Grades 9, 10, 11, 12

Standards for Mathematical Practice

Literacy Skills for Mathematical Proficiency

Algebra I

Geometry

Algebra II

Integrated Math I

Integrated Math II

Integrated Math III

Bridge Math

Precalculus

Statistics

Applied Mathematical Concepts

Calculus

Make sense of problems and persevere in solving them.

Reason abstractly and quantitatively.

Construct viable arguments and critique the reasoning of others.

Model with mathematics.

Use appropriate tools strategically.

Attend to precision.

Look for and make use of structure.

Look for and express regularity in repeated reasoning.

Use multiple reading strategies.

Understand and use correct mathematical vocabulary.

Discuss and articulate mathematical ideas.

Write mathematical arguments.

Number and Quantity

Algebra

Functions

Statistics and Probability

Geometry

Number and Quantity

Algebra

Functions

Statistics and Probability

Number and Quantity

Algebra

Functions

Geometry

Statistics and Probability

Number and Quantity

Algebra

Functions

Geometry

Statistics and Probability

Number and Quantity

Algebra

Functions

Geometry

Statistics and Probability

Number and Quantity

Algebra

Functions

Geometry

Statistics and Probability

Number and Quantity

Algebra

Functions

Geometry

Statistics and Probability

Exploring Data

Probability

Probability Distributions

Sampling and Experimentation

Number and Quantity

Algebra

Geometry and Measurement

Data Analysis, Statistics, and Probability

Functions, Graphs, and Limits

Derivatives

Integrals

Quantities

Seeing Structure in Expressions

Arithmetic with Polynomials and Rational Expressions

Creating Equations

Reasoning with Equations and Inequalities

Interpreting Functions

Building Functions

Linear, Quadratic, and Exponential Models

Interpreting Categorical and Quantitative Data

Congruence

Similarity, Right Triangles, and Trigonometry

Circles

Expressing Geometric Properties with Equations

Geometric Measurement and Dimension

Modeling with Geometry

The Real Number System

Quantities

The Complex Number System

Seeing Structure in Expressions

Arithmetic with Polynomials and Rational Expressions

Creating Equations

Reasoning with Equations and Inequalities

Interpreting Functions

Building Functions

Linear, Quadratic, and Exponential Models

Trigonometric Functions

Interpreting Categorical and Quantitative Data

Making Inferences and Justifying Conclusions

Conditional Probability and the Rules of Probability

Quantities

Seeing Structure in Expressions

Creating Equations

Reasoning with Equations and Inequalities

Interpreting Functions

Building Functions

Linear and Exponential Models

Congruence

Interpreting Categorical and Quantitative Data

The Real Number System

Quantities

The Complex Number System

Seeing Structure in Expressions

Arithmetic with Polynomials and Rational Expressions

Creating Equations

Reasoning with Equations and Inequalities

Interpreting Functions

Building Functions

Similarity, Right Triangles, and Trigonometry

Geometric Measurement and Dimension

Interpreting Categorical and Quantitative Data

Quantities

Seeing Structure in Expressions

Arithmetic with Polynomials and Rational Expressions

Creating Equations

Reasoning with Equations and Inequalities

Interpreting Functions

Building Functions

Linear, Quadratic, and Exponential Models

Trigonometric Functions

Congruence

Circles

Expressing Geometric Properties with Equations

Modeling with Geometry

Interpreting Categorical and Quantitative Data

Making Inferences and Justifying Conclusions

The Real Number System

Quantities

The Complex Number System

Seeing Structure in Expressions

Arithmetic with Polynomials and Rational Expressions

Creating Equations

Reasoning with Equations and Inequalities

Interpreting Functions

Similarity, Right Triangles and Trigonometry

Circles

Geometric Measurement and Dimension

Modeling with Geometry

Interpreting Categorical and Quantitative Data

Conditional Probability and the Rules of Probability

Number Expressions

The Complex Number System

Vector and Matrix Quantities

Sequences and Series

Reasoning with Equations and Inequalities

Parametric Equations

Conic Sections

Building Functions

Interpreting Functions

Trigonometric Functions

Graphing Trigonometric Functions

Applied Trigonometry

Trigonometric Identities

Polar Coordinates

Model with Data

Interpreting Categorical and Quantitative Data

Conditional Probability and the Rules of Probability

Using Probability to Make Decisions

Making Inferences and Justifying Conclusions

Financial Mathematics

Linear Programming

Logic and Boolean Algebra

Problem Solving

Investigate Logic

Organize and Interpret Data

Counting and Combinatorial Reasoning

Normal Probability Distribution

Understand and Use Confidence Intervals

Limits of Functions

Behavior of Functions

Continuity

Understand the Concept of the Derivative

Computing and Applying Derivatives

Understanding Integrals

Calculate and Apply Integrals

Reason quantitatively and use units to solve problems.

Interpret the structure of expressions.

Write expressions in equivalent forms to solve problems.

Perform arithmetic operations on polynomials.

Understand the relationship between zeros and factors of polynomials.

Create equations that describe numbers or relationships.

Understand solving equations as a process of reasoning and explain the reasoning.

Solve equations and inequalities in one variable.

Solve systems of equations.

Represent and solve equations and inequalities graphically.

Understand the concept of function and use function notation.

Interpret functions that arise in applications in terms of the context.

Analyze functions using different representations.

Build a function that models a relationship between two quantities.

Build new functions from existing functions.

Construct and compare linear, quadratic, and exponential models and solve problems.

Interpret expressions for functions in terms of the situation they model.

Summarize, represent, and interpret data on a single count or measurement variable.

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

Interpret linear models.

Experiment with transformations in the plane.

Understand congruence in terms of rigid motions.

Prove geometric theorems.

Make geometric constructions.

Understand similarity in terms of similarity transformations.

Prove theorems involving similarity.

Define trigonometric ratios and solve problems involving triangles.

Understand and apply theorems about circles.

Find areas of sectors of circles.

Translate between the geometric description and the equation for a circle.

Use coordinates to prove simple geometric theorems algebraically.

Explain volume and surface area formulas and use them to solve problems.

Apply geometric concepts in modeling situations.

Extend the properties of exponents to rational exponents.

Explain how the definition of the meaning of rational exponents follows from extending the properties of integer exponents to those values, allowing for a notation for radicals in terms of rational exponents.

Rewrite expressions involving radicals and rational exponents using the properties of exponents.

Reason quantitatively and use units to solve problems.

Perform arithmetic operations with complex numbers.

Use complex numbers in quadratic equations.

Interpret the structure of expressions.

Use expressions in equivalent forms to solve problems.

Understand the relationship between zeros and factors of polynomials.

Use polynomial identities to solve problems.

Rewrite rational expressions.

Create equations that describe numbers or relationships.

Understand solving equations as a process of reasoning and explain the reasoning.

Solve equations and inequalities in one variable.

Solve systems of equations.

Represent and solve equations graphically.

Interpret functions that arise in applications in terms of the context.

For a function that models a relationship between two quantities, interpret key features of graphs and tables in terms of the quantities, and sketch graphs showing key features given a verbal description of the relationship.

Calculate and interpret the average rate of change of a function (presented symbolically or as a table) over a specified interval. Estimate the rate of change from a graph.

Analyze functions using different representations.

Build a function that models a relationship between two quantities.

Build new functions from existing functions.

Construct and compare linear, quadratic, and exponential models and solve problems.

Interpret expressions for functions in terms of the situation they model.

Extend the domain of trigonometric functions using the unit circle.

Prove and apply trigonometric identities.

Summarize, represent, and interpret data on a single count or measurement variable.

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

Make inferences and justify conclusions from sample surveys, experiments, and observational studies.

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

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

Reason quantitatively and use units to solve problems.

Interpret the structure of expressions.

Write expressions in equivalent forms to solve problems.

Create equations that describe numbers or relationships

Solve equations and inequalities in one variable.

Solve systems of equations.

Represent and solve equations and inequalities graphically.

Understand the concept of a function and use function notation.

Interpret functions that arise in applications in terms of the context.

Analyze functions using different representations.

Compare properties of two functions each represented in a different way (algebraically, graphically, numerically in tables, or by verbal descriptions).

Build a function that models a relationship between two quantities.

Construct and compare linear and exponential models and solve problems.

Interpret expressions for functions in terms of the situation they model.

Experiment with transformations in the plane.

Understand congruence in terms of rigid motions.

Prove geometric theorems.

