#### A.1: The student understands that a function represents a dependence of one quantity on another and can be described in a variety of ways.

A.1.A: describe independent and dependent quantities in functional relationships;

A.1.C: describe functional relationships for given problem situations and write equations or inequalities to answer questions arising from the situations;

A.1.D: represent relationships among quantities using concrete models, tables, graphs, diagrams, verbal descriptions, equations, and inequalities; and

A.1.E: interpret and make decisions, predictions, and critical judgments from functional relationships.

#### A.2: The student uses the properties and attributes of functions.

A.2.A: identify and sketch the general forms of linear (y = x) and quadratic (y = x²) parent functions;

A.2.B: identify mathematical domains and ranges and determine reasonable domain and range values for given situations, both continuous and discrete;

A.2.D: collect and organize data, make and interpret scatterplots (including recognizing positive, negative, or no correlation for data approximating linear situations), and model, predict, and make decisions and critical judgments in problem situations.

#### A.3: The student understands how algebra can be used to express generalizations and recognizes and uses the power of symbols to represent situations.

A.3.A: use symbols to represent unknowns and variables; and

A.3.B: look for patterns and represent generalizations algebraically.

#### A.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.

A.4.A: find specific function values, simplify polynomial expressions, transform and solve equations, and factor as necessary in problem situations;

A.4.C: connect equation notation with function notation, such as y = x + 1 and f(x) = x + 1.

#### A.5: The student understands that linear functions can be represented in different ways and translates among their various representations.

A.5.A: determine whether or not given situations can be represented by linear functions;

A.5.B: determine the domain and range for linear functions in given situations; and

A.5.C: use, translate, and make connections among algebraic, tabular, graphical, or verbal descriptions of linear functions.

#### A.6: The student understands the meaning of the slope and intercepts of the graphs of linear functions and zeros of linear functions and interprets and describes the effects of changes in parameters of linear functions in real-world and mathematical situations

A.6.A: develop the concept of slope as rate of change and determine slopes from graphs, tables, and algebraic representations;

A.6.B: interpret the meaning of slope and intercepts in situations using data, symbolic representations, or graphs;

A.6.C: investigate, describe, and predict the effects of changes in m and b on the graph of y = mx + b;

A.6.D: graph and write equations of lines given characteristics such as two points, a point and a slope, or a slope and y-intercept;

A.6.E: determine the intercepts of the graphs of linear functions and zeros of linear functions from graphs, tables, and algebraic representations;

A.6.F: interpret and predict the effects of changing slope and y-intercept in applied situations; and

A.6.G: relate direct variation to linear functions and solve problems involving proportional change.

#### A.7: 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.

A.7.A: analyze situations involving linear functions and formulate linear equations or inequalities to solve problems;

A.7.B: investigate methods for solving linear equations and inequalities using concrete models, graphs, and the properties of equality, select a method, and solve the equations and inequalities; and

A.7.C: interpret and determine the reasonableness of solutions to linear equations and inequalities.

#### A.8: 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.

A.8.B: solve systems of linear equations using concrete models, graphs, tables, and algebraic methods; and

#### A.9: 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.

A.9.A: determine the domain and range for quadratic functions in given situations;

A.9.B: investigate, describe, and predict the effects of changes in a on the graph of y = ax² + c;

A.9.C: investigate, describe, and predict the effects of changes in c on the graph of y = ax² + c; and

A.9.D: analyze graphs of quadratic functions and draw conclusions.

#### A.10: The student understands there is more than one way to solve a quadratic equation and solves them using appropriate methods.

A.10.A: solve quadratic equations using concrete models, tables, graphs, and algebraic methods; and

A.10.B: make connections among the solutions (roots) of quadratic equations, the zeros of their related functions, and the horizontal intercepts (x-intercepts) of the graph of the function.

#### A.11: The student understands there are situations modeled by functions that are neither linear nor quadratic and models the situations.

A.11.A: use patterns to generate the laws of exponents and apply them in problem-solving situations;

A.11.B: analyze data and represent situations involving inverse variation using concrete models, tables, graphs, or algebraic methods; and

A.11.C: analyze data and represent situations involving exponential growth and decay using concrete models, tables, graphs, or algebraic methods.

