High School Math Arizona Standards

Standards for Mathematical Practice

Conditional Probability and the rules of Probability

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

Statistics and Probability

Linear, Quadratic, and Exponential Models

Building Functions

Interpreting Functions

Functions

Reasoning with Equations and Inequalities

Creating Equations

Arithmetic with Polynomials and Rational Expressions

Seeing Structure in Expressions

Algebra

The Real Number System

Number and Quantity

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.

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. Focus on quadratic and cubic polynomials in which linear and quadratic factors are available.

Create equations that describe numbers or relationships.

Create equations and inequalities in one variable and use them to solve problems. Include problem-solving opportunities utilizing real-world context. Focus on equations and inequalities that are linear, quadratic, or exponential.

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 non-viable 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 linear and quadratic equations as following from the equality of numbers asserted at the previous step, starting from the assumption that the original equation has a solution. Construct a viable argument to justify a solution method.

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

Solve quadratic equations in one variable.

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

Solve quadratic equations by inspection (e.g., x² = 49), taking square roots, completing the square, the quadratic formula and factoring, as appropriate to the initial form of the equation. Focus on solutions for quadratic equations that have real roots. Include cases that recognize when a quadratic equation has no real solutions.

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, focusing on pairs of linear equations in two variables. Include problem solving opportunities utilizing real-world context.

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). Focus on cases where f(x) and/or g(x) are linear, absolute value, quadratic, and exponential 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 expressions by viewing one or more of their parts as a single entity.

Use structure to identify ways to rewrite numerical and polynomial expressions. Focus on polynomial multiplication and factoring patterns.

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.

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 real-world context. Focus on linear, absolute value, quadratic, exponential, and piecewise-defined functions (limited to the aforementioned functions).

Build new functions from existing functions.

Identify the effect on the graph of replacing f(x) by f(x) + k, k f(x), 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. Focus on linear, absolute value, quadratic, exponential and piecewise-defined functions (limited to the aforementioned functions).

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).

Evaluate a function for inputs in the domain, 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. Include problem-solving opportunities utilizing real-world context. Key features include: intercepts; intervals where the function is increasing, decreasing, positive, or negative; relative maximums and minimums. Focus on linear, absolute value, quadratic, exponential and piecewise-defined functions (limited to the aforementioned functions).

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 continuous function (presented symbolically or as a table) on a closed interval. Estimate the rate of change from a graph. Include problem-solving opportunities utilizing real-world context. Focus on linear, absolute value, quadratic, and exponential functions.

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. Functions include linear, exponential, quadratic, and piecewise-defined functions (limited to the aforementioned functions).

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

Compare properties of two functions each represented in a different way (algebraically, graphically, numerically in tables, or by verbal descriptions). Focus on linear, absolute value, quadratic, exponential and piecewise-defined functions (limited to the aforementioned functions).

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 input/output pairs.

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

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

Interpret the parameters in a linear or exponential function with integer exponents utilizing real world context.

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.

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, include utilizing real-world context.

Define appropriate quantities for the purpose of descriptive modeling. Include problem-solving opportunities utilizing real-world context.

Choose a level of accuracy appropriate to limitations on measurement when reporting quantities utilizing real-world context.

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.

Solve equations and inequalities in one variable.

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

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

Use the Multiplication Rule for independent events to 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.

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

Represent real-value data with plots for the purpose of comparing two or more data sets.

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 outliers if present.

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 quantities are related.

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

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

Interpret linear models.

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

Compute and interpret the correlation coefficient of a linear relationship.

Distinguish between correlation and causation.

Standards for Mathematical Practice

Conditional Probability and the Rules of Probability

Making Inferences and Justifying Conclusions

Interpreting Categorical and Quantitative Data

Statistics and Probability

Trigonometric Functions

Linear, Quadratic, and Exponential Models

Building Functions

Interpreting Functions

Functions

Reasoning with Equations and Inequalities

Creating Equations

Arithmetic with Polynomials and Rational Expressions

Seeing Structure in Expressions

Algebra

The Complex Number System

Quantities

The Real Number System

Number and Quantity

Solve equations and inequalities in one variable.