Summarize, represent, and interpret data on a single count or measurement variable.

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

Interpret linear models.

Extend the properties of exponents to rational exponents.

Reason quantitatively and use units to solve problems.

Perform arithmetic operations with complex numbers.

Use complex numbers in polynomial identities and equations.

Interpret the structure of expressions.

Write expressions in equivalent forms to solve problems.

Perform arithmetic operations on polynomials.

Create equations that describe numbers or relationships.

Understand solving equations as a process of reasoning and explain the reasoning.

Solve equations and inequalities in one variable.

Solve systems of equations.

Interpret functions that arise in applications in terms of the context.

Analyze functions using different representation.

Build a function that models a relationship between two quantities.

Build new functions from existing functions.

Understand similarity in terms of similarity transformations.

Prove theorems involving similarity.

Define trigonometric ratios and solve problems involving triangles.

Explain volume and surface area formulas and use them to solve problems.

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

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

Reason quantitatively and use units to solve problems.

Interpret the structure of expressions.

Write expressions in equivalent forms to solve problems.

Understand the relationship between zeros and factors of polynomials.

Use polynomial identities to solve problems.

Rewrite rational expressions.

Create equations that describe numbers or relationships.

Understand solving equations as a process of reasoning and explain the reasoning.

Represent and solve equations graphically.

Interpret functions that arise in applications in terms of the context.

Analyze functions using different representations.

Compare properties of two functions each represented in a different way (algebraically, graphically, numerically in tables, or by verbal descriptions).

Build new functions from existing functions.

Construct and compare linear, quadratic, and exponential models and solve problems.

Extend the domain of trigonometric functions using the unit circle.

Prove and apply trigonometric identities.

Make geometric constructions.

Understand and apply theorems about circles.

Find areas of sectors of circles.

Translate between the geometric description and the equation for a circle.

Use coordinates to prove simple geometric theorems algebraically.

Apply geometric concepts in modeling situations.

Summarize, represent, and interpret data on a single count or measurement variable.

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

Understand and evaluate random processes underlying statistical experiments.

Make inferences and justify conclusions from sample surveys, experiments, and observational studies.

Use properties of rational and irrational numbers.

Reason quantitatively and use units to solve problems.

Perform arithmetic operations with complex numbers.

Write expressions in equivalent forms to solve problems.

Perform arithmetic operations on polynomials.

Understand the relationship between zeros and factors of polynomials.

Create equations that describe numbers or relationships.

Understand solving equations as a process of reasoning and explain the reasoning.

Solve equations and inequalities in one variable.

Solve systems of equations.

Represent and solve equations and inequalities graphically.

Understand the concept of a function and use function notation.

Interpret functions that arise in applications in terms of the context.

Analyze functions using different representations.

Understand similarity in terms of similarity transformations.

Define trigonometric ratios and solve problems involving right triangles.

Find arc lengths and areas of sectors of circles.

Visualize relationships between two-dimensional and three-dimensional objects.

Apply geometric concepts in modeling situations.

Summarize, represent, and interpret data on a single count or measurement variable.

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

Interpret linear models.

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

Represent, interpret, compare, and simplify number expressions.

Perform complex number arithmetic and understand the representation on the complex plane.

Use complex numbers in polynomial identities and equations.

Represent and model with vector quantities.

Understand the graphic representation of vectors and vector arithmetic.

Perform operations on matrices and use matrices in applications.

Understand and use sequences and series.

Determine whether a given arithmetic or geometric series converges or diverges.

Find the sum of a given geometric series (both infinite and finite).

Find the sum of a finite arithmetic series.

Solve systems of equations and nonlinear inequalities.

Describe and use parametric equations.

Understand the properties of conic sections and model real-world phenomena.

Build new functions from existing functions.

Analyze functions using different representations.

Extend the domain of trigonometric functions using the unit circle.

Model periodic phenomena with trigonometric functions.

Use trigonometry to solve problems.

Apply trigonometric identities to rewrite expressions and solve equations.

Use polar coordinates.

Model data using regressions equations.

Understand, represent, and use univariate data.

Understand, represent, and use bivariate data.

Understand and apply basic concepts of probability.

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

Understand and use discrete probability distributions.

Understand the normal probability distribution.

Know the characteristics of well-designed studies.

Design and conduct a statistical experiment to study a problem, then interpret and communicate the outcomes.

Make inferences about population parameters based on a random sample from that population.

Understand and use confidence intervals.

Use distributions to make inferences about a data set.

Use financial mathematics to solve problems.

Use financial mathematics to make decisions.

Determine appropriate models to solve contextual problems.

Use linear programming techniques to solve real-world problems.

Solve real-world optimization problems.

Use logic and Boolean Algebra in real-world situations.

Apply Boolean Algebra to real-world network problems.

Apply problem solving techniques to real-world situations.

Use logic to make arguments and solve problems.

Determine the validity of arguments.

Analyze data from multiple viewpoints and perspectives.

Apply probability and counting principles to real world situations.

Use combinatorial reasoning to solve real-world problems.

Work with thenormal distribution in real-world situations.

Work with confidence intervals in real-world situations.

Understand the concept of the limit of a function.

Calculate limits (including limits at infinity) using algebra.

Estimate limits of functions (including one-sided limits) from graphs or tables of data. Apply the definition of a limit to a variety of functions, including piecewise functions.

Draw a sketch that illustrates the definition of the limit; develop multiple real-world scenarios that illustrate the definition of the limit.

Describe the asymptotic and unbounded behavior of functions.

Develop an understanding of continuity as a property of functions

Demonstrate an understanding of the derivative.

Understand the derivative at a point.

Apply differentiation techniques.

Use first and second derivatives to analyze a function.

Apply derivatives to solve problems.

Demonstrate understanding of a definite integral.

Understand and apply the Fundamental Theorem of Calculus.

Apply techniques of anti-differentiation.

Apply integrals to solve problems.

Use units as a way to understand problems and to guide the solution of multi-step problems; choose and interpret units consistently in formulas; choose and interpret the scale and the origin in graphs and data displays.

Identify, interpret, and justify appropriate quantities for the purpose of descriptive modeling.

Choose a level of accuracy appropriate to limitations on measurement when reporting quantities.

Interpret expressions that represent a quantity in terms of its context.

Use the structure of an expression to identify ways to rewrite it.

Choose and produce an equivalent form of an expression to reveal and explain properties of the quantity represented by the expression.

Understand that polynomials form a system analogous to the integers, namely, they are closed under the operations of addition, subtraction, and multiplication; add, subtract, and multiply polynomials.

Identify zeros of polynomials when suitable factorizations are available, and use the zeros to construct a rough graph of the function defined by the polynomial.

Create equations and inequalities in one variable and use them to solve problems.

Create equations in two or more variables to represent relationships between quantities; graph equations with two variables on coordinate axes with labels and scales.

Represent constraints by equations or inequalities and by systems of equations and/or inequalities, and interpret solutions as viable or nonviable options in a modeling context.

Rearrange formulas to highlight a quantity of interest, using the same reasoning as in solving equations.

Explain each step in solving an equation as following from the equality of numbers asserted at the previous step, starting from the assumption that the original equation has a solution. Construct a viable argument to justify a solution method.

Solve linear equations and inequalities in one variable, including equations with coefficients represented by letters.

Solve quadratic equations and inequalities in one variable.

Write and solve a system of linear equations in context.

Understand that the graph of an equation in two variables is the set of all its solutions plotted in the coordinate plane, often forming a curve (which could be a line).

Explain why the x-coordinates of the points where the graphs of the equations y = f(x) and y = g(x) intersect are the solutions of the equation f(x) = g(x); find the approximate solutions using technology.

Graph the solutions to a linear inequality in two variables as a half-plane (excluding the boundary in the case of a strict inequality), and graph the solution set to a system of linear inequalities in two variables as the intersection of the corresponding half-planes.

Understand that a function from one set (called the domain) to another set (called the range) assigns to each element of the domain exactly one element of the range. If f is a function and x is an element of its domain, then f(x) denotes the output of f corresponding to the input x. The graph of f is the graph of the equation y = f(x).

Use function notation, evaluate functions for inputs in their domains, and interpret statements that use function notation in terms of a context.

For a function that models a relationship between two quantities, interpret key features of graphs and tables in terms of the quantities, and sketch graphs showing key features given a verbal description of the relationship.

Relate the domain of a function to its graph and, where applicable, to the quantitative relationship it describes.

