High School Math Washington Standards

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.

Flexibly, efficiently, and accurately demonstrate that polynomials form a system similar to the integers, namely, they are closed under the operations of addition, subtraction, and multiplication; add, subtract, and multiply polynomials.

Flexibly, efficiently, and accurately create equations and inequalities in one variable and use them to solve problems. Include equations arising from linear, quadratic, and exponential functions.

Flexibly, efficiently, and accurately create linear, quadratic, exponential equations to represent relationships between quantities; graph equations 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 within linear, quadratic, and exponential equations.

Flexibly, efficiently, and accurately rearrange formulas to highlight a quantity of interest, using the same reasoning as in solving equations within linear, quadratic, and exponential equations.

Explain each step in solving an equation as following from the equality of numbers asserted at the previous step flexibly, efficiently, and accurately selecting and demonstrating use of strategies to solve equations, 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 in one variable by inspection, taking square roots, and factoring as appropriate to the initial form of the equation.

Demonstrate using a variety of strategies that, given a system of two equations in two variables, replacing one equation by the sum of that equation and a multiple of the other produces a system with the same solutions.

Flexibly, efficiently, and accurately solve systems of linear equations exactly and approximately (e.g., with graphs), focusing on pairs of linear equations in two variables.

Flexibly, efficiently, and accurately solve a simple system consisting of a linear equation and a quadratic equation in two variables algebraically and graphically.

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).

Using a variety of strategies explain the x-coordinates of the points where the graphs of the equations 𝑦𝑦 = 𝑓𝑓(𝑥𝑥) and 𝑦𝑦 = 𝑔𝑔(𝑥𝑥) intersect are the solutions of the equation 𝑓𝑓(𝑥𝑥) = 𝑔𝑔(𝑥𝑥); find the solutions approximately, e.g., using technology to graph the functions, make tables of values, or find successive approximations. Include cases where 𝑓𝑓(𝑥𝑥) and/or 𝑔𝑔(𝑥𝑥) are linear, exponential, and quadratic.

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.

Interpret expressions that represent a quantity in terms of its context within linear, exponential, and quadratic functions.

Use the structure of an expression to identify ways to rewrite it within exponential and quadratic functions.

Flexibly, efficiently, and accurately create an equivalent form of an expression to reveal and explain properties of the quantity represented by the expression including factoring quadratic expressions and using properties of exponents to create equivalent forms of exponential expressions to reveal properties of interest in the function.

Flexibly, efficiently, and accurately write a function that describes a relationship between two quantities, including linear and exponential arithmetic and geometric sequences in context.

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

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. Using a variety of strategies, experiment with cases and illustrate an explanation of the effects on the graph using technology.

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 𝑓𝑓 is a function and x is an element of its domain, then 𝑓𝑓(𝑥𝑥) denotes the output of f corresponding to the input 𝑥𝑥. The graph of f is the graph of the equation 𝑦𝑦 = 𝑓𝑓(𝑥𝑥).

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

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

For a function that models a relationship between two quantities in context, 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. Key features include intercepts; intervals where the function is increasing, decreasing, positive, or negative; relative maximums and minimums; symmetries for functions including linear, exponential, and quadratic.

Relate the domain of a function to its graph and, where applicable, to the quantitative relationship it describes in linear, exponential, or quadratic contexts.

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

Graph linear, exponential, and quadratic functions expressed symbolically and show key features of the graph, including intercepts, maximum, minimum, and interpreting end behavior for exponential functions by hand in simple cases and using technology for more complicated cases.

Flexibly, efficiently, and accurately write a function defined by an expression in different but equivalent forms to reveal and explain different properties of the function including zeros and symmetry, using factoring for quadratic functions and integer constants for time with exponential growth and decay.

Compare properties of two functions each represented in a different way (algebraically, graphically, numerically in tables, or by verbal descriptions). Functions could be linear, exponential, or quadratic.

Distinguish between situations that can be modeled with linear functions (equal differences over equal intervals) and with exponential functions (equal factors over equal intervals), recognizing constant rates per unit interval, and growth or decay by a constant percent rate per unit interval.

Flexibly, efficiently, and accurately construct linear and exponential functions given a graph, a description of a relationship, or two input-output pairs (include reading these from a table).

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

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

Formulate multivariable statistical investigative questions and determine how data can be collected and provide an answer, consider causality and prediction when posing the question.

Understand the issues of bias and confounding variables when collecting data and their impact on interpretation. Understand practices for collecting and handling data, including sensitive information and concerns for privacy and how that may affect data collection.

Create and analyze data sets and data displays, including but not limited to scatter plots, regressions, histograms and boxplots using technology to sort or filter data, summarize, and describe relationships between quantitative variables.

Acknowledge the presence of missing data values and understand how missing values may add bias to analysis and interpretation. Examine and discuss competing explanations for data trends observed such as confounding variables. Respond to competing arguments or interpretations of the data of different community groups, paying careful attention to what conclusions the data supports, taking into account correlation versus causation.

Collect and consider data.

Write expressions in equivalent forms to solve problems.

Linear, Quadratic, and Exponential Models

Use properties of rational and irrational numbers.

Creating Equations

Build new functions from existing functions.

Interpreting Categorical and Quantitative Data

Seeing Structure in Expressions

Quantities

Arithmetic with Polynomials and Rational Expressions

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

Analyze the data.

Create equations that describe numbers or relationships.

Functions

The Real Number System

Data Science

Solve systems of equations.

Number & Quantity

Interpret results.

Standards for Mathematical Practice

Algebra

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

Represent and solve equations and inequalities graphically.

Build a function that models a relationship between two quantities.

Interpret linear models.

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

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

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

Extend the properties of exponents to rational exponents.

Reason quantitatively and use units to solve problems.

Statistics and Probability

Reason with Equations and Inequalities

Formulate statistical investigative questions.

Solve equations and inequalities in one variable.

Perform arithmetic operations on polynomials.

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

Interpreting Functions

Interpret the structure of expressions.

Understand the concept of a function and use function notation.

Analyze functions using different representations.

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.

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

Flexibly, efficiently, and accurately explain how the definition of the meaning of rational exponents follows from extending the properties of integer exponents to those values using a variety of strategies, allowing for a notation for radicals in terms of rational exponents.

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

Explain why the sum or product of two rational numbers is rational; that the sum of a rational number and an irrational number is irrational; and that the product of a nonzero rational number and an irrational number is irrational.

Represent data with plots on the real number line (dot plots, histograms, 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).

Summarize categorical data for two categories in two-way frequency tables. Interpret relative frequencies in the context of the data (including joint, marginal, and conditional relative frequencies). Recognize possible associations and trends in the data.

Represent data on two quantitative variables on a scatter plot, and describe how the variables are related to solve problems in context by fitting functions to the data and explaining trends and relationships within the data.

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.

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.

Flexibly, efficiently, and accurately demonstrate that polynomials form a system similar to the integers, namely, they are closed under the operations of addition, subtraction, and multiplication; add, subtract, and multiply polynomials.

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.

Prove polynomial identities and use them to describe numerical relationships.

Rewrite simple rational expressions in different forms; write a(x)/b(x) in the form q(x) + r(x)/b(x), where a(x), b(x), q(x), and r(x) are polynomials with the degree of r(x) less than the degree of b(x), using inspection, long division, or, for the more complicated examples, a computer algebra system.

Flexibly, efficiently, and accurately create equations and inequalities in one variable and use them to solve problems.

Flexibly, efficiently, and accurately create equations in two or more variables to represent relationships between quantities; graph equations 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.

Flexibly, efficiently, and accurately rearrange formulas to highlight a quantity of interest, using the same reasoning as in solving equations.

Solve rational and radical equations in one variable, and give examples showing how extraneous solutions may arise.

Solve quadratic equations in one variable by inspection, factoring, completing the square and derive the quadratic formula from this form. Recognize when the quadratic formula give complex solutions and write them as a ± bi for real numbers a and b.

Using a variety of strategies explain why the x-coordinates of the points where the graphs of the equations 𝑦𝑦 = 𝑓𝑓(𝑥𝑥) and 𝑦𝑦 = 𝑔𝑔(𝑥𝑥) intersect are the solutions of the equation 𝑓𝑓(𝑥𝑥) = 𝑔𝑔(𝑥𝑥) find the solutions approximately, e.g., using technology to graph the functions, make tables of values, or find successive approximations. Include cases where 𝑓𝑓(𝑥𝑥) and/or 𝑔𝑔(𝑥𝑥) are linear, polynomial, rational, absolute value, exponential, and logarithmic functions.

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

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

Flexibly, efficiently, and accurately create an equivalent form of an expression to reveal and explain properties of the quantity represented by the expression including factoring quadratic expressions, completing the square in a quadratic expression to reveal maximums or minimums, and using properties of exponents to create equivalent forms of exponential expressions to reveal properties of interest in the function.

Derive the formula for the sum of a finite geometric series (when the common ratio is not 1), and use the formula to solve problems.

Write a function that describes a relationship between two quantities including determining an explicit expression, recursive process, or steps for calculation from a context, and combining standard function types using arithmetic operations.

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

Identify the effect on the graph of replacing 𝑓𝑓(𝑥𝑥) 𝑏𝑏𝑏𝑏 𝑓𝑓(𝑥𝑥) + 𝑘𝑘, 𝑘𝑘 𝑓𝑓(𝑥𝑥), 𝑓𝑓(𝑘𝑘𝑘𝑘), 𝑎𝑎𝑎𝑎𝑎𝑎 𝑓𝑓(𝑥𝑥 + 𝑘𝑘) for specific values of 𝑘𝑘 (both positive and negative); find the value of 𝑘𝑘 given the graphs. Experiment with cases and illustrate an explanation of the effects on the graph using technology.

Find inverse functions through focus on relationships between inputs and outputs.

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. Key features include intercepts; intervals where the function is increasing, decreasing, positive, or negative; relative maximums and minimums; symmetries. Functions can include: polynomial, radical, rational, logarithms, absolute value, piecewise, and trigonometric. Linear, exponential, and quadratic relationships in increased complexity.

Relate the domain of a function to its graph and, where applicable, to the quantitative relationship it describes in context. Functions can include: polynomial, radical, rational, logarithms, absolute value, piecewise, and trigonometric. Linear, exponential, and quadratic relationships in increased complexity.

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 in simple cases and using technology for more complicated cases including square root, cube root, and piecewise-defined functions, including step functions and absolute value functions, polynomial functions, identifying zeros when suitable factorizations are available, and showing end behavior, and exponential and logarithmic functions, showing intercepts and end behavior, and trigonometric functions, showing period, midline, and amplitude.

