High School Math South Carolina Standards

Describe the effect of the transformations 𝑘𝑓(𝑥), 𝑓(𝑥)+𝑘, 𝑓(𝑥 − 𝑘), and combinations of such transformations on the graph of parent function 𝑦 = 𝑓(𝑥𝑥) for any real number 𝑘; find the value of 𝑘 given the graphs; and write the equation of a transformed parent function given its graph.

Data, Probability, and Statistical Reasoning

Summarize categorical data in two-way frequency tables, interpret relative frequencies in real-world situations, and informally determine possible associations and trends in the data.

Summarize quantitative data in a table and on a scatter plot and describe how the variables are associated. Limit to linear data.

Find a linear function for a scatter plot that suggests a linear association.

For linear associations, use technology to determine the correlation coefficient, evaluate the strength of the association, and find the line of best fit.

Use statistical reasoning to summarize, represent, and interpret data on two categorical and quantitative variables.

Analyze and interpret models for two categorical and quantitative variables

Use two-way frequency tables to make inferences and interpret the data in terms of real-world or mathematical situations.

Interpret the slope and the intercept of a linear model in the context of the data.

Use a linear model to interpolate and extrapolate unknown values close to the data set.

Measurement, Geometry, and Spatial Reasoning

Use geometric concepts and measurement opportunities to model mathematical and real-world situations.

Identify any limitations specific to a real-world situation.

Numerical Reasoning

Rewrite numerical and algebraic expressions of irrational and rational numbers involving radicals, including addition, subtraction, multiplication, and division. Limit to square and cube roots.

Represent all points on the number line as irrational and rational numbers in the real number system.

Represent exponents and radical expressions in different ways.

Translate between rational exponents and radical expressions of irrational and rational numbers. Use properties of addition, subtraction, multiplication, and division to simplify radical and rational expressions. Limit to square and cube roots.

Patterns, Algebra, and Functional Reasoning

Transform an equation in one variable to create new equations that have the same solution as the original and justify the steps taken.

Solve literal equations and formulas for a specified variable including equations and formulas that arise in a variety of disciplines.

Solve mathematical and real-world situations using linear, quadratic, exponential (same bases), and linear absolute value equations in one variable.

Add, subtract, and multiply polynomials with initial terms up to a degree of 2.

Transform and/or solve equations and expressions in one variable that model real-world and mathematical situations, interpret the solutions, and determine whether they are reasonable.

Create, solve, and transform equations and inequalities in two or more variables to represent relationships between quantities and graph the equations on coordinate axes using appropriate labels, units, and scales.

Transform linear, quadratic, exponential, and linear absolute value functions to equivalent forms to identify slope and y-intercept for linear, vertex, and roots (if any) for quadratic and linear absolute value, and y-intercept for exponential.

Analyze the growth/decay rate between linear and exponential functions specifically between consecutive integers.

Solve quadratic equations by completing the square, factoring, and the quadratic formula, explaining the connection between the zeros of the function derived from the equation, its linear factors (if it factors), the x-intercepts of its graph (if they exist), and the solutions (if any) to the corresponding quadratic equation.

Solve and graph linear, quadratic, exponential, and linear absolute value equations given in tabular, symbolic, and/or verbal forms using intercepts, domain and range, intervals of increasing and decreasing, vertex (maximum and minimum), end-behavior, and symmetry, and interpret these in terms of mathematical and real-world situations.

Create, solve, and graph linear inequalities in two variables.

Write arithmetic and geometric sequences both recursively and with an explicit formula, use them to model situations, and translate between the two forms.

Create symbolic representations of linear and exponential functions, including arithmetic and geometric sequences, given graphs, verbal descriptions, and tables.

Use graphs to obtain exact and/or approximate solutions of equations, inequalities, and systems of linear equations in two variables (given or obtained by using technology).

Solve systems of linear equations algebraically and graphically

Represent and interpret functions symbolically and graphically.

Recognize that 𝑓(𝑥) denotes the output of function 𝑓 that corresponds to the input 𝑥, and this corresponds to the set of all the ordered pairs (x, y) that satisfy the equation y = f(x) both tabularly and graphically.

Use the definition of a function to analyze the domain and range of a function in relation to its graph, mapping, table, verbal, and/or symbolic description and, where applicable, using interval and set notation.

Translate among graphical, tabular, verbal, and symbolic representations in function notation, to identify intercepts, intervals where the function is increasing, decreasing, constant, maximums and minimums, and symmetries and explain their meanings in real-world and mathematical situations.

Interpret how lead coefficients impact the shape of a function’s graph.

Given a real-world or mathematical situation, determine the parent graph that best models the situation.

Given different representations of two different functions, compare key features including intercepts, domain and range, intervals of increasing and decreasing, constant, average rate of change, and maximum and minimum values.

Reason with parent functions in varying representations to find families of functions that all have similar distinguishing attributes common to the family and use common characteristics to aid in rewriting and identifying linear, linear absolute value, quadratic, and exponential functions.

Solve an equation of the form 𝑓(𝑥) = 𝑔(𝑥) graphically by identifying the 𝑥- coordinate(s) of the point(s) of intersection of the graphs of 𝑦 = 𝑓(𝑥) and 𝑦𝑦 = 𝑔(𝑥).

Mathematical Process Standards

Analyze & Justify

Use critical thinking skills to reason both abstractly and quantitatively.Identify and apply regularity in repeated reasoning to make generalizations.

Connections

Demonstrate a deep and flexible conceptual understanding of mathematical ideas, operations, and relationships while making real-world connections.

Problem Solving

Make sense of problems and persevere in solving them strategically.Explain ideas using precise and contextually appropriate mathematical language, tools, and models.

