Courses of Study: Mathematics

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-CN1]

2 ) Use the relation i² = -1 and the commutative, associative, and distributive properties to add, subtract, and multiply complex numbers. [N-CN2]

3 ) (+) Find the conjugate of a complex number; use conjugates to find moduli and quotients of complex numbers. [N-CN3]

4 ) Solve quadratic equations with real coefficients that have complex solutions. [N-CN7]

5 ) (+) Extend polynomial identities to the complex numbers.

6 ) (+) Know the Fundamental Theorem of Algebra; show that it is true for quadratic polynomials. [N-CN9]

7 ) (+) Use matrices to represent and manipulate data, e.g., to represent payoffs or incidence relationships in a network. (Use technology to approximate roots.) [N-VM6] (Alabama)

8 ) (+) Multiply matrices by scalars to produce new matrices, e.g., as when all of the payoffs in a game are doubled. [N-VM7]

9 ) (+) Add, subtract, and multiply matrices of appropriate dimensions. [N-VM8]

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. [N-VM9]

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. [N-VM10]

12 ) Interpret expressions that represent a quantity in terms of its context.* [A-SSE1]

13 ) Use the structure of an expression to identify ways to rewrite it. [A-SSE2]

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.* [A-SSE4]

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. [A-APR1]

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). [A-APR2]

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. [A-APR3]

18 ) Prove polynomial identities and use them to describe numerical relationships. [A-APR4]

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. [A-APR6]

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. [A-CED1]

21 ) Create equations in two or more variables to represent relationships between quantities; graph equations on coordinate axes with labels and scales. [A-CED2]

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. [A-CED3]

23 ) Rearrange formulas to highlight a quantity of interest, using the same reasoning as in solving equations. [A-CED4]

24 ) Solve simple rational and radical equations in one variable, and give examples showing how extraneous solutions may arise. [A-REI2]

25 ) Recognize when the quadratic formula gives complex solutions, and write them as a ± bi for real numbers a and b. [A-REI4b] (Alabama)

26 ) (+) Find the inverse of a matrix if it exists and use it to solve systems of linear equations (using technology for matrices of demensions 3 x 3 or greater). [A-REI9]

27 ) 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 linear, polynomial, rational, absolute value, exponential, and logarithmic functions.* [A-REI11]

28 ) Create graphs of conic sections, including parabolas, hyperbolas, ellipses, circles, and degenerate conics, from second-degree equations. (Alabama)

29 ) Relate the domain of a function to its graph and, where applicable, to the quantitative relationship it describes.* [F-IF5]

30 ) Graph functions expressed symbolically, and show key features of the graph, by hand in simple cases and using technology for more complicated cases.* [F-IF7]

31 ) Write a function defined by an expression in different but equivalent forms to reveal and explain different properties of the function. [F-IF8]

32 ) Compare properties of two functions each represented in a different way (algebraically, graphically, numerically in tables, or by verbal descriptions). [F-IF9]

33 ) Write a function that describes a relationship between two quantities.* [F-BF1]

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. [F-BF3]

35 ) Find inverse functions. [F-BF4]

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. [F-LE4]

37 ) (+) Use probabilities to make fair decisions (e.g., drawing by lots, using a random number generator). [S-MD6]

38 ) (+) Analyze decisions and strategies using probability concepts (e.g., product testing, medical testing, pulling a hockey goalie at the end of a game). [S-MD7]

39 ) 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-CP1]

40 ) 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-CP3]

41 ) 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. [S-CP4]

42 ) Recognize and explain the concepts of conditional probability and independence in everyday language and everyday situations. [S-CP5]

43 ) 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. [S-CP6]

44 ) 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. [S-CP7]

45 ) (+) Apply the general Multiplication Rule in a uniform probability model, P(A and B) = P(A)P(B|A) = P(B)P(A|B), and interpret the answer in terms of the model. [S-CP8]

46 ) (+) Use permutations and combinations to compute probabilities of compound events and solve problems. [S-CP9]