Write a function defined by an expression in different but equivalent forms to reveal and explain different properties of the function, including factoring and completing the square to reveal zeros, symmetry, and extreme values of a quadratic functions and non-integer constants for time with exponential growth and decay in context.

Compare properties of two functions each represented in a different way (algebraically, graphically, numerically in tables, or by verbal descriptions). Functions can include: polynomial, radical, rational, logarithms, absolute value, piecewise, and trigonometric. Linear, exponential, and quadratic relationships in increased complexity.

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

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.

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

Prove the Pythagorean identity sin2(θ) + cos2(θ) = 1 and use it to find sin(θ), cos(θ), or tan(θ) given sin(θ), cos(θ), or tan(θ) and the quadrant of the angle.

Formulate multivariable statistical investigative questions and determine how data can be collected and provide an answer, consider causality and prediction when posing the question.

Understand the issues of bias and confounding variables when collecting data and their impact on interpretation. Understand practices for collecting and handling data, including sensitive information and concerns for privacy and how that may affect data collection.

Create and analyze data sets and data displays, including but not limited to scatter plots, regressions, histograms and boxplots using technology to sort or filter data, summarize, and describe relationships between quantitative variables.

Acknowledge the presence of missing data values and understand how missing values may add bias to analysis and interpretation. Examine and discuss competing explanations for data trends observed such as confounding variables. Respond to competing arguments or interpretations of the data of different community groups, paying careful attention to what conclusions the data supports, taking into account correlation versus causation.

Represent and solve equations and inequalities graphically.

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

Making Inferences and Justifying Conclusions.

Building Functions

Number & Quantity

Data Science

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

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

Arithmetic with Polynomials and Rational Expressions

Complex Numbers

Reason with Equations and Inequalities

Understand and evaluate random processes underlying statistical experiments.

Build new functions from existing functions.

Build a function that models a relationship between two quantities.

Trigonometric Functions

Collect and consider data.

Creating Equations

Create equations that describe numbers or relationships.

Interpreting Functions

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

Prove and apply trigonometric identities.

Linear, Quadratic, and Exponential Models

Interpret results.

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

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

Algebra

Use complex numbers in polynomial identities and equations.

Interpret the structure of expressions.

Seeing Structure in Expressions

Perform arithmetic operations on polynomials.

Write expressions in equivalent forms to solve problems.

Solve equations and inequalities in one variable.

Formulate statistical investigative questions.

Analyze functions using different representations.

Perform arithmetic operations with complex numbers.

Interpreting Categorical and Quantitative Data

Extend the domain of trigonometric functions using the unit circle.

Standards for Mathematical Practice

Analyze the data.

Statistics and Probability

Functions

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.

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.

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; develop a margin of error through the use of simulation models for random sampling.

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

Evaluate reports based on data.

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.

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.

Flexibly, efficiently, and accurately prove that all circles are similar.

Identify and describe relationships among inscribed angles, radii, and chords, including how angles formed inside the circle, the circle's radius, and line segments within the circle are related. Understand special cases including angles formed by diameters and how the circle's edge interacts with its radius.

Construct the inscribed and circumscribed circles of a triangle and flexibly, efficiently, and accurately prove properties of angles for a quadrilateral inscribed in a circle.

Derive using similarity the fact that the length of the arc intercepted by an angle is proportional to the radius, and define the radian measure of the angle as the constant of proportionality; derive the formula for the area of a sector.

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

Flexibly, efficiently, and accurately represent transformations in the plane, e.g., transparencies and geometry software; describe transformations as functions that take points in the plane as inputs and give other points as outputs. Compare transformations that preserve distance and angle 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 it 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 rotation, reflection, or translation, draw the transformed figure using, e.g., graph paper, tracing paper, or geometry software. Flexibly, efficiently, and accurately specify a sequence of transformations 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 decide 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, and SSS) follow from the definition of congruence in terms of rigid motions.

Flexibly, efficiently, and accurately prove theorems about triangles: interior angles, base angles, segments joining midpoint of two sides, and medians of a triangle.

Flexibly, efficiently, and accurately prove theorems about parallelograms: congruence of opposite sides and opposite angles, properties of diagonals.

Flexibly, efficiently, and accurately prove theorems about lines and angles: vertical, transversals, alternate interior and exterior, perpendicular bisectors, etc.

Make formal geometric constructions with a variety of tools and methods.

Construct an equilateral triangle, a square, and a regular hexagon inscribed in a circle.

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

Use volume formulas for cylinders, pyramids, cones, and spheres to solve problems.

Identify the shapes of two-dimensional cross-sections of three-dimensional objects, and identify three-dimensional objects generated by rotations of two-dimensional objects.

Derive 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 (e.g., find the equation of a line parallel or perpendicular to a given line that passes through a given point).

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

Use coordinates to compute perimeters of polygons and areas of triangles and rectangles, e.g., using the distance formula.

Use geometric shapes, their measures, and their properties to describe objects (e.g., modeling a tree trunk or a human torso as a cylinder).

Apply concepts of density based on area and volume in modeling situations (e.g., persons per square mile, BTUs per cubic foot).

Apply geometric methods to solve design problems (e.g., designing an object or structure to satisfy physical constraints or minimize cost; working with typographic grid systems based on ratios).

Verify experimentally the properties of dilations given by a center and a scale factor by seeing what happens to lines affected by a center of dilation and how scale factor affects line segments.

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.

Flexibly, efficiently, and accurately prove theorems about triangles: proportionality, triangle similarity, and the Pythagorean Theorem.

Flexibly, efficiently, and accurately use congruence and similarity criteria for triangles to solve problems and to prove 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.

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

Formulate multivariable statistical investigative questions and determine how data can be collected and provide an answer, consider causality and prediction when posing the question.

Understand the issues of bias and confounding variables when collecting data and their impact on interpretation. Understand practices for collecting and handling data, including sensitive information and concerns for privacy and how that may affect data collection.

Create and analyze data sets and data displays, including but not limited to scatter plots, regressions, histograms and boxplots using technology to sort or filter data, summarize, and describe relationships between quantitative variables.

Acknowledge the presence of missing data values and understand how missing values may add bias to analysis and interpretation. Examine and discuss competing explanations for data trends observed such as confounding variables. Respond to competing arguments or interpretations of the data of different community groups, paying careful attention to what conclusions the data supports, taking into account correlation versus causation.

Collect and consider data.

Use coordinates to prove simple geometric theorems algebraically.

Expressing Geometric Properties with Equations

Apply geometric concepts in modeling situations.

Conditional Probability and the Rules of Probability

Similarity, Right Triangles, and Trigonometry

Experiment with transformations in the plane.

Interpreting Categorical and Quantitative Data

Use the rules of probability to compute probabilities of compound events.

Understand congruence in terms of rigid motions.

Prove theorems involving similarity

Understand and apply theorems about circles.

Congruence

Interpret linear models.

Geometric Measurement and Dimension

Data Science

Make geometric constructions.

Explain volume formulas and use them to solve problems.

Interpret results.

Define trigonometric ratios and solve problems involving right triangles.

Modeling with Geometry

Standards for Mathematical Practice

Solve real-world and mathematical problems involving area, surface area, and volume.

Statistics and Probability

Find arc lengths and areas of sectors of circles.

Translate between the geometric description and the equation for a conic section.

Understand similarity in terms of similarity transformations.

Circles

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

Geometry

Formulate statistical investigative questions.

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

Analyze the data.

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

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

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.

Understand the conditional probability of 𝐴𝐴 given 𝐵𝐵 as 𝑃𝑃(𝐴𝐴 𝑎𝑎𝑎𝑎𝑎𝑎 𝐵𝐵), 𝑃𝑃(𝐵𝐵) and interpret independence of 𝐴𝐴 and 𝐵𝐵 as saying that the conditional probability of 𝐴𝐴 given 𝐵𝐵 is the same as the probability of 𝐴𝐴, and the conditional probability of 𝐵𝐵 given 𝐴𝐴 is the same as the probability of 𝐵𝐵.

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.

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.

Apply the Addition Rule, 𝑃𝑃(𝐴𝐴 𝑜𝑜𝑜𝑜 𝐵𝐵) = 𝑃𝑃(𝐴𝐴) + 𝑃𝑃(𝐵𝐵) − 𝑃𝑃(𝐴𝐴 𝑎𝑎𝑎𝑎𝑎𝑎 𝐵𝐵), and interpret the answer in terms of the model.

Represent data with plots on the real number line (dot plots, histograms, 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).

Summarize categorical data for two categories in two-way frequency tables. Interpret relative frequencies in the context of the data (including joint, marginal, and conditional relative frequencies). Recognize possible associations and trends in the data.

Represent data on two quantitative variables on a scatter plot, and describe how the variables are related to solve problems in context by fitting functions to the data and explaining trends and relationships within the data.

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.

Perform arithmetic operations on polynomials

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.

Understand the relationship between zeros and factors of polynomials

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.

Use polynomial identities to solve problems

Prove polynomial identities and use them to describe numerical relationships.

(+) 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.

Rewrite rational expressions

Rewrite simple rational expressions in different forms; write ^a(x/_b(x) in the form q(x) + ^r(x)/_b(x), where a(x), b(x), q(x), and r(x) are polynomials with the degree of r(x) less than the degree of b(x), using inspection, long division, or, for the more complicated examples, a computer algebra system.

(+) 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.

Create equations that describe numbers or relationships

Create equations and inequalities in one variable and use them to solve problems. Include equations arising from linear and quadratic functions, and simple rational and exponential functions.

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

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

Explain each step in solving a simple 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 simple rational and radical equations in one variable, and give examples showing how extraneous solutions may arise.

Solve equations and inequalities in one variable

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

Solve quadratic equations in one variable.

Use the method of completing the square to transform 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, 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 systems of equations

Prove that, given a system of two equations in two variables, replacing one equation by the sum of that equation and a multiple of the other produces a system with the same solutions.

Solve systems of linear equations exactly and approximately (e.g., with graphs), focusing on pairs of linear equations in two variables.

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

(+) 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).

Represent and solve equations and inequalities graphically

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 solutions approximately, e.g., using technology to graph the functions, make tables of values, or find successive approximations. Include cases where f(x) and/or g(x) are linear, polynomial, rational, absolute value, exponential, and logarithmic functions.

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.

Interpret the structure of expressions

Interpret expressions that represent a quantity in terms of its context

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 structure of an expression to identify ways to rewrite it.

Write expressions in equivalent forms to solve problems

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

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

Complete the square in a quadratic expression to reveal the maximum or minimum value of the function it defines.

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

Derive the formula for the sum of a finite geometric series (when the common ratio is not 1), and use the formula to solve problems.