Representation & Communication

Explain ideas using precise and contextually appropriate mathematical language, tools, and models.

Structure & Patterns

Identify and apply regularity in repeated reasoning to make generalizations.

Data, Probability, and Statistical Reasoning

Understand independence and conditional probability and use them to interpret data.

Describe events as subsets of a sample space using characteristics or categories of the outcomes, or as unions, intersections, or complements of other events.

Explain whether two events, A and B, are independent if and only if the probability of A and B occurring together is the product of their probabilities and use this characterization to determine if they are independent.

Determine whether the conditional probability of A given B as P(A and B)/P(B) and interpret independence of A and B as saying that the conditional probability of A given B is the same as the probability of A, and the conditional probability of B given A is the same as the probability of B in mathematical and real-world situations.

Recognize and explain the concepts of conditional probability and independence.

Use the rules of probability to compute probabilities of compound events in a uniform probability model.

Find the conditional probability of A given B as the fraction of B’s outcomes that also belong to A and interpret the answer in terms of the model.

Apply the Addition Rule, P(A or B) = P(A) + P(B) − P(A and B) and interpret the answer in terms of the model.

Apply the general Multiplication Rule in a uniform probability model, P(A and B) = P(A)⋅P(B|A) = P(B)⋅P(A|B) and interpret the answer in terms of the model.

Use permutations and combinations to determine the number of possible outcomes in a sample space.

Measurement, Geometry, and Spatial Reasoning

Explore and analyze sine and cosine functions using the unit circle, right triangle definitions, and models of periodic phenomena.

Build the unit circle for sine and cosine functions using right triangle definitions.

Use models of periodic phenomena to evaluate and analyze the graph of sine and cosine functions.

Numerical Reasoning

Recognize that the complex number system extends the real number system to allow for solution to all polynomial equations.

Understand that there is an imaginary unit i such that i^2 = -1 and explain the structure of a complex number as a + bi, where a and b are real.

Add, subtract, and multiply complex numbers.

Represent and manipulate data using matrices.

Perform operations with matrices including addition, subtraction, and scalar multiplication.

Patterns, Algebra, and Functional Reasoning

Explore and analyze quadratic and polynomial functions and inequalities and use them to model real-world situations.

Graph, identify roots, and analyze quadratic functions in mathematical and real-world situations.

Solve quadratic inequalities that model mathematical and real-world situations.

Graph and analyze polynomial functions in mathematical and real-world situations.

Solve polynomial inequalities that model mathematical and real-world situations.

Recognize perfect squares and perfect cubes and use them to describe the structure of polynomials.

Explore and analyze rational and radical functions and use them to model real-world phenomena.

Graph rational and radical functions and describe their key features. Limit to square roots and cube roots only.

Perform arithmetic operations on rational expressions, including problems in context, and express rational expressions in irreducible form.

Create and solve rational and radical equations in one variable, including those that model real-life situations, and verify solutions to identify extraneous solutions if they appear.

Explore and analyze exponential functions and use them to model real-world phenomena.

Create, solve, and graph exponential functions, including those that model real-life situations.

Find the sum of the terms of arithmetic and geometric sequences.

Reason with parent functions to find families of functions that all have similar distinguishing attributes common to the family and use common characteristics to aid in rewriting and identifying functions.

Identify the effect on the graph of replacing 𝑓(x) by 𝑘𝑓 (𝑥), 𝑓(𝑥)+𝑘, 𝑓(𝑥-𝑘), 𝑓(𝑘𝑥) for any real number k including multiple transformations; write an equation of a transformed parent function given its graph. Extend to equations involving rational, polynomial, radical, exponential, and piecewise.

Explore and analyze piecewise functions and linear absolute value inequalities and use them to model real-world phenomena.

Graph piecewise functions and describe their key features.

Solve linear absolute value inequalities.

Represent and interpret functions symbolically and graphically.

Find the inverse of functions and verify graphically.

Calculate and interpret the average rate of change of the function over a specified interval, given a function in graphical, symbolic, or numerical form.

Use linear programming to solve systems of equations and inequalities by addressing the constraints that arise in real-world situations.

Mathematical Process Standards

Analyze & Justify

Use critical thinking skills to reason both abstractly and quantitatively.

Demonstrate a deep and flexible conceptual understanding of mathematical ideas, operations, and relationships while making real-world connections.

Connections

Problem Solving

Make sense of problems and persevere in solving them strategically.

Representation & Communication

Explain ideas using precise and contextually appropriate mathematical language, tools, and models.

Structure & Patterns

Identify and apply regularity in repeated reasoning to make generalizations.

Data, Probability, and Statistical Reasoning

Summarize and interpret data represented in tables or graphs to make predictions.

Summarize and interpret trends to make predictions in real-world situations.

Calculate and explain pay scale based on occupational outlook projections.

Calculate and explain operating costs, including cost of materials, supplies, equipment, license fees, and insurance fees.

Construct and analyze charts that reflect current demographics in various industries.

Solve problems involving probability and probability models, and use expected value to make informed decisions in real-world situations.

Determine the probability of simple and compound events in real-world situations.

Use probabilities to make and justify decisions about risk in real-world situations.

Calculate and analyze the expected value of a probability model (binominal, normal, and Poisson distributions) for a real-world situation to make decisions about fairness, payoff, and risk.

Measurement, Geometry, and Spatial Reasoning

Apply trigonometric principles to solve real-world geometric situations involving inaccessible distances.

Apply sine, cosine, and tangent ratios and the Law of Sines and the Law of Cosines to discover distances.

Critique the appropriateness of measurements in terms of precision, accuracy, and approximate error.