Solve systems of equations.

Represent and solve equations and inequalities graphically.

Write expressions in equivalent forms to solve problems.

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, include utilizing real-world context.

Understand the relationship between zeros and factors of polynomials.

Know and apply the Remainder and Factor 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. Focus on quadratic, cubic, and quartic polynomials including polynomials for which factors are not provided

Use polynomial identities to solve problems.

Prove polynomial identities and use them to describe numerical relationships.

Rewrite rational expressions.

Rewrite 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.

Create equations that describe numbers or relationships.

Create equations and inequalities in one variable and use them to solve problems. Include problem-solving opportunities utilizing real-world context. Focus on equations and inequalities arising from linear, quadratic, rational, and exponential functions.

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

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

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

Fluently solve quadratic equations in one variable. 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 a system consisting of a linear equation and a quadratic equation in two variables algebraically and graphically.

Explain why the x-coordinates of the points where the graphs of the equations y = f(x) and y = g(x) intersect are the solutions of the equation f(x) =g(x); find the solutions approximately (e.g., using technology to graph the functions, make tables of values, or find successive approximations). Include problems in real-world context. Extend from linear, quadratic, and exponential functions to cases where f(x) and/or g(x) are polynomial, rational, exponential, and logarithmic functions.

Interpret the structure of expressions.

Use structure to identify ways to rewrite polynomial and rational expressions. Focus on polynomial operations and factoring patterns.

Choose and produce an equivalent form of an expression to reveal and explain properties of the quantity represented by the expression. Include problem-solving opportunities utilizing real-world context and focus on expressions with rational exponents.

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. Extend from linear, quadratic and exponential functions to include polynomial, radical, logarithmic, rational, sine, cosine, exponential, and piecewise-defined functions. Include problem-solving opportunities utilizing real-world context.

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

Combine function types using arithmetic operations and function composition.

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, kf(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. Extend from linear, quadratic and exponential functions to include polynomial, radical, logarithmic, rational, sine, cosine, and exponential functions, and piecewise-defined functions.

Find inverse functions.

Understand that an inverse function can be obtained by expressing the dependent variable of one function as the independent variable of another, recognizing that functions f and g are inverse functions if and only if f(x) = y and g(y) = x for all values of x in the domain of f and all values of y in the domain of g.

Understand that if a function contains a point (a,b), then the graph of the inverse relation of the function contains the point (b,a).

Interpret the meaning of and relationship between a function and its inverse utilizing real-world context.

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. Include problem-solving opportunities utilizing a real-world context. Key features include: intercepts; intervals where the function is increasing, decreasing, positive, or negative; relative maximums and minimums; symmetries; end behavior; and periodicity. Extend from linear, quadratic and exponential to include polynomial, radical, logarithmic, rational, sine, cosine, tangent, exponential, and piecewise-defined functions.

Calculate and interpret the average rate of change of a continuous function (presented symbolically or as a table) on a closed interval. Estimate the rate of change from a graph. Include problem-solving opportunities utilizing real-world context. Extend from linear, quadratic and exponential functions to include polynomial, radical, logarithmic, rational, sine, cosine, tangent, exponential, and piecewise-defined functions.

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. Extend from linear, quadratic and exponential functions to include square root, cube root, polynomial, exponential, logarithmic, sine, cosine, tangent and piecewise-defined functions.

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

Use the properties of exponents to interpret expressions for exponential functions and classify those functions as exponential growth or decay.

Compare properties of two functions each represented in a different way (algebraically, graphically, numerically in tables, or by verbal descriptions.). Extend from linear, quadratic and exponential functions to include polynomial, radical, logarithmic, rational, trigonometric, exponential, and piecewise-defined functions

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

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 logarithms that are not readily found by hand or observation using technology.

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

Interpret the parameters in an exponential function with rational exponents utilizing real-world context.

Extend the domain of trigonometric functions using the unit circle.

