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Number & Quantity


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.