Determine dimensions by scaling plans or blueprints.

Apply knowledge of fractions for reading a ruler to 1/16 inch to interpreting blueprints and measuring materials.

Compare the metric and the British imperial systems of measurements used in industry.

Apply two- and three-dimensional representations, geometric transformations, and scale models in planning, designing, and constructing solutions to real-world situations.

Calculate lengths utilizing the Pythagorean Theorem.

Apply the concepts of area, volume, scale factors, and scale drawings to applied problems for a specific project.

Determine the level of precision and the appropriate tools for taking the measurements in constructing a two-dimensional visual representation of a three-dimensional object or structure.

Apply Heron’s Formula for finding the area of a triangular region.

Apply two- and three-dimensional representations in coordinate systems to find solutions to real-world situations.

Plot coordinates on a three-dimensional Cartesian coordinate system and use relationships between coordinates to solve design problems.

Use technology and other tools to explore the results of simple transformations using three-dimensional coordinates, including translations in the x, y, and/or z directions; rotations of 90º, 180º, or 270º about the x, y, and z axes; reflections over the xy, yz, and xz planes; and dilations from the origin.

Use vectors and matrices to represent, organize, and describe data to solve problems in mathematical and real-world situations.

Apply vectors to mathematical and real-world situations by recognizing vectors as mathematical objects having both magnitude and direction.

Use and apply matrices to represent geometric transformations in real-world situations.

Numerical Reasoning

Solve problems using fractions, percents, and ratios for real-world situations involving linear, quadratic, exponential, and absolute functions.

Apply numerical reasoning to real-world situations involving percent increase and decrease.

Patterns, Algebra, and Functional Reasoning

Create and analyze mathematical models to make decisions on real-world situations.

Use exponential functions to model change in a variety of financial situations.

Compare the various means of paying for an automobile, including leasing, purchasing by cash, and purchasing by loan.

Use sequences to represent simple and compound interest and depreciation.

Analyze and solve application-based problems relating to direct, inverse, and joint variation.

Apply variations to mathematical and real-world situations to describe troubleshooting in business and industrial applications.

Utilize mathematical skills for troubleshooting in business and industrial applications.

Analyze and apply linear programming to mathematical and real-world situations.

Calculate the values of the variables that maximize or minimize the objective function, given four or more constraints.

Mathematical Process Standards

Analyze & Justify

Use critical thinking skills to reason both abstractly and quantitatively

Connections

Demonstrate a deep and flexible conceptual understanding of mathematical ideas, operations, and relationships while making real-world connections.

Problem Solving

Make sense of problems and persevere in solving them strategically.

Representation & Communication

Explain ideas using precise and contextually appropriate mathematical language, tools, and models.

Structure & Patterns

Identify and apply regularity in repeated reasoning to make generalizations.

Measurement, Geometry, and Spatial Reasoning

Explain the concept of the integral of a function geometrically, numerically, analytically, and contextually.

Explain how the definite integral is used to solve area problems.

Approximate definite integrals by a finite sum.

Interpret the definite integral as a limit of Riemann sums.

Explain the relationship between the integral and derivative as expressed in both parts of the Fundamental Theorem of Calculus. Interpret the relationship in terms of rates of change.

Apply theorems and rules of integration to solve mathematical and real-world situations.

Apply the Fundamental Theorem of Calculus to solve mathematical and real-world situations.

Explain graphically and verbally the properties of the definite integral. Apply these properties to evaluate basic definite integrals.

Evaluate integrals using substitution.

Numerical Reasoning

Apply the concepts of a limit graphically, numerically, analytically, and contextually.

Estimate and verify limits using tables, graphs of functions, and technology.

Calculate limits—including one-sided limits—algebraically, using direct substitution, simplification, rationalization, and the limit laws for constant multiples, sums, differences, products, and quotients.

Calculate infinite limits and limits at infinity and use those limits to identify the asymptotes.

Patterns, Algebra, and Functional Reasoning

Apply the definition and graphical interpretation of continuity of a function.

Apply the definition of continuity of a function at a point to solve problems.

Classify discontinuities as removable, jump, or infinite. Justify that classification using the definition of continuity.

Understand the Intermediate Value Theorem and apply the theorem to prove the existence of solutions of equations arising in mathematical and real-world situations

Understand the concept of the derivative of a function geometrically, numerically, analytically, and verbally.

Interpret the value of the derivative of a function as the slope of the corresponding tangent line.

Explain graphically and analytically the relationship between differentiability and continuity.

Interpret the value of the derivative as an instantaneous rate of change in a variety of real-world contexts such as velocity and population growth.

Approximate the derivative graphically by finding the slope of the tangent line drawn to a curve at a given point and numerically by using the difference quotient.

Explain graphically and analytically the relationship between the average rate of change and the instantaneous rate of change

Use the definition of the derivative to determine the derivatives of various functions.

Apply the rules of differentiation to functions.

Identify and apply the derivatives of constant, power, trigonometric, inverse trigonometric, exponential, and logarithmic functions.

Use the constant multiple, sum, difference, product, quotient, and chain rules to find the derivatives of functions.

Apply the methods of implicit and logarithmic differentiation.

Apply theorems and rules of differentiation to solve mathematical and realworld situations.

Explain the mathematical and real-world meanings of the Extreme Value Theorem and the Mean Value Theorem.

Write an equation of a line tangent to the graph of a function at a point.

Explain the relationship between the increasing/decreasing behavior of 𝑓 and the signs of 𝑓′. Use the relationship to generate a graph of 𝑓 given the graph of 𝑓′, and vice versa, and to identify relative and absolute extrema of 𝑓.

