The Real Number System
Extend the properties of exponents to rational exponents.
N-RN.1 Explain the definition of the meaning of rational exponents.
N-RN.2 Rewrite expressions involving radicals and rational exponents using the properties of exponents.
Use properties of rational and irrational numbers.
N-RN.3 The sum or product of two rational numbers is rational; that
the sum of a rational number and irrational number is irrational; and
that the product of a nonzero rational number and an irrational
number is irrational.
Reason quantitatively and use units to solve problems.
N-Q.1 Use labels to understand problems; use labels with formulas;
choose the appropriate scale in graphs and data displays.
N-Q.2 Define appropriate quantities for the purpose of descriptive modeling.
N-Q.3 Choose a level of accuracy appropriate to limitations on measurement when reporting quantities.
(IA) Understand and apply the mathematical of voting.
IA.3 (+) Understand, analyze, apply, and evaluate some common voting and analysis methods in addition to majority and plurality, such as runoff, approval, the so-called instant-runoff voting (IRV) method, the Borda method and the Condorcet method.
(IA) Understand and apply some basic mathematics of information processing and the Internet.
IA.4 (+) Describe the role of mathematics in information processing, particularly with respect to the Internet.
IA.5 (+) Understand and apply elementary set theory and logic as used in simple Internet searches.
IA.6 (+) Understand and apply basic number theory, including modular arithmetic, for example, as used in keeping information secure through public-key cryptography.
The Complex Number System
Perform arithmetic operations with complex numbers.
N-CN.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.2 Use the relation i2 = -1 and the commutative, associative, and the distributive properties to add, subtract, and multiply complex numbers.
N-CN.3 Find the conjugate of a complex number; use conjugates to find moduli and quotients of complex numbers.
Represent complex numbers and their operations on the complex plane.
N-CN.4 Represent complex numbers on the complex plane in rectangular and polar form, 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 + has a modulus 2 and an 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.
Use complex numbers in polynomial identities and equations.
N-CN.7 Solve quadratic equations with real coefficients that have complex solutions.
N-CN.8 Extend polynomial identities to the complex numbers. For example, rewrite x2 + 4 as (x + 2i)(x – 2i)
N-CN.9 Know the Fundamental Theorem of Algebra; show that it is true for quadratic polynomials.
Vector and Matrix Quantities
Represent and model with vector quantities.
(N-VM.1) Recognize vector quantities as having both magnitude and direction.
(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 solved by vectors.
Perform operations on vectors.
(N-VM.4) Add and subtract vectors.
a. Add vectors end-to-end, component-wise, and by the parallelogram rule.
b. Given two vectors in magnitude and direction form, determine the magnitude and direction of their sum.
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.
(N-VM.5) Multiply a vector by a scalar.
a. Represent scalar multiplication graphically by scaling vectors and possibly reversing their direction; perform scalar multiplication component-wise.
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).
Perform operations on matrices and use matrices in applications.
(N-VM.6) Use matrices to represent and manipulate data.
(N-VM.7) Multiply matrices by scalars to produce new matrices.
(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.10) 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-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 2x2 matrices as transformations of the plane, and interpret the absolute value of the determinant in terms of area.