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Math Educational Standards for Texas as Correlated to MathTrax
Texas Essential Knowledge and Skills for Mathematics, Grades 9-12
http://www.tea.state.tx.us/rules/tac/chapter111/ch111c.html
Algebra I
(1) The student understands that a function represents a dependence of one quantity on another and can be described in a variety of ways. Following are performance descriptions.
(A) The student describes independent and dependent quantities in functional relationships.
(B) The student gathers and records data, or uses data sets, to determine functional (systematic) relationships between quantities.
(C) The student describes functional relationships for given problem situations and writes equations or inequalities to answer questions arising from the situations.
(D) The student represents relationships among quantities using concrete models, tables, graphs, diagrams, verbal descriptions, equations, and inequalities.
(E) The student interprets and makes inferences from functional relationships.
(2) The student uses the properties and attributes of functions. Following are performance descriptions.
(A) The student identifies and sketches the general forms of linear (y = x) and quadratic (y = x 2) parent functions.
(B) For a variety of situations, the student identifies the mathematical domains and ranges and determines reasonable domain and range values for given situations.
(C) The student interprets situations in terms of given graphs or creates situations that fit given graphs.
(D) In solving problems, the student collects and organizes data, makes and interprets scatterplots, and models, predicts, and makes decisions and critical judgments.
(3) The student understands how algebra can be used to express generalizations and recognizes and uses the power of symbols to represent situations. Following are performance descriptions.
(A) The student uses symbols to represent unknowns and variables.
(B) Given situations, the student looks for patterns and represents generalizations algebraically.
(4) The student understands the importance of the skills required to manipulate symbols in order to solve problems and uses the necessary algebraic skills required to simplify algebraic expressions and solve equations and inequalities in problem situations. Following are performance descriptions.
(A) The student finds specific function values, simplifies polynomial expressions, transforms and solves equations, and factors as necessary in problem situations.
(B) The student uses the commutative, associative, and distributive properties to simplify algebraic expressions.
(c) Linear functions: knowledge and skills and performance descriptions.
(1) The student understands that linear functions can be represented in different ways and translates among their various representations. Following are performance descriptions.
(A) The student determines whether or not given situations can be represented by linear functions.
(B) The student determines the domain and range values for which linear functions make sense for given situations.
(C) The student translates among and uses algebraic, tabular, graphical, or verbal descriptions of linear functions.
(2) The student understands the meaning of the slope and intercepts of linear functions and interprets and describes the effects of changes in parameters of linear functions in real-world and mathematical situations. Following are performance descriptions.
(A) The student develops the concept of slope as rate of change and determines slopes from graphs, tables, and algebraic representations.
(B) The student interprets the meaning of slope and intercepts in situations using data, symbolic representations, or graphs.
(C) The student investigates, describes, and predicts the effects of changes in m and b on the graph of y = mx + b.
(D) The student graphs and writes equations of lines given characteristics such as two points, a point and a slope, or a slope and y-intercept.
(E) The student determines the intercepts of linear functions from graphs, tables, and algebraic representations.
(F) The student interprets and predicts the effects of changing slope and y-intercept in applied situations.
(G) The student relates direct variation to linear functions and solves problems involving proportional change.
(3) The student formulates equations and inequalities based on linear functions, uses a variety of methods to solve them, and analyzes the solutions in terms of the situation. Following are performance descriptions.
(A) The student analyzes situations involving linear functions and formulates linear equations or inequalities to solve problems.
(B) The student investigates methods for solving linear equations and inequalities using concrete models, graphs, and the properties of equality, selects a method, and solves the equations and inequalities.
(C) For given contexts, the student interprets and determines the reasonableness of solutions to linear equations and inequalities.
(4) The student formulates systems of linear equations from problem situations, uses a variety of methods to solve them, and analyzes the solutions in terms of the situation. Following are performance descriptions.
(A) The student analyzes situations and formulates systems of linear equations to solve problems.
