Indiana State Standards for Mathematics: Grade 9

IN.A1. Algebra I

A1.1. Operations with Real Numbers

A1.1.1. Determine whether a relation represented by a table, graph, words or equation is a function or not a function and translate among tables, graphs, words and equations.

A1.1.2. Identify the domain and range of relations represented by tables, graphs, words, and equations.

A1.2. Linear Equations and Inequalities

A1.2.1. Translate among various representations of linear functions including tables, graphs, words and equations.

A1.2.2. Graph linear equations and show that they have constant rates of change.

A1.2.3. Determine the slope, x-intercept, and y-intercept of a line given its graph, its equation, or two points on the line and determine the equation of a line given sufficient information.

A1.2.4. Write, interpret, and translate among equivalent forms of equations for linear functions (slope-intercept, point-slope, and standard), recognizing that equivalent forms reveal more or less information about a given situation.

A1.2.5. Solve problems that can be modeled using linear equations and inequalities, interpret the solutions, and determine whether the solutions are reasonable.

A1.2.6. Graph a linear inequality in two variables.

A1.3. Relations and Functions

A1.3.1. Understand the relationship between a solution of a pair of linear equations in two variables and the graphs of the corresponding lines and solve pairs of linear equations in two variables by graphing, substitution or elimination.

A1.3.2. Graph the solution set for a pair of linear inequalities in two variables with and without technology and use the graph to find the solution set.

A1.3.3. Solve problems that can be modeled using pairs of linear equations in two variables, interpret the solutions, and determine whether the solutions are reasonable.

A1.4. Linear Equations and Inequalities

A1.4.1. Use the laws of exponents for variables with exponents and multiply, divide, and find powers of variables with exponents.

A1.4.2. Add, subtract and multiply polynomials and divide polynomials by monomials.

A1.4.3. Factor common terms from polynomials and factor quadratic expressions.

A1.5. Pairs of Linear Equations and Inequalities

A1.5.1. Graph quadratic functions.

A1.5.2. Solve quadratic equations in the real number system with real number solutions by factoring, by completing the square, and by using the quadratic formula.

A1.5.3. Solve problems that can be modeled using quadratic equations, interpret the solutions, and determine whether the solutions are reasonable.

A1.5.4. Analyze and describe the relationships among the solutions of a quadratic equation, the zeros of a quadratic function, the x-intercepts of the graph of a quadratic function, and the factors of a quadratic expression.

A1.5.5. Sketch and interpret linear and non-linear graphs representing given situations and identify independent and dependent variables.

A1.6. Polynomials

A1.6.1. Add, subtract, multiply, divide, reduce, and evaluate rational expressions with polynomial denominators. Simplify rational expressions with linear and quadratic denominators, including denominators with negative exponents.

A1.6.2. Solve equations involving rational and common irrational expressions.

A1.6.3. Simplify radical expressions involving square roots.

A1.6.4. Solve equations that contain radical expressions on only one side of the equation and identify extraneous roots when they occur.

A1.7. Algebraic Fractions

A1.7.1. Organize and display data using appropriate methods to detect patterns and departures from patterns. Summarize the data using measures of center (mean, median) and spread (range, percentiles, variance, standard deviation). Compare data sets using graphs and summary statistics.

A1.7.2. Distinguish between random and non-random sampling methods, identify possible sources of bias in sampling, describe how such bias can be controlled and reduced, evaluate the characteristics of a good survey and well-designed experiment, design simple experiments or investigations to collect data to answer questions of interest, and make inferences from sample results.

A1.7.3. Evaluate reports based on data published in the media by considering the source of the data, the design of the study, the way the data are analyzed and displayed and whether the report confuses correlation with causation.

IN.A2. Algebra II

A2.1. Relations and Functions

A2.1.1. Find the zeros, domain, and range of a function.

A2.1.2. Use and interpret function notation, including evaluation of functions represented by tables, graphs, words, equations or a set of ordered pairs.

A2.1.3. Recognize and describe the relationships among the solutions of an equation, the zeros of a function, the x-intercepts of a graph, and the factors of a polynomial expression.

A2.2. Linear and Absolute Value Equations and Inequalities

A2.2.1. Solve systems of linear equations and inequalities in three variables by substitution and elimination.

A2.2.2. Solve problems that can be modeled using systems of linear equations up to three variables, interpret the solutions, and determine whether the solutions are reasonable.

A2.2.3. Graph piecewise-defined functions.

A2.2.4. Solve equations and inequalities involving the absolute value of a linear function.

A2.3. Quadratic Equations and Functions

A2.3.1. Define, add, subtract, multiply and divide complex numbers. Represent complex numbers, and the addition, subtraction and absolute value of complex numbers, in the complex plane.

A2.3.2. Solve quadratic equations in the complex number system.

A2.3.3. Analyze, describe, and sketch graphs of quadratic functions including the lines of symmetry.

A2.3.4. Determine how the graph of a parabola changes if a, b, and c changes in the equation y = a(x - b)^2 + c. Find an equation for a parabola given sufficient information.

A2.3.5. Solve problems that can be modeled using quadratic equations and functions, interpret the solutions, and determine whether the solutions are reasonable.

A2.4. Conic Sections

A2.4.1. Analyze, describe, and sketch graphs of polynomial functions by examining intercepts, zeros, domain and range, and end behavior.