Calculate and interpret the average rate of change of a function (presented symbolically or as a table) over a specified interval. Estimate the rate of change from a graph.

Graph functions expressed symbolically and show key features of the graph, by hand and using technology.

Write a function defined by an expression in different but equivalent forms to reveal and explain different properties of the function.

Compare properties of two functions each represented in a different way (algebraically, graphically, numerically in tables, or by verbal descriptions).

Write a function that describes a relationship between two quantities.

Identify the effect on the graph of replacing f(x) by f(x) + k, k f(x), f(kx), and f(x + k) for specific values of k (both positive and negative); find the value of k given the graphs. Experiment with cases and illustrate an explanation of the effects on the graph using technology.

Distinguish between situations that can be modeled with linear functions and with exponential functions.

Construct linear and exponential functions, including arithmetic and geometric sequences, given a graph, a table, a description of a relationship, or input-output pairs.

Observe using graphs and tables that a quantity increasing exponentially eventually exceeds a quantity increasing linearly, quadratically, or (more generally) as a polynomial function.

Interpret the parameters in a linear or exponential function in terms of a context.

Represent single or multiple data sets with dot plots, histograms, stem plots (stem and leaf), and box plots.

Use statistics appropriate to the shape of the data distribution to compare center (median, mean) and spread (interquartile range, standard deviation) of two or more different data sets.

Interpret differences in shape, center, and spread in the context of the data sets, accounting for possible effects of extreme data points (outliers).

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

Interpret the slope (rate of change) and the intercept (constant term) of a linear model in the context of the data.

Use technology to compute and interpret the correlation coefficient of a linear fit.

Distinguish between correlation and causation.

Know precise definitions of angle, circle, perpendicular line, parallel line, and line segment, based on the undefined notions of point, line, plane, distance along a line, and distance around a circular arc.

Represent transformations in the plane in multiple ways, including technology. Describe transformations as functions that take points in the plane (pre-image) as inputs and give other points (image) as outputs. Compare transformations that preserve distance and angle measure to those that do not (e.g., translation versus horizontal stretch).

Given a rectangle, parallelogram, trapezoid, or regular polygon, describe the rotations and reflections that carry the shape onto itself.

Develop definitions of rotations, reflections, and translations in terms of angles, circles, perpendicular lines, parallel lines, and line segments.

Given a geometric figure and a rigid motion, draw the image of the figure in multiple ways, including technology. Specify a sequence of rigid motions that will carry a given figure onto another.

Use geometric descriptions of rigid motions to transform figures and to predict the effect of a given rigid motion on a given figure; given two figures, use the definition of congruence in terms of rigid motions to determine informally if they are congruent.

Use the definition of congruence in terms of rigid motions to show that two triangles are congruent if and only if corresponding pairs of sides and corresponding pairs of angles are congruent.

Explain how the criteria for triangle congruence (ASA, SAS, AAS, and SSS) follow from the definition of congruence in terms of rigid motions.

Prove theorems about lines and angles.

Prove theorems about triangles.

Prove theorems about parallelograms.

Make formal geometric constructions with a variety of tools and methods (compass and straightedge, string, reflective devices, paper folding, dynamic geometric software, etc.).

Verify informally the properties of dilations given by a center and a scale factor.

Given two figures, use the definition of similarity in terms of similarity transformations to decide if they are similar; explain using similarity transformations the meaning of similarity for triangles as the equality of all corresponding pairs of angles and the proportionality of all corresponding pairs of sides.

Use the properties of similarity transformations to establish the AA criterion for two triangles to be similar.

Prove theorems about similar triangles.

Use congruence and similarity criteria for triangles to solve problems and to justify relationships in geometric figures.

Understand that by similarity, side ratios in right triangles are properties of the angles in the triangle, leading to definitions of trigonometric ratios for acute angles.

Explain and use the relationship between the sine and cosine of complementary angles.

Solve triangles.

Recognize that all circles are similar.

Identify and describe relationships among inscribed angles, radii, and chords.

Construct the incenter and circumcenter of a triangle and use their properties to solve problems in context.

Know the formula and find the area of a sector of a circle in a real-world context.

Know and write the equation of a circle of given center and radius using the Pythagorean Theorem.

Use coordinates to prove simple geometric theorems algebraically.

Prove the slope criteria for parallel and perpendicular lines and use them to solve geometric problems.

Find the point on a directed line segment between two given points that partitions the segment in a given ratio.

Know and use coordinates to compute perimeters of polygons and areas of triangles and rectangles.

Give an informal argument for the formulas for the circumference of a circle and the volume and surface area of a cylinder, cone, prism, and pyramid.

Know and use volume and surface area formulas for cylinders, cones, prisms, pyramids, and spheres to solve problems.

Use geometric shapes, their measures, and their properties to describe objects.

Apply geometric methods to solve realworld problems.

Identify, interpret, and justify appropriate quantities for the purpose of descriptive modeling.

Know there is a complex number i such that i² = –1, and every complex number has the form a + bi with a and b real.

Know and use the relation i² = –1 and the commutative, associative, and distributive properties to add, subtract, and multiply complex numbers.

Solve quadratic equations with real coefficients that have complex solutions.

Use the structure of an expression to identify ways to rewrite it.

Choose and produce an equivalent form of an expression to reveal and explain properties of the quantity represented by the expression.

Recognize a finite geometric series (when the common ratio is not 1), and know and use the sum formula to solve problems in context.

Know and apply the Remainder Theorem: For a polynomial p(x) and a number a, the remainder on division by x – a is p(a), so p(a) = 0 if and only if (x – a) is a factor of p(x).

Identify zeros of polynomials when suitable factorizations are available, and use the zeros to construct a rough graph of the function defined by the polynomial.

Know and use polynomial identities to describe numerical relationships.

Rewrite rational expressions in different forms.

Create equations and inequalities in one variable and use them to solve problems.

Rearrange formulas to highlight a quantity of interest, using the same reasoning as in solving equations.

Explain each step in solving an equation as following from the equality of numbers asserted at the previous step, starting from the assumption that the original equation has a solution. Construct a viable argument to justify a solution method.

Solve rational and radical equations in one variable, and identify extraneous solutions when they exist.

Solve quadratic equations and inequalities in one variable.

Write and solve a system of linear equations in context.

Solve a system consisting of a linear equation and a quadratic equation in two variables algebraically and graphically.

Explain why the x-coordinates of the points where the graphs of the equations y = f(x) and y = g(x) intersect are the solutions of the equation f(x) = g(x); find the approximate solutions using technology.

Graph functions expressed symbolically and show key features of the graph, by hand and using technology.

Write a function defined by an expression in different but equivalent forms to reveal and explain different properties of the function.

Compare properties of two functions each represented in a different way (algebraically, graphically, numerically in tables, or by verbal descriptions).

Write a function that describes a relationship between two quantities.

Know and write arithmetic and geometric sequences with an explicit formula and use them to model situations.

Identify the effect on the graph of replacing f(x) by f(x) + k, k f(x), f(kx), and f(x + k) for specific values of k (both positive and negative); find the value of k given the graphs. Experiment with cases and illustrate an explanation of the effects on the graph using technology.

Find inverse functions.

Construct linear and exponential functions, including arithmetic and geometric sequences, given a graph, a table, a description of a relationship, or input-output pairs.

For exponential models, express as a logarithm the solution to ab^ct = d where a, c, and d are numbers and the base b is 2, 10, or e; evaluate the logarithm using technology.

Interpret the parameters in a linear or exponential function in terms of a context.

Understand and use radian measure of an angle.

Explain how the unit circle in the coordinate plane enables the extension of trigonometric functions to all real numbers, interpreted as radian measures of angles traversed counterclockwise around the unit circle.

Know and use trigonometric identities to to find values of trig functions.

Use the mean and standard deviation of a data set to fit it to a normal distribution and to estimate population percentages using the Empirical Rule.

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

Recognize the purposes of and differences among sample surveys, experiments, and observational studies; explain how randomization relates to each.

Use data from a sample survey to estimate a population mean or proportion; use a given margin of error to solve a problem in context.

Describe events as subsets of a sample space (the set of outcomes) using characteristics (or categories) of the outcomes, or as unions, intersections, or complements of other events ("or," "and," "not").

Understand that two events A and B are independent 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.

Know and understand 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.

Recognize and explain the concepts of conditional probability and independence in everyday language and everyday situations.

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.

Know and 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.

