


Mathematics (2015) 
Grade(s): 9  12 
Algebra II with Trigonometry 
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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. [NCN1]

Mathematics (2015) 
Grade(s): 9  12 
Algebra II with Trigonometry 
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2 ) Use the relation i^{2} = 1 and the commutative, associative, and distributive properties to add, subtract, and multiply complex numbers. [NCN2]

Mathematics (2015) 
Grade(s): 9  12 
Algebra II with Trigonometry 
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3 ) (+) Find the conjugate of a complex number; use conjugates to find moduli and quotients of complex numbers. [NCN3]


Mathematics (2015) 
Grade(s): 9  12 
Algebra II with Trigonometry 
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4 ) Solve quadratic equations with real coefficients that have complex solutions. [NCN7]

Mathematics (2015) 
Grade(s): 9  12 
Algebra II with Trigonometry 
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5 ) (+) Extend polynomial identities to the complex numbers. [NCN8]
Example: Rewrite x^{2} + 4 as (x + 2i)(x  2i).

Mathematics (2015) 
Grade(s): 9  12 
Algebra II with Trigonometry 
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6 ) (+) Know the Fundamental Theorem of Algebra; show that it is true for quadratic polynomials. [NCN9]



Mathematics (2015) 
Grade(s): 9  12 
Algebra II with Trigonometry 
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7 ) (+) Use matrices to represent and manipulate data, e.g., to represent payoffs or incidence relationships in a network. (Use technology to approximate roots.) [NVM6] (Alabama)

Mathematics (2015) 
Grade(s): 9  12 
Algebra II with Trigonometry 
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8 ) (+) Multiply matrices by scalars to produce new matrices, e.g., as when all of the payoffs in a game are doubled. [NVM7]

Mathematics (2015) 
Grade(s): 9  12 
Algebra II with Trigonometry 
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3 
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3 
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9 ) (+) Add, subtract, and multiply matrices of appropriate dimensions. [NVM8]

Mathematics (2015) 
Grade(s): 9  12 
Algebra II with Trigonometry 
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10 ) (+) Understand that, unlike multiplication of numbers, matrix multiplication for square matrices is not a commutative operation, but still satisfies the associative and distributive properties. [NVM9]

Mathematics (2015) 
Grade(s): 9  12 
Algebra II with Trigonometry 
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11 ) (+) Understand that the zero and identity matrices play a role in matrix addition and multiplication similar to the role of 0 and 1 in the real numbers. The determinant of a square matrix is nonzero if and only if the matrix has a multiplicative inverse. [NVM10]




Mathematics (2015) 
Grade(s): 9  12 
Algebra II with Trigonometry 
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12 ) Interpret expressions that represent a quantity in terms of its context.* [ASSE1]
a. Interpret parts of an expression such as terms, factors, and coefficients. [ASSE1a]
b. Interpret complicated expressions by viewing one or more of their parts as a single entity. [ASSE1b]
Example: Interpret P(1+r)^{n} as the product of P and a factor not depending on P.

Mathematics (2015) 
Grade(s): 9  12 
Algebra II with Trigonometry 
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13 ) Use the structure of an expression to identify ways to rewrite it. [ASSE2]
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}).


Mathematics (2015) 
Grade(s): 9  12 
Algebra II with Trigonometry 
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14 ) 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.* [ASSE4]
Example: Calculate mortgage payments.



Mathematics (2015) 
Grade(s): 9  12 
Algebra II with Trigonometry 
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15 ) Understand that polynomials form a system analogous to the integers; namely, they are closed under the operations of addition, subtraction, and multiplication; add, subtract, and multiply polynomials. [AAPR1]


Mathematics (2015) 
Grade(s): 9  12 
Algebra II with Trigonometry 
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16 ) 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). [AAPR2]

Mathematics (2015) 
Grade(s): 9  12 
Algebra II with Trigonometry 
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17 ) 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. [AAPR3]


Mathematics (2015) 
Grade(s): 9  12 
Algebra II with Trigonometry 
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18 ) Prove polynomial identities and use them to describe numerical relationships. [AAPR4]
Example: The polynomial identity (x^{2} + y^{2})^{2} = (x^{2}  y^{2})^{2} + (2xy)^{2} can be used to generate Pythagorean triples.