A2.4.2. Use the binomial theorem to expand binomial expressions raised to positive integer powers.

A2.4.3. Perform arithmetic operations, including long division and division with remainders, on polynomials by others of equal or lower degree.

A2.4.4. Factor polynomials completely and solve polynomial equations by factoring.

A2.4.5. Use graphing technology to find approximate solutions for polynomial equations.

A2.4.6. Solve problems that can be represented or modeled using polynomial equations, interpret the solutions, and determine whether the solutions are reasonable.

A2.4.7. Find a polynomial function of lowest degree with real coefficients given its roots and use the relationship between solutions of an equation, zeros of a function, x-intercepts of a graph and factors of a polynomial expression to solve problems.

A2.5. Polynomials

A2.5.1. Analyze, describe, and sketch graphs of rational functions by examining intercepts, zeros, domain and range, and asymptotic and end behavior.

A2.5.2. Add, subtract, multiply, divide, reduce and evaluate rational expressions with polynomial denominators. Simplify rational expressions, including expressions with negative exponents in the denominator.

A2.5.3. Understand the properties of rational exponents and use the properties to simplify, multiply, divide, and find powers of expressions containing negative and fractional exponents. Relate expressions containing rational exponents to the corresponding radical expressions.

A2.5.4. Analyze, describe, and sketch graphs of square root and cube root functions by examining intercepts, zeros, domain and range, and end behavior.

A2.5.5. Solve equations that contain radical expressions and identify extraneous roots when they occur.

A2.5.6. Solve problems that can be modeled using equations involving rational and radical functions, including problems of direct and inverse variation. Interpret the solutions, and determine whether the solutions are reasonable.

A2.6. Algebraic Fractions

A2.6.1. Analyze, describe, and sketch graphs of exponential functions by examining intercepts, zeros, domain and range, and asymptotic and end behavior.

A2.6.2. Know that the inverse of an exponential function is a logarithm, use laws of exponents to derive laws of logarithms, and use the inverse relationship between exponential functions and logarithms and the laws of logarithms to solve problems.

A2.6.3. Solve exponential and logarithmic equations.

A2.6.4. Solve problems that can be modeled using exponential and logarithmic equations, interpret the solutions, and determine whether the solutions are reasonable using technology as appropriate.

A2.7. Logarithmic and Exponential Functions

A2.7.1. Write the recursive formula for arithmetic and geometric sequences and find specific terms of arithmetic and geometric sequences.

A2.7.2. Write the formula for the general term for arithmetic and geometric sequences and make connections to linear and exponential functions.

A2.7.4. Solve problems involving applications that can be modeled using sequences and finite arithmetic and geometric series, interpret the solutions, and determine whether the solutions are reasonable using spreadsheets as appropriate.

A2.8. Sequences and Series

A2.8.1. Use the relative frequency of a specified outcome of an event to estimate the probability of the outcome and apply the law of large numbers in simple examples.

A2.8.2. Determine the probability of simple events involving independent and dependent events and conditional probability. Analyze probabilities to interpret odds and risk of events.

A2.8.3. Know and apply the characteristics of the normal distribution.

A2.8.3.a. Identify settings in which the normal distribution may be useful.

A2.8.3.b. Determine whether a set of data appears to be uniform, skewed or normally distributed.

A2.8.3.c. Use the empirical rule to find probabilities that an event will occur in a specific interval that can be described in terms of one, two or three standard deviations about the mean.

A2.8.4. Use permutations, combinations, and other counting methods to determine the number of ways that events can occur and to calculate probabilities, including the probability of compound events.

IN.G. Geometry

G.1. Points, Lines, Angles, and Planes

G.1.1. Find the length of line segments in one- or two-dimensional coordinate systems, the slopes of line segments in two-dimensional coordinate systems, and find the point that is a given fractional distance from one end of the segment to another.

G.1.2. Construct congruent segments and angles, angle bisectors, perpendicular bisectors, and parallel and perpendicular lines using appropriate geometric construction tools, explaining and justifying the process used.

G.1.3. Recognize, use, and justify the relationships between special angles created by parallel lines and transversals.

G.1.4. Identify and apply properties of and theorems about parallel and perpendicular lines, and write equations of parallel and perpendicular lines, and develop simple geometric proofs involving parallel and perpendicular lines.

G.1.5. Identify, justify and apply properties of planes.

G.1.6. Represent geometric objects and figures algebraically using coordinates, use algebra to solve geometric problems, and develop simple coordinate proofs involving geometric objects in the coordinate plane.

G.1.7. Describe the intersection of two or more geometric figures in the plane.

G.2. Polygons

G.2.1. General: Find and use the sum of the measures of interior and exterior angles of convex polygons, justifying the method used.

G.2.2. General: Identify types of symmetry (line, point, rotational, self-congruences) of polygons.

G.2.3. General: Solve problems involving congruent and similar polygons.

G.2.4. General: Predict and describe the results of translations, reflections, and rotations on polygons and describe a motion or series of motions that will show that two shapes are congruent.

G.2.5. General: Deduce formulas relating lengths and sides, perimeters, and areas of regular polygons and understand how limiting cases of such formulas lead to expressions for the circumference and the area of a circle.

G.2.6. General: Recognize and use coordinate geometry to verify properties of polygons such as regularity, congruence and similarity.