Use units as a way to understand problems and to guide the solution of multi-step problems; choose and interpret units consistently in formulas; choose and interpret the scale and the origin in graphs and data displays.

Identify, interpret, and justify appropriate quantities for the purpose of descriptive modeling.

Choose a level of accuracy appropriate to limitations on measurement when reporting quantities.

Interpret expressions that represent a quantity in terms of its context.

Choose and produce an equivalent form of an expression to reveal and explain properties of the quantity represented by the expression.

Create equations and inequalities in one variable and use them to solve problems.

Create equations in two or more variables to represent relationships between quantities; graph equations with two variables on coordinate axes with labels and scales.

Represent constraints by equations or inequalities and by systems of equations and/or inequalities, and interpret solutions as viable or nonviable options in a modeling context.

Rearrange formulas to highlight a quantity of interest, using the same reasoning as in solving equations.

Solve linear equations and inequalities in one variable, including equations with coefficients represented by letters.

Write and solve a system of linear equations in context.

Understand that the graph of an equation in two variables is the set of all its solutions plotted in the coordinate plane, often forming a curve (which could be a line).

Explain why the x-coordinates of the points where the graphs of the equations y = f(x) and y = g(x) intersect are the solutions of the equation f(x) = g(x); find the approximate solutions using technology.

Graph the solutions to a linear inequality in two variables as a half-plane (excluding the boundary in the case of a strict inequality), and graph the solution set to a system of linear inequalities in two variables as the intersection of the corresponding half-planes.

Understand that a function from one set (called the domain) to another set (called the range) assigns to each element of the domain exactly one element of the range. If f is a function and x is an element of its domain, then f(x) denotes the output of f corresponding to the input x. The graph of f is the graph of the equation y = f(x).

Use function notation, evaluate functions for inputs in their domains, and interpret statements that use function notation in terms of a context.

For a function that models a relationship between two quantities, interpret key features of graphs and tables in terms of the quantities, and sketch graphs showing key features given a verbal description of the relationship.

Relate the domain of a function to its graph and, where applicable, to the quantitative relationship it describes.

Calculate and interpret the average rate of change of a function (presented symbolically or as a table) over a specified interval. Estimate the rate of change from a graph.

Graph functions expressed symbolically and show key features of the graph, by hand and using technology.

Write a function that describes a relationship between two quantities.

Write arithmetic and geometric sequences with an explicit formula and use them to model situations.

Distinguish between situations that can be modeled with linear functions and with exponential functions.

Construct linear and exponential functions, including arithmetic and geometric sequences, given a graph, a table, a description of a relationship, or input-output pairs.

Observe using graphs and tables that a quantity increasing exponentially eventually exceeds a quantity increasing linearly.

Interpret the parameters in a linear or exponential function in terms of a context.

Know precise definitions of angle, circle, perpendicular line, parallel line, and line segment, based on the undefined notions of point, line, plane, distance along a line, and distance around a circular arc.

Represent transformations in the plane in multiple ways, including technology. Describe transformations as functions that take points in the plane (pre-image) as inputs and give other points (image) as outputs. Compare transformations that preserve distance and angle measure to those that do not (e.g., translation versus horizontal stretch).

Given a rectangle, parallelogram, trapezoid, or regular polygon, describe the rotations and reflections that carry the shape onto itself.

Develop definitions of rotations, reflections, and translations in terms of angles, circles, perpendicular lines, parallel lines, and line segments.

Given a geometric figure and a rigid motion, draw the image of the figure in multiple ways, including technology. Specify a sequence of rigid motions that will carry a given figure onto another.

Use geometric descriptions of rigid motions to transform figures and to predict the effect of a given rigid motion on a given figure; given two figures, use the definition of congruence in terms of rigid motions to determine informally if they are congruent.

Use the definition of congruence in terms of rigid motions to show that two triangles are congruent if and only if corresponding pairs of sides and corresponding pairs of angles are congruent.

Explain how the criteria for triangle congruence (ASA, SAS, AAS, and SSS) follow from the definition of congruence in terms of rigid motions.

Prove theorems about lines and angles.

Prove theorems about triangles.

Prove theorems about parallelograms.

Represent single or multiple data sets with dot plots, histograms, stem plots (stem and leaf), and box plots.

Use statistics appropriate to the shape of the data distribution to compare center (median, mean) and spread (interquartile range, standard deviation) of two or more different data sets.

Interpret differences in shape, center, and spread in the context of the data sets, accounting for possible effects of extreme data points (outliers).

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

Interpret the slope (rate of change) and the intercept (constant term) of a linear model in the context of the data.

Compute (using technology) and interpret the correlation coefficient of a linear fit.

Distinguish between correlation and causation.

Explain how the definition of the meaning of rational exponents follows from extending the properties of integer exponents to those values, allowing for a notation for radicals in terms of rational exponents.

Rewrite expressions involving radicals and rational exponents using the properties of exponents.

Identify, interpret, and justify appropriate quantities for the purpose of descriptive modeling.

Know there is a complex number i such that i² = –1, and every complex number has the form a + bi with a and b real.

Know and use the relation i² = –1 and the commutative, associative, and distributive properties to add, subtract, and multiply complex numbers.

Solve quadratic equations with real coefficients that have complex solutions.

Interpret expressions that represent a quantity in terms of its context.

Use the structure of an expression to identify ways to rewrite it.

Choose and produce an equivalent form of an expression to reveal and explain properties of the quantity represented by the expression.

Understand that polynomials form a system analogous to the integers, namely, they are closed under the operations of addition, subtraction, and multiplication; add, subtract, and multiply polynomials.

Create equations and inequalities in one variable and use them to solve problems.

Create equations in two or more variables to represent relationships between quantities; graph equations with two variables on coordinate axes with labels and scales.

Rearrange formulas to highlight a quantity of interest, using the same reasoning as in solving equations.

Explain each step in solving an equation as following from the equality of numbers asserted at the previous step, starting from the assumption that the original equation has a solution. Construct a viable argument to justify a solution method.

Solve quadratic equations and inequalities in one variable.

Write and solve a system of linear equations in context.

For a function that models a relationship between two quantities, interpret key features of graphs and tables in terms of the quantities and sketch graphs showing key features given a verbal description of the relationship.

Relate the domain of a function to its graph and, where applicable, to the quantitative relationship it describes.

Calculate and interpret the average rate of change of a function (presented symbolically or as a table) over a specified interval. Estimate the rate of change from a graph.

Graph functions expressed symbolically and show key features of the graph, by hand and using technology.

Write a function defined by an expression in different but equivalent forms to reveal and explain different properties of the function.

Compare properties of two functions each represented in a different way (algebraically, graphically, numerically in tables, or by verbal descriptions).

Write a function that describes a relationship between two quantities.

Identify the effect on the graph of replacing f(x) by f(x) + k, k f(x), f(kx), and f(x + k) for specific values of k (both positive and negative); find the value of k given the graphs. Experiment with cases and illustrate an explanation of the effects on the graph using technology.

Verify informally the properties of dilations given by a center and a scale factor.

Given two figures, use the definition of similarity in terms of similarity transformations to decide if they are similar; explain using similarity transformations the meaning of similarity for triangles as the equality of all corresponding pairs of angles and the proportionality of all corresponding pairs of sides.

Use the properties of similarity transformations to establish the AA criterion for two triangles to be similar.

Prove theorems about similar triangles.

Use congruence and similarity criteria for triangles to solve problems and to justify relationships in geometric figures.

Understand that by similarity, side ratios in right triangles are properties of the angles in the triangle, leading to definitions of trigonometric ratios for acute angles.

Explain and use the relationship between the sine and cosine of complementary angles.

Solve triangles.

Give an informal argument for the formulas for the circumference of a circle and the volume and surface area of a cylinder, cone, prism, and pyramid.

Know and use volume and surface area formulas for cylinders, cones, prisms, pyramids, and spheres to solve problems.

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

Describe events as subsets of a sample space (the set of outcomes) using characteristics (or categories) of the outcomes, or as unions, intersections, or complements of other events ("or," "and," "not").

Understand that two events A and B are independent 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.

Know and understand 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.

Recognize and explain the concepts of conditional probability and independence in everyday language and everyday situations.

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.

Know and 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.

Identify, interpret, and justify appropriate quantities for the purpose of descriptive modeling.

Use the structure of an expression to identify ways to rewrite it.

Choose and produce an equivalent form of an expression to reveal and explain properties of the quantity represented by the expression.