#### 2A.1: The student uses properties and attributes of functions and applies functions to problem situations.

2A.1.A: identify the mathematical domains and ranges of functions and determine reasonable domain and range values for continuous and discrete situations; and

2A.1.B: collect and organize data, make and interpret scatterplots, fit the graph of a function to the data, interpret the results, and proceed to model, predict, and make decisions and critical judgments.

#### 2A.2: 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.

2A.2.A: use tools including factoring and properties of exponents to simplify expressions and to transform and solve equations; and

2A.2.B: use complex numbers to describe the solutions of quadratic equations.

#### 2A.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.

2A.3.A: analyze situations and formulate systems of equations in two or more unknowns or inequalities in two unknowns to solve problems;

2A.3.B: use algebraic methods, graphs, tables, or matrices, to solve systems of equations or inequalities; and

2A.3.C: interpret and determine the reasonableness of solutions to systems of equations or inequalities for given contexts.

#### 2A.4: The student connects algebraic and geometric representations of functions.

2A.4.A: identify and sketch graphs of parent functions, including linear (f(x) = x), quadratic (f(x) = x²), exponential (f(x) = a to the x power), and logarithmic (f(x) = log of a(x)) functions, absolute value of x (f(x) = |x|), square root of x (f(x) = square root of x), and reciprocal of x (f(x) = 1/x);

2A.4.B: extend parent functions with parameters such as a in f(x) = a/x and describe the effects of the parameter changes on the graph of parent functions; and

#### 2A.5: The student knows the relationship between the geometric and algebraic descriptions of conic sections.

2A.5.A: describe a conic section as the intersection of a plane and a cone;

2A.5.B: sketch graphs of conic sections to relate simple parameter changes in the equation to corresponding changes in the graph;

2A.5.C: identify symmetries from graphs of conic sections;

2A.5.D: identify the conic section from a given equation; and

#### 2A.6: The student understands that quadratic functions can be represented in different ways and translates among their various representations.

2A.6.A: determine the reasonable domain and range values of quadratic functions, as well as interpret and determine the reasonableness of solutions to quadratic equations and inequalities;

2A.6.B: relate representations of quadratic functions, such as algebraic, tabular, graphical, and verbal descriptions; and

2A.6.C: determine a quadratic function from its roots or a graph.

#### 2A.7: The student interprets and describes the effects of changes in the parameters of quadratic functions in applied and mathematical situations.

2A.7.B: use 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.

#### 2A.8: 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.

2A.8.A: analyze situations involving quadratic functions and formulate quadratic equations or inequalities to solve problems;

2A.8.B: analyze and interpret the solutions of quadratic equations using discriminants and solve quadratic equations using the quadratic formula;

2A.8.D: solve quadratic equations and inequalities using graphs, tables, and algebraic methods.

#### 2A.9: 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.

2A.9.A: use the parent function to investigate, describe, and predict the effects of parameter changes on the graphs of square root functions and describe limitations on the domains and ranges;

2A.9.B: relate representations of square root functions, such as algebraic, tabular, graphical, and verbal descriptions;

2A.9.C: determine the reasonable domain and range values of square root functions, as well as interpret and determine the reasonableness of solutions to square root equations and inequalities;

2A.9.D: determine solutions of square root equations using graphs, tables, and algebraic methods;

2A.9.E: determine solutions of square root inequalities using graphs and tables;

2A.9.F: analyze situations modeled by square root functions, formulate equations or inequalities, select a method, and solve problems; and

2A.9.G: connect inverses of square root functions with quadratic functions.

#### 2A.10: 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.

2A.10.A: use quotients of polynomials to describe the graphs of rational functions, predict the effects of parameter changes, describe limitations on the domains and ranges, and examine asymptotic behavior;

2A.10.B: analyze various representations of rational functions with respect to problem situations;

2A.10.C: determine the reasonable domain and range values of rational functions, as well as interpret and determine the reasonableness of solutions to rational equations and inequalities;

2A.10.D: determine the solutions of rational equations using graphs, tables, and algebraic methods;

2A.10.F: analyze a situation modeled by a rational function, formulate an equation or inequality composed of a linear or quadratic function, and solve the problem; and

2A.10.G: use functions to model and make predictions in problem situations involving direct and inverse variation.

#### 2A.11: The student formulates equations and inequalities based on exponential and logarithmic functions, uses a variety of methods to solve them, and analyzes the solutions in terms of the situation.