Mathematics (2015) 
Grade(s): 9  12 
Algebra II with Trigonometry 
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19 ) 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. [AAPR6]



Mathematics (2015) 
Grade(s): 9  12 
Algebra II with Trigonometry 
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20 ) Create equations and inequalities in one variable and use them to solve problems. Include equations arising from linear and quadratic functions, and simple rational and exponential functions. [ACED1]

Mathematics (2015) 
Grade(s): 9  12 
Algebra II with Trigonometry 
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14 
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21 ) Create equations in two or more variables to represent relationships between quantities; graph equations on coordinate axes with labels and scales. [ACED2]

Mathematics (2015) 
Grade(s): 9  12 
Algebra II with Trigonometry 
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22 ) Represent constraints by equations or inequalities, and by systems of equations and/or inequalities, and interpret solutions as viable or nonviable options in a modeling context. [ACED3]
Example: Represent inequalities describing nutritional and cost constraints on combinations of different foods.

Mathematics (2015) 
Grade(s): 9  12 
Algebra II with Trigonometry 
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23 ) Rearrange formulas to highlight a quantity of interest, using the same reasoning as in solving equations. [ACED4]
Example: Rearrange Ohm's law V = IR to highlight resistance R.



Mathematics (2015) 
Grade(s): 9  12 
Algebra II with Trigonometry 
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24 ) Solve simple rational and radical equations in one variable, and give examples showing how extraneous solutions may arise. [AREI2]


Mathematics (2015) 
Grade(s): 9  12 
Algebra II with Trigonometry 
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25 ) Recognize when the quadratic formula gives complex solutions, and write them as a ± bi for real numbers a and b. [AREI4b] (Alabama)


Mathematics (2015) 
Grade(s): 9  12 
Algebra II with Trigonometry 
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26 ) (+) Find the inverse of a matrix if it exists and use it to solve systems of linear equations (using technology for matrices of dimension 3 x 3 or greater). [AREI9] (Alabama)


Mathematics (2015) 
Grade(s): 9  12 
Algebra II with Trigonometry 
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27 ) Explain why the xcoordinates 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 linear, polynomial, rational, absolute value, exponential, and logarithmic functions.* [AREI11]




Mathematics (2015) 
Grade(s): 9  12 
Algebra II with Trigonometry 
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28 ) Create graphs of conic sections, including parabolas, hyperbolas, ellipses, circles, and degenerate conics, from seconddegree equations. (Alabama)
Example: Graph x^{2}  6x + y^{2}  12y + 41 = 0 or y^{2}  4x + 2y + 5 = 0.
a. Formulate equations of conic sections from their determining characteristics. (Alabama)
Example: Write the equation of an ellipse with center (5, 3), a horizontal major axis of length 10, and a minor axis of length 4.



Mathematics (2015) 
Grade(s): 9  12 
Algebra II with Trigonometry 
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29 ) Relate the domain of a function to its graph and, where applicable, to the quantitative relationship it describes.* [FIF5]
Example: If the function h(n) gives the number of personhours it takes to assemble n engines in a factory, then the positive integers would be an appropriate domain for the function.


Mathematics (2015) 
Grade(s): 9  12 
Algebra II with Trigonometry 
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30 ) Graph functions expressed symbolically, and show key features of the graph, by hand in simple cases and using technology for more complicated cases.* [FIF7]
a. Graph square root, cube root, and piecewisedefined functions, including step functions and absolute value functions. [FIF7b]
b. Graph polynomial functions, identifying zeros when suitable factorizations are available, and showing end behavior. [FIF7c]
c. Graph exponential and logarithmic functions, showing intercepts and end behavior, and trigonometric functions, showing period, midline, and amplitude. [FIF7e]

Mathematics (2015) 
Grade(s): 9  12 
Algebra II with Trigonometry 
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31 ) Write a function defined by an expression in different but equivalent forms to reveal and explain different properties of the function. [FIF8]

Mathematics (2015) 
Grade(s): 9  12 
Algebra II with Trigonometry 
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32 ) Compare properties of two functions each represented in a different way (algebraically, graphically, numerically in tables, or by verbal descriptions). [FIF9]
Example: Given a graph of one quadratic function and an algebraic expression for another, say which has the larger maximum.



