**10.53.512** MONTANA HIGH SCHOOL MATHEMATICS NUMBER AND QUANTITY STANDARDS

(1) Mathematics number and quantity: the real number system content standards for high school are:

(a) explain how the definition of the meaning of rational exponents follows from extending the properties of integer exponents to those values, allowing for a notation for radicals in terms of rational exponents; for example, we define 5^{1/3} to be the cube root of 5 because we want (5^{1/3})^{3} = 5^{(1/3)3} to hold, so (5^{1/3})^{3} must equal 5;

(b) rewrite expressions involving radicals and rational exponents using the properties of exponents; and

(c) explain why the sum or product of two rational numbers is rational; that the sum of a rational number and an irrational number is irrational; and that the product of a nonzero rational number and an irrational number is irrational.

(2) Mathematics number and quantity: quantities content standards for high school are:

(a) use units as a way to understand problems from a variety of contexts (e.g., science, history, and culture), including those of Montana American Indians, and to guide the solution of multistep problems; choose and interpret units consistently in formulas; and choose and interpret the scale and the origin in graphs and data displays;

(b) define appropriate quantities for the purpose of descriptive modeling; and

(c) choose a level of accuracy appropriate to limitations on measurement when reporting quantities.

(3) Mathematics number and quantity: the complex number system content standards for high school are:

(a) know there is a complex number i such that i^{2} = 每1 and every complex number has the form a + bi with a and b real;

(b) use the relation i^{2} = 每1 and the commutative, associative, and distributive properties to add, subtract, and multiply complex numbers;

(c) (+) find the conjugate of a complex number and use conjugates to find moduli and quotients of complex numbers;

(d) (+) 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;

(e) (+) represent addition, subtraction, multiplication, and conjugation of complex numbers geometrically on the complex plane; use properties of this representation for computation; for example, (-1 + ﹟3 i)^{3} = 8 because (-1 + ﹟3 i) has modulus 2 and argument 120∼;

(f) (+) 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;

(g) solve quadratic equations with real coefficients that have complex solutions;

(h) (+) extend polynomial identities to the complex numbers and for example, rewrite x^{2} + 4 as (x + 2i)(x 每 2i); and

(i) (+) know the Fundamental Theorem of Algebra and show that it is true for quadratic polynomials.

(4) Mathematics number and quantity: vector and matrix quantities content standards for high school are:

(a) (+) 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);

(b) (+) find the components of a vector by subtracting the coordinates of an initial point from the coordinates of a terminal point;

(c) (+) solve problems from a variety of contexts (e.g., science, history, and culture), including those of Montana American Indians, involving velocity and other quantities that can be represented by vectors;

(d) (+) add and subtract vectors;

(i) add vectors end-to-end, component-wise, and by the parallelogram rule and understand that the magnitude of a sum of two vectors is typically not the sum of the magnitudes;

(ii) given two vectors in magnitude and direction form, determine the magnitude and direction of their sum; and

(iii) understand vector subtraction v 每 w as v + (每w) where 每w is the additive inverse of w, with the same magnitude as w and pointing in the opposite direction and represent vector subtraction graphically by connecting the tips in the appropriate order and perform vector subtraction component-wise;

(e) (+) multiply a vector by a scalar;

(i) represent scalar multiplication graphically by scaling vectors and possibly reversing their direction and perform scalar multiplication component-wise, e.g., as c(v_{x}, v_{y}) = (cv_{x}, cv_{y}); and

(ii) compute the magnitude of a scalar multiple cv using ||cv|| = |c|v and 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);

(f) (+) use matrices to represent and manipulate data, e.g., to represent payoffs or incidence relationships in a network;

(g) (+) multiply matrices by scalars to produce new matrices, e.g., as when all of the payoffs in a game are doubled;

(h) (+) add, subtract, and multiply matrices of appropriate dimensions;

(i) (+) understand that, unlike multiplication of numbers, matrix multiplication for square matrices is not a commutative operation, but still satisfies the associative and distributive properties;

(j) (+) 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 and the determinant of a square matrix is nonzero if and only if the matrix has a multiplicative inverse;

(k) (+) multiply a vector (regarded as a matrix with one column) by a matrix of suitable dimensions to produce another vector and work with matrices as transformations of vectors; and

(l) (+) work with 2 ℅ 2 matrices as transformations of the plane and interpret the absolute value of the determinant in terms of area.

History: 20-2-114, MCA; __IMP__, 20-2-121, 20-3-106, 20-7-101, MCA; __NEW__, 2011 MAR p. 2522, Eff. 11/26/11.