Explain the relationships among the concavity of the graph of 𝑓, the increasing/decreasing behavior of 𝑓′ and the signs of 𝑓′′. Use those relationships to generate graphs of 𝑓, 𝑓′, and 𝑓′′ given any one of them and identify the points of inflection of 𝑓.

Solve a variety of real-world situations involving related rates, optimization, linear approximation, and rates of change.

Mathematical Process Standards

Analyze & Justify

Use critical thinking skills to reason both abstractly and quantitatively.

Demonstrate a deep and flexible conceptual understanding of mathematical ideas, operations, and relationships while making real-world connections.

Connections

Problem Solving

Make sense of problems and persevere in solving them strategically.

Representation & Communication

Explain ideas using precise and contextually appropriate mathematical language, tools, and models.

Identify and apply regularity in repeated reasoning to make generalizations.

Structure & Patterns

Data, Probability, and Statistical Reasoning

Analyze, model, and solve problems involving fair outcomes.

Investigate and describe the results of various election methods.

Explain fairness and equity in relation to the paradoxes of voting.

Solve apportionment problems using a variety of methods.

Compare voting methods to determine the method most appropriate for the situation.

Determine power indexes for weighted voting systems.

Measurement, Geometry, and Spatial Reasoning

Use graph theory to model relationships and solve problems.

Distinguish between inductive and deductive reasoning.

Determine statements and rephrase them symbolically.

Use negation, disjunction, and conjunction to determine if statements are logically equivalent.

Write statements in words and symbolically using converse, inverse, and contrapositive.

Verify arguments and syllogisms.

Represent real-world situations using a vertex-edge graph.

Test graphs and digraphs for paths and circuits.

Numerical Reasoning

Investigate principles of set theory.

Define basic terms and concepts in set theory.

Compare sets with appropriate language and notation.

Determine and explain the cardinality of sets.

Analyze numbers with different bases in real-world situations.

Perform arithmetic operations using modular arithmetic properties.

Solve problems involving modular arithmetic in real-world situations.

Explain and apply binary and hexadecimal number systems

Determine the number of ways an event can occur.

Calculate combinations and permutations.

Patterns, Algebra, and Functional Reasoning

Classify and compare objects using estimation and sets for real-world situations.

Use estimation to get an approximate answer in real-world situations.

Perform operations on sets.

Develop an understanding of and carry out proofs by mathematical induction using the Principle of Mathematical Induction.

Create mathematical induction proofs using the Principle of Mathematical Induction.

Use matrices to model and solve mathematical and real-world situations.

Manipulate matrices using addition, subtraction, multiplication, inverse, and power properties.

Write and evaluate matrices drawn from real-world situations.

Mathematical Process Standards

Analyze & Justify

Use critical thinking skills to reason both abstractly and quantitatively.

Connections

Demonstrate a deep and flexible conceptual understanding of mathematical ideas, operations, and relationships while making real-world connections.

Problem Solving

Make sense of problems and persevere in solving them strategically.

Representation & Communication

Explain ideas using precise and contextually appropriate mathematical language, tools, and models.

Structure & Patterns

Identify and apply regularity in repeated reasoning to make generalizations.

Patterns, Algebra, and Functional Reasoning

Numerical Reasoning

Measurement, Geometry, and Spatial Reasoning

Data, Probability, and Statistical Reasoning

STRUCTURE & PATTERNS

ANALYZE & JUSTIFY

CONNECTIONS

REPRESENTATION & COMMUNICATION

PROBLEM SOLVING

Mathematical Process Standards

Analyze data sets to identify their statistical elements.

Create and analyze scatter plots to represent numerical data sets in mathematical and real-world situations.

Draw inferences about data sets from two populations using the shape of the distribution, measures of center, and measures of variability. Limit measures to mean, median, mode, range, mean absolute deviation, and interquartile range.

Describe how adding and deleting data throughout the data set can affect the mean, median, mode, and distribution of the data set.

For two data sets (numerical or graphical), compare and interpret the centers, spreads, and overlap of data to draw inferences about data in mathematical and real-world situations. Limit displays to double line graphs, backto-back stem-and-leaf plots, and double box plots.

Calculate and interpret probability.

Determine the sample space for a compound event.

Calculate and interpret the probability of compound independent and dependent events.

Determine the measurements of geometric figures.

Find the distance between any two points in the coordinate plane using the Pythagorean Theorem.

Given the Pythagorean Theorem, determine unknown side lengths in right triangles in mathematical and real-world situations.

Determine if a given set of sides forms a right triangle.

Determine angle and/or side relationships.

Discover and apply the Exterior Angle Theorem of triangles to find a missing angle.

Graph on a coordinate plane.

Identify congruent angles and congruent line segments of a preimage and its image.

Determine if a table, graph, verbal description, or equation represents a function and describe its characteristics.

Use multiple representations including mappings, tables, graphs, verbal description, and equations (only when linear) of two functions to compare the functions and draw conclusions.

Write, simplify, and evaluate algebraic expressions; write and solve algebraic equations and inequalities.

Describe single-variable equations as having one solution, no solution, or an infinite number of solutions.

Explain why the slope, m, is the same between any two distinct points on a linear graph.

Given a table or a graph, identify the slope and the y-intercept of a line and write a linear equation to express that line.

Apply mathematical patterns, properties, and algorithms to the set of rational numbers to find sums, differences, products, and quotients and to write equivalent expressions.

Analyze patterns of perfect squares and perfect cubes to evaluate square roots and cube roots. Limit to square roots less than or equal to 400 and cube roots less than or equal to 1,000.

Approximate non-perfect square roots and cube roots to nearest tenth. Limit to square roots less than or equal to 400 and cube roots less than or equal to 1,000.

