seeing Structure in Expressions |
| Interpret the structure of expressions |
|
A.SSE.A.2 |
Use the structure of an expression to identify ways
to rewrite it (polynomial, rational). Includes factoring by
grouping. |
| 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 a quadratic expression to reveal the zeros of
the function it defines. Includes trinomials with leading
coefficients other than 1. |
| c |
Use the properties of exponents to transform
expressions for exponential functions. For example the
expression 1.15t can be rewritten as (1.151/12)12t
≈ 1.01212t to reveal the approximate equivalent monthly
interest rate if the annual rate is 15%. |
|
A.SSE.B.4 |
Derive the formula for the sum of a finite geometric
series (when the common ratio is not 1), and use the formula to solve
problems. For example, calculate mortgage payments. Includes
summation notation. |
| 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
and use them to solve problems (linear, quadratic, exponential (integer
inputs only), simple roots). |
| 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 simple rational and radical equations in one
variable, and give examples showing how extraneous solutions may 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 |
Explain why the x-coordinates of the points where
the graphs of the equations y = f(x) and y = g(x)
intersect are the solutions of the equation f(x) = g(x); find
the solutions approximately, e.g., using technology to graph the
functions, make tables of values, or find successive approximations.
Include cases where f(x) and/or g(x) are polynomial,
rational, radical, and exponential functions. |
| FUNCTIONS |
| Interpreting Functions |
| Understand the concept of a function and
use function notation |
|
F.IF.A.3 |
Recognize that sequences are functions, sometimes
defined recursively, whose domain is a subset of the integers. For
example, the Fibonacci sequence is defined recursively by f(0) = f(1) =
1, f(n+1) = f(n) + f(n-1) for n ≥ 1. |
| 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 (presented symbolically or as a table) over a specified
interval. Estimate the rate of change from a graph (emphasize
selection of appropriate models). |
| Analyze functions using different
representations |
|
F.IF.C.7 |
Graph functions expressed symbolically and show key
features of the graph, by hand in simple cases and using technology for
more complicated cases. |
| c |
Graph polynomial functions, identifying zeros when
suitable factorizations are available, and showing end behavior. |
| e |
Graph exponential and logarithmic functions, showing
intercepts and end behavior, and trigonometric functions, showing
period, midline, and amplitude. Focus on using key features to
guide selection of appropriate type of model function. |
|
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 an explicit expression, a recursive
process, or steps for calculation from a context. |
| 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 (all
types of functions studies). |
|
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
(linear, exponential, quadratic). |
| Build new functions from existing
functions |
|
F.BF.B.3 |
Identify the effect on the graph of replacing
f(x) by f(x) + k, k f(x), f(kx), and
f(x + k) for specific values of k (both positive and
negative); find the value of k given the graphs.
Experiment with cases and illustrate an explanation of the effects on
the graph using technology. Include recognizing even and odd
functions from their graphs and algebraic expressions for them (simple
radical, rational and exponential functions; emphasize common effect of
each transformation across function types). |
|
F.BF.B.4 |
Find inverse functions. |
|
F.BF.B.6 |
Represent and evaluate the sum of a finite arithmetic or finite geometric
series, using summation (sigma) notation. |
| 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 |
For exponential models, express as a logarithm the
solution to abct = d where a, c,
and d are numbers and the base b is 2, 10, or e;
evaluate the logarithm using technology (logarithms as solutions for
exponentials). |
| 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 (linear and exponential of form
f(x)=bx+k). |
| 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 and use it to find sin(θ), cos(θ), or tan(θ)
given sin(θ), cos(θ), or tan(θ) and 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 |
Decide if a specified model is consistent with
results from a given data-generating process, e.g., using simulation.
For example, a model says a spinning coin falls heads up with
probability 0.5. Would a result of 5 tails in a row cause you to
question the model? |
| 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 |
Construct and 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 to approximate conditional probabilities.
For example, collect data from a random sample of students in your
school on their favorite subject among math, science, and English.
Estimate the probability that a randomly selected student from your
school will favor science given that the student is in tenth grade.
Do the same for other subjects and compare the results. |
| 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. |