Topic: Mathematics for Advanced Courses |
Common Core Mathematics 9-12 |
| Cluster Perform arithmetic operations with complex numbers. |
| | Grade 9-12 |
| | | N.CN.3. | (+) Find the conjugate of a complex number; use conjugates to find
moduli and quotients of complex numbers. |
| Cluster Represent complex numbers and their operations on the complex
plane. |
| | Grade 9-12 |
| | | N.CN.4. | (+) Represent complex numbers on the complex plane in rectangular
and polar form (including real and imaginary numbers), and explain
why the rectangular and polar forms of a given complex number
represent the same number. |
| | | N.CN.5. | (+) Represent addition, subtraction, multiplication, and conjugation of
complex numbers geometrically on the complex plane; use properties
of this representation for computation. For example, (-1 + √ 3i)^{3} = 8
because (-1 + √ 3i) has modulus 2 and argument 120°. |
| | | N.CN.6. | (+) Calculate the distance between numbers in the complex plane as
the modulus of the difference, and the midpoint of a segment as the
average of the numbers at its endpoints. |
| Cluster Represent and model with vector quantities. |
| | Grade 9-12 |
| | | N.VM.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||, v). |
| | | N.VM.2. | (+) Find the components of a vector by subtracting the coordinates of
an initial point from the coordinates of a terminal point. |
| | | N.VM.3. | (+) Solve problems involving velocity and other quantities that can be
represented by vectors. |
| Cluster Perform operations on vectors. |
| | Grade 9-12 |
| | | N.VM.4. | (+) Add and subtract vectors. |
| | | N.VM.4.a. | 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. |
| | | N.VM.4.b. | Given two vectors in magnitude and direction form, determine the
magnitude and direction of their sum. |
| | | N.VM.4.c. | 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. |
| | | N.VM.5. | (+) Multiply a vector by a scalar. |
| | | N.VM.5.a. | Represent scalar multiplication graphically by scaling vectors and possibly reversing their direction; perform scalar multiplication component-wise, e.g., as c(v_{x}, v_{y}) = (cv_{x}, cv_{y}). |
| | | N.VM.5.b. | Compute the magnitude of a scalar multiple cv using ||cv|| = |c|v.
Compute the direction of cv knowing that when |c|v ≠ 0, the
direction of cv is either along v (for c > 0) or against v (for c < 0). |
| Cluster Perform operations on matrices and use matrices in applications. |
| | Grade 9-12 |
| | | N.VM.6. | (+) Use matrices to represent and manipulate data, e.g., to represent
payoffs or incidence relationships in a network. |
| | | N.VM.7. | (+) Multiply matrices by scalars to produce new matrices, e.g., as when
all of the payoffs in a game are doubled. |
| | | N.VM.8. | (+) Add, subtract, and multiply matrices of appropriate dimensions. |
| | | N.VM.9. | (+) 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.VM.11. | (+) 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. |
| | | N.VM.12. | (+) Work with 2 × 2 matrices as transformations of the plane, and
interpret the absolute value of the determinant in terms of area. |
| Cluster Solve systems of equations |
| | Grade 9-12 |
| | | A.REI.8. | (+) Represent a system of linear equations as a single matrix equation
in a vector variable. |
| | | A.REI.9. | (+) 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 × 3
or greater). |
| Cluster Analyze functions using different representations |
| | Grade 9-12 |
| | | F.IF.7.d. | (+) Graph rational functions, identifying zeros and asymptotes
when suitable factorizations are available, and showing end
behavior. |
| Cluster Build a function that models a relationship between two quantities |
| | Grade 9-12 |
| | | F.BF.1.c. | (+) Compose functions. For example, if T(y) is the temperature in the atmosphere as a function of height, and h(t) is the height of a weather balloon as a function of time, then T(h(t)) is the temperature
at the location of the weather balloon as a function of time. |