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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.  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

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 (xa) is a factor of p(x).

A.APR.B.3

Identify zeros of polynomial functions 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 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). This standard is a fluency expectation. This standard sets an expectation that students will divide polynomials with remainders by inspection in simple cases. For example, one can view the rational expression (x+4)/(x+3) as ((x+3)+1)/(x+3) which is 1+1/(x+3).
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

b. Explain each step when solving a rational or radical 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.

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.
b. Solve quadratic equations by: i) inspection (An example for inspection would be x2=-81, where a student should know that the solutions would include 9i and -9i), ii) taking square roots, iii) factoring, iv) completing the square (An example where students need to factor out a leading coefficient while completing the square would be 4x2+8x-9=0), v) the quadratic formula, and vi) graphing.
Write complex solutions in a+bi form.

Solve systems of equations
A.REI.C.6 Solve systems of linear equations in three variables.
A.REI.C.7

b. 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:
i) interpret key features of graphs and tables in terms of the quantities, and
ii) sketch graphs showing key features given a verbal description of the relationship.
Key features include: intercepts, zeros; intervals where the function is increasing, decreasing, positive, or negative; relative maximums and minimums; symmetries; end behavior; and periodicity.  Tasks may involve real-world context and may include polynomial, square root, cube root, exponential, logarithmic, and trigonometric functions.

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.8

Write a function in different but equivalent forms to reveal and explain different properties of the function.
b. Use the properties of exponents to interpret exponential functions, and classify them as representing exponential growth or decay.
Tasks also include real world problems that involve compounding growth/decay (A=P(1+(r/n))nt) and continuous compounding growth/decay (A=Pert).

F.IF.C.9

Compare properties of two functions each represented in a different way (algebraically, graphically, numerically in tables, or by verbal descriptions).  Tasks may involve polynomial, square root, cube root, exponential, logarithmic, and trigonometric functions.

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

b. 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

a. 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 a linear or exponential function symbolically given: i) a graph; ii) a description of the relationship; and iii) 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

Apply concepts of the unit circle in the coordinate plane to calculate the values of the six trigonometric functions given angles in radian measure.

F.TF.A.4

Use the unit circle to explain symmetry (odd and even) and periodicity of trigonometric functions.  Focus of this standard is on cos(x), sin(x) and tan(x).

F.TF.B.5

Choose trigonometric functions to model periodic phenomena with specified amplitude, frequency, horizontal shift, 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 a. Recognize whether or not a normal curve is appropriate for a given data set.
b. If appropriate, determine population percentages using a graphing calculator for an appropriate normal curve.
Summarize, represent, and interpret data on two categorical and quantitative variables
S.ID.B.6

Represent bivariate data on a scatter plot, and describe how the variables’ values are related.  It’s important to keep in mind that the data must be linked to the same “subjects”, not just two unrelated quantitative variables. Do not assume that an association between two variables implies that one causes another to change.
a. Fit a function to real-world data; use functions fitted to data to solve problems in the context of the data.  Emphasis is on quadratic, exponential, and power models, including the regression capabilities of the calculator.

Making Inferences & Justifying Conclusions
Understand and evaluate random processes underlying statistical experiments
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

Given a simulation model based on a sample proportion or mean, construct the 95% interval centered on the statistic (+/- two standard deviations) and determine if a suggested parameter is plausible.

S.IC.B.6

a. Use the tools of statistics to draw conclusions from numerical summaries.
b. Use the language of statistics to critique claims from informational texts.  For example, causation vs correlation, bias, measures of center and spread.

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.

Use the rules of probability to compute probabilities of compound events in a uniform probability 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.

 
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