G.2.7. General: Develop simple geometric proofs involving congruent and similar polygons and provide reasons for each statement.

G.2.8. Quadrilaterals: Describe, classify, and recognize relationships among the quadrilaterals such as squares, rectangles, rhombuses, parallelograms, trapezoids and kites.

G.2.9. Quadrilaterals: Prove and apply theorems about parallelograms and trapezoids (including isosceles trapezoids) involving their angles, sides, and diagonals and prove that given quadrilaterals are parallelograms, rhombuses, rectangles, squares, or trapezoids (as appropriate).

G.2.10. Triangles: Define, identify, construct, and solve problems involving perpendicular bisectors, angle bisectors, medians and altitudes in triangles.

G.2.11. Triangles: Construct triangles congruent to given triangles, explaining and justifying the process used.

G.2.12. Triangles: Use theorems to show whether two triangles are congruent (SSS, SAS, ASA) or similar (AA, SAS, SSS).

G.2.13. Triangles: Apply the triangle inequality theorem.

G.2.14. Triangles: Develop simple geometric proofs involving triangles and provide reasons for each statement.

G.2.15. Triangles: Prove and apply the isosceles triangle theorem and its converse.

G.2.16. Right Triangles: Prove the Pythagorean Theorem and its converse and use them to solve problems, including problems involving the length of a segment in the coordinate plane.

G.2.17. Right Triangles: Prove and apply the relationships that exist when the altitude is drawn to the hypotenuse of a right triangle.

G.2.18. Right Triangles: Use special right triangles (30 degrees - 60 degrees and 45 degrees - 45 degrees) to solve problems.

G.2.19. Right Triangles: Define and use the trigonometric functions (sine, cosine, tangent) in terms of angles of right triangles.

G.2.20. Right Triangles: Deduce and apply the area formula A = 1/2 ab sinC, where a and b are the lengths of two sides of a triangle and C is the measure of the included angle formed by the two sides.

G.2.21. Right Triangles: Solve problems that can be modeled using right triangles, including problems that can be modeled using trigonometric functions. Interpret the solutions, and determine whether the solutions are reasonable, using technology as when appropriate.

G.3. Quadrilaterals

G.3.1. Construct the circle that passes through three given points not on a line and construct tangents to circles and circumscribe and inscribe circles, justifying the processes used.

G.3.2. Define, deduce and use formulas for, and prove theorems for radius, diameter, chord, secant, and tangent.

G.3.3. Define, deduce and use formulas for, and prove theorems for measures of arcs and related angles (central, inscribed, and intersections of secants and tangents).

G.3.4. Define, deduce and use formulas for, and prove theorems for measures of circumference, arc length, and areas of circles and sectors.

G.3.5. Find the equation of a circle in the coordinate plane in terms of its center and radius and determine how the graph of a circle changes if a, b, and r are changed in the equation (x - a)^2 + (y - b)^2 = r^2.

G.3.6. Develop simple geometric proofs involving circles and provide reasons for each statement.

G.4. Triangles

G.4.1. Identify, justify and apply properties of prisms, regular pyramids, cylinders, right circular cones and spheres.

G.4.2. Solve problems involving congruent and similar solids.

G.4.3. Find and use measures of sides, volumes, and surface areas of prisms, regular pyramids, cylinders, right circular cones and spheres. Relate these measures to each other using formulas.

G.4.4. Visualize solids and surfaces in three-dimensional space when given two-dimensional representations and create two-dimensional representations for the surfaces of three-dimensional objects.

G.5. Right Triangles

G.5.1. Describe the structure of and relationships within an axiomatic system (undefined terms, definitions, axioms/postulates, methods of reasoning, and theorems).

G.5.2. Recognize that there are geometries, other than Euclidean geometry, in which the parallel postulate is not true and illustrate its counterparts in other geometries.

G.5.3. Understand the difference between supporting evidence, counterexamples, and actual proofs.

G.5.4. Develop simple geometric proofs (direct proofs, indirect proofs, proofs by contradiction and proofs involving coordinate geometry) using two-column, paragraphs, and flow charts formats and providing reasons for each statement in the proofs.

IN.IM1. Integrated Mathematics I

IM1.1. Number Sense and Computation

IM1.1.1. Relations and Functions: Determine whether a relation represented by a table, graph, words, or equation is a function or not a function.

IM1.1.2. Relations and Functions: Identify the domain and range of relations represented by tables, graphs, words, and equations.

IM1.1.3. Relations and Functions: Solve problems that can be modeled using linear equations and inequalities, interpret the solutions, and determine whether the solutions are reasonable.

IM1.1.4. Linear Functions, Equations and Inequalities: Translate among various representations of linear functions including tables, graphs, words, and equations.

IM1.1.5. Linear Functions, Equations and Inequalities: Graph linear equations and show that they have constant rates of change.

IM1.1.6. Linear Functions, Equations and Inequalities: Determine the slope, x-intercept, and y-intercept of a line given its graph, its equation, or two points on the line.

IM1.1.7. Linear Functions, Equations and Inequalities: Write, interpret, and translate among equivalent forms of equations for linear functions (slope-intercept, point-slope, and standard), recognizing that equivalent forms reveal more or less information about a given situation.

IM1.1.8. Linear Functions, Equations and Inequalities: Solve problems that can be modeled using linear equations and inequalities, interpret the solutions, and determine whether the solutions are reasonable.