(B) The student solves systems of linear equations using concrete models, graphs, tables, and algebraic methods.
(C) For given contexts, the student interprets and determines the reasonableness of solutions to systems of linear equations.
(d) Quadratic and other nonlinear functions: knowledge and skills and performance descriptions.
(1) The student understands that the graphs of quadratic functions are affected by the parameters of the function and can interpret and describe the effects of changes in the parameters of quadratic functions. Following are performance descriptions.
(A) The student determines the domain and range values for which quadratic functions make sense for given situations.
(B) The student investigates, describes, and predicts the effects of changes in a on the graph of y = ax 2.
(C) The student investigates, describes, and predicts the effects of changes in c on the graph of y = x 2 + c.
(D) For problem situations, the student analyzes graphs of quadratic functions and draws conclusions.
(2) The student understands there is more than one way to solve a quadratic equation and solves them using appropriate methods. Following are performance descriptions.
(A) The student solves quadratic equations using concrete models, tables, graphs, and algebraic methods.
(B) The student relates the solutions of quadratic equations to the roots of their functions.
(3) The student understands there are situations modeled by functions that are neither linear nor quadratic and models the situations. Following are performance descriptions.
(A) The student uses patterns to generate the laws of exponents and applies them in problem-solving situations.
(B) The student analyzes data and represents situations involving inverse variation using concrete models, tables, graphs, or algebraic methods.
(C) The student analyzes data and represents situations involving exponential growth and decay using concrete models, tables, graphs, or algebraic methods.
Algebra II
(b) Foundations for functions: knowledge and skills and performance descriptions.
(1) The student uses properties and attributes of functions and applies functions to problem situations. Following are performance descriptions.
(A) For a variety of situations, the student identifies the mathematical domains and ranges and determines reasonable domain and range values for given situations.
(B) In solving problems, the student collects data and records results, organizes the data, makes scatterplots, fits the curves to the appropriate parent function, interprets the results, and proceeds to model, predict, and make decisions and critical judgments.
(3) The student formulates systems of equations and inequalities from problem situations, uses a variety of methods to solve them, and analyzes the solutions in terms of the situations. Following are performance descriptions.
(A) The student analyzes situations and formulates systems of equations or inequalities in two or more unknowns to solve problems.
(B) The student uses algebraic methods, graphs, tables, or matrices, to solve systems of equations or inequalities.
(C) For given contexts, the student interprets and determines the reasonableness of solutions to systems of equations or inequalities.
(2) The student knows the relationship between the geometric and algebraic descriptions of conic sections. Following are performance descriptions.
(A) The student describes a conic section as the intersection of a plane and a cone.
(B) In order to sketch graphs of conic sections, the student relates simple parameter changes in the equation to corresponding changes in the graph.
(C) The student identifies symmetries from graphs of conic sections.
(D) The student identifies the conic section from a given equation.
(E) The student uses the method of completing the square.
(d) Quadratic and square root functions: knowledge and skills and performance descriptions.
(1) The student understands that quadratic functions can be represented in different ways and translates among their various representations. Following are performance descriptions.
(A) For given contexts, the student determines the reasonable domain and range values of quadratic functions, as well as interprets and determines the reasonableness of solutions to quadratic equations and inequalities.
(B) The student relates representations of quadratic functions, such as algebraic, tabular, graphical, and verbal descriptions.
(C) The student determines a quadratic function from its roots or a graph.
(2) The student interprets and describes the effects of changes in the parameters of quadratic functions in applied and mathematical situations. Following are performance descriptions.
(A) The student uses characteristics of the quadratic parent function to sketch the related graphs and connects between the y = ax 2 + bx + c and the y = a(x - h) 2 + k symbolic representations of quadratic functions.
(B) The student uses the parent function to investigate, describe, and predict the effects of changes in a, h, and k on the graphs of y = a(x - h) 2 + k form of a function in applied and purely mathematical situations.