Investigate and apply relationships among segments and angles in circles.

Use angle and segment relationships in circles to solve mathematical and realworld situations.

Investigate and apply relationships in circles, inscribed angles, radii, secants, and chords; among inscribed angles, central angles, and circumscribed angles; and between radii and tangents to circles.

Summarize, represent, and interpret data on two categorical and quantitative variables.

Represent data for two quantitative variables on a scatter plot and describe how the variables are related.

Use two representative points from the data to find an approximate line of fit and compare it to the line of best fit.

Conduct an investigation for a statistical question, interpret statistical significance in the context of a situation, and answer investigative questions appropriately.

Analyze and interpret models for two quantitative variables.

Distinguish between correlation and causation.

Solve problems involving the probability of compound events in real-world situations.

Describe categories of events as subsets of a sample space using unions, intersections, or complements of other events.

Apply the Addition Rule to find the probability of both mutually exclusive and not mutually exclusive events and interpret the answers in context.

Apply the Multiplication Rule to determine the probability of independent events and interpret the answers in context.

Compute area and volume of figures by determining how the figure might be obtained from simpler figures by dissection and recombination.

Apply area and volume formulas of two and three-dimensional figures to solve real-world situations.

Identify the shape of a two-dimensional cross-section of a three-dimensional figure.

Use cross-sections of three-dimensional figures to model and solve mathematical and real-world situations.

Apply rigid geometric transformations to figures, describing their attributes and symmetries.

Describe the results of transformations on a given figure using geometric terminology from the definitions of the transformations.

Describe and apply a sequence of transformations that maps a preimage onto its image.

Determine that two figures are congruent by demonstrating that a rigid motion or a sequence of rigid motions maps one figure onto the other.

Identify types of symmetry of polygons, including line, point, rotational, and self congruence, and use symmetry to analyze mathematical situations.

Demonstrate that triangles and quadrilaterals are congruent by a combination of translations, rotations, and reflections.

Recognize the criteria for showing triangles are congruent using a sequence of rigid motions that map one triangle to another and justify that the two triangles are congruent by applying the Side-SideSide, Side-Angle-Side, Angle-Side-Angle, Angle-Angle-Side, and Hypotenuse-Leg congruence conditions.

Determine that two figures are similar by demonstrating a similarity transformation or a sequence of similarity transformations that maps one figure onto the other.

Demonstrate experimentally the properties of dilations given by a center and a scale factor.

Justify experimentally that a dilation of a line segment is longer or shorter, given the ratio.

Recognize that the criteria for showing triangles are similar using a similarity transformation that maps one figure to the other and justify the two triangles are similar by applying the Angle-Angle, SideSide-Side, and Side-Angle-Side similarity conditions.

Demonstrate whether a conjecture or theorem is true or false using a variety of algebraic and geometric explanations.

Justify and apply the attributes of angle relationships/lines in mathematical and real-world situations.

Apply the attributes of triangles in mathematical and real-world situations.

Apply the attributes of quadrilaterals, including diagonals, sides, and angles, to prove that a given quadrilateral is a parallelogram in mathematical and real world situations.

Discover and apply relationships in similar right triangles.

Discover and apply the converse of the Pythagorean Theorem.

Discover and apply the constant ratios of the sides in 30-60-90 and 45-45-90 right triangles.

Define the trigonometric ratios using the properties of similar right triangles.

Determine the sine, cosine, and tangent of an acute angle in a right triangle in the context of mathematical and real-world situations.

Apply trigonometric ratios (sine, cosine, tangent) and the Pythagorean Theorem to solve right triangle problems in real-life situations.

Represent all points on the number line as irrational and rational numbers in the real number system.

Rewrite numerical expressions of irrational and rational numbers involving radicals, including addition, subtraction, multiplication, and division, to recognize geometric patterns.

Analyze the structure of an equation or inequality to determine an efficient strategy to find a solution, if one exists, then justify the solution.

Discover and apply the formulas for the length of an arc and the area of a sector in a circle to develop mathematical models and solve mathematical and real world situations.

Analyze and apply the derivations of the formulas for the circumference of a circle, area of a circle, and volume of a cylinder, pyramid, and cone to model real phenomena and solve mathematical and real-world situations.

Interpret the structure of expressions, equations, and inequalities to analyze and make predictions in different contexts.

Apply surface area and volume formulas for prisms, cylinders, pyramids, cones, spheres, and/or compositions of figures to solve problems and justify results.

Analyze slopes of lines to determine whether lines are parallel, perpendicular, or neither.

Determine the equation of a line passing through a given point that is parallel or perpendicular to a given line.

Determine the exact or approximate solutions of equations and inequalities using graphs on the coordinate plane.

Use coordinates to prove simple geometric theorems algebraically.

Determine distance and midpoint of segments in a coordinate plane to find areas of triangles and quadrilaterals, when given coordinates.

Use critical thinking skills to reason both abstractly and quantitatively.

Demonstrate a deep and flexible conceptual understanding of mathematical ideas, operations, and relationships while making real-world connections.

Make sense of problems and persevere in solving them strategically.

Explain ideas using precise and contextually appropriate mathematical language, tools, and models.

Identify and apply regularity in repeated reasoning to make generalizations.

Data, Probability, and Statistical Reasoning

Summarize, represent, and interpret data on two categorical and quantitative variables.

Represent data for two quantitative variables on a scatter plot and describe how the variables are related.

Use two representative points from the data to find an approximate line of fit and compare it to the line of best fit.

Conduct an investigation for a statistical question, interpret statistical significance in the context of a situation, and answer investigative questions appropriately.

Analyze and interpret models for two quantitative variables.

Distinguish between correlation and causation.