IM1.1.9. Linear Functions, Equations and Inequalities: Graph a linear inequality in two variables.

IM.1.10. Pairs of Linear Equations and Inequalities: Understand the relationship between a solution of a pair of linear equations in two variables and the graphs of the corresponding lines and solve pairs of linear equations in two variables by graphing, substitution or elimination.

IM1.1.11. Pairs of Linear Equations and Inequalities: Graph the solution set for a pair of linear inequalities in two variables with and without technology and use the graph to find the solution, including systems with no solution or infinitely many solutions.

IM1.1.12. Pairs of Linear Equations and Inequalities: Solve problems that can be modeled using systems of linear equations containing two variables, interpret the solutions, and determine whether the solutions are reasonable.

IM1.1.13. Polynomials: Use the laws of exponents for variables with exponents. Multiply, divide, and find powers of variables with exponents.

IM1.1.14. Polynomials: Add, subtract and multiply polynomials and divide polynomials by monomials.

IM1.1.15. Polynomials: Factor common terms from polynomials and factor quadratic expressions.

IM1.1.16. Quadratic Equations and Functions: Graph quadratic functions.

IM1.1.17. Quadratic Equations and Functions: Solve quadratic equations in the real number system with real number solutions by factoring, by completing the square, and by using the quadratic formula.

IM1.1.18. Quadratic Equations and Functions: Solve problems that can be modeled using quadratic equations, interpret the solutions, and determine whether the solutions are reasonable.

IM1.1.19. Quadratic Equations and Functions: Analyze and describe the relationships among the solutions of a quadratic equation, the zeros of a quadratic function, the x-intercepts of a graph, and the factors of a quadratic expression.

IM1.1.20. Quadratic Equations and Functions: Sketch and interpret linear and non-linear graphs representing given situations and identify independent and dependent variables.

IM1.1.21. Rational and Radical Expressions and Equations: Add, subtract, multiply, divide, reduce, and evaluate rational expressions with polynomial denominators. Simplify rational expressions with linear and quadratic denominators, including denominators with negative exponents.

IM1.1.22. Rational and Radical Expressions and Equations: Solve equations involving rational expressions.

IM1.1.23. Rational and Radical Expressions and Equations: Simplify radical expressions involving square roots

IM1.1.24. Rational and Radical Expressions and Equations: Solve equations that contain radical expressions on only one side of the equation and identify extraneous roots when they occur.

IM1.2. Algebra and Functions

IM1.2.1. Find the length of line segments in one- or two-dimensional coordinate systems, the slopes of line segments in two-dimensional coordinate systems, and find the point that is a given fractional distance from one end of the segment to another.

IM1.2.2. Find and use measures of interior and exterior angles of polygons, justifying the method used.

IM1.2.3. Solve problems involving congruent and similar polygons.

IM1.2.4. Predict and describe the results of translations, reflections, and rotations on polygons. Describe a motion or series of motions that will show that two shapes are congruent.

IM1.2.5. Deduce formulas relating lengths and sides, perimeters and areas of regular polygons and understand how limiting cases of such formulas lead to expressions for the circumference and area of a circle.

IM1.2.6. Develop simple geometric proofs (direct proofs, indirect proofs, proofs by contradiction and proofs involving coordinate geometry) using two-column, paragraphs, and flow charts formats and providing reasons for each statement in the proofs.

IM1.3. Geometry and Measurement

IM1.3.1. Organize and display data using appropriate methods to detect patterns and departures from patterns. Summarize the data using measures of center (mean, median) and spread (range, percentiles, variance, standard deviation). Compare data sets using graphs and summary statistics.

IM1.3.2. Distinguish between random and non-random sampling methods, identify possible sources of bias in sampling, describe how such bias can be controlled and reduced, evaluate the characteristics of a good survey and well-designed experiment, design simple experiments or investigations to collect data to answer questions of interest, and make inferences from sample results.

IM1.3.3. Evaluated reports based on data published in the media by considering the source of the data, the design of the study, and the way the data are analyzed and displayed.

IM1.4. Data Analysis and Statistics

IM1.4.1. Analyze and apply algorithms for searching (sequential, binary), for sorting (bubble sort, quick sort, bin sort) and for solving optimization problems.

IM1.4.2. Analyze and interpret relationships defined iteratively and recursively.

IM1.4.3. Define arithmetic and geometric sequences recursively.

IM1.4.4. Determine the number of ways events can occur using permutations, combinations, and the Fundamental Counting Principle.

IM1.4.5. Determine whether two propositions are logically equivalent.

IM1.4.6. Distinguish between inductive and deductive reasoning. Identify inductive reasoning as central to the scientific method and deductive reasoning as characteristic of mathematics.

IN.IM2. Integrated Mathematics II

IM2.1. Algebra and Functions

IM2.1.1. Use and interpret function notation, including evaluation of functions represented by tables, graphs, words, equations or a set of ordered pairs.

IM2.1.2. Recognize and describe the relationships among the solutions of an equation, the zeros of a function, the x-intercepts of a graph, and the factors of a polynomial expression.

IM2.1.3. Analyze, describe, and sketch graphs of quadratic functions including the lines of symmetry.