Build a function that models a relationship between two quantities

Write a function that describes a relationship between two quantities

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

Combine standard function types using arithmetic operations.

(+) Compose functions.

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

Build new functions from existing functions

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. Include recognizing even and odd functions from their graphs and algebraic expressions for them.

Find inverse functions.

Solve an equation of the form f(x) = c for a simple function f that has an inverse and write an expression for the inverse.

(+) 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.

(+) Produce an invertible function from a non-invertible function by restricting the domain.

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

Understand the concept of a function and use function notation

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

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.

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.

Analyze functions using different representations

Graph functions expressed symbolically and show key features of the graph, by hand in simple cases and using technology for more complicated cases.

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 rational functions, identifying zeros and asymptotes when suitable factorizations are available, and showing end behavior.

Graph exponential and logarithmic functions, showing intercepts and end behavior, and trigonometric functions, showing period, midline, and amplitude.

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

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.

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

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

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

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

Prove 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 percent rate per unit interval relative to another.

Construct linear and exponential functions, including arithmetic and geometric sequences, given a graph, a description of a relationship, or two input-output pairs (include reading these from a table).

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.

Interpret expressions for functions in terms of the situation they model

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

Extend the domain of trigonometric functions using the unit circle

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

Model periodic phenomena with trigonometric functions

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

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

(+) 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.

Prove and apply trigonometric identities

Prove the Pythagorean identity sin²(θ) + cos²(θ) = 1 and use it to find sin(θ), cos(θ), or tan(θ) given sin(θ), cos(θ), or tan(θ) and the quadrant of the angle.

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

Understand and apply theorems about circles

Prove that all circles are similar.

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

Construct the inscribed and circumscribed circles of a triangle, and prove properties of angles for a quadrilateral inscribed in a circle.

(+) Construct a tangent line from a point outside a given circle to the circle.

Find arc lengths and areas of sectors of circles

Derive using similarity the fact that the length of the arc intercepted by an angle is proportional to the radius, and define the radian measure of the angle as the constant of proportionality; derive the formula for the area of a sector.

Experiment with transformations in the plane

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

Represent transformations in the plane using, e.g., transparencies and geometry software; describe transformations as functions that take points in the plane as inputs and give other points as outputs. Compare transformations that preserve distance and angle 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 it 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 rotation, reflection, or translation, draw the transformed figure using, e.g., graph paper, tracing paper, or geometry software. Specify a sequence of transformations that will carry a given figure onto another.

Understand congruence in terms of rigid motions

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 decide 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, and SSS) follow from the definition of congruence in terms of rigid motions.

Prove geometric theorems

Prove theorems about triangles.

Prove theorems about parallelograms.

Prove theorems about lines and angles.

Make geometric constructions

Make formal geometric constructions with a variety of tools and methods (compass and straightedge, string, reflective devices, paper folding, dynamic geometric software, etc.). Copying a segment; copying an angle; bisecting a segment; bisecting an angle; constructing perpendicular lines, including the perpendicular bisector of a line segment; and constructing a line parallel to a given line through a point not on the line.

Construct an equilateral triangle, a square, and a regular hexagon inscribed in a circle.

Explain volume formulas and use them to solve problems

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

(+) Give an informal argument using Cavalieri's principle for the formulas for the volume of a sphere and other solid figures.

Use volume formulas for cylinders, pyramids, cones, and spheres to solve problems.

Visualize relationships between two-dimensional and three-dimensional objects

Identify the shapes of two-dimensional cross-sections of three-dimensional objects, and identify three-dimensional objects generated by rotations of two-dimensional objects.

Translate between the geometric description and the equation for a conic section

Derive the equation of a circle of given center and radius using the Pythagorean Theorem; complete the square to find the center and radius of a circle given by an equation.

Derive the equation of a parabola given a focus and directrix.

(+) 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.

Use coordinates to prove simple geometric theorems algebraically

Use coordinates to prove simple geometric theorems algebraically.

Prove the slope criteria for parallel and perpendicular lines and use them to solve geometric problems (e.g., find the equation of a line parallel or perpendicular to a given line that passes through a given point).

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

Use coordinates to compute perimeters of polygons and areas of triangles and rectangles, e.g., using the distance formula.

Apply geometric concepts in modeling situations

Use geometric shapes, their measures, and their properties to describe objects (e.g., modeling a tree trunk or a human torso as a cylinder).

Apply concepts of density based on area and volume in modeling situations (e.g., persons per square mile, BTUs per cubic foot).

Apply geometric methods to solve design problems (e.g., designing an object or structure to satisfy physical constraints or minimize cost; working with typographic grid systems based on ratios).

Understand similarity in terms of similarity transformations

Verify experimentally the properties of dilations given by a center and a scale factor:

A dilation takes a line not passing through the center of the dilation to a parallel line, and leaves a line passing through the center unchanged.

The dilation of a line segment is longer or shorter in the ratio given by the 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 involving similarity

Prove theorems about triangles.

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

Define trigonometric ratios and solve problems involving right triangles

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.

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

Apply trigonometry to general triangles

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

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

(+) Derive the formula A = 1/2 ab sin(C) for the area of a triangle by drawing an auxiliary line from a vertex perpendicular to the opposite side.

Perform arithmetic operations with complex 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.

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

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

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.

(+) 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.

Use complex numbers in polynomial identities and equations.

Solve quadratic equations with real coefficients that have complex solutions.

(+) Extend polynomial identities to the complex numbers.

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

Reason quantitatively and use units 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.

Define appropriate quantities for the purpose of descriptive modeling.

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

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.

Use properties of rational and irrational numbers.

Explain why the sum or product of two rational numbers is rational; that the sum of a rational number and an irrational number is irrational; and that the product of a nonzero rational number and an irrational number is irrational.

Represent and model with vector quantities.

(+) 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.

Perform operations on vectors.

(+) Add and subtract vectors.

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 perform vector subtraction component-wise.

(+) Multiply a vector by a scalar.

Represent scalar multiplication graphically by scaling vectors and possibly reversing their direction; perform scalar multiplication component-wise, e.g., as c(v_x, v_y) = (cv_x, cv_y).

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).

Perform operations on matrices and use matrices in applications.

(+) 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.

(+) 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 independence and conditional probability and use them to interpret data

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.

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.

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.

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

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 compute probabilities of compound events and solve problems.

Understand and evaluate random processes underlying statistical experiments

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.

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

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; develop a margin of error through the use of simulation models for random sampling.

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

Evaluate reports based on data.

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

Represent data with plots on the real number line (dot plots, histograms, 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).

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.

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

Summarize categorical data for two categories in two-way frequency tables. Interpret relative frequencies in the context of the data (including joint, marginal, and conditional relative frequencies). Recognize possible associations and trends in the data.

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

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

Informally assess the fit of a function by plotting and analyzing residuals.

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

Interpret linear models

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.

Calculate expected values and use them to solve problems

(+) 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.

(+) 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.

Use probability to evaluate outcomes of decisions

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

Find the expected payoff for a game of chance.

Evaluate and compare strategies on the basis of 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).

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.

High School — Functions

High School — Statistics and Probability

The Real Number System

Circles

Interpreting Functions

Modeling with Geometry

Congruence

Trigonometric Functions

High School — Geometry

Creating Equations

Standards for Mathematical Practice

Vector and Matrix Quantities

High School — Algebra

Geometric Measurement and Dimension

Arithmetic with Polynomials and Rational Expressions

Making Inferences and Justifying Conclusions

Quantities

Seeing Structure in Expressions

Building Functions

High School — Number and Quantity

Conditional Probability and the Rules of Probability

The Complex Number System

Similarity, Right Triangles, and Trigonometry

Expressing Geometric Properties with Equations

Linear, Quadratic, and Exponential Models

Using Probability to Make Decisions

Interpreting Categorical and Quantitative Data

Reasoning with Equations and Inequalities

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.

Flexibly, efficiently, and accurately demonstrate that polynomials form a system similar to the integers, namely, they are closed under the operations of addition, subtraction, and multiplication; add, subtract, and multiply polynomials.

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.

Prove polynomial identities and use them to describe numerical relationships.

Rewrite simple rational expressions in different forms; write a(x)/b(x) in the form q(x) + r(x)/b(x), where a(x), b(x), q(x), and r(x) are polynomials with the degree of r(x) less than the degree of b(x), using inspection, long division, or, for the more complicated examples, a computer algebra system.

Flexibly, efficiently, and accurately create equations and inequalities in one variable and use them to solve problems.

Flexibly, efficiently, and accurately create equations in two or more variables to represent relationships between quantities; graph equations 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.

Flexibly, efficiently, and accurately rearrange formulas to highlight a quantity of interest, using the same reasoning as in solving equations.

Explain each step in solving a simple 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 give examples showing how extraneous solutions may arise.

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

Solve quadratic equations in one variable by inspection, factoring, completing the square and derive the quadratic formula from this form. Recognize when the quadratic formula give complex solutions and write them as a ± bi for real numbers a and b.

Demonstrate using a variety of strategies that, given a system of two equations in two variables, replacing one equation by the sum of that equation and a multiple of the other produces a system with the same solutions.

Flexibly, efficiently, and accurately solve systems of linear equations exactly and approximately (e.g., with graphs), focusing on pairs of linear equations in two variables.

Flexibly, efficiently, and accurately solve a simple system consisting of a linear equation and a quadratic equation in two variables algebraically and graphically.

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).

Using a variety of strategies explain why the x-coordinates of the points where the graphs of the equations 𝑦𝑦 = 𝑓𝑓(𝑥𝑥) and 𝑦𝑦 = 𝑔𝑔(𝑥𝑥) intersect are the solutions of the equation 𝑓𝑓(𝑥𝑥) = 𝑔𝑔(𝑥𝑥) find the solutions approximately, e.g., using technology to graph the functions, make tables of values, or find successive approximations. Include cases where 𝑓𝑓(𝑥𝑥) and/or 𝑔𝑔(𝑥𝑥) are linear, polynomial, rational, absolute value, exponential, and logarithmic functions.

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.

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

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

Flexibly, efficiently, and accurately create an equivalent form of an expression to reveal and explain properties of the quantity represented by the expression including factoring quadratic expressions, completing the square in a quadratic expression to reveal maximums or minimums, and using properties of exponents to create equivalent forms of exponential expressions to reveal properties of interest in the function.

Derive the formula for the sum of a finite geometric series (when the common ratio is not 1), and use the formula to solve problems.

Write a function that describes a relationship between two quantities including determining an explicit expression, recursive process, or steps for calculation from a context, and combining standard function types using arithmetic operations.