Recognize a finite geometric series (when the common ratio is not 1), and know and use the sum formula to solve problems in context.

Know and apply the Remainder Theorem: For a polynomial p(x) and a number a, the remainder on division by x – a is p(a), so p(a) = 0 if and only if (x – a) is a factor of p(x).

Identify zeros of polynomials when suitable factorizations are available, and use the zeros to construct a rough graph of the function defined by the polynomial.

Know and use polynomial identities to describe numerical relationships.

Rewrite rational expressions in different forms.

Create equations and inequalities in one variable and use them to solve problems.

Create equations in two or more variables to represent relationships between quantities; graph equations with two variables on coordinate axes with labels and scales.

Rearrange formulas to highlight a quantity of interest, using the same reasoning as in solving equations.

Explain each step in solving an equation as following from the equality of numbers asserted at the previous step, starting from the assumption that the original equation has a solution. Construct a viable argument to justify a solution method.

Solve rational and radical equations in one variable, and identify extraneous solutions when they exist.

Explain why the x-coordinates of the points where the graphs of the equations y = f(x) and y = g(x) intersect are the solutions of the equation f(x) = g(x); find the approximate solutions using technology.

For a function that models a relationship between two quantities, interpret key features of graphs and tables in terms of the quantities, and sketch graphs showing key features given a verbal description of the relationship.

Calculate and interpret the average rate of change of a function (presented symbolically or as a table) over a specified interval. Estimate the rate of change from a graph.

Graph functions expressed symbolically and show key features of the graph, by hand and using technology.

Identify the effect on the graph of replacing f(x) by f(x) + k, k f(x), f(kx), and f(x + k) for specific values of k (both positive and negative); find the value of k given the graphs. Experiment with cases and illustrate an explanation of the effects on the graph using technology.

Find inverse functions.

Observe using graphs and tables that a quantity increasing exponentially eventually exceeds a quantity increasing linearly, quadratically, or (more generally) as a polynomial function.

For exponential models, express as a logarithm the solution to ab^ct = d where a, c, and d are numbers and the base b is 2, 10, or e; evaluate the logarithm using technology.

Understand and use radian measure of an angle.

Explain how the unit circle in the coordinate plane enables the extension of trigonometric functions to all real numbers, interpreted as radian measures of angles traversed counterclockwise around the unit circle.

Use trigonometric identities to find values of trig functions.

Make formal geometric constructions with a variety of tools and methods (compass and straightedge, string, reflective devices, paper folding, dynamic geometric software, etc.).

Recognize that all circles are similar.

Identify and describe relationships among inscribed angles, radii, and chords.

Construct the incenter and circumcenter of a triangle and use their properties to solve problems in context.

Find the area of a sector of a circle in a real-world context.

Know and write the equation of a circle of given center and radius using the Pythagorean Theorem.

Use coordinates to prove simple geometric theorems algebraically.

Prove the slope criteria for parallel and perpendicular lines and use them to solve geometric problems.

Find the point on a directed line segment between two given points that partitions the segment in a given ratio.

Know and use coordinates to compute perimeters of polygons and areas of triangles and rectangles.

Use geometric shapes, their measures, and their properties to describe objects.

Apply geometric methods to solve real-world problems.

Use the mean and standard deviation of a data set to fit it to a normal distribution and to estimate population percentages using the Empirical Rule.

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

Understand statistics as a process for making inferences about population parameters based on a random sample from that population.

Decide if a specified model is consistent with results from a given data generating process (e.g., using simulation).

Recognize the purposes of and differences among sample surveys, experiments, and observational studies; explain how randomization relates to each.

Use data from a sample survey to estimate a population mean or proportion; use a given margin of error to solve a problem in context.

Use rational and irrational numbers in calculations and in real-world context.

Use units as a way to understand problems and to guide the solution of multi-step problems; choose and interpret units consistently in formulas; choose and interpret the scale and the origin in graphs and data displays.

Define appropriate quantities for the purpose of descriptive modeling.

Solve problems involving squares, square roots of numbers, cubes, and cube roots of numbers.

Know there is a complex number i such that i² = -1, and every complex number has the form a + bi with a and b real.

Know and use the relation i² = -1 and the commutative, associative, and distributive properties to add, subtract, and multiply complex numbers.

Use properties of multiplication and division to solve problems containing scientific notation.

Use algebraic structures to solve problems involving proportional reasoning in real-world context.

Understand that polynomials form a system analogous to the integers, namely, they are closed under the operations of addition, subtraction, and multiplication; add, subtract, and multiply polynomials.

Identify zeros of polynomials when suitable factorizations are available, and use the zeros to construct a rough graph of the function defined by the polynomial.

Create equations and inequalities in one variable and use them to solve real-world problems.

Create equations in two or more variables to represent relationships between quantities.

Rearrange formulas to highlight a quantity of interest, using the same reasoning as in solving equations.

Build functions and write expressions, equations, and inequalities for common algebra settings leading to a solution in context (e.g., rate and distance problems and problems that can be solved using proportions).

Solve quadratic equations in one variable. Solve quadratic equations by inspection (e.g., for x² = 49), taking square roots, completing the square, knowing and applying the quadratic formula, and factoring, as appropriate to the initial form of the equation. Recognize when the quadratic formula gives complex solutions and write them as a ± bi for real numbers a and b.

Solve and explain the solutions to a system of equations using a variety of representations including combinations of linear and non-linear equations.

Use algebra and geometry to solve problems involving midpoints and distances.

Solve a linear inequality using multiple methods and interpret the solution as it applies to the context.

Understand that a function from one set (called the domain) to another set (called the range) assigns to each element of the domain exactly one element of the range. If f is a function and x is an element of its domain, then f(x) denotes the output of f corresponding to the input x. The graph of f is the graph of the equation y = f(x).

Use function notation, evaluate functions for inputs in their domains, and interpret statements that use function notation in terms of a context.

Recognize functions as mappings of an independent variable into a dependent variable.

Graph linear, quadratic, absolute value, and piecewise functions expressed symbolically and show key features of the graph, by hand in simple cases and using technology for more complicated ones.

Write a function defined by an expression in different but equivalent forms to reveal and explain different properties of the function.

Use the properties of exponents to interpret expressions for exponential functions.

Apply similar triangles to solve problems, such as finding heights and distances.

Apply basic trigonometric ratios to solve right triangle problems.

Apply properties of 30° 60° 90°, 45° 45° 90°, similar, and congruent triangles.

Solve problems involving angles of elevation and angles of depression.

Apply a variety of strategies to determine the area and circumference of circles after identifying necessary information.

Use relationships involving area, perimeter, and volume of geometric figures to compute another measure.

Use several angle properties to find an unknown angle measure.

Apply a variety of strategies using relationships between perimeter, area, and volume to calculate desired measures in composite figures (i.e., combinations of basic figures).

Use appropriate technology to find the mathematical model for a set of non-linear data.

Solve problems involving surface area and volume in real-world context.

Use statistics appropriate to the shape of the data distribution to compare center (median, mean) and spread (interquartile range, standard deviation) of two or more different data sets.

Interpret and use data from tables, charts, and graphs.

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

Interpret the slope (rate of change) and the intercept (constant term) of a linear model in the context of the data.

Understand and use basic counting techniques in contextual settings.

Compute a probability when the event and/or sample space are not given or obvious.

Recognize the concepts of conditional and joint probability expressed in real-world contexts.

Recognize the concept of independence expressed in real-world contexts.

Use the laws of exponents and logarithms to expand or collect terms in expressions; simplify expressions or modify them in order to analyze them or compare them.

Understand the inverse relationship between exponents and logarithms and use this relationship to solve problems involving logarithms and exponents.

Classify real numbers and order real numbers that include transcendental expressions, including roots and fractions of π and e.

Simplify complex radical and rational expressions; discuss and display understanding that rational numbers are dense in the real numbers and the integers are not.

Understand that rational expressions form a system analogous to the rational numbers, closed under addition, subtraction, multiplication, and division by a nonzero rational expression; add, subtract, multiply, and divide rational expressions.

Perform arithmetic operations with complex numbers expressing answers in the form a + bi.

Find the conjugate of a complex number; use conjugates to find moduli and quotients of complex numbers.

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.

Calculate the distance between numbers in the complex plane as the modulus of the difference, and the midpoint of a segment as the average of the numbers at its endpoints.

Extend polynomial identities to the complex numbers.