IM2.2. Geometry and Measurement

IM2.2.1. Find the lengths and midpoints of line segments in one- or two-dimensional coordinate systems and find the point that is a given fractional distance from one end of the segment to another.

IM2.2.2. Construct congruent segments and angles, angle bisectors, perpendicular bisectors, and parallel and perpendicular lines using appropriate geometric construction tools, explaining and justifying the process used.

IM2.2.3. Recognize, use, and justify the relationships between special pairs of angles formed by parallel lines and transversals.

IM2.2.4. Identify and apply properties of and theorems about parallel and perpendicular lines, and write equations of parallel and perpendicular lines, and develop simple geometric proofs involving parallel and perpendicular lines.

IM2.2.5. Identify, justify and apply properties of planes.

IM2.2.6. Deduce and apply the area formula A = 1/2 ab sinC, where a and b are the lengths of two sides of a triangle and C is the measure of the included angle formed by the two sides.

IM2.2.7. Prove and apply the triangle inequality theorem.

IM2.2.8. Prove and apply the isosceles triangle theorem and its converse.

IM2.2.9. Describe, classify, and recognize relationships among the quadrilaterals such as squares, rectangles, rhombuses, parallelograms, trapezoids, and kites.

IM2.2.10. Develop simple geometric proofs involving congruent and similar polygons and provide reasons for each statement.

IM2.2.11. Define, identify, construct and solve problems involving altitudes, medians, angle bisectors, and perpendicular bisectors in triangles.

IM2.2.12. Prove the Pythagorean Theorem and its converse to solve problems, including problems involving the length of a segment in the coordinate plane.

IM2.2.13. Use special right triangles (30 degrees - 60 degrees and 45 degrees) to solve problems.

IM2.2.14. Define and use the trigonometric functions (sine, cosine, tangent) in terms of angles of right triangles.

IM2.2.15. Solve problems that can be modeled using right triangles, including problems that can be modeled using trigonometric functions. Interpret the solutions, and determine whether the solutions are reasonable using technology as appropriate.

IM2.2.16. Construct the circle that passes through three given points not on a line and construct tangents to circles and circumscribe and inscribe circles, justifying the process used.

IM2.2.17. Define, deduce and use formulas for, and prove theorems for radius, diameter, chord, secant, and tangent.

IM2.2.18. Develop simple geometric proofs involving circles and provide reasons for each statement.

IM2.2.19. Define, deduce and use formulas for, and prove theorems for measures of arcs and related angles (central, inscribed, and intersections of secants and tangents).

IM2.2.20. Define, deduce and use formulas for, and prove theorems for measures of circumference, arc length, and areas of circles and sectors.

IM2.2.21. Identify, justify and apply properties of prisms, regular pyramids, cylinders, right circular cones and spheres.

IM2.3. Data Analysis and Statistics

IM2.3.1. For bivariate measurement data, create a scatter plot, describe its shape, and determine regression coefficients, regression equations, and correlation coefficients using technological tools.

IM2.3.2. Display and analyze bivariate data where at least one variable is categorical.

IM2.3.3. Recognize how linear transformations of univariate data affect shape, center, and spread.

IM2.3.4. Calculate and interpret the correlation coefficient. Use the correlation coefficient and residuals to evaluate a ''best-fit'' line.

IM2.3.5. Construct sample spaces and probability distributions in simple cases and use them to solve problems.

IM2.3.6. Determine the probability of simple events involving independent and dependent events and conditional probability. Analyze probabilities to interpret odds and risk of events.

IM2.3.7. Use permutations, combinations, and other counting methods to determine the number of ways that events can occur.

IM2.4. Probability

IM2.4.1. Use the properties of matrix addition, subtraction, and scalar multiplication to solve problems.

IM2.4.2. Create matrices to organize and store data categorized by two variables and interpret the meaning of a particular entry in a matrix.

IM2.4.3. Use the properties of matrix multiplication, including identity and inverse matrices, to solve problems.

IM2.4.4. Represent a system of equations in two or three variables as a matrix equation Ax = b and use technology to find x = A^-1b.

IM2.4.5. Model and solve problems using matrices.

IM2.4.6. Use and interpret relational conjunctions (and, or, not), terms of causation (if...then) and equivalence (if and only if). Distinguish between the common uses of such terms in everyday language and their use in mathematics.

IM2.4.7. Use truth tables to determine the truth values of propositional statements.

IM2.4.8. Recognize syllogisms, tautologies, flawed reasoning, and circular reasoning.

IM2.4.9. Construct and interpret directed and undirected graphs, decision trees, networks, and flow charts.

IM2.4.10. Use critical-path analysis to solve scheduling problems.

IN.IM3. Integrated Mathematics III

IM3.1. Algebra and Functions

IM3.1.1. Functions: Find the zeros, domain and range of a function.

IM3.1.2. Functions: Use and interpret function notation, including evaluation of functions represented by tables, graphs, words, equations or a set of ordered pairs.

IM3.1.3. Linear and Absolute Value Equations, Inequalities and Functions: Solve systems of linear equations and inequalities in three variables by substitution and elimination.

IM3.1.4. Linear and Absolute Value Equations, Inequalities and Functions: Solve problems that can be modeled using systems of linear equations in up to three variables, interpret the solutions, and determine whether the solutions are reasonable.

IM3.1.5. Linear and Absolute Value Equations, Inequalities and Functions: Graph piecewise-defined functions.