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

Identify the effect on the graph of replacing 𝑓𝑓(𝑥𝑥) 𝑏𝑏𝑏𝑏 𝑓𝑓(𝑥𝑥) + 𝑘𝑘, 𝑘𝑘 𝑓𝑓(𝑥𝑥), 𝑓𝑓(𝑘𝑘𝑘𝑘), 𝑎𝑎𝑎𝑎𝑎𝑎 𝑓𝑓(𝑥𝑥 + 𝑘𝑘) for specific values of 𝑘𝑘 (both positive and negative); find the value of 𝑘𝑘 given the graphs. Experiment with cases and illustrate an explanation of the effects on the graph using technology.

Find inverse functions through focus on relationships between inputs and outputs.

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 𝑓𝑓 is a function and x is an element of its domain, then 𝑓𝑓(𝑥𝑥) denotes the output of f corresponding to the input 𝑥𝑥. The graph of f is the graph of the equation 𝑦𝑦 = 𝑓𝑓(𝑥𝑥).

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

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

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. Key features include intercepts; intervals where the function is increasing, decreasing, positive, or negative; relative maximums and minimums; symmetries. Functions can include: polynomial, radical, rational, logarithms, absolute value, piecewise, and trigonometric. Linear, exponential, and quadratic relationships in increased complexity.

Relate the domain of a function to its graph and, where applicable, to the quantitative relationship it describes in context. Functions can include: polynomial, radical, rational, logarithms, absolute value, piecewise, and trigonometric. Linear, exponential, and quadratic relationships in increased complexity.

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 in simple cases and using technology for more complicated cases including linear, quadratic, exponential, square root, cube root, and piecewise-defined functions, including step functions and absolute value functions, polynomial functions, identifying zeros when suitable factorizations are available, and showing end behavior, and exponential and logarithmic functions, showing intercepts and end behavior, and trigonometric functions, showing period, midline, and amplitude.

Write a function defined by an expression in different but equivalent forms to reveal and explain different properties of the function, including factoring and completing the square to reveal zeros, symmetry, and extreme values of a quadratic functions and non-integer constants for time with exponential growth and decay in context.

Compare properties of two functions each represented in a different way (algebraically, graphically, numerically in tables, or by verbal descriptions). Functions can include: polynomial, radical, rational, logarithms, absolute value, piecewise, and trigonometric. Linear, exponential, and quadratic relationships in increased complexity.

Distinguish between situations that can be modeled with linear functions (equal differences over equal intervals) and with exponential functions (equal factors over equal intervals), recognizing constant rates per unit interval, and growth or decay by a constant percent rate per unit interval.

Flexibly, efficiently, and accurately construct linear and exponential functions given a graph, a description of a relationship, or two input-output pairs (include reading these from a table).

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

For exponential models, express as a logarithm the solution to abct = 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.

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.

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

Prove the Pythagorean identity sin2(θ) + cos2(θ) = 1 and use it to find sin(θ), cos(θ), or tan(θ) given sin(θ), cos(θ), or tan(θ) and the quadrant of the angle.

Flexibly, efficiently, and accurately prove that all circles are similar.

Identify and describe relationships among inscribed angles, radii, and chords, including how angles formed inside the circle, the circle's radius, and line segments within the circle are related. Understand special cases including angles formed by diameters and how the circle's edge interacts with its radius.

Construct the inscribed and circumscribed circles of a triangle and flexibly, efficiently, and accurately prove properties of angles for a quadrilateral inscribed in a circle.

Derive using similarity the fact that the length of the arc intercepted by an angle is proportional to the radius, and define the radian measure of the angle as the constant of proportionality; derive the formula for the area of a sector.

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

Flexibly, efficiently, and accurately represent transformations in the plane, e.g., transparencies and geometry software; describe transformations as functions that take points in the plane as inputs and give other points as outputs. Compare transformations that preserve distance and angle 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 it 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 rotation, reflection, or translation, draw the transformed figure using, e.g., graph paper, tracing paper, or geometry software. Specify a sequence of transformations 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 decide 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, and SSS) follow from the definition of congruence in terms of rigid motions.

Flexibly, efficiently, and accurately prove theorems about triangles: interior angles, base angles, segments joining midpoint of two sides, and medians of a triangle.

Flexibly, efficiently, and accurately prove theorems about parallelograms: congruence of opposite sides and opposite angles, properties of diagonals.

Flexibly, efficiently, and accurately prove theorems about lines and angles: vertical, transversals, alternate interior and exterior, perpendicular bisectors, etc.

Make formal geometric constructions with a variety of tools and methods.

Construct an equilateral triangle, a square, and a regular hexagon inscribed in a circle.

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

Use volume formulas for cylinders, pyramids, cones, and spheres to solve problems.

Identify the shapes of two-dimensional cross-sections of three-dimensional objects, and identify three-dimensional objects generated by rotations of two-dimensional objects.

Derive the equation of a circle of given center and radius using the Pythagorean Theorem; complete the square to find the center and radius of a circle given by an equation.

Use coordinates to prove simple geometric theorems algebraically.

Prove the slope criteria for parallel and perpendicular lines and use them to solve geometric problems (e.g., find the equation of a line parallel or perpendicular to a given line that passes through a given point).

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

Use coordinates to compute perimeters of polygons and areas of triangles and rectangles, e.g., using the distance formula.

Use geometric shapes, their measures, and their properties to describe objects (e.g., modeling a tree trunk or a human torso as a cylinder).

Apply concepts of density based on area and volume in modeling situations (e.g., persons per square mile, BTUs per cubic foot).

Apply geometric methods to solve design problems (e.g., designing an object or structure to satisfy physical constraints or minimize cost; working with typographic grid systems based on ratios).

Verify experimentally the properties of dilations given by a center and a scale factor by seeing what happens to lines affected by a center of dilation and how scale factor affects line segments.

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.

Flexibly, efficiently, and accurately prove theorems about triangles: proportionality, triangle similarity, and the Pythagorean Theorem.

Flexibly, efficiently, and accurately use congruence and similarity criteria for triangles to solve problems and to prove 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.

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

Formulate multivariable statistical investigative questions and determine how data can be collected and provide an answer, consider causality and prediction when posing the question.

Understand the issues of bias and confounding variables when collecting data and their impact on interpretation. Understand practices for collecting and handling data, including sensitive information and concerns for privacy and how that may affect data collection.

Create and analyze data sets and data displays, including but not limited to scatter plots, regressions, histograms and boxplots using technology to sort or filter data, summarize, and describe relationships between quantitative variables.

Acknowledge the presence of missing data values and understand how missing values may add bias to analysis and interpretation. Examine and discuss competing explanations for data trends observed such as confounding variables. Respond to competing arguments or interpretations of the data of different community groups, paying careful attention to what conclusions the data supports, taking into account correlation versus causation.

Write expressions in equivalent forms to solve problems.

Define trigonometric ratios and solve problems involving right triangles.

Analyze functions using different representations.

Functions

Understand the concept of a function and use function notation.

Translate between the geometric description and the equation for a conic section.

Use complex numbers in polynomial identities and equations.

Make geometric constructions.

Data Science

Geometric Measurement and Dimension

Use coordinates to prove simple geometric theorems algebraically.

Use the rules of probability to compute probabilities of compound events.

Solve systems of equations.

Solve equations and inequalities in one variable.

Use properties of rational and irrational numbers.

Arithmetic with Polynomials and Rational Expressions

Extend the properties of exponents to rational exponents.

Understand similarity in terms of similarity transformations.

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

Congruence

Analyze the data.

Extend the domain of trigonometric functions using the unit circle.

Conditional Probability and the Rules of Probability

Perform arithmetic operations with complex numbers.

Create equations that describe numbers or relationships.

Linear, Quadratic, and Exponential Models

Explain volume formulas and use them to solve problems.

Interpreting Categorical and Quantitative Data

Find arc lengths and areas of sectors of circles.

Modeling with Geometry

Formulate statistical investigative questions.

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

Perform arithmetic operations on polynomials.

Solve real-world and mathematical problems involving area, surface area, and volume.

Build a function that models a relationship between two quantities.

Interpreting Functions

Circles

Building Functions

Algebra

Seeing Structure in Expressions

Represent and solve equations and inequalities graphically.

Expressing Geometric Properties with Equations

Understand and evaluate random processes underlying statistical experiments.

Collect and consider data.

Interpret results.

Prove and apply trigonometric identities.

Reason quantitatively and use units to solve problems.

Interpret linear models.

Creating Equations

Prove theorems involving similarity

Complex Numbers

The Real Number System

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

Interpret the structure of expressions.

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

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

Experiment with transformations in the plane.

Geometry

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

Trigonometric Functions

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

Statistics and Probability

Understand congruence in terms of rigid motions.

Making Inferences and Justifying Conclusions.

Standards for Mathematical Practice

Understand and apply theorems about circles.

Build new functions from existing functions.

Similarity, Right Triangles, and Trigonometry

Apply geometric concepts in modeling situations.

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

Reason with Equations and Inequalities

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

Number & Quantity

Quantities

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

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.

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 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.

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

Flexibly, efficiently, and accurately explain how the definition of the meaning of rational exponents follows from extending the properties of integer exponents to those values using a variety of strategies, allowing for a notation for radicals in terms of rational exponents.

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

Explain why the sum or product of two rational numbers is rational; that the sum of a rational number and an irrational number is irrational; and that the product of a nonzero rational number and an irrational number is irrational.

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.

Understand the conditional probability of 𝐴𝐴 given 𝐵𝐵 as 𝑃𝑃(𝐴𝐴 𝑎𝑎𝑎𝑎𝑎𝑎 𝐵𝐵), 𝑃𝑃(𝐵𝐵) and interpret independence of 𝐴𝐴 and 𝐵𝐵 as saying that the conditional probability of 𝐴𝐴 given 𝐵𝐵 is the same as the probability of 𝐴𝐴, and the conditional probability of 𝐵𝐵 given 𝐴𝐴 is the same as the probability of 𝐵𝐵.

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.

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.

Apply the Addition Rule, 𝑃𝑃(𝐴𝐴 𝑜𝑜𝑜𝑜 𝐵𝐵) = 𝑃𝑃(𝐴𝐴) + 𝑃𝑃(𝐵𝐵) − 𝑃𝑃(𝐴𝐴 𝑎𝑎𝑎𝑎𝑎𝑎 𝐵𝐵), and interpret the answer in terms of the model.

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; develop a margin of error through the use of simulation models for random sampling.

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

Evaluate reports based on data.

Represent data with plots on the real number line (dot plots, histograms, 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).

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.

Summarize categorical data for two categories in two-way frequency tables. Interpret relative frequencies in the context of the data (including joint, marginal, and conditional relative frequencies). Recognize possible associations and trends in the data.