Solve problems involving the probability of compound events in real-world situations.

Describe categories of events as subsets of a sample space using unions, intersections, or complements of other events.

Apply the Addition Rule to find the probability of both mutually exclusive and not mutually exclusive events and interpret the answers in context.

Apply the Multiplication Rule to determine the probability of independent events and interpret the answers in context.

Measurement, Geometry, and Spatial Reasoning

Compute area and volume of figures by determining how the figure might be obtained from simpler figures by dissection and recombination.

Apply area and volume formulas of two- and three-dimensional figures to solve real-world situations.

Identify the shape of a two-dimensional cross-section of a three-dimensional figure.

Use cross-sections of three-dimensional figures to model and solve mathematical and real-world situations.

Apply rigid geometric transformations to figures, describing their attributes and symmetries.

Describe the results of transformations on a given figure using geometric terminology from the definitions of the transformations.

Describe and apply a sequence of transformations that maps a pre-image onto its image.

Determine that two figures are congruent by demonstrating that a rigid motion or a sequence of rigid motions maps one figure onto the other.

Identify types of symmetry of polygons, including line, point, rotational, and self-congruence, and use symmetry to analyze mathematical situations.

Demonstrate that triangles and quadrilaterals are congruent by a combination of translations, rotations, and reflections.

Recognize the criteria for showing triangles are congruent using a sequence of rigid motions that map one triangle to another and justify that the two triangles are congruent by applying the Side-Side-Side, Side-Angle-Side, Angle-Side-Angle, Angle-Angle-Side, and Hypotenuse-Leg congruence conditions.

Determine that two figures are similar by demonstrating a similarity transformation or a sequence of similarity transformations that maps one figure onto the other.

Demonstrate experimentally the properties of dilations given by a center and a scale factor.

Justify experimentally that a dilation of a line segment is longer or shorter, given the ratio.

Recognize that the criteria for showing triangles are similar using a similarity transformation that maps one figure to the other and justify the two triangles are similar by applying the Angle-Angle, Side-Side-Side, and Side-Angle-Side similarity conditions.

Demonstrate whether a conjecture or theorem is true or false using a variety of algebraic and geometric explanations.

Justify and apply the attributes of angle relationships/lines in mathematical and real-world situations.

Apply the attributes of triangles in mathematical and real-world situations.

Apply the attributes of quadrilaterals, including diagonals, sides, and angles, to prove that a given quadrilateral is a parallelogram in mathematical and real-world situations.

Discover and apply relationships in similar right triangles.

Discover and apply the converse of the Pythagorean Theorem.

Discover and apply the constant ratios of the sides in 30-60-90 and 45-45-90 right triangles.

Define the trigonometric ratios using the properties of similar right triangles.

Determine the sine, cosine, and tangent of an acute angle in a right triangle in the context of mathematical and real-world situations.

Apply trigonometric ratios (sine, cosine, tangent) and the Pythagorean Theorem to solve right triangle problems in real-life situations.

Investigate and apply relationships among segments and angles in circles.

Use angle and segment relationships in circles to solve mathematical and real-world situations.

Investigate and apply relationships in circles, inscribed angles, radii, secants, and chords; among inscribed angles, central angles, and circumscribed angles; and between radii and tangents to circles.

Numerical Reasoning

Represent all points on the number line as irrational and rational numbers in the real number system.

Rewrite numerical expressions of irrational and rational numbers involving radicals, including addition, subtraction, multiplication, and division, to recognize geometric patterns.

Patterns, Algebra, and Functional Reasoning

Analyze the structure of an equation or inequality to determine an efficient strategy to find a solution, if one exists, then justify the solution.

Discover and apply the formulas for the length of an arc and the area of a sector in a circle to develop mathematical models and solve mathematical and real-world situations.

Analyze and apply the derivations of the formulas for the circumference of a circle, area of a circle, and volume of a cylinder, pyramid, and cone to model real phenomena and solve mathematical and real-world situations.

Interpret the structure of expressions, equations, and inequalities to analyze and make predictions in different contexts.

Apply surface area and volume formulas for prisms, cylinders, pyramids, cones, spheres, and/or compositions of figures to solve problems and justify results.

Analyze slopes of lines to determine whether lines are parallel, perpendicular, or neither.

Determine the equation of a line passing through a given point that is parallel or perpendicular to a given line.

Determine the exact or approximate solutions of equations and inequalities using graphs on the coordinate plane.

Use coordinates to prove simple geometric theorems algebraically.

Determine distance and midpoint of segments in a coordinate plane to find areas of triangles and quadrilaterals, when given coordinates.

Analyze & Justify

Use critical thinking skills to reason both abstractly and quantitatively.

Connections

Demonstrate a deep and flexible conceptual understanding of mathematical ideas, operations, and relationships while making real-world connections.

Problem Solving

Make sense of problems and persevere in solving them strategically.

Representation & Communication

Explain ideas using precise and contextually appropriate mathematical language, tools, and models.

Structure & Patterns

Identify and apply regularity in repeated reasoning to make generalizations.

Mathematical Process Standards

Mathematical Process Standards

Analyze & Justify

Use critical thinking skills to reason both abstractly and quantitatively.

Connections

Demonstrate a deep and flexible conceptual understanding of mathematical ideas, operations, and relationships while making real-world connections.

Problem Solving

Make sense of problems and persevere in solving them strategically

Explain ideas using precise and contextually appropriate mathematical language, tools, and models.

Representation & Communication

Structure & Patterns

Identify and apply regularity in repeated reasoning to make generalizations.