Know the Fundamental Theorem of Algebra; show that it is true for quadratic polynomials.

Recognize vector quantities as having both magnitude and direction. Represent vector quantities by directed line segments, and use appropriate symbols for vectors and their magnitudes (e.g., v, |v|, ||v||, v).

Find the components of a vector by subtracting the coordinates of an initial point from the coordinates of a terminal point.

Solve problems involving velocity and other quantities that can be represented by vectors.

Add and subtract vectors.

Multiply a vector by a scalar.

Calculate and interpret the dot product of two vectors.

Use matrices to represent and manipulate data, e.g., to represent payoffs or incidence relationships in a network.

Multiply matrices by scalars to produce new matrices, e.g., as when all of the payoffs in a game are doubled.

Add, subtract, and multiply matrices of appropriate dimensions.

Understand that, unlike multiplication of numbers, matrix multiplication for square matrices is not a commutative operation, but still satisfies the associative and distributive properties.

Understand that the zero and identity matrices play a role in matrix addition and multiplication similar to the role of 0 and 1 in the real numbers. The determinant of a square matrix is nonzero if and only if the matrix has a multiplicative inverse.

Multiply a vector (regarded as a matrix with one column) by a matrix of suitable dimensions to produce another vector. Work with matrices as transformations of vectors.

Work with 2 × 2 matrices as transformations of the plane, and interpret the absolute value of the determinant in terms of area.

Demonstrate an understanding of sequences by representing them recursively and explicitly.

Use sigma notation to represent a series; expand and collect expressions in both finite and infinite settings.

Derive and use the formulas for the general term and summation of finite or infinite arithmetic and geometric series, if they exist.

Understand that series represent the approximation of a number when truncated; estimate truncation error in specific examples.

Know and apply the Binomial Theorem for the expansion of (x + y)ⁿ in powers of x and y for a positive integer n, where x and y are any numbers, with coefficients determined for example by Pascal's Triangle.

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

Find the inverse of a matrix if it exists and use it to solve systems of linear equations (using technology for matrices of dimension 3 × 3 or greater).

Solve nonlinear inequalities (quadratic, trigonometric, conic, exponential, logarithmic, and rational) by graphing (solutions in interval notation if one-variable), by hand and with appropriate technology.

Solve systems of nonlinear inequalities by graphing.

Graph curves parametrically (by hand and with appropriate technology).

Eliminate parameters by rewriting parametric equations as a single equation.

Display all of the conic sections as portions of a cone.

Derive the equations of ellipses and hyperbolas given the foci, using the fact that the sum or difference of distances from the foci is constant.

From an equation in standard form, graph the appropriate conic section: ellipses, hyperbolas, circles, and parabolas. Demonstrate an understanding of the relationship between their standard algebraic form and the graphical characteristics.

Transform equations of conic sections to convert between general and standard form.

Understand how the algebraic properties of an equation transform the geometric properties of its graph.

Develop an understanding of functions as elements that can be operated upon to get new functions: addition, subtraction, multiplication, division, and composition of functions.

Compose functions.

Construct the difference quotient for a given function and simplify the resulting expression.

Find inverse functions (including exponential, logarithmic, and trigonometric).

Explain why the graph of a function and its inverse are reflections of one another over the line y = x.

Determine whether a function is even, odd, or neither.

Analyze qualities of exponential, polynomial, logarithmic, trigonometric, and rational functions and solve real-world problems that can be modeled with these functions (by hand and with appropriate technology).

Identify the real zeros of a function and explain the relationship between the real zeros and the x-intercepts of the graph of a function (exponential, polynomial, logarithmic, trigonometric, and rational).

Identify characteristics of graphs based on a set of conditions or on a general equation such as y = ax² + c.

Visually locate critical points on the graphs of functions and determine if each critical point is a minimum, amaximum, or point of inflection. Describe intervals where the function is increasing or decreasing and where different types of concavity occur.

Graph rational functions, identifying zeros, asymptotes (including slant), and holes (when suitable factorizations are available) and showing end-behavior.

Recognize that sequences are functions, sometimes defined recursively, whose domain is a subset of the integers.

Convert from radians to degrees and from degrees to radians.

Use special triangles to determine geometrically the values of sine, cosine, tangent for π/3, π/4 and π/6, 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.

Use the unit circle to explain symmetry (odd and even) and periodicity of trigonometric functions.

Choose trigonometric functions to model periodic phenomena with specified amplitude, frequency, and midline.

Interpret transformations of trigonometric functions.

Determine the difference made by choice of units for angle measurement when graphing a trigonometric function.

Graph the six trigonometric functions and identify characteristics such as period, amplitude, phase shift, and asymptotes.

Find values of inverse trigonometric expressions (including compositions), applying appropriate domain and range restrictions.

Understand that restricting a trigonometric function to a domain on which it is always increasing or always decreasing allows its inverse to be constructed.

Determine the appropriate domain and corresponding range for each of the inverse trigonometric functions.

Graph the inverse trigonometric functions and identify their key characteristics.

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

Use the definitions of the six trigonometric ratios as ratios of sides in a right triangle to solve problems about lengths of sides and measures of angles.

Derive the formula A = ½ ab sin(C) for the area of a triangle by drawing an auxiliary line from a vertex perpendicular to the opposite side.

Derive and apply the formulas for the area of sector of a circle.

Calculate the arc length of a circle subtended by a central angle.

Prove the Laws of Sines and Cosines and use them to solve problems.

Understand and apply the Law of Sines (including the ambiguous case) and the Law of Cosines to find unknown measurements in right and non-right triangles (e.g., surveying problems, resultant forces).

Apply trigonometric identities to verify identities and solve equations. Identities include: Pythagorean, reciprocal, quotient, sum/difference, double-angle, and half-angle.

Prove the addition and subtraction formulas for sine, cosine, and tangent and use them to solve problems.

Graph functions in polar coordinates.

Convert between rectangular and polar coordinates.

Represent situations and solve problems involving polar coordinates.

Create scatter plots, analyze patterns, and describe relationships for bivariate data (linear, polynomial, trigonometric, or exponential) tomodel real-world phenomena and tomake predictions.

Determine a regression equation to model a set of bivariate data. Justify why this equation best fits the data.

Use a regression equation, modeling bivariate data, to make predictions. Identify possible considerations regarding the accuracy of predictions when interpolating or extrapolating.

Understand the term 'variable' and differentiate between the data types: measurement, categorical, univariate, and bivariate.

Understand histograms, parallel box plots, and scatterplots, and use them to display and compare data.

Summarize distributions of univariate data.

Compute basic statistics and understand the distinction between a statistic and a parameter.

For univariate measurement data, be able to display the distribution and describe its shape; select and calculate summary statistics.

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

Analyze the effect of changing units on summary measures.

Construct and interpret two-way frequency tables of data when two categories are associated with each object being classified. Use the two-way table as a sample space to decide if events are independent and to approximate conditional probabilities.

Describe individual performances in terms of percentiles, z-scores, and t-scores.

Represent and analyze categorical data.

Display and discuss bivariate data where at least one variable is categorical.

For bivariate measurement data, be able to display a scatterplot and describe its shape; use technological tools to determineregression equations and correlation coefficients.

Identify trends in bivariate data; find functions that model the data and that transform the data so that they can bemodeled.

Describe events as subsets of a sample space (the set of outcomes) using characteristics (or categories) of the outcomes, or as unions, intersections, or complements of other events ("or," "and," "not").

Use permutations and combinations to compute probabilities of compound events and solve problems.

Demonstrate an understanding of the Law of Large Numbers (Strong and Weak).

Demonstrate an understanding of the addition rule, the multiplication rule, conditional probability, and independence.

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.

Define a random variable for a quantity of interest by assigning a numerical value to each event in a sample space; graph the corresponding probability distribution using the same graphical displays as for data distributions.

Calculate the expected value of a random variable; interpret it as the mean of the probability distribution.

Design a simulation of random behavior and probability distributions (e.g., drawing by lots, using a random number generator, and using the results to make fair decisions).

Analyze discrete random variables and their probability distributions, including binomial and geometric.

Develop a probability distribution for a random variable defined for a sample space in which theoretical probabilities can be calculated; find the expected value.

Develop a probability distribution for a random variable defined for a sample space in which probabilities are assigned empirically; find the expected value.

Weigh the possible outcomes of a decision by assigning probabilities to payoff values and finding expected values.