Represent data on two quantitative variables on a scatter plot, and describe how the variables are related to solve problems in context by fitting functions to the data and explaining trends and relationships within the data.

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.

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.

Flexibly, efficiently, and accurately demonstrate that polynomials form a system similar to the integers, namely, they are closed under the operations of addition, subtraction, and multiplication; add, subtract, and multiply polynomials.

Flexibly, efficiently, and accurately create equations and inequalities in one variable and use them to solve problems. Include equations arising from linear, quadratic, and exponential functions.

Flexibly, efficiently, and accurately create linear, quadratic, exponential equations to represent relationships between quantities; graph equations 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 within linear, quadratic, and exponential equations.

Flexibly, efficiently, and accurately rearrange formulas to highlight a quantity of interest, using the same reasoning as in solving equations within linear, quadratic, and exponential equations.

Explain each step in solving an equation as following from the equality of numbers asserted at the previous step flexibly, efficiently, and accurately selecting and demonstrating use of strategies to solve equations, 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 in one variable by inspection, taking square roots, and factoring as appropriate to the initial form of the equation.

Demonstrate using a variety of strategies that, given a system of two equations in two variables, replacing one equation by the sum of that equation and a multiple of the other produces a system with the same solutions.

Flexibly, efficiently, and accurately solve systems of linear equations exactly and approximately (e.g., with graphs), focusing on pairs of linear equations in two variables.

Flexibly, efficiently, and accurately solve a simple system consisting of a linear equation and a quadratic equation in two variables algebraically and graphically.

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).

Using a variety of strategies explain the x-coordinates of the points where the graphs of the equations 𝑦𝑦 = 𝑓𝑓(𝑥𝑥) and 𝑦𝑦 = 𝑔𝑔(𝑥𝑥) intersect are the solutions of the equation 𝑓𝑓(𝑥𝑥) = 𝑔𝑔(𝑥𝑥); find the solutions approximately, e.g., using technology to graph the functions, make tables of values, or find successive approximations. Include cases where 𝑓𝑓(𝑥𝑥) and/or 𝑔𝑔(𝑥𝑥) are linear, exponential, and quadratic.

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.

Interpret expressions that represent a quantity in terms of its context within linear, exponential, and quadratic functions.

Use the structure of an expression to identify ways to rewrite it within exponential and quadratic functions.

Flexibly, efficiently, and accurately create an equivalent form of an expression to reveal and explain properties of the quantity represented by the expression including factoring quadratic expressions and using properties of exponents to create equivalent forms of exponential expressions to reveal properties of interest in the function.

Flexibly, efficiently, and accurately write a function that describes a relationship between two quantities, including linear and exponential arithmetic and geometric sequences in context.

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

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. Using a variety of strategies, experiment with cases and illustrate an explanation of the effects on the graph using technology.

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 𝑓𝑓 is a function and x is an element of its domain, then 𝑓𝑓(𝑥𝑥) denotes the output of f corresponding to the input 𝑥𝑥. The graph of f is the graph of the equation 𝑦𝑦 = 𝑓𝑓(𝑥𝑥).

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

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

For a function that models a relationship between two quantities in context, 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. Key features include intercepts; intervals where the function is increasing, decreasing, positive, or negative; relative maximums and minimums; symmetries for functions including linear, exponential, and quadratic.

Relate the domain of a function to its graph and, where applicable, to the quantitative relationship it describes in linear, exponential, or quadratic contexts.

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

Graph linear, exponential, and quadratic functions expressed symbolically and show key features of the graph, including intercepts, maximum, minimum, and interpreting end behavior for exponential functions by hand in simple cases and using technology for more complicated cases.

Flexibly, efficiently, and accurately write a function defined by an expression in different but equivalent forms to reveal and explain different properties of the function including zeros and symmetry, using factoring for quadratic functions and integer constants for time with exponential growth and decay.

Compare properties of two functions each represented in a different way (algebraically, graphically, numerically in tables, or by verbal descriptions). Functions could be linear, exponential, or quadratic.

Distinguish between situations that can be modeled with linear functions (equal differences over equal intervals) and with exponential functions (equal factors over equal intervals), recognizing constant rates per unit interval, and growth or decay by a constant percent rate per unit interval.

Flexibly, efficiently, and accurately construct linear and exponential functions given a graph, a description of a relationship, or two input-output pairs (include reading these from a table).

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

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

Flexibly, efficiently, and accurately prove that all circles are similar.

Identify and describe relationships among inscribed angles, radii, and chords, including how angles formed inside the circle, the circle's radius, and line segments within the circle are related. Understand special cases including angles formed by diameters and how the circle's edge interacts with its radius.

Construct the inscribed and circumscribed circles of a triangle and flexibly, efficiently, and accurately prove properties of angles for a quadrilateral inscribed in a circle.

Derive using similarity the fact that the length of the arc intercepted by an angle is proportional to the radius, and define the radian measure of the angle as the constant of proportionality; derive the formula for the area of a sector.

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

Flexibly, efficiently, and accurately represent transformations in the plane, e.g., transparencies and geometry software; describe transformations as functions that take points in the plane as inputs and give other points as outputs. Compare transformations that preserve distance and angle 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 it 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 rotation, reflection, or translation, draw the transformed figure using, e.g., graph paper, tracing paper, or geometry software. Flexibly, efficiently, and accurately specify a sequence of transformations 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 decide 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, and SSS) follow from the definition of congruence in terms of rigid motions.

Flexibly, efficiently, and accurately prove theorems about triangles: interior angles, base angles, segments joining midpoint of two sides, and medians of a triangle.

Flexibly, efficiently, and accurately prove theorems about parallelograms: congruence of opposite sides and opposite angles, properties of diagonals.

Flexibly, efficiently, and accurately prove theorems about lines and angles: vertical, transversals, alternate interior and exterior, perpendicular bisectors, etc.

Make formal geometric constructions with a variety of tools and methods.

Construct an equilateral triangle, a square, and a regular hexagon inscribed in a circle.

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

Use volume formulas for cylinders, pyramids, cones, and spheres to solve problems.

Identify the shapes of two-dimensional cross-sections of three-dimensional objects, and identify three-dimensional objects generated by rotations of two-dimensional objects.

Derive 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 (e.g., find the equation of a line parallel or perpendicular to a given line that passes through a given point).

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

Use coordinates to compute perimeters of polygons and areas of triangles and rectangles, e.g., using the distance formula.

Use geometric shapes, their measures, and their properties to describe objects (e.g., modeling a tree trunk or a human torso as a cylinder).

Apply concepts of density based on area and volume in modeling situations (e.g., persons per square mile, BTUs per cubic foot).

Apply geometric methods to solve design problems (e.g., designing an object or structure to satisfy physical constraints or minimize cost; working with typographic grid systems based on ratios).

Verify experimentally the properties of dilations given by a center and a scale factor by seeing what happens to lines affected by a center of dilation and how scale factor affects line segments.

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.

Flexibly, efficiently, and accurately prove theorems about triangles: proportionality, triangle similarity, and the Pythagorean Theorem.

Flexibly, efficiently, and accurately use congruence and similarity criteria for triangles to solve problems and to prove 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.

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

Formulate multivariable statistical investigative questions and determine how data can be collected and provide an answer, consider causality and prediction when posing the question.

Understand the issues of bias and confounding variables when collecting data and their impact on interpretation. Understand practices for collecting and handling data, including sensitive information and concerns for privacy and how that may affect data collection.

Create and analyze data sets and data displays, including but not limited to scatter plots, regressions, histograms and boxplots using technology to sort or filter data, summarize, and describe relationships between quantitative variables.

Acknowledge the presence of missing data values and understand how missing values may add bias to analysis and interpretation. Examine and discuss competing explanations for data trends observed such as confounding variables. Respond to competing arguments or interpretations of the data of different community groups, paying careful attention to what conclusions the data supports, taking into account correlation versus causation.

Reason quantitatively and use units to solve problems.

Standards for Mathematical Practice

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

Find arc lengths and areas of sectors of circles.

Circles

Experiment with transformations in the plane.

Represent and solve equations and inequalities graphically.

Interpret results.

Seeing Structure in Expressions

Expressing Geometric Properties with Equations

Write expressions in equivalent forms to solve problems.

Perform arithmetic operations on polynomials.

Build new functions from existing functions.

Prove theorems involving similarity

Understand and apply theorems about circles.

Understand congruence in terms of rigid motions.

Functions

Define trigonometric ratios and solve problems involving right triangles.

Analyze the data.

Statistics and Probability

Creating Equations

Build a function that models a relationship between two quantities.

Linear, Quadratic, and Exponential Models

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

Conditional Probability and the Rules of Probability

Arithmetic with Polynomials and Rational Expressions

Use coordinates to prove simple geometric theorems algebraically.

Modeling with Geometry

Data Science

Interpret linear models.

Geometric Measurement and Dimension

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

Explain volume formulas and use them to solve problems.

Apply geometric concepts in modeling situations.

Reason with Equations and Inequalities

Analyze functions using different representations.

Translate between the geometric description and the equation for a conic section.

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

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

Use the rules of probability to compute probabilities of compound events.

Formulate statistical investigative questions.

Collect and consider data.

Interpret the structure of expressions.

Algebra

Make geometric constructions.

Solve systems of equations.

Congruence

Quantities

Understand the concept of a function and use function notation.

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

Similarity, Right Triangles, and Trigonometry

Number & Quantity

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

Use properties of rational and irrational numbers.

Interpreting Functions

Interpreting Categorical and Quantitative Data

Extend the properties of exponents to rational exponents.

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

Solve equations and inequalities in one variable.

Solve real-world and mathematical problems involving area, surface area, and volume.

Understand similarity in terms of similarity transformations.

The Real Number System

Create equations that describe numbers or relationships.

Geometry

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.

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

Flexibly, efficiently, and accurately explain how the definition of the meaning of rational exponents follows from extending the properties of integer exponents to those values using a variety of strategies, allowing for a notation for radicals in terms of rational exponents.

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

Explain why the sum or product of two rational numbers is rational; that the sum of a rational number and an irrational number is irrational; and that the product of a nonzero rational number and an irrational number is irrational.

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.

Understand the conditional probability of 𝐴𝐴 given 𝐵𝐵 as 𝑃𝑃(𝐴𝐴 𝑎𝑎𝑎𝑎𝑎𝑎 𝐵𝐵), 𝑃𝑃(𝐵𝐵) and interpret independence of 𝐴𝐴 and 𝐵𝐵 as saying that the conditional probability of 𝐴𝐴 given 𝐵𝐵 is the same as the probability of 𝐴𝐴, and the conditional probability of 𝐵𝐵 given 𝐴𝐴 is the same as the probability of 𝐵𝐵.