2A.11.A: develop the definition of logarithms by exploring and describing the relationship between exponential functions and their inverses;

2A.11.B: use the parent functions to investigate, describe, and predict the effects of parameter changes on the graphs of exponential and logarithmic functions, describe limitations on the domains and ranges, and examine asymptotic behavior;

2A.11.C: determine the reasonable domain and range values of exponential and logarithmic functions, as well as interpret and determine the reasonableness of solutions to exponential and logarithmic equations and inequalities;

2A.11.D: determine solutions of exponential and logarithmic equations using graphs, tables, and algebraic methods;

2A.11.E: determine solutions of exponential and logarithmic inequalities using graphs and tables; and

2A.11.F: analyze a situation modeled by an exponential function, formulate an equation or inequality, and solve the problem.

#### G.1: The student understands the structure of, and relationships within, an axiomatic system.

G.1.A: develop an awareness of the structure of a mathematical system, connecting definitions, postulates, logical reasoning, and theorems;

#### G.2: The student analyzes geometric relationships in order to make and verify conjectures.

G.2.A: use constructions to explore attributes of geometric figures and to make conjectures about geometric relationships; and

G.2.B: make conjectures about angles, lines, polygons, circles, and three-dimensional figures and determine the validity of the conjectures, choosing from a variety of approaches such as coordinate, transformational, or axiomatic.

#### G.3: The student applies logical reasoning to justify and prove mathematical statements.

G.3.A: determine the validity of a conditional statement, its converse, inverse, and contrapositive;

G.3.B: construct and justify statements about geometric figures and their properties;

G.3.E: use deductive reasoning to prove a statement.

#### G.5: The student uses a variety of representations to describe geometric relationships and solve problems.

G.5.A: use numeric and geometric patterns to develop algebraic expressions representing geometric properties;

G.5.B: use numeric and geometric patterns to make generalizations about geometric properties, including properties of polygons, ratios in similar figures and solids, and angle relationships in polygons and circles;

G.5.D: identify and apply patterns from right triangles to solve meaningful problems, including special right triangles (45-45-90 and 30-60-90) and triangles whose sides are Pythagorean triples.

#### G.6: The student analyzes the relationship between three-dimensional geometric figures and related two-dimensional representations and uses these representations to solve problems.

G.6.A: describe and draw the intersection of a given plane with various three-dimensional geometric figures;

G.6.B: use nets to represent and construct three-dimensional geometric figures; and

G.6.C: use orthographic and isometric views of three-dimensional geometric figures to represent and construct three-dimensional geometric figures and solve problems.

#### G.7: The student understands that coordinate systems provide convenient and efficient ways of representing geometric figures and uses them accordingly.

G.7.A: use one- and two-dimensional coordinate systems to represent points, lines, rays, line segments, and figures;

G.7.B: use slopes and equations of lines to investigate geometric relationships, including parallel lines, perpendicular lines, and special segments of triangles and other polygons; and

G.7.C: derive and use formulas involving length, slope, and midpoint.

#### G.8: The student uses tools to determine measurements of geometric figures and extends measurement concepts to find perimeter, area, and volume in problem situations.

G.8.A: find areas of regular polygons, circles, and composite figures;

G.8.B: find areas of sectors and arc lengths of circles using proportional reasoning;

G.8.C: derive, extend, and use the Pythagorean Theorem; and

G.8.D: find surface areas and volumes of prisms, pyramids, spheres, cones, cylinders, and composites of these figures in problem situations.

#### G.9: The student analyzes properties and describes relationships in geometric figures.

G.9.A: formulate and test conjectures about the properties of parallel and perpendicular lines based on explorations and concrete models;

G.9.C: formulate and test conjectures about the properties and attributes of circles and the lines that intersect them based on explorations and concrete models; and

#### G.10: The student applies the concept of congruence to justify properties of figures and solve problems.

G.10.A: use congruence transformations to make conjectures and justify properties of geometric figures including figures represented on a coordinate plane; and

G.10.B: justify and apply triangle congruence relationships.

#### G.11: The student applies the concepts of similarity to justify properties of figures and solve problems.

G.11.A: use and extend similarity properties and transformations to explore and justify conjectures about geometric figures;

G.11.B: use ratios to solve problems involving similar figures;

G.11.C: develop, apply, and justify triangle similarity relationships, such as right triangle ratios, trigonometric ratios, and Pythagorean triples using a variety of methods; and

G.11.D: describe the effect on perimeter, area, and volume when one or more dimensions of a figure are changed and apply this idea in solving problems.

