Administrative Rules of Montana
Department 10 - EDUCATION
Chapter 10.53 - CONTENT STANDARDS
Subchapter 10.53.5 - Mathematics Content Standards
Rule 10.53.513 - MONTANA HIGH SCHOOL MATHEMATICS ALGEBRA CONTENT STANDARDS

(a) interpret expressions that represent a quantity in terms of its context; *

(b) use the structure of an expression to identify ways to rewrite it; f or example, see x4 - y4 as (x2)2 - (y2)2, thus recognizing it as a difference of squares that can be factored as (x2 - y2)(x2 + y2);

(c) choose and produce an equivalent form of an expression to reveal and explain properties of the quantity represented by the expression; *

(d) derive the formula for the sum of a finite geometric series (when the common ratio is not 1) and use the formula to solve problems; f or example, calculate mortgage payments. *

(a) understand that polynomials form a system analogous to the integers, namely, they are closed under the operations of addition, subtraction, and multiplication and add, subtract, and multiply polynomials;

(b) know and apply the Remainder Theorem: for a polynomial p (x) and a number a, the remainder on division by x - a is p (a), so p (a) = 0 if and only if (x - a) is a factor of p (x);

(c) identify zeros of polynomials when suitable factorizations are available and use the zeros to construct a rough graph of the function defined by the polynomial;

(d) prove polynomial identities and use them to describe numerical relationships; f or example, the polynomial identity (x2 + y2)2 = (x2 - y2)2 + (2xy)2 can be used to generate Pythagorean triples;

(e) (+) know and apply the Binomial Theorem for the expansion of (x + y) n in powers of x and y for a positive integer n, where x and y are any numbers, with coefficients determined for example by Pascal's Triangle;

(f) rewrite simple rational expressions in different forms; write a (x)/ b (x) in the form q (x) + r (x)/ b (x), where a (x), b (x), q (x), and r (x) are polynomials with the degree of r (x) less than the degree of b (x), using inspection, long division, or, for the more complicated examples, a computer algebra system; and

(g) (+) understand that rational expressions form a system analogous to the rational numbers, closed under addition, subtraction, multiplication, and division by a nonzero rational expression and add, subtract, multiply, and divide rational expressions.

(a) create equations and inequalities in one variable and use them to solve problems from a variety of contexts (e.g., science, history, and culture, including those of Montana American Indians) and i nclude equations arising from linear and quadratic functions, and simple rational and exponential functions;

(b) create equations in two or more variables to represent relationships between quantities and graph equations on coordinate axes with labels and scales;

(c) represent constraints by equations or inequalities and by systems of equations and/or inequalities and interpret solutions as viable or nonviable options in a modeling context; f or example, represent inequalities describing nutritional and cost constraints on combinations of different foods; and

(d) rearrange formulas to highlight a quantity of interest using the same reasoning as in solving equations; f or example, rearrange Ohm's law V = IR to highlight resistance R.

(a) explain each step in solving a simple equation as following from the equality of numbers asserted at the previous step, starting from the assumption that the original equation has a solution and construct a viable argument to justify a solution method;

(b) solve simple rational and radical equations in one variable and give examples showing how extraneous solutions may arise;

(c) solve linear equations and inequalities in one variable, including equations with coefficients represented by letters;

(d) solve quadratic equations in one variable;

(e) prove that given a system of two equations in two variables, replacing one equation by the sum of that equation and a multiple of the other produces a system with the same solutions;

(f) solve systems of linear equations exactly and approximately (e.g., with graphs) focusing on pairs of linear equations in two variables;

(g) solve a simple system consisting of a linear equation and a quadratic equation in two variables algebraically and graphically; for example, find the points of intersection between the line y = -3 x and the circle x 2 + y 2 = 3;

(h) (+) represent a system of linear equations as a single matrix equation in a vector variable;

(i) (+) 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);

(j) understand that the graph of an equation in two variables is the set of all its solutions plotted in the coordinate plane, often forming a curve (which could be a line);

(k) explain why the x -coordinates of the points where the graphs of the equations y = f (x) and y = g (x) intersect are the solutions of the equation f (x) = g (x); find the solutions approximately, e.g., using technology to graph the functions, make tables of values or find successive approximations and include cases where f (x) and/or g (x) are linear, polynomial, rational, absolute value, exponential, and logarithmic functions; * and

(l) graph the solutions to a linear inequality in two variables as a half-plane (excluding the boundary in the case of a strict inequality) and graph the solution set to a system of linear inequalities in two variables as the intersection of the corresponding half-planes.