IM3.1.6. Linear and Absolute Value Equations, Inequalities and Functions: Solve equations and inequalities involving the absolute value of a linear function.

IM3.1.7. Quadratic Equations and Functions: Define, add, subtract, multiply and divide complex numbers. Represent complex numbers, and the addition, subtraction and absolute value of complex numbers, in the complex plane.

IM3.1.8. Quadratic Equations and Functions: Solve quadratic equations in the complex number system.

IM3.1.9. Quadratic Equations and Functions: Analyze, describe, and sketch graphs of quadratic functions including the lines of symmetry.

IM3.1.10. Quadratic Equations and Functions: Determine how the graph of a parabola changes if a, b, and c changes in the equation y = a(x - b)^2 + c. Find an equation for a parabola given sufficient information.

IM3.1.11. Polynomial Expressions, Equations and Functions: Analyze, describe, and sketch graphs of polynomial functions by examining intercepts, zeros, domain and range, and end behavior.

IM3.1.12. Polynomial Expressions, Equations and Functions: Use the binomial theorem to expand binomial expressions raised to positive integer powers.

IM3.1.13. Polynomial Expressions, Equations and Functions: Perform arithmetic operations, including long division and division with remainders, on polynomials by others of equal or lower degree.

IM3.1.14. Polynomial Expressions, Equations and Functions: Factor polynomials completely and solve polynomial equations by factoring.

IM3.1.15. Polynomial Expressions, Equations and Functions: Use graphing technology to find approximate solutions for polynomial equations.

IM3.1.16. Polynomial Expressions, Equations and Functions: Solve problems that can be represented or modeled using polynomial equations, interpret the solutions, and determine whether the solutions are reasonable.

IM3.1.17. Rational and Radical Expressions, Equations and Functions: Write a polynomial function of the lowest degree with real coefficients given its zeros.

IM3.1.18. Rational and Radical Expressions, Equations and Functions: Analyze, describe, and sketch graphs of rational functions by examining intercepts, zeros, domain and range, and asymptotic and end behavior.

IM3.1.19. Rational and Radical Expressions, Equations and Functions: Add, subtract, multiply, divide, reduce and evaluate rational expressions with polynomial denominators. Simplify rational expressions, including expressions with negative exponents in the denominator.

IM3.1.20. Rational and Radical Expressions, Equations and Functions: Understand the properties of rational exponents and use the properties to simplify, multiply, divide, and find powers of expressions containing negative and fractional exponents. Relate expressions containing rational exponents to the corresponding radical expressions.

IM3.1.21. Rational and Radical Expressions, Equations and Functions: Analyze, describe, and sketch graphs of square root and cube root functions by examining intercepts, zeros, domain and range, and end behavior.

IM3.1.22. Rational and Radical Expressions, Equations and Functions: Solve equations that contain radical expressions and identify extraneous roots when they occur.

IM3.1.23. Rational and Radical Expressions, Equations and Functions: Solve problems that can be modeled using equations involving rational and radical functions, including problems of direct and inverse variation. Interpret the solutions, and determine whether the solutions are reasonable.

IM3.1.24. Exponential and Logarithmic Functions: Analyze, describe, and sketch graphs of exponential functions by examining intercepts, zeros, domain and range, and asymptotic and end behavior.

IM3.1.25. Exponential and Logarithmic Functions: Know that the inverse of an exponential function is a logarithm, use laws of exponents to derive laws of logarithms, and use the inverse relationship between exponential functions and logarithms and the laws of logarithms to solve problems.

IM3.1.26. Exponential and Logarithmic Functions: Solve exponential and logarithmic equations.

IM3.1.27. Exponential and Logarithmic Functions: Solve problems that can be modeled using exponential and logarithmic equations, interpret the solutions, and determine whether the solutions are reasonable using technology as appropriate.

IM3.1.28. Sequences and Series: Write the recursive formula for arithmetic and geometric sequences and find specified terms of arithmetic and geometric sequences.

IM3.1.29. Sequences and Series: Write the formula for the general term for arithmetic and geometric sequences and make connections to linear and exponential functions.

IM3.1.30. Sequences and Series: Find partial sums of arithmetic and geometric series.

IM3.1.31. Sequences and Series: Solve problems involving applications that can be modeled using sequences and finite arithmetic and geometric series. Interpret the solutions and determine whether the solutions are reasonable using spreadsheets as appropriate.

IM3.2. Geometry and Measurement

IM3.2.1. Identify and apply properties of and theorems about parallel and perpendicular lines and write equations of parallel and perpendicular lines.

IM3.2.2. Identify, justify and apply properties of planes.

IM3.2.3. Represent geometric objects and figures algebraically using coordinates, use algebra to solve geometric problems, and develop simple coordinate proofs involving geometric objects in the coordinate plane.

IM3.2.4. Describe the intersection of two or more geometric figures in the plane.

IM3.2.5. Construct triangles congruent to given triangles, explaining and justifying the process used.

IM3.2.6. Prove and apply the triangle inequality theorem.

IM3.2.7. Develop simple geometric proofs involving triangles and provide reasons for each statement.

IM3.2.8. Find the equation of a circle in the coordinate plane in terms of its center and radius and determine how the graph of a circle changes if a, b, and r are changed in the equation (x - a)^2 + (y - b)^2 = r^2.