Solve an equation of the form 𝑓(𝑥) = 𝑔(𝑥) graphically by identifying the 𝑥- coordinate(s) of the point(s) of intersection of the graphs of 𝑦 = 𝑓(𝑥) and 𝑦 = 𝑔(𝑥).

Measurement, Geometry, and Spatial Reasoning

Analyze the behaviors of conic sections and polar coordinates to model mathematical and real-world situations.

Identify and graph different conic sections given the equations in standard form.

Identify different conic sections in general form and complete the square to convert the equation of a conic section into standard form.

Define polar coordinates and relate polar coordinates to Cartesian coordinates.

Solve problems and model periodic phenomena with trigonometric expressions and functions.

Determine the area of a triangle to solve problems.

Prove and apply the Law of Sines and the Law of Cosines to find unknown measurements in right and non-right triangles.

Derive the formulas for the length of an arc and the area of a sector in a circle and apply these formulas to solve mathematical and real-world situations.

Determine geometrically the values of the sine, cosine, and tangent for 𝜋/6 , 𝜋/4 , and 𝜋/3 by special triangles, and use the unit circle to express the values of sine, cosine, and tangent for 𝜋 - 𝑥, 𝜋 + 𝑥, and 2𝜋 - 𝑥 in terms of their values for 𝑥, where 𝑥 is any real number.

Define the six trigonometric ratios in terms of x, y, and r using the unit circle centered at the origin of the coordinate plane and interpret radian measures of angles as a rotation both counterclockwise and clockwise around the unit circle.

Explain symmetry, both odd and even, and periodicity of trigonometric functions

Numerical Reasoning

Represent and manipulate data using matrices.

Identify the identity and zero matrices for any dimension and add, subtract, and multiply matrices.

Find the additive and multiplicative inverses of square matrices.

Explain the role of the determinant in determining if a square matrix has a multiplicative inverse.

Find the determinant of a square matrix if and only if the matrix has a multiplicative inverse.

Represent and model with vector quantities.

Represent vector quantities as directed line segments and represent magnitude and direction of vectors in component form.

Find the components of a vector by adding and subtracting vectors on a coordinate plane using a variety of methods.

Solve problems, including real-life situations, that can be represented by vectors.

Add and subtract vectors and multiply vectors by a scalar to find the resultant vector.

Represent complex numbers and their operations on the complex plane.

Represent complex numbers on the complex plane in rectangular and polar form, including real and imaginary numbers, and explain why the rectangular and polar forms of a given complex number represent the same number.

Represent addition, subtraction, multiplication, and conjugation of complex numbers geometrically on the complex plane; use properties of this representation for computation.

Patterns, Algebra, and Functional Reasoning

Build new functions from existing functions to solve mathematical and realworld situations.

Combine and compose functions algebraically, tabularly, and graphically.

Find the inverse of functions and verify algebraically, numerically, and graphically.

Compare the key features of a function and its inverse function and use the relationship to model real-world situations and solve problems.

Graph and describe the effect on the graph 𝑓(𝑥) of 𝑓(𝑥)+𝑘, 𝑓(𝑥-𝑘), 𝑘 (𝑥), and 𝑓(𝑘 ), for specific values of both negative and positive values of k.

Explore and analyze the behaviors of rational and piecewise functions to model contextual mathematical problems.

Graph rational functions and describe their key features.

Solve rational equations and inequalities in one variable and explain when extraneous solutions may arise.

Transform rational expressions in different forms.

Graph piecewise-defined functions, including step functions and absolute value functions, and describe their key features.

Explore and analyze structures and patterns for radical functions and use radical expressions, equations, and functions to model real-life phenomena.

Transform radical expressions as expressions with rational exponents and extend the properties of integer exponents to rational exponents.

Solve radical equations and describe how extraneous solutions may arise.

Analyze and graph radical functions.

Explore and analyze structures and patterns for exponential and logarithmic functions and use exponential and logarithmic expressions, equations, and functions to model real-life phenomena.

Graph logarithmic functions and describe their key features.

Use the definition of a logarithm, logarithmic properties, and the inverse relationship between exponential and logarithmic functions to solve problems, including real-life context.

Model real-life situations and solve problems involving exponential and logarithmic functions.

Explore and analyze structures and patterns of trigonometric functions and use trigonometric functions to model real-life phenomena.

Graph trigonometric functions and their inverses and describe their key features.

Restrict the domain of a trigonometric function to define the six inverse trigonometric functions, graph the inverse function, and evaluate inverse trigonometric expressions.

Use inverse functions to solve trigonometric equations that arise in modeling contexts; evaluate the solutions and interpret them in terms of the context.

Manipulate, prove, and apply trigonometric identities and equations to solve contextual mathematical problems.

Apply the fundamental trigonometric identities to simplify expressions and verify other identities.

Apply the sum, difference, double-angle, and half-angle formulas for sine, cosine, and tangent and use them to solve problems.

Model real-life situations and solve problems involving trigonometric equations.

Represent data with matrices, perform mathematical operations, and solve systems of linear equations for mathematical problems.

Solve a simple system consisting of a linear equation and a quadratic equation in two variables algebraically and graphically. Understand that such systems may have zero, one, or two solutions.

Represent a system of linear equations as a single matrix equation in a vector variable.

Mathematical Process Standards

Analyze & Justify

Use critical thinking skills to reason both abstractly and quantitatively.

Connections

Demonstrate a deep and flexible conceptual understanding of mathematical ideas, operations, and relationships while making real-world connections.

Problem Solving

Make sense of problems and persevere in solving them strategically.

Representation & Communication

Explain ideas using precise and contextually appropriate mathematical language, tools, and models.

Structure & Patterns

Identify and apply regularity in repeated reasoning to make generalizations.