#### P.1: The student defines functions, describes characteristics of functions, and translates among verbal, numerical, graphical, and symbolic representations of functions, including polynomial, rational, power (including radical), exponential, logarithmic, trigo

P.1.A: describe parent functions symbolically and graphically, including f(x) = x to the n power, f(x) = 1n x, f(x) = loga x, f(x) = 1/x, f(x) = e to the x power, f(x) = |x|, f(x) = a to the x power, f(x) = sin x, f(x) = arcsin x, etc.;

P.1.B: determine the domain and range of functions using graphs, tables, and symbols;

P.1.D: recognize and use connections among significant values of a function (zeros, maximum values, minimum values, etc.), points on the graph of a function, and the symbolic representation of a function; and

#### P.2: The student interprets the meaning of the symbolic representations of functions and operations on functions to solve meaningful problems.

P.2.A: apply basic transformations, including a • f(x), f(x) + d, f(x - c), f(b • x), and compositions with absolute value functions, including |f(x)|, and f(|x|), to the parent functions;

P.2.C: investigate identities graphically and verify them symbolically, including logarithmic properties, trigonometric identities, and exponential properties.

#### P.3: The student uses functions and their properties, tools and technology, to model and solve meaningful problems.

P.3.A: investigate properties of trigonometric and polynomial functions;

P.3.B: use functions such as logarithmic, exponential, trigonometric, polynomial, etc. to model real-life data;

P.3.C: use regression to determine the appropriateness of a linear function to model real-life data (including using technology to determine the correlation coefficient);

P.3.D: use properties of functions to analyze and solve problems and make predictions; and

P.3.E: solve problems from physical situations using trigonometry, including the use of Law of Sines, Law of Cosines, and area formulas and incorporate radian measure where needed.

#### P.4: The student uses sequences and series as well as tools and technology to represent, analyze, and solve real-life problems.

P.4.A: represent patterns using arithmetic and geometric sequences and series;

P.4.B: use arithmetic, geometric, and other sequences and series to solve real-life problems;

#### P.5: The student uses conic sections, their properties, and parametric representations, as well as tools and technology, to model physical situations.

P.5.A: use conic sections to model motion, such as the graph of velocity vs. position of a pendulum and motions of planets;

#### P.6: The student uses vectors to model physical situations.

P.6.A: use the concept of vectors to model situations defined by magnitude and direction; and

P.6.B: analyze and solve vector problems generated by real-life situations.

#### M.2: The student uses graphical and numerical techniques to study patterns and analyze data.

M.2.A: interpret information from various graphs, including line graphs, bar graphs, circle graphs, histograms, scatterplots, line plots, stem and leaf plots, and box and whisker plots to draw conclusions from the data;

M.2.B: analyze numerical data using measures of central tendency, variability, and correlation in order to make inferences;

M.2.D: use regression methods available through technology to describe various models for data such as linear, quadratic, exponential, etc., select the most appropriate model, and use the model to interpret information.

#### M.4: The student uses probability models to describe everyday situations involving chance.

M.4.A: compare theoretical and empirical probability; and

#### M.5: The student uses functional relationships to solve problems related to personal income.

M.5.A: use rates, linear functions, and direct variation to solve problems involving personal finance and budgeting, including compensations and deductions;

#### M.7: The student uses algebraic formulas, numerical techniques, and graphs to solve problems related to financial planning.

M.7.A: analyze types of savings options involving simple and compound interest and compare relative advantages of these options;

#### M.8: The student uses algebraic and geometric models to describe situations and solve problems.

M.8.B: use trigonometric ratios and functions available through technology to calculate distances and model periodic motion; and

M.8.C: use direct and inverse variation to describe physical laws such as Hook's, Newton's, and Boyle's laws.

#### M.9: The student uses algebraic and geometric models to represent patterns and structures.

M.9.A: use geometric transformations, symmetry, and perspective drawings to describe mathematical patterns and structure in art and architecture; and

M.9.B: use geometric transformations, proportions, and periodic motion to describe mathematical patterns and structure in music.

Correlation last revised: 10/30/2009

This correlation lists the recommended Gizmos for this state's curriculum standards. Click any Gizmo title below for more information.