IM3.2.9. Visualize solids and surfaces in three-dimensional space when given two-dimensional representations and create two-dimensional representations for the surfaces of three-dimensional objects.

IM3.2.10. Find and use measures of sides, volumes, and surface areas of prisms, regular pyramids, cylinders, right circular cones and spheres. Relate these measures to each other using formulas.

IM3.3. Data Analysis and Statistics

IM3.3.1. Use simulations to explore the variability of sample statistics from a known population and to construct sampling distributions.

IM3.3.2. Evaluate published reports that are based on data by examining the design of the study, the appropriateness of the data analysis, and the validity of conclusions. Interpret confidence levels and ''margin of error.''

IM3.3.3. Compare the difference among surveys, experiments and observational studies and recognize which types of inferences can legitimately be drawn from each.

IM3.3.4. Compute basic statistics (mean, median, weighted mean, range, variance, standard deviation) and understand the distinction between a statistic and a parameter.

IM3 3.5. Understand the meaning of measurement data and categorical data, of univariate and bivariate data, and of the term variable.

IM3.3.6. Use simulations to construct empirical probability distributions.

IM3.3.7. Apply the properties of the normal distribution.

IM3.3.8. Compute and interpret the expected value of random variables in simple cases.

IM3.3.9. Compute the probability of compound events.

IM3.3.10. Model and solve problems, including probability problems, using counting techniques.

IM3.4. Probability

IM3.4.1. Know and use the concepts of sets, elements, and subsets.

IM3.4.2. Perform operations on sets (union, intersection, complement, cross product).

IM3.4.3. Identify and give examples of undefined terms, axioms, and theorems.

IM3.4.4. Describe logical statements using the terms assumption, hypothesis, conclusion, converse, inverse, and contrapositive. Find the converse, inverse, and contrapositive of statements.

IM3.4.5. Explain and illustrate the role of definitions, conjectures, theorems, proofs and counterexamples in mathematical reasoning. Construct logical arguments, assess the validity of logical arguments and give counterexamples to disprove statements.

IM3.4.6. Model and solve problems involving patterns using recursion and iteration, growth and decay, and compound interest.

IM3.4.7. Use mathematical induction to prove simple propositions.

IM3.4.8. Use graph-coloring techniques to solve problems.

IM3.4.9. Use bin-packing techniques to solve problems of optimizing resource usage.

IN.PC. Pre-Calculus/Trigonometry

PC.1. Relations and Functions

PC.1.1. Use paper and pencil methods and technology to graph polynomial, absolute value, rational, algebraic, exponential, logarithmic, trigonometric, inverse trigonometric and piecewise-defined functions, use these graphs to solve problems, and translate among verbal, tabular, graphical, and symbolic representations of functions using technology as appropriate.

PC.1.2. Identify domain, range, intercepts, zeros, asymptotes, and points of discontinuity of functions represented symbolically or graphically, using technology as appropriate.

PC.1.3. Solve word problems that can be modeled using functions and equations.

PC.1.4. Recognize and describe continuity, end behavior, asymptotes, symmetry, and limits and connect these concepts to graphs of functions.

PC.1.5. Find, interpret, and graph the sum, difference, product, and quotient (when it exists) of two functions, indicating the relevant domain and range of the resulting function.

PC.1.6. Find the composition of two functions, and determine the domain and the range of the composite function. Conversely, given a function, find two other functions the composition of which is the given one.

PC.1.7. Define and find inverse functions, their domains and ranges, and verify whether two given functions are inverses of each other, symbolically and graphically.

PC.1.8. Apply transformations to functions and interpret the results of these transformations verbally, graphically, and numerically.

PC.2. Logarithmic and Exponential Functions

PC.2.1. Derive equations for conic sections and use the equations that have been found.

PC.2.2. Graph conic sections with axes of symmetry parallel to the coordinate axes by hand, by completing the square, and find the foci, center, asymptotes, eccentricity, axes, and vertices (as appropriate).

PC.3. Trigonometry in Triangles

PC.3.1. Compare and contrast y = e^x with other exponential functions, symbolically and graphically.

PC.3.2. Define the logarithmic function g(x) = log base a of x as the inverse of the exponential function f(x) = a^x. Apply the inverse relationship between exponential and logarithmic functions and the laws of logarithms to solve problems.

PC.3.3. Analyze, describe, and sketch graphs of logarithmic and exponential functions by examining intercepts, zeros, domain and range, and asymptotic and end behavior.

PC.3.4. Solve problems that can be modeled using logarithmic and exponential functions. Interpret the solutions, and determine whether the solutions are reasonable.

PC.4. Trigonometric Functions

PC.4.1. Define and use the trigonometric ratios cotangent, secant, and cosecant in terms of angles of right triangles.

PC.4.2. Model and solve problems involving triangles using trigonometric ratios.

PC.4.3. Develop and use the laws of sines and cosines to solve problems.

PC.4.4. Define sine and cosine using the unit circle.

PC.4.5. Develop and use radian measures of angles, measure angles in degrees and radians, and convert between degree and radian measures.

PC.4.6. Deduce geometrically and use the value of the sine, cosine, and tangent functions at 0, pi/6, pi/4, pi/3, and pi/2 radians and their multiples.