(3) The student formulates equations and inequalities based on quadratic functions, uses a variety of methods to solve them, and analyzes the solutions in terms of the situation. Following are performance descriptions.
(A) The student analyzes situations involving quadratic functions and formulates quadratic equations or inequalities to solve problems.
(B) The student analyzes and interprets the solutions of quadratic equations using discriminants and solves quadratic equations using the quadratic formula.
(C) The student compares and translates between algebraic and graphical solutions of quadratic equations.
(D) The student solves quadratic equations and inequalities.
(4) The student formulates equations and inequalities based on square root functions, uses a variety of methods to solve them, and analyzes the solutions in terms of the situation. Following are performance descriptions.
(A) The student uses the parent function to investigate, describe, and predict the effects of parameter changes on the graphs of square root functions and describes limitations on the domains and ranges.
(B) The student relates representations of square root functions, such as algebraic, tabular, graphical, and verbal descriptions.
(e) Rational functions: knowledge and skills and performance descriptions. The student formulates equations and inequalities based on rational functions, uses a variety of methods to solve them, and analyzes the solutions in terms of the situation. Following are performance descriptions.
(1) The student uses quotients to describe the graphs of rational functions, describes limitations on the domains and ranges, and examines asymptotic behavior.
(2) The student analyzes various representations of rational functions with respect to problem situations.
(3) For given contexts, the student determines the reasonable domain and range values of rational functions, as well as interprets and determines the reasonableness of solutions to rational equations and inequalities.
(4) The student solves rational equations and inequalities using graphs, tables, and algebraic methods.
(5) The student analyzes a situation modeled by a rational function, formulates an equation or inequality composed of a linear or quadratic function, and solves the problem.
Pre-Calculus
(c) Knowledge and skills.
(1) The student defines functions, describes characteristics of functions, and translates among verbal, numerical, graphical, and symbolic representations of functions, including polynomial, rational, radical, exponential, logarithmic, trigonometric, and piecewise-defined functions. The student is expected to:
(A) describe parent functions symbolically and graphically, including y = x n, y = ln x, y = log a x, y =
, y = e x, y = a x, y = sin x, etc.;
(B) determine the domain and range of functions using graphs, tables, and symbols;
(C) describe symmetry of graphs of even and odd functions;
(D) recognize and use connections among significant points of a function (roots, maximum points, and minimum points), the graph of a function, and the symbolic representation of a function; and
(E) investigate continuity, end behavior, vertical and horizontal asymptotes, and limits and connect these characteristics to the graph of a function.
(2) The student interprets the meaning of the symbolic representations of functions and operations on functions within a context. The student is expected to:
(A) apply basic transformations, including a • f(x), f(x) + d, f(x - c), f(b • x), |f(x)|, f(|x|), to the parent functions;
(B) perform operations including composition on functions, find inverses, and describe these procedures and results verbally, numerically, symbolically, and graphically; and
(C) investigate identities graphically and verify them symbolically, including logarithmic properties, trigonometric identities, and exponential properties.
(4) The student uses sequences and series to represent, analyze, and solve real-life problems. The student is expected to:
(A) represent patterns using arithmetic and geometric sequences and series;
(B) use arithmetic, geometric, and other sequences and series to solve real-life problems;
(C) describe limits of sequences and apply their properties to investigate convergent and divergent series; and
(D) apply sequences and series to solve problems including sums and binomial expansion.
(5) The student uses conic sections, their properties, and parametric representations to model physical situations. The student is expected to:
(A) use conic sections to model motion, such as the graph of velocity vs. position of a pendulum and motions of planets;
(B) use properties of conic sections to describe physical phenomena such as the reflective properties of light and sound;
(C) convert between parametric and rectangular forms of functions and equations to graph them; and
(D) use parametric functions to simulate problems involving motion.
(6) The student uses vectors to model physical situations. The student is expected to:
(A) use the concept of vectors to model situations defined by magnitude and direction; and
(B) analyze and solve vector problems generated by real-life situations.
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