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.

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.

Apply the Addition Rule, 𝑃𝑃(𝐴𝐴 𝑜𝑜𝑜𝑜 𝐵𝐵) = 𝑃𝑃(𝐴𝐴) + 𝑃𝑃(𝐵𝐵) − 𝑃𝑃(𝐴𝐴 𝑎𝑎𝑎𝑎𝑎𝑎 𝐵𝐵), and interpret the answer in terms of the model.

Represent data with plots on the real number line (dot plots, histograms, 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).

Summarize categorical data for two categories in two-way frequency tables. Interpret relative frequencies in the context of the data (including joint, marginal, and conditional relative frequencies). Recognize possible associations and trends in the data.

Represent data on two quantitative variables on a scatter plot, and describe how the variables are related to solve problems in context by fitting functions to the data and explaining trends and relationships within the data.

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.

Perform arithmetic operations on polynomials

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.

Understand the relationship between zeros and factors of polynomials

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.

Use polynomial identities to solve problems

Prove polynomial identities and use them to describe numerical relationships.

(+) 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.

Rewrite rational expressions

Rewrite simple rational expressions in different forms; write ^a(x/_b(x) in the form q(x) + ^r(x)/_b(x), where a(x), b(x), q(x), and r(x) are polynomials with the degree of r(x) less than the degree of b(x), using inspection, long division, or, for the more complicated examples, a computer algebra system.

(+) 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.

Create equations that describe numbers or relationships

Create equations and inequalities in one variable and use them to solve problems. Include equations arising from linear and quadratic functions, and simple rational and exponential functions.

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

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

Explain each step in solving a simple 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 simple rational and radical equations in one variable, and give examples showing how extraneous solutions may arise.

Solve equations and inequalities in one variable

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

Solve quadratic equations in one variable.

Use the method of completing the square to transform 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, 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 systems of equations

Prove that, given a system of two equations in two variables, replacing one equation by the sum of that equation and a multiple of the other produces a system with the same solutions.

Solve systems of linear equations exactly and approximately (e.g., with graphs), focusing on pairs of linear equations in two variables.

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

(+) 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).

Represent and solve equations and inequalities graphically

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 solutions approximately, e.g., using technology to graph the functions, make tables of values, or find successive approximations. Include cases where f(x) and/or g(x) are linear, polynomial, rational, absolute value, exponential, and logarithmic functions.

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.

Interpret the structure of expressions

Interpret expressions that represent a quantity in terms of its context

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 structure of an expression to identify ways to rewrite it.

Write expressions in equivalent forms to solve problems

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

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

Complete the square in a quadratic expression to reveal the maximum or minimum value of the function it defines.

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

Derive the formula for the sum of a finite geometric series (when the common ratio is not 1), and use the formula to solve problems.

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.

Standards for Mathematical Practice

Reasoning with Equations and Inequalities

Seeing Structure in Expressions

Arithmetic with Polynomials and Rational Expressions

Creating Equations

Build a function that models a relationship between two quantities

Write a function that describes a relationship between two quantities

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

Combine standard function types using arithmetic operations.

(+) Compose functions.

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

Build new functions from existing functions

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. Include recognizing even and odd functions from their graphs and algebraic expressions for them.

Find inverse functions.

Solve an equation of the form f(x) = c for a simple function f that has an inverse and write an expression for the inverse.

(+) 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.

(+) Produce an invertible function from a non-invertible function by restricting the domain.

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

Understand the concept of a function and use function notation

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

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.

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.

Analyze functions using different representations

Graph functions expressed symbolically and show key features of the graph, by hand in simple cases and using technology for more complicated cases.

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 rational functions, identifying zeros and asymptotes when suitable factorizations are available, and showing end behavior.

Graph exponential and logarithmic functions, showing intercepts and end behavior, and trigonometric functions, showing period, midline, and amplitude.

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

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.

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

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

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

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

Prove 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 percent rate per unit interval relative to another.

Construct linear and exponential functions, including arithmetic and geometric sequences, given a graph, a description of a relationship, or two input-output pairs (include reading these from a table).

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.

Interpret expressions for functions in terms of the situation they model

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

Extend the domain of trigonometric functions using the unit circle

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

Model periodic phenomena with trigonometric functions

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

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

(+) 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.

Prove and apply trigonometric identities

Prove the Pythagorean identity sin²(θ) + cos²(θ) = 1 and use it to find sin(θ), cos(θ), or tan(θ) given sin(θ), cos(θ), or tan(θ) and the quadrant of the angle.

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

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.

Interpreting Functions

Standards for Mathematical Practice

Trigonometric Functions

Building Functions

Linear, Quadratic, and Exponential Models

Understand and apply theorems about circles

Prove that all circles are similar.

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

Construct the inscribed and circumscribed circles of a triangle, and prove properties of angles for a quadrilateral inscribed in a circle.

(+) Construct a tangent line from a point outside a given circle to the circle.

Find arc lengths and areas of sectors of circles

Derive using similarity the fact that the length of the arc intercepted by an angle is proportional to the radius, and define the radian measure of the angle as the constant of proportionality; derive the formula for the area of a sector.

Experiment with transformations in the plane

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

Represent transformations in the plane using, e.g., transparencies and geometry software; describe transformations as functions that take points in the plane as inputs and give other points as outputs. Compare transformations that preserve distance and angle 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 it 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 rotation, reflection, or translation, draw the transformed figure using, e.g., graph paper, tracing paper, or geometry software. Specify a sequence of transformations that will carry a given figure onto another.

Understand congruence in terms of rigid motions

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 decide 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, and SSS) follow from the definition of congruence in terms of rigid motions.

Prove geometric theorems

Prove theorems about triangles.

Prove theorems about parallelograms.

Prove theorems about lines and angles.

Make geometric constructions

Make formal geometric constructions with a variety of tools and methods (compass and straightedge, string, reflective devices, paper folding, dynamic geometric software, etc.). Copying a segment; copying an angle; bisecting a segment; bisecting an angle; constructing perpendicular lines, including the perpendicular bisector of a line segment; and constructing a line parallel to a given line through a point not on the line.

Construct an equilateral triangle, a square, and a regular hexagon inscribed in a circle.

Explain volume formulas and use them to solve problems

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

(+) Give an informal argument using Cavalieri's principle for the formulas for the volume of a sphere and other solid figures.

Use volume formulas for cylinders, pyramids, cones, and spheres to solve problems.

Visualize relationships between two-dimensional and three-dimensional objects

Identify the shapes of two-dimensional cross-sections of three-dimensional objects, and identify three-dimensional objects generated by rotations of two-dimensional objects.

Translate between the geometric description and the equation for a conic section

Derive the equation of a circle of given center and radius using the Pythagorean Theorem; complete the square to find the center and radius of a circle given by an equation.

Derive the equation of a parabola given a focus and directrix.

(+) 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.

Use coordinates to prove simple geometric theorems algebraically

Use coordinates to prove simple geometric theorems algebraically.

Prove the slope criteria for parallel and perpendicular lines and use them to solve geometric problems (e.g., find the equation of a line parallel or perpendicular to a given line that passes through a given point).

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

Use coordinates to compute perimeters of polygons and areas of triangles and rectangles, e.g., using the distance formula.

Apply geometric concepts in modeling situations

Use geometric shapes, their measures, and their properties to describe objects (e.g., modeling a tree trunk or a human torso as a cylinder).

Apply concepts of density based on area and volume in modeling situations (e.g., persons per square mile, BTUs per cubic foot).

Apply geometric methods to solve design problems (e.g., designing an object or structure to satisfy physical constraints or minimize cost; working with typographic grid systems based on ratios).

Understand similarity in terms of similarity transformations

Verify experimentally the properties of dilations given by a center and a scale factor:

A dilation takes a line not passing through the center of the dilation to a parallel line, and leaves a line passing through the center unchanged.

The dilation of a line segment is longer or shorter in the ratio given by the 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 involving similarity

Prove theorems about triangles.

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

Define trigonometric ratios and solve problems involving right triangles

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.

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

Apply trigonometry to general triangles

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

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

(+) Derive the formula A = 1/2 ab sin(C) for the area of a triangle by drawing an auxiliary line from a vertex perpendicular to the opposite side.

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.

Standards for Mathematical Practice

Expressing Geometric Properties with Equations

Similarity, Right Triangles, and Trigonometry

Congruence

Circles

Geometric Measurement and Dimension

Modeling with Geometry

Perform arithmetic operations with complex 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.

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

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

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.

(+) 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.

Use complex numbers in polynomial identities and equations.

Solve quadratic equations with real coefficients that have complex solutions.

(+) Extend polynomial identities to the complex numbers.

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

Reason quantitatively and use units 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.

Define appropriate quantities for the purpose of descriptive modeling.

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

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.

Use properties of rational and irrational numbers.

Explain why the sum or product of two rational numbers is rational; that the sum of a rational number and an irrational number is irrational; and that the product of a nonzero rational number and an irrational number is irrational.

Represent and model with vector quantities.

(+) 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.

Perform operations on vectors.

(+) Add and subtract vectors.

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 perform vector subtraction component-wise.

(+) Multiply a vector by a scalar.

Represent scalar multiplication graphically by scaling vectors and possibly reversing their direction; perform scalar multiplication component-wise, e.g., as c(v_x, v_y) = (cv_x, cv_y).

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).

Perform operations on matrices and use matrices in applications.

(+) 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.

(+) 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.

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.

Standards for Mathematical Practice

The Complex Number System

Quantities

Vector and Matrix Quantities

The Real Number System

Understand independence and conditional probability and use them to interpret data

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.

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.

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.

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

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 compute probabilities of compound events and solve problems.

Understand and evaluate random processes underlying statistical experiments

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.

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

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; develop a margin of error through the use of simulation models for random sampling.

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

Evaluate reports based on data.

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

Represent data with plots on the real number line (dot plots, histograms, 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).

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.

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

Summarize categorical data for two categories in two-way frequency tables. Interpret relative frequencies in the context of the data (including joint, marginal, and conditional relative frequencies). Recognize possible associations and trends in the data.

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

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

Informally assess the fit of a function by plotting and analyzing residuals.

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

Interpret linear models

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.

Calculate expected values and use them to solve problems

(+) 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.

(+) 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.

Use probability to evaluate outcomes of decisions

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

Find the expected payoff for a game of chance.

Evaluate and compare strategies on the basis of 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).

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.

Standards for Mathematical Practice

Making Inferences and Justifying Conclusions

Using Probability to Make Decisions

Conditional Probability and the Rules of Probability

Interpreting Categorical and Quantitative Data

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.

