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Mathematics, Grade 9 - 12, Analytical Mathematics, 2013

Vector and Matrix Quantities
Represent and model with vector quantities.
1.) (+) Recognize vector quantities as having both magnitude and direction. Represent vector quantities by directed line segments, and use appropriate symbols for vectors and their magnitudes (e.g., v, | v |, | |v| |), including the use of eigen-values and eigen-vectors. [N-VM1] (Alabama)

2.) (+) Solve problems involving velocity and other quantities that can be represented by vectors, including navigation (e.g., airplane, aerospace, oceanic). [N-VM3] (Alabama)

3.) (+) Add vectors end-to-end, component-wise, and by the parallelogram rule. Understand that the magnitude of a sum of two vectors is typically not the sum of the magnitudes. Find the dot product and the cross product of vectors. [N-VM4a] (Alabama)

4.) (+) Given two vectors in magnitude and direction form, determine the magnitude and direction of their sum, including vectors in complex vector spaces. [N-VM4b] (Alabama)

5.) (+) Understand vector subtraction v - w as v + (-w), where (-w) is the additive inverse of w, with the same magnitude as w and pointing in the opposite direction. Represent vector subtraction graphically by connecting the tips in the appropriate order, and perform vector subtraction component-wise, including vectors in complex vector spaces. [N-VM4c] (Alabama)

Perform operations on matrices and use matrices in applications.
6.) (+) Use matrices to represent and manipulate data, e.g., to represent payoffs or incidence relationships in a network, including linear programming. [N-VM6] (Alabama)

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

8.) (+) 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. Solve matrix equations using augmented matrices. [N-VM10] (Alabama)

9.) (+) Multiply a vector (regarded as a matrix with one column) by a matrix of suitable dimensions to produce another vector. Work with matrices as transformations of vectors, including matrices larger than 2 x 2. [N-VM11] (Alabama)

10.) (+) Work with 2 x 2 matrices as transformations of the plane, and interpret the absolute value of the determinant in terms of area. Solve matrix application problems using reduced row echelon form. [N-VM12] (Alabama)

Complex Numbers
Use complex numbers in polynomial identities and equations.
11.) (+) Know the Fundamental Theorem of Algebra; show that it is true for quadratic polynomials. Understand the importance of using complex numbers in graphing functions on the Cartesian or complex plane. [N-CN9] (Alabama)

Understand limits of functions. (Alabama)
12.) Calculate the limit of a sequence, of a function, and of an infinite series. (Alabama)

Seeing Structure in Expressions
13.) Use the laws of Boolean Algebra to describe true/false circuits. Simplify Boolean expressions using the relationships between conjunction, disjunction, and negation operations. (Alabama)

14.) Use logic symbols to write truth tables. (Alabama)

Arithmetic With Polynomials and Rational Functions
15.) Reduce the degree of either the numerator or denominator of a rational function by using partial fraction decomposition or partial fraction expansion. (Alabama)

Trigonometric Functions
Extend the domain of trigonometric functions using the unit circle.
16.) (+) Use the unit circle to explain symmetry (odd and even) and periodicity of trigonometric functions. [F-TF4].

Apply trigonometry to general triangles.
17.) (+) Prove the Law of Sines and the Law of Cosines and use them to solve problems. Understand Law of Sines = 2r, where r is the radius of the circumscribed circle of the triangle. Apply the Law of Tangents. [G-SRT10] (Alabama)

18.) Apply Euler's and deMoivre's formulas as links between complex numbers and trigonometry. (Alabama)

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