Analyze decisions and strategies using probability concepts (e.g., product testing, medical testing, pulling a hockey goalie at the end of a game).

Calculate the mean (expected value) and standard deviation of both a random variable and a linear transformation of a random variable.

Use the mean and standard deviation of a data set to fit it to a normal distribution and to estimate population percentages. Recognize that there are data sets for which such a procedure is not appropriate. Use calculators, spreadsheets, and tables to estimate areas under the normal curve.

Understand the differences among various kinds of studies and which types of inferences can be legitimately drawn from each.

Compare census, sample survey, experiment, and observational study.

Describe the role of randomization in surveys and experiments.

Describe the role of experimental control and its effect on confounding.

Identify bias in sampling and determine ways to reduce it to improve results.

Describe the sampling distribution of a statistic and define the standard error of a statistic.

Demonstrate an understanding of the Central Limit Theorem.

Select a method to collect data and plan and conduct surveys and experiments.

Compare and use sampling methods, including simple random sampling, stratified random sampling, and cluster sampling.

Test hypotheses using appropriate statistics.

Analyze results andmake conclusions from observational studies, experiments, and surveys.

Use data from a randomized experiment to compare two treatments; use simulations to decide if differences between parameters are significant.

Develop and evaluate inferences and predictions that are based on data.

Use properties of point estimators, including biased/unbiased, and variability.

Understand the meaning of confidence level, of confidence intervals, and the properties of confidence intervals.

Construct and interpret a large sample confidence interval for a proportion and for a difference between two proportions.

Construct the confidence interval for a mean and for a difference between two means.

Apply the properties of a Chi-square distribution in appropriate situations in order to make inferences about a data set.

Apply the properties of the normal distribution in appropriate situations in order to make inferences about a data set.

Interpret the t-distribution and determine the appropriate degrees of freedom.

Define interest, compound interest, annuities, sinking funds, amortizations, annuities, future value, and present value.

Recognize the importance of applying a financial model to business.

Determine future value and present value of an annuity.

Determine the amortization schedule for an annuity and a home mortgage.

Apply financial mathematics to depreciation schedules.

Solve contextual problems involving financial decision-making.

Apply arithmetic and geometric sequences to simple and compound interest, annuities, loans, and amortization.

Solve problems in mathematics of finance involving compound interest using exponential and logarithmic techniques.

Know when to use transcendental functions to accomplish various application purposes such as predicting population growth.

Use orders of magnitude estimates for determining an appropriate model for a contextual situation.

Use mathematical models involving equations and systems of equations to represent, interpret, and analyze quantitative relationships, change in various contexts, and other real-world phenomena.

Read, interpret, and solve linear programming problems graphically and by computational methods.

Use linear programming to solve optimization problems.

Interpret the meaning of the maximum or minimum value in terms of the objective function.

Develop the symbols and properties of Boolean algebra; connect Boolean algebra to standard logic.

Construct truth tables to determine the validity of an argument.

Analyze basic electrical networks; compare the networks to Boolean Algebra configurations.

Develop electrical networks and translate them into Boolean Algebra equations.

Apply problem solving strategies to real-world situations.

Define the order of operations for the logical operators.

Define conjunction, disjunction, negation, conditional, and biconditional.

Solve a variety of logic puzzles.

Construct and use a truth table to draw conclusions about a statement.

Apply the laws of logic to judge the validity of arguments.

Give counterexamples to disprove statements.

Analyze arguments with quantifiers through the use of Venn diagrams.

Represent logical statements with networks.

Organize data for problem solving.

Use a variety of counting methods to organize information, determine probabilities, and solve problems.

Translate from one representation of data to another, e.g., a bar graph to a circle graph.

Calculate and interpret statistical problems using measures of central tendency and graphs.

Calculate expected value, e.g., to determine the fair price of an investment.

Analyze survey data using Venn diagrams.

Evaluate and compare two investments or strategies, where one investment or strategy is safer but has lower expected value. Include large and small investments and situations with serious consequences.

Use permutations, combinations, and the multiplication principle to solve counting problems.

Design and interpret simple experiments using tree-diagrams, permutations, and combinations.

Apply counting principles to probabilistic situations involving equally likely outcomes.

Solve counting problems by using Venn diagrams and show relationships modeled by the Venn diagram.

Use permutations and combinations to compute probabilities of compound events and solve problems.

Apply the Law of Large Numbers to contextual situations.

Discuss the various examples and consequences of innumeracy; consider poor estimation, improper experimental design, inappropriate comparisons, and scientific notation comparisons.

Weigh the possible outcomes of a decision by assigning probabilities to payoff values and finding expected values.

Use probabilities to make fair decisions (e.g., drawing by lots, using a random number generator).

Analyze decisions and strategies using probability concepts (e.g., product testing, medical testing, pulling a hockey goalie at the end of a game).

Calculate the mean (expected value) and standard deviation of both a random variable and a linear transformation of a random variable.

Use the mean and standard deviation of a data set to fit it to a normal distribution and to estimate population percentages. Recognize that there are data sets for which such a procedure is not appropriate. Use calculators, spreadsheets, and tables to estimate areas under the normal curve.

Understand the meaning of confidence level, of confidence intervals, and the properties of confidence intervals.

Construct and interpret a large sample confidence interval for a proportion and for a difference between two proportions.

Construct the confidence interval for a mean and for a difference between two means.

Describe asymptotic behavior (analytically and graphically) in terms of infinite limits and limits at infinity.

Discuss the various types of end behavior of functions; identify prototypical functions for each type of end behavior.

Define continuity at a point using limits; define continuous functions.

Determine whether a given function is continuous at a specific point.

Determine and define different types of discontinuity (point, jump, infinite) in terms of limits.

Apply the Intermediate Value Theorem and Extreme Value Theorem to continuous functions.

Represent and interpret the derivative of a function graphically, numerically, and analytically.

Interpret the derivative as an instantaneous rate of change.

Define the derivative as the limit of the difference quotient; illustrate with the sketch of a graph.

Demonstrate the relationship between differentiability and continuity.

Interpret the derivative as the slope of a curve (which could be a line) at a point, including points at which there are vertical tangents and points at which there are no tangents (i.e., where a function is not locally linear).

Approximate both the instantaneous rate of change and the average rate of change given a graph or table of values.

Write the equation of the line tangent to a curve at a given point.

Apply the Mean Value Theorem.

Understand Rolle's Theorem as a special case of the Mean Value Theorem.

Describe in detail how the basic derivative rules are used to differentiate a function; discuss the difference between using the limit definition of the derivative and using the derivative rules.

Calculate the derivative of basic functions (power, exponential, logarithmic, and trigonometric).

Calculate the derivatives of sums, products, and quotients of basic functions.

Apply the chain rule to find the derivative of a composite function.

Implicitly differentiate an equation in two or more variables.

Use implicit differentiation to find the derivative of the inverse of a function.

Relate the increasing and decreasing behavior of f to the sign of f' both analytically and graphically.

Use the first derivative to find extrema (local and global).

Analytically locate the intervals on which a function is increasing, decreasing, or neither.

Relate the concavity of f to the sign of f" both analytically and graphically.

Use the second derivative to find points of inflection as points where concavity changes.

Analytically locate intervals on which a function is concave up, concave down, or neither.

Relate corresponding characteristics of the graphs of f, f', and f".

Translate verbal descriptions into equations involving derivatives and vice versa.

Model rates of change, including related rates problems. In each case, include a discussion of units.

Solve optimization problems to find a desired maximum or minimum value.

Use differentiation to solve problems involving velocity, speed, and acceleration.

Use tangent lines to approximate function values and changes in function values when inputs change (linearization).

Define the definite integral as the limit of Riemann sums and as the net accumulation of change.

Correctly write a Riemann sum that represents the definition of a definite integral.

Use Riemann sums (left, right, and midpoint evaluation points) and trapezoid sums to approximate definite integrals of functions represented graphically, numerically, and by tables of values.

Recognize differentiation and anti-differentiation as inverse operations.

Evaluate definite integrals using the Fundamental Theorem of Calculus.

Use the Fundamental Theorem of Calculus to represent a particular anti-derivative of a function and to understand when the antiderivative so represented is continuous and differentiable.

Apply basic properties of definite integrals (e.g., additive, constant multiple, translations).

Develop facility with finding anti-derivatives that follow directly from derivatives of basic functions (power, exponential, logarithmic, and trigonometric).