Flexibly, efficiently, and accurately demonstrate that polynomials form a system similar to the integers, namely, they are closed under the operations of addition, subtraction, and multiplication; add, subtract, and multiply polynomials.

Flexibly, efficiently, and accurately create equations and inequalities in one variable and use them to solve problems. Include equations arising from linear, quadratic, and exponential functions.

Flexibly, efficiently, and accurately create linear, quadratic, exponential equations to represent relationships between quantities; graph equations on coordinate axes with labels and scales.

Flexibly, efficiently, and accurately rearrange formulas to highlight a quantity of interest, using the same reasoning as in solving equations within linear, quadratic, and exponential equations.

Solve quadratic equations in one variable by inspection, taking square roots, and factoring as appropriate to the initial form of the equation.

Flexibly, efficiently, and accurately solve a simple system consisting of a linear equation and a quadratic equation in two variables algebraically and graphically.

Interpret expressions that represent a quantity in terms of its context within linear, exponential, and quadratic functions.

Use the structure of an expression to identify ways to rewrite it within exponential and quadratic functions.

Flexibly, efficiently, and accurately create an equivalent form of an expression to reveal and explain properties of the quantity represented by the expression including factoring quadratic expressions and using properties of exponents to create equivalent forms of exponential expressions to reveal properties of interest in the function.

Flexibly, efficiently, and accurately write a function that describes a relationship between two quantities, including linear and exponential arithmetic and geometric sequences in context.

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. Using a variety of strategies, experiment with cases and illustrate an explanation of the effects on the graph using technology.

For a function that models a relationship between two quantities in context, 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. Key features include intercepts; intervals where the function is increasing, decreasing, positive, or negative; relative maximums and minimums; symmetries for functions including linear, exponential, and quadratic.

Relate the domain of a function to its graph and, where applicable, to the quantitative relationship it describes in linear, exponential, or quadratic contexts.

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

Graph linear, exponential, and quadratic functions expressed symbolically and show key features of the graph, including intercepts, maximum, minimum, and interpreting end behavior for exponential functions by hand in simple cases and using technology for more complicated cases.

Flexibly, efficiently, and accurately write a function defined by an expression in different but equivalent forms to reveal and explain different properties of the function including zeros and symmetry, using factoring for quadratic functions and integer constants for time with exponential growth and decay.

Compare properties of two functions each represented in a different way (algebraically, graphically, numerically in tables, or by verbal descriptions). Functions could be linear, exponential, or quadratic.

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

Flexibly, efficiently, and accurately prove that all circles are similar.

Identify and describe relationships among inscribed angles, radii, and chords, including how angles formed inside the circle, the circle's radius, and line segments within the circle are related. Understand special cases including angles formed by diameters and how the circle's edge interacts with its radius.

Construct the inscribed and circumscribed circles of a triangle and flexibly, efficiently, and accurately prove properties of angles for a quadrilateral inscribed in a circle.

Derive using similarity the fact that the length of the arc intercepted by an angle is proportional to the radius, and define the radian measure of the angle as the constant of proportionality; derive the formula for the area of a sector.

Flexibly, efficiently, and accurately prove theorems about triangles: interior angles, base angles, segments joining midpoint of two sides, and medians of a triangle.

Flexibly, efficiently, and accurately prove theorems about parallelograms: congruence of opposite sides and opposite angles, properties of diagonals.

Flexibly, efficiently, and accurately prove theorems about lines and angles: vertical, transversals, alternate interior and exterior, perpendicular bisectors, etc.

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

Use volume formulas for cylinders, pyramids, cones, and spheres to solve problems.

Identify the shapes of two-dimensional cross-sections of three-dimensional objects, and identify three-dimensional objects generated by rotations of two-dimensional objects.

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

Use coordinates to prove simple geometric theorems algebraically.

Use geometric shapes, their measures, and their properties to describe objects (e.g., modeling a tree trunk or a human torso as a cylinder).

Apply concepts of density based on area and volume in modeling situations (e.g., persons per square mile, BTUs per cubic foot).

Apply geometric methods to solve design problems (e.g., designing an object or structure to satisfy physical constraints or minimize cost; working with typographic grid systems based on ratios).

Verify experimentally the properties of dilations given by a center and a scale factor by seeing what happens to lines affected by a center of dilation and how scale factor affects line segments.

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.

Flexibly, efficiently, and accurately prove theorems about triangles: proportionality, triangle similarity, and the Pythagorean Theorem.

Flexibly, efficiently, and accurately use congruence and similarity criteria for triangles to solve problems and to prove 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.

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

Formulate multivariable statistical investigative questions and determine how data can be collected and provide an answer, consider causality and prediction when posing the question.

Understand the issues of bias and confounding variables when collecting data and their impact on interpretation. Understand practices for collecting and handling data, including sensitive information and concerns for privacy and how that may affect data collection.

Create and analyze data sets and data displays, including but not limited to scatter plots, regressions, histograms and boxplots using technology to sort or filter data, summarize, and describe relationships between quantitative variables.

Acknowledge the presence of missing data values and understand how missing values may add bias to analysis and interpretation. Examine and discuss competing explanations for data trends observed such as confounding variables. Respond to competing arguments or interpretations of the data of different community groups, paying careful attention to what conclusions the data supports, taking into account correlation versus causation.

Circles

Use complex numbers in polynomial identities and equations.

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

Modeling with Geometry

Creating Equations

Use coordinates to prove simple geometric theorems algebraically.

Standards for Mathematical Practice

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

Analyze the data.

Understand and apply theorems about circles.

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

Functions

Solve real-world and mathematical problems involving area, surface area, and volume.

Create equations that describe numbers or relationships.

Interpret the structure of expressions.

Seeing Structure in Expressions

Complex Numbers

Reason with Equations and Inequalities

Extend the properties of exponents to rational exponents.

Formulate statistical investigative questions.

Apply geometric concepts in modeling situations.

Analyze functions using different representations.

Statistics and Probability

Congruence

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

Define trigonometric ratios and solve problems involving right triangles.

Use properties of rational and irrational numbers.

Geometry

Interpreting Functions

Collect and consider data.

Solve systems of equations.

Perform arithmetic operations with complex numbers.

Expressing Geometric Properties with Equations

Similarity, Right Triangles, and Trigonometry

Arithmetic with Polynomials and Rational Expressions

Prove theorems involving similarity

Solve equations and inequalities in one variable.

Geometric Measurement and Dimension

Use the rules of probability to compute probabilities of compound events.

Data Science

Interpret results.

Perform arithmetic operations on polynomials.

Conditional Probability and the Rules of Probability

Understand similarity in terms of similarity transformations.

Algebra

Write expressions in equivalent forms to solve problems.

Find arc lengths and areas of sectors of circles.

Explain volume formulas and use them to solve problems.

Translate between the geometric description and the equation for a conic section.

Number & Quantity

Build new functions from existing functions.

The Real Number System

Linear, Quadratic, and Exponential Models

Build a function that models a relationship between two quantities.

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.

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

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

Flexibly, efficiently, and accurately explain how the definition of the meaning of rational exponents follows from extending the properties of integer exponents to those values using a variety of strategies, allowing for a notation for radicals in terms of rational exponents.

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

Explain why the sum or product of two rational numbers is rational; that the sum of a rational number and an irrational number is irrational; and that the product of a nonzero rational number and an irrational number is irrational.

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.

Understand the conditional probability of 𝐴𝐴 given 𝐵𝐵 as 𝑃𝑃(𝐴𝐴 𝑎𝑎𝑎𝑎𝑎𝑎 𝐵𝐵), 𝑃𝑃(𝐵𝐵) and interpret independence of 𝐴𝐴 and 𝐵𝐵 as saying that the conditional probability of 𝐴𝐴 given 𝐵𝐵 is the same as the probability of 𝐴𝐴, and the conditional probability of 𝐵𝐵 given 𝐴𝐴 is the same as the probability of 𝐵𝐵.

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.

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.

Apply the Addition Rule, 𝑃𝑃(𝐴𝐴 𝑜𝑜𝑜𝑜 𝐵𝐵) = 𝑃𝑃(𝐴𝐴) + 𝑃𝑃(𝐵𝐵) − 𝑃𝑃(𝐴𝐴 𝑎𝑎𝑎𝑎𝑎𝑎 𝐵𝐵), and interpret the answer in terms of the model.

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.

Flexibly, efficiently, and accurately create equations and inequalities in one variable and use them to solve problems. Include equations arising from linear, quadratic, and exponential functions.

Flexibly, efficiently, and accurately create linear, quadratic, exponential equations to represent relationships between quantities; graph equations 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 within linear, quadratic, and exponential equations.

Flexibly, efficiently, and accurately rearrange formulas to highlight a quantity of interest, using the same reasoning as in solving equations within linear, quadratic, and exponential equations.

Explain each step in solving an equation as following from the equality of numbers asserted at the previous step flexibly, efficiently, and accurately selecting and demonstrating use of strategies to solve equations, 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.

Demonstrate using a variety of strategies that, given a system of two equations in two variables, replacing one equation by the sum of that equation and a multiple of the other produces a system with the same solutions.

Flexibly, efficiently, and accurately solve systems of linear equations exactly and approximately (e.g., with graphs), focusing on pairs of linear equations in two variables.

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).

Using a variety of strategies explain the x-coordinates of the points where the graphs of the equations 𝑦𝑦 = 𝑓𝑓(𝑥𝑥) and 𝑦𝑦 = 𝑔𝑔(𝑥𝑥) intersect are the solutions of the equation 𝑓𝑓(𝑥𝑥) = 𝑔𝑔(𝑥𝑥); find the solutions approximately, e.g., using technology to graph the functions, make tables of values, or find successive approximations. Include cases where 𝑓𝑓(𝑥𝑥) and/or 𝑔𝑔(𝑥𝑥) are linear, exponential, and quadratic.

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.

Interpret expressions that represent a quantity in terms of its context within linear, exponential, and quadratic functions.

Flexibly, efficiently, and accurately write a function that describes a relationship between two quantities, including linear and exponential arithmetic and geometric sequences in context.

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

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. Using a variety of strategies, experiment with cases and illustrate an explanation of the effects on the graph using technology.

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 𝑓𝑓 is a function and x is an element of its domain, then 𝑓𝑓(𝑥𝑥) denotes the output of f corresponding to the input 𝑥𝑥. The graph of f is the graph of the equation 𝑦𝑦 = 𝑓𝑓(𝑥𝑥).

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

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

For a function that models a relationship between two quantities in context, 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. Key features include intercepts; intervals where the function is increasing, decreasing, positive, or negative; relative maximums and minimums; symmetries for functions including linear, exponential, and quadratic.