Understand radian measure of an angle as the length of the arc on any circle subtended by the angle, measured in units of the circle's radius.

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

Model periodic phenomena with trigonometric functions.

Create and interpret trigonometric functions that model periodic phenomena with specified amplitude, frequency, and midline.

Apply trigonometric identities.

Use 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.

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.

Perform arithmetic operations with complex numbers.

Apply the relation i² = –1 and the commutative, associative, and distributive properties to add, subtract, and multiply complex numbers. Write complex numbers in the form ( a+bi ) with a and b real.

Use complex numbers in polynomial identities and equations.

Solve quadratic equations with real coefficients that have complex solutions.

Define appropriate quantities for the purpose of descriptive modeling. Include problem-solving opportunities utilizing real-world context.

Choose a level of accuracy appropriate to limitations on measurement when reporting quantities utilizing real-world context.

Extend the properties of exponents to rational exponents.

Explain how the definition 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.

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

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 utilizing real-world context.

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

Use Bayes Rule to 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.

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.

Explain whether a specified model is consistent with results from a given data-generating process

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

Recognize the purposes of and differences between designed experiments, sample surveys and observational studies.

Use data from a sample survey to estimate a population mean or proportion; recognize that estimates are unlikely to be correct and the estimates will be more precise with larger sample sizes.

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

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

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

Represent data of two quantitative variables on a scatter plot, and describe how the quantities are related. Extend to polynomial and exponential models.

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

Interpret models.

Interpret parameters of exponential models.

Standards for Mathematical Practice

Modeling with Geometry

Geometric Measurement and Dimension

Expressing Geometric Properties with Equations

Circles

Similarity, Right Triangles, and Trigonometry

Congruence

Geometry

Quantities

Number and Quantity

Understand and apply theorems about circles.

Prove that all circles are similar.

Identify and describe relationships among inscribed angles, radii, and chords. Include the relationship between central, inscribed, and circumscribed angles; inscribed angles on a diameter are right angles; the radius of a circle is perpendicular to the tangent where the radius intersects the circle.

Construct the inscribed and circumscribed circles of a triangle, and prove properties of angles for a quadrilateral inscribed in a 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. Convert between degrees and radians.

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 and describe transformations in the plane 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.

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. Specify a sequence of transformations that will carry a given figure onto another.

Understand congruence in terms of rigid motions.

Use geometric definitions 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, AAS, SAS, and SSS) follow from the definition of congruence in terms of rigid motions.

Prove geometric theorems.

Prove theorems about triangles. Theorems include: measures of interior angles of a triangle sum to 180°; base angles of isosceles triangle are congruent; the segment joining midpoints of two sides of a triangle is parallel to the third side and half the length; the medians of a triangle meet at a point.

Prove theorems about parallelograms. Theorems include: opposite sides are congruent, opposite angles are congruent, the diagonals of a parallelogram bisect each other, and rectangles are parallelograms with congruent diagonals.

Prove theorems about lines and angles. Theorems include: vertical angles are congruent; when a transversal crosses parallel lines, alternate interior angles are congruent and corresponding angles are congruent; points on a perpendicular bisector of a line segment are exactly those equidistant from the segment's endpoints.

Make geometric constructions.

Make formal geometric constructions with a variety of tools and methods. Constructions include: copying segments; copying angles; bisecting segments; bisecting angles; 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; with a variety of tools and methods.

Explain volume formulas and use them to solve problems.

Analyze and verify the formulas for the volume of a cylinder, pyramid, and cone.

Use volume formulas for cylinders, pyramids, cones, and spheres to solve problems utilizing real-world context.

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.

Use coordinates to prove geometric theorems algebraically.

Use coordinates to algebraically prove or disprove geometric relationships algebraically. Relationships include: proving or disproving geometric figures given specific points in the coordinate plane; and proving or disproving if a specific point lies on a given circle.

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

Apply geometric concepts in modeling situations.

Use geometric shapes, their measures, and their properties to describe objects utilizing