(1) Mathematics algebra: seeing structure in expressions content standards for high school are: (a) interpret expressions that represent a quantity in terms of its context; (i) interpret parts of an expression, such as terms, factors, and coefficients; and (ii) interpret complicated expressions by viewing one or more of their parts as a single entity; for example, interpret P(1+r) (b) use the structure of an expression to identify ways to rewrite it; for example, see x (c) choose and produce an equivalent form of an expression to reveal and explain properties of the quantity represented by the expression; (i) factor a quadratic expression to reveal the zeros of the function it defines; (ii) complete the square in a quadratic expression to reveal the maximum or minimum value of the function it defines; and (iii) use the properties of exponents to transform expressions for exponential functions; for example the expression 1.15 (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; for example, calculate mortgage payments. (2) Mathematics algebra: arithmetic with polynomials and rational expressions content standards for high school are: (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; for example, the polynomial identity (x (e) (+) know and apply the Binomial Theorem for the expansion of (x + y) (f) rewrite simple rational expressions in different forms; write (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. (3) Mathematics algebra: creating equations content standards for high school are: (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 include 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; for 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; for example, rearrange Ohm's law V = IR to highlight resistance R. (4) Mathematics algebra: reasoning with equations and inequalities content standards for high school are: (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; (i) use the method of completing the square to transform any quadratic equation in x into an equation of the form (x p) (ii) solve quadratic equations by inspection (e.g., for x (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 = 3x and the circle x (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 Χ 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; (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. History: 20-2-114, MCA; |