Relate the domain of a function to its graph and, where applicable, to the quantitative relationship it describes in linear, exponential, or quadratic contexts.

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

Graph linear, exponential, and quadratic functions expressed symbolically and show key features of the graph, including intercepts, maximum, minimum, and interpreting end behavior for exponential functions by hand in simple cases and using technology for more complicated cases.

Compare properties of two functions each represented in a different way (algebraically, graphically, numerically in tables, or by verbal descriptions). Functions could be linear, exponential, or quadratic.

Distinguish between situations that can be modeled with linear functions (equal differences over equal intervals) and with exponential functions (equal factors over equal intervals), recognizing constant rates per unit interval, and growth or decay by a constant percent rate per unit interval.

Flexibly, efficiently, and accurately construct linear and exponential functions given a graph, a description of a relationship, or two input-output pairs (include reading these from a table).

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

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, distance along a line, and distance around a circular arc.

Flexibly, efficiently, and accurately represent transformations in the plane, e.g., transparencies and geometry software; describe transformations as functions that take points in the plane as inputs and give other points as outputs. Compare transformations that preserve distance and angle 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 it 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 rotation, reflection, or translation, draw the transformed figure using, e.g., graph paper, tracing paper, or geometry software. Flexibly, efficiently, and accurately specify a sequence of transformations 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 decide 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, and SSS) follow from the definition of congruence in terms of rigid motions.

Make formal geometric constructions with a variety of tools and methods.

Construct an equilateral triangle, a square, and a regular hexagon inscribed in a circle.

Use coordinates to prove simple geometric theorems algebraically.

Prove the slope criteria for parallel and perpendicular lines and use them to solve geometric problems (e.g., find the equation of a line parallel or perpendicular to a given line that passes through a given point).

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

Use coordinates to compute perimeters of polygons and areas of triangles and rectangles, e.g., using the distance formula.

Formulate multivariable statistical investigative questions and determine how data can be collected and provide an answer, consider causality and prediction when posing the question.

Understand the issues of bias and confounding variables when collecting data and their impact on interpretation. Understand practices for collecting and handling data, including sensitive information and concerns for privacy and how that may affect data collection.

Create and analyze data sets and data displays, including but not limited to scatter plots, regressions, histograms and boxplots using technology to sort or filter data, summarize, and describe relationships between quantitative variables.

Acknowledge the presence of missing data values and understand how missing values may add bias to analysis and interpretation. Examine and discuss competing explanations for data trends observed such as confounding variables. Respond to competing arguments or interpretations of the data of different community groups, paying careful attention to what conclusions the data supports, taking into account correlation versus causation.

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

Formulate statistical investigative questions.

Analyze functions using different representations.

Represent and solve equations and inequalities graphically.

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

Analyze the data.

Standards for Mathematical Practice

Number & Quantity

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

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

Solve systems of equations.

Interpreting Categorical and Quantitative Data

Expressing Geometric Properties with Equations

Functions

Reason quantitatively and use units to solve problems.

Interpreting Functions

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

Creating Equations

Understand the concept of a function and use function notation.

Interpret linear models.

Create equations that describe numbers or relationships.

Algebra

Understand congruence in terms of rigid motions.

Experiment with transformations in the plane.

Build a function that models a relationship between two quantities.

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

Interpret results.

Build new functions from existing functions.

Linear, Quadratic, and Exponential Models

Seeing Structure in Expressions

Solve equations and inequalities in one variable.

Congruence

Collect and consider data.

Make geometric constructions.

Statistics and Probability

Use coordinates to prove simple geometric theorems algebraically.

Geometry

Quantities

Reason with Equations and Inequalities

Data Science

Interpret the structure of expressions.

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.

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

Represent data with plots on the real number line (dot plots, histograms, 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).

Summarize categorical data for two categories in two-way frequency tables. Interpret relative frequencies in the context of the data (including joint, marginal, and conditional relative frequencies). Recognize possible associations and trends in the data.

Represent data on two quantitative variables on a scatter plot, and describe how the variables are related to solve problems in context by fitting functions to the data and explaining trends and relationships within the data.

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.

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.

Flexibly, efficiently, and accurately demonstrate that polynomials form a system similar to the integers, namely, they are closed under the operations of addition, subtraction, and multiplication; add, subtract, and multiply polynomials.

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.

Prove polynomial identities and use them to describe numerical relationships.

Rewrite simple rational expressions in different forms; write a(x)/b(x) in the form q(x) + r(x)/b(x), where a(x), b(x), q(x), and r(x) are polynomials with the degree of r(x) less than the degree of b(x), using inspection, long division, or, for the more complicated examples, a computer algebra system.

Flexibly, efficiently, and accurately create equations and inequalities in one variable and use them to solve problems.

Flexibly, efficiently, and accurately create equations in two or more variables to represent relationships between quantities; graph equations 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.

Flexibly, efficiently, and accurately rearrange formulas to highlight a quantity of interest, using the same reasoning as in solving equations.

Solve rational and radical equations in one variable, and give examples showing how extraneous solutions may arise.

Solve quadratic equations in one variable by inspection, factoring, completing the square and derive the quadratic formula from this form. Recognize when the quadratic formula give complex solutions and write them as a ± bi for real numbers a and b.

Using a variety of strategies explain why the x-coordinates of the points where the graphs of the equations 𝑦𝑦 = 𝑓𝑓(𝑥𝑥) and 𝑦𝑦 = 𝑔𝑔(𝑥𝑥) intersect are the solutions of the equation 𝑓𝑓(𝑥𝑥) = 𝑔𝑔(𝑥𝑥) find the solutions approximately, e.g., using technology to graph the functions, make tables of values, or find successive approximations. Include cases where 𝑓𝑓(𝑥𝑥) and/or 𝑔𝑔(𝑥𝑥) are linear, polynomial, rational, absolute value, exponential, and logarithmic functions.

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

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

Flexibly, efficiently, and accurately create an equivalent form of an expression to reveal and explain properties of the quantity represented by the expression including factoring quadratic expressions, completing the square in a quadratic expression to reveal maximums or minimums, and using properties of exponents to create equivalent forms of exponential expressions to reveal properties of interest in the function.

Derive the formula for the sum of a finite geometric series (when the common ratio is not 1), and use the formula to solve problems.

Write a function that describes a relationship between two quantities including determining an explicit expression, recursive process, or steps for calculation from a context, and combining standard function types using arithmetic operations.

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

Identify the effect on the graph of replacing 𝑓𝑓(𝑥𝑥) 𝑏𝑏𝑏𝑏 𝑓𝑓(𝑥𝑥) + 𝑘𝑘, 𝑘𝑘 𝑓𝑓(𝑥𝑥), 𝑓𝑓(𝑘𝑘𝑘𝑘), 𝑎𝑎𝑎𝑎𝑎𝑎 𝑓𝑓(𝑥𝑥 + 𝑘𝑘) for specific values of 𝑘𝑘 (both positive and negative); find the value of 𝑘𝑘 given the graphs. Experiment with cases and illustrate an explanation of the effects on the graph using technology.

Find inverse functions through focus on relationships between inputs and outputs.

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. Key features include intercepts; intervals where the function is increasing, decreasing, positive, or negative; relative maximums and minimums; symmetries. Functions can include: polynomial, radical, rational, logarithms, absolute value, piecewise, and trigonometric. Linear, exponential, and quadratic relationships in increased complexity.

Relate the domain of a function to its graph and, where applicable, to the quantitative relationship it describes in context. Functions can include: polynomial, radical, rational, logarithms, absolute value, piecewise, and trigonometric. Linear, exponential, and quadratic relationships in increased complexity.

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 in simple cases and using technology for more complicated cases including linear, quadratic, exponential, square root, cube root, and piecewise-defined functions, including step functions and absolute value functions, polynomial functions, identifying zeros when suitable factorizations are available, and showing end behavior, and exponential and logarithmic functions, showing intercepts and end behavior, and trigonometric functions, showing period, midline, and amplitude.

Write a function defined by an expression in different but equivalent forms to reveal and explain different properties of the function, including factoring and completing the square to reveal zeros, symmetry, and extreme values of a quadratic functions and non-integer constants for time with exponential growth and decay in context.

Compare properties of two functions each represented in a different way (algebraically, graphically, numerically in tables, or by verbal descriptions). Functions can include: polynomial, radical, rational, logarithms, absolute value, piecewise, and trigonometric. Linear, exponential, and quadratic relationships in increased complexity.

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

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

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

Prove the Pythagorean identity sin2(θ) + cos2(θ) = 1 and use it to find sin(θ), cos(θ), or tan(θ) given sin(θ), cos(θ), or tan(θ) and the quadrant of the angle.

Identify the shapes of two-dimensional cross-sections of three-dimensional objects, and identify three-dimensional objects generated by rotations of two-dimensional objects.

Formulate multivariable statistical investigative questions and determine how data can be collected and provide an answer, consider causality and prediction when posing the question.

Understand the issues of bias and confounding variables when collecting data and their impact on interpretation. Understand practices for collecting and handling data, including sensitive information and concerns for privacy and how that may affect data collection.

Create and analyze data sets and data displays, including but not limited to scatter plots, regressions, histograms and boxplots using technology to sort or filter data, summarize, and describe relationships between quantitative variables.

Acknowledge the presence of missing data values and understand how missing values may add bias to analysis and interpretation. Examine and discuss competing explanations for data trends observed such as confounding variables. Respond to competing arguments or interpretations of the data of different community groups, paying careful attention to what conclusions the data supports, taking into account correlation versus causation.

Seeing Structure in Expressions

Functions

Write expressions in equivalent forms to solve problems.

Understand and evaluate random processes underlying statistical experiments.

Analyze functions using different representations.

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

Geometric Measurement and Dimension

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

Represent and solve equations and inequalities graphically.

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

Interpret results.

Perform arithmetic operations on polynomials.

Analyze the data.

Formulate statistical investigative questions.

Linear, Quadratic, and Exponential Models

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

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

Interpreting Functions

Create equations that describe numbers or relationships.

Collect and consider data.

Building Functions

Statistics and Probability

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

Prove and apply trigonometric identities.

Interpreting Categorical and Quantitative Data

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

Arithmetic with Polynomials and Rational Expressions

Data Science

Creating Equations

Interpret the structure of expressions.

Solve equations and inequalities in one variable.

Extend the domain of trigonometric functions using the unit circle.

Build new functions from existing functions.

Geometry

Trigonometric Functions

Reason with Equations and Inequalities

Making Inferences and Justifying Conclusions.

Build a function that models a relationship between two quantities.

Standards for Mathematical Practice

Algebra

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; develop a margin of error through the use of simulation models for random sampling.

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

Evaluate reports based on data.

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.