Mathematics (2015) 
Grade(s): 9  12 
Algebra II with Trigonometry 
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33 ) Write a function that describes a relationship between two quantities.* [FBF1]
a. Combine standard function types using arithmetic operations. [FBF1b]
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.


Mathematics (2015) 
Grade(s): 9  12 
Algebra II with Trigonometry 
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34 ) 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.
[FBF3]

Mathematics (2015) 
Grade(s): 9  12 
Algebra II with Trigonometry 
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35 ) Find inverse functions. [FBF4]
Solve an equation of the form f(x) = c for a simple function f that has an inverse, and write an expression for the inverse. [FBF4a]
Example f(x) = 2x^{3} or f(x) = (x+1)/(x1) for x ≠ 1.



Mathematics (2015) 
Grade(s): 9  12 
Algebra II with Trigonometry 
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36 ) For exponential models, express as a logarithm the solution to ab^{ct} = d where a, c, and d are numbers, and the base b is 2, 10, or e; evaluate the logarithm using technology. [FLE4]



Mathematics (2015) 
Grade(s): 9  12 
Algebra II with Trigonometry 
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37 ) Understand radian measure of an angle as the length of the arc on the unit circle subtended by the angle. [FTF1]

Mathematics (2015) 
Grade(s): 9  12 
Algebra II with Trigonometry 
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38 ) 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. [FTF2]

Mathematics (2015) 
Grade(s): 9  12 
Algebra II with Trigonometry 
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39 ) Define the six trigonometric functions using ratios of the sides of a right triangle, coordinates on the unit circle, and the reciprocal of other functions. (Alabama)


Mathematics (2015) 
Grade(s): 9  12 
Algebra II with Trigonometry 
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40 ) Choose trigonometric functions to model periodic phenomena with specified amplitude, frequency, and midline.* [FTF5]




Mathematics (2015) 
Grade(s): 9  12 
Algebra II with Trigonometry 
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41 ) (+) Use probabilities to make fair decisions (e.g., drawing by lots, using a random number generator). [SMD6]

Mathematics (2015) 
Grade(s): 9  12 
Algebra II with Trigonometry 
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42 ) (+) Analyze decisions and strategies using probability concepts (e.g., product testing, medical testing, pulling a hockey goalie at the end of a game). [SMD7]



Mathematics (2015) 
Grade(s): 9  12 
Algebra II with Trigonometry 
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43 ) 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"). [SCP1]

Mathematics (2015) 
Grade(s): 9  12 
Algebra II with Trigonometry 
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44 ) 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. [SCP3]

Mathematics (2015) 
Grade(s): 9  12 
Algebra II with Trigonometry 
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45 ) Construct and interpret twoway frequency tables of data when two categories are associated with each object being classified. Use the twoway table as a sample space to decide if events are independent and to approximate conditional probabilities. [SCP4]
Example: Collect data from a random sample of students in your school on their favorite subject among mathematics, 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.

Mathematics (2015) 
Grade(s): 9  12 
Algebra II with Trigonometry 
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46 ) Recognize and explain the concepts of conditional probability and independence in everyday language and everyday situations. [SCP5]
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.


Mathematics (2015) 
Grade(s): 9  12 
Algebra II with Trigonometry 
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47 ) Find the conditional probability of A given B as the fraction of B's outcomes that also belong to A, and interpret the answer in terms of the model. [SCP6]

Mathematics (2015) 
Grade(s): 9  12 
Algebra II with Trigonometry 
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48 ) 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. [SCP7]

Mathematics (2015) 
Grade(s): 9  12 
Algebra II with Trigonometry 
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49 ) (+) Apply the general Multiplication Rule in a uniform probability model, P(A and B) = P(A)P(BA) = P(B)P(AB), and interpret the answer in terms of the model. [SCP8]

Mathematics (2015) 
Grade(s): 9  12 
Algebra II with Trigonometry 
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50 ) (+) Use permutations and combinations to compute probabilities of compound events and solve problems. [SCP9]
