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 51/3 to be the cube root of 5 because we want (51/3)3 = 5(1/3)3 to hold, so (51/3)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 i2 = 每1 and every complex number has the form a + bi with a and b real; (b) use the relation i2 = 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 x2 + 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(vx, vy) = (cvx, cvy); 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.