Use substitution of variables to calculate anti-derivatives (including changing limits for definite integrals).

Find specific anti-derivatives using initial conditions.

Use a definite integral to find the area of a region.

Use a definite integral to find the volume of a solid formed by rotating a region around a given axis.

Use integrals to solve a variety of problems (e.g., distance traveled by a particle along a line, exponential growth/decay).

Interpret parts of an expression, such as terms, factors, and coefficients.

Interpret complicated expressions by viewing one or more of their parts as a single entity.

Factor a quadratic expression to reveal the zeros of the function it defines.

Complete the square in a quadratic expression in the form Ax² + Bx + C where A = 1 to reveal the maximum or minimum value of the function it defines.

Use the properties of exponents to rewrite exponential expressions.

Use the method of completing the square to rewrite any quadratic equation in x into an equation of the form (x – p)² = q that has the same solutions. Derive the quadratic formula from this form.

Solve quadratic equations by inspection (e.g., for x² = 49), taking square roots, completing the square, knowing and applying the quadratic formula, and factoring, as appropriate to the initial form of the equation. Recognize when the quadratic formula gives complex solutions.

Graph linear and quadratic functions and show intercepts, maxima, and minima.

Graph square root, cube root, and piecewise-defined functions, including step functions and absolute value functions.

Use the process of factoring and completing the square in a quadratic function to show zeros, extreme values, and symmetry of the graph, and interpret these in terms of a context.

Determine an explicit expression, a recursive process, or steps for calculation from a context.

Recognize that linear functions grow by equal differences over equal intervals and that exponential functions grow by equal factors over equal intervals.

Recognize situations in which one quantity changes at a constant rate per unit interval relative to another.

Recognize situations in which a quantity grows or decays by a constant factor per unit interval relative to another.

Fit a function to the data; use functions fitted to data to solve problems in the context of the data. Use given functions or choose a function suggested by the context.

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

Know and use trigonometric ratios and the Pythagorean Theorem to solve right triangles in applied problems.

Know and use the Law of Sines and Law of Cosines to solve problems in real life situations. Recognize when it is appropriate to use each.

Use the properties of exponents to rewrite expressions for exponential functions.

Solve quadratic equations by inspection (e.g., for x² = 49), taking square roots, completing the square, knowing and applying the quadratic formula, and factoring, as appropriate to the initial form of the equation. Recognize when the quadratic formula gives complex solutions and write them as a ± bi for real numbers a and b.

Graph square root, cube root, and piecewise defined functions, including step functions and absolute value functions.

Graph polynomial functions, identifying zeros when suitable factorizations are available and showing end behavior.

Graph exponential and logarithmic functions, showing intercepts and end behavior.

Know and use the properties of exponents to interpret expressions for exponential functions.

Determine an explicit expression, a recursive process, or steps for calculation from a context.

Combine standard function types using arithmetic operations.

Find the inverse of a function when the given function is one-to-one.

Understand radian measure of an angle as the length of the arc on the unit circle subtended by the angle.

Use the unit circle to find sin θ, cos θ, and tan θ when θ is a commonly recognized angle between 0 and 2π.

Given a point on a circle centered at the origin, recognize and use the right triangle ratio definitions of sin θ, cos θ, and tan θ to evaluate the trigonometric functions.

Given the quadrant of the angle, use the identity sin² θ + cos² θ = 1 to find sin θ given cos θ, or vice versa.

Fit a function to the data; use functions fitted to data to solve problems in the context of the data.

Interpret parts of an expression, such as terms, factors, and coefficients.

Interpret complicated expressions by viewing one or more of their parts as a single entity.

Use the properties of exponents to rewrite exponential expressions.

Graph linear and quadratic functions and show intercepts, maxima, and minima.

Determine an explicit expression, a recursive process, or steps for calculation from a context.

Recognize that linear functions grow by equal differences over equal intervals and that exponential functions grow by equal factors over equal intervals.

Recognize situations in which one quantity changes at a constant rate per unit interval relative to another.

Recognize situations in which a quantity grows or decays by a constant factor per unit interval relative to another.

Fit a function to the data; use functions fitted to data to solve problems in the context of the data. Use given functions or choose a function suggested by the context.

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

Interpret complicated expressions by viewing one or more of their parts as a single entity.

Factor a quadratic expression to reveal the zeros of the function it defines.

Complete the square in a quadratic expression in the form Ax² + Bx + C where A = 1 to reveal the maximum or minimum value of the function it defines.

Use the method of completing the square to rewrite any quadratic equation in x into an equation of the form (x – p)² = q that has the same solutions. Derive the quadratic formula from this form.

Solve quadratic equations by inspection (e.g., for x² = 49), taking square roots, completing the square, knowing and applying the quadratic formula, and factoring, as appropriate to the initial form of the equation. Recognize when the quadratic formula gives complex solutions and write them as a ± bi for real numbers a and b.

Solve a system consisting of a linear equation and a quadratic equation in two variables algebraically and graphically.

Graph linear and quadratic functions and show intercepts, maxima, and minima.

Graph square root, cube root, and piecewise-defined functions, including step functions and absolute value functions.

Graph exponential and logarithmic functions, showing intercepts and end behavior.

Use the process of factoring and completing the square in a quadratic function to show zeros, extreme values, and symmetry of the graph, and interpret these in terms of a context.

Know and use the properties of exponents to interpret expressions for exponential functions.

Determine an explicit expression, a recursive process, or steps for calculation from a context.

Combine standard function types using arithmetic operations.

Know and use trigonometric ratios and the Pythagorean Theorem to solve right triangles in applied problems.

Know and use the Law of Sines and the Law of Cosines to solve triangles in applied problems. Recognize when it is appropriate to use each.

Fit a function to the data; use functions fitted to data to solve problems in the context of the data.

Use the properties of exponents to rewrite expressions for exponential functions.

Graph linear and quadratic functions and show intercepts, maxima, and minima.

Graph square root, cube root, and piecewise-defined functions, including step functions and absolute value functions.

Graph polynomial functions, identifying zeros when suitable factorizations are available and showing end behavior.

Graph exponential and logarithmic functions, showing intercepts and end behavior.

Find the inverse of a function when the given function is one-to-one.

Understand radian measure of an angle as the length of the arc on the unit circle subtended by the angle.

Use the unit circle to find sin θ, cos θ, and tan θ when θ is a commonly recognized angle between 0 and 2π.

Given a point on a circle centered at the origin, recognize and use the right triangle ratio definitions of sin θ, cos θ, and tan θ to evaluate the trigonometric functions.

Given the quadrant of the angle, use the identity sin² θ + cos² θ = 1 to find sin θ given cos θ, or vice versa.

Fit a function to the data; use functions fitted to data to solve problems in the context of the data. Use given functions or choose a function suggested by the context.

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

Fit a function to the data; use functions fitted to data to solve problems in the context of the data. Use given functions or choose a function suggested by the context. Emphasize linear, quadratic, and exponential models.

Add vectors end-to-end, component-wise, and by the parallelogram rule. Understand that the magnitude of a sum of two vectors is typically not the sum of the magnitudes.

Given two vectors in magnitude and direction form, determine the magnitude and direction of their sum.

Understand vector subtraction v – w as v + (–w), where –w is the additive inverse of w, with the same magnitude as w and pointing in the opposite direction. Represent vector subtraction graphically by connecting the tips in the appropriate order, and performvector subtraction component-wise.

Represent scalar multiplication graphically by scaling vectors and possibly reversing their direction; perform scalar multiplication component-wise, e.g., as c(vx, vy) = (cvx, cvy).

Compute the magnitude of a scalar multiple cv using ||cv|| = |c|v. Compute the direction of cv knowing that when |c|v ≠ 0, the direction of cv is either along v (for c > 0) or against v (for c < 0).

Calculate the inverse of a function, f^-1 (x), with respect to each of the functional operations; in other words, the additive inverse, −f(x) , the multiplicative inverse, 1/f(x), and the inverse with respect to composition, f − 1(x) . Understand the algebraic and graphical implications of each type.

Verify by composition that one function is the inverse of another.

Read values of an inverse function from a graph or a table, given that the function has an inverse.

Recognize a function is invertible if and only if it is one-to-one. Produce an invertible function from a non-invertible function by restricting the domain.

Find the expected payoff for a game of chance.

Evaluate and compare strategies on the basis of expected values.

Find the expected payoff for a game of chance.

Evaluate and compare strategies on the basis of expected values.