PC.4.7. Make connections between right triangle ratios, trigonometric functions, and the coordinate function on the unit circle.

PC.4.8. Analyze and graph trigonometric functions, including the translation of these trigonometric functions. Describe their characteristics (spread, amplitude, zeros, symmetry, phase, shift, vertical shift, frequency).

PC.4.9. Define, analyze and graph inverse trigonometric functions and find the values of inverse trigonometric functions.

PC.4.10. Solve problems that can be modeled using trigonometric functions, interpret the solutions, and determine whether the solutions are reasonable.

PC.4.11. Derive the fundamental Pythagorean trigonometric identities, sum and difference identities, half-angle and double-angle identities and the secant, cosecant, and cotangent functions and use these identities to verify other identities and simplify trigonometric expressions.

PC.4.12. Solve trigonometric equations and interpret solutions graphically.

PC.5. Trigonometric Identities and Equations

PC.5.1. Define and use polar coordinates and relate polar coordinates to Cartesian coordinates.

PC.5.2. Represent equations given in Cartesian coordinates in terms of polar coordinates.

PC.5.3. Graph equations in the polar coordinate plane.

PC.5.4. Define complex numbers, convert complex numbers to polar form, and multiply complex numbers in polar form.

PC.5.5. Prove and use De Moivre's Theorem.

PC.6. Polar Coordinates and Complex Numbers

PC.6.1. Define arithmetic and geometric sequences and series.

PC.6.2. Derive and use formulas for finding the general term for arithmetic and geometric sequences.

PC.6.3. Develop, prove and use sum formulas for arithmetic series and for finite and infinite geometric series.

PC.6.4. Generate a sequence using recursion.

PC.6.5. Describe the concept of the limit of a sequence and a limit of a function. Decide whether simple sequences converge or diverge, and recognize an infinite series as the limit of a sequence of partial sums.

PC.6.6. Model and solve word problems involving applications of sequences and series, interpret the solutions and determine whether the solutions are reasonable.

PC.6.7. Derive the binomial theorem by combinatorics.

PC.7. Sequences and Series

PC.7.1. Define vectors as objects having magnitude and direction and represent vectors geometrically.

PC.7.2. Use parametric equations to represent situations involving motion in the plane.

PC.7.3. Convert between a pair of parametric equations and an equation in x and y

PC.7.4. Analyze planar curves, including those given in parametric form.

PC.7.5. Model and solve problems using parametric equations.

PC.8. Data Analysis

PC.8.1. Use linear models using the median fit and least squares regression methods. Decide which among several linear models gives a better fit. Interpret the slope in terms of the original context

PC.8.2. Calculate and interpret the correlation coefficient. Use the correlation coefficient and residuals to evaluate a ''best-fit'' line.

IN.PS. Probability and Statistics

PS.1. Descriptive Statistics

PS.1.1. Build new mathematical knowledge through problem solving.

PS.1.2. Solve problems that arise in mathematics and in other contexts.

PS.1.3. Apply and adapt a variety of appropriate strategies to solve problems.

PS.1.4. Monitor and reflect on the process of mathematical problem solving.

PS.2. Probability

PS.2.1. Recognize reasoning and proof as fundamental aspects of mathematics.

PS.2.2. Make and investigate mathematical conjectures.

PS.2.3. Develop and evaluate mathematical arguments and proofs.

PS.2.4. Select and use various types of reasoning and methods of proof.

PS.3. Statistical Inference

PS.3.1. Organize and consolidate their mathematical thinking through communication.

PS.3.2. Communicate their mathematical thinking coherently and clearly to peers, teachers, and others.

PS.3.3. Analyze and evaluate the mathematical thinking and strategies of others.

PS.3.4. Use the language of mathematics to express mathematical ideas precisely.

PS.4. Connections

PS.4.1. Recognize and use connections among mathematical ideas.

PS.4.2. Understand how mathematical ideas interconnect and build on one another to produce a coherent whole.

PS.4.3. Recognize and apply mathematics in contexts outside of mathematics.

PS.5. Representation

PS.5.1. Create and use representations to organize, record, and communicate mathematical ideas.

PS.5.2. Select, apply, and translate among mathematical representations to solve problems.

PS.5.3. Use representations to model and interpret physical, social, and mathematical phenomena.

PS.6. Estimation and Mental Computation

PS.6.1. Know and apply appropriate methods for estimating the results of computations.

PS.6.2. Round numbers to a specified place value.

PS.6.3. Use estimation to decide whether answers are reasonable.

PS.6.4. Decide when estimation is an appropriate strategy for solving a problem.

PS.6.5. Determine appropriate accuracy and precision of measurements in problem situations.

PS.6.6. Use properties of numbers and operations to perform mental computation.

PS.6.7. Recognize when the numbers involved in a computation allow for a mental computation strategy.

PS.7. Technology

PS.7.1. Technology should be used as a tool in mathematics education to support and extend the mathematics curriculum.

PS.7.2. Technology can contribute to concept development, simulation, representation, communication, and problem solving.

PS.7.3. The challenge is to ensure that technology supports-but is not a substitute for- the development of skills with basic operations, quantitative reasoning, and problem-solving skills.

PS.7.3.a. Graphing calculators should be used to enhance middle school and high school students' understanding and skills.

PS.7.3.b. The focus must be on learning mathematics, using technology as a tool rather than as an end in itself.