Data, Probability, and Statistical Reasoning

Apply statistical reasoning to complete investigations.

Formulate a statistical question and develop a statistical method to address questions/studies through exploration of the research cycle.

Write and identify a null hypothesis and an alternative hypothesis, as well as what makes up an experimental study.

Identify the population of interest and the variables to be used in each study. Determine the appropriate sampling design, sampling technique, and statistical analysis for each research question.

Analyze data of a statistical experiment.

Compare and contrast categorical and quantitative data.

Identify the variable of interest, interpret a variety of graphical displays (particularly histograms and box plots), and estimate center, spread, shape, outliers, and unusual features.

Analyze histograms in depth, analyzing the effect of changing the bin size (also known as interval width).

Analyze the appropriateness and usefulness of the chosen measure of center and of the graphical display.

Analyze the shape, spread, and unusual features of data sets and identify limitations based on data collection.

Explore the sources of variability in sampling methods.

Analyze possible sources of variability in the data, including biased sampling methods (such as non-representative sampling and under coverage) and biased statistics, as well as natural and induced variability.

Identify and explore various possible sources of statistical bias (such as response bias, nonresponse bias, and observer effect) and examine the effects of statistical bias on the generalizability of results.

Measurement, Geometry, and Spatial Reasoning

Identify transformations using matrices.

Use matrices to organize information and identify matrices that can be used to describe geometric transformations.

Represent figures using matrices and explore ways of determining different transformations (translations, reflection, rotations, dilations, or combinations).

Analyze truth tables to validate real-world situations.

Analyze truth tables to determine and verify the validity of arguments.

Create arguments and statements to validate arguments.

Numerical Reasoning

Analyze numerical data through estimation and approximation in real-world situations.

Use various numerical techniques when estimating and calculating very large and small values.

Apply proportional reasoning with aspect ratios.

Use weighted averages and sums.

Investigate and validate identification numbers.

Analyze present and future value of investments involving interest.

Compare and contrast the nominal interest rate with the annual percentage rate (APR).

Determine the future value of an investment, given the present value.

Analyze real-world scenarios involving credit card debt and loans.

Determine the monthly payment to retire a debt at a fixed rate.

Compare and contrast different credit card offers using minimum payments.

Patterns, Algebra, and Functional Reasoning

Analyze regression of linear functions.

Compute and analyze the correlation coefficient of data to determine the strength of the linear model.

Analyze data that follow a linear pattern using recursively defined rules and compare those rules to explicit function rules.

Analyze step and piecewise functions in real-world situations.

Explore step and piecewise functions to make predictions and decisions about a variety of mathematical and real-world situations.

Analyze data that follow an exponential pattern using the idea of a common ratio between consecutive values.

Find recursive rules to model data and make connections between the recursive rule and the explicit function rule of the exponential relationship.

Mathematical Process Standards

Analyze & Justify

Use critical thinking skills to reason both abstractly and quantitatively

Connections

Demonstrate a deep and flexible conceptual understanding of mathematical ideas, operations, and relationships while making real-world connections.

Problem Solving

Make sense of problems and persevere in solving them strategically.

Representation & Communication

Explain ideas using precise and contextually appropriate mathematical language, tools, and models.

Structure & Patterns

Identify and apply regularity in repeated reasoning to make generalizations.

Data, Probability, and Statistical Reasoning

Communicate using descriptive and inferential statistics by collecting, critiquing, analyzing, and interpreting real-world data.

Calculate and interpret z-scores as a measure of relative standing to standardize units.

Approximate percentages using the Empirical Rule and z-scores for normally distributed data.

Using simulations taken from a given population, model sample-to-sample variability in sampling distributions of a statistic.

Construct and compare confidence intervals of different models to make conclusions about reliability given a margin of error.

Summarize and evaluate reports based on data for appropriateness of study design, analysis methods, and statistical measures used.

Formulate investigative statistical questions that can be answered using data.

Formulate investigative statistical questions about a population using samples taken from the population.

Formulate comparative and associative investigative statistical questions for surveys and observational studies to compare two or more groups or to investigate the association of two or more variables.

Formulate comparative and associative investigative statistical questions for experiments to compare two or more groups or to investigate the association of two or more variables.

Formulate inferential investigative statistical questions regarding association and prediction.

Formulate investigative statistical questions for two variables.

Design and implement a plan to collect data to address the investigative statistical question.

Apply an appropriate data-collection plan when collecting data for the investigative statistical question of interest.

Distinguish between sample surveys, observational studies, and experiments.

Design sample surveys, experiments, and observational studies using statistical methods.

Differentiate between random selection and random assignment and identify their impact on generalizing.

Examine potential sources and effects of bias and confounding variables.

Describe and comply with the ethical use of data.

Identify when data can be generalized to a target population.

Use appropriate graphical and numerical methods to analyze data.

Describe quantitative and categorical data.

Summarize and describe relationships between two variables.

Describe the relationship between two quantitative variables by interpreting correlation (r) and a least-square regression line (using technology).

Assess the fit of a linear model by plotting and analyzing residuals, including the squares of the residuals, to improve its fit.

Calculate and interpret the p-value for a population proportion and/or population mean.

Use simulated sampling distributions to describe the sample to-sample variability of sample statistics.

Use simulations to investigate associations between two categorical variables and to compare groups.

Interpret the results of the analysis by making connections to the investigative statistical question.

Use statistical evidence from analyses to answer investigative statistical questions.

Determine the possible impact of extreme data points, missing values, or incorrect values on the results.

Use and interpret the p-value to determine whether the estimate for a population parameter is reasonable.

Interpret a given margin of error corresponding to an estimate of a population parameter.

Explain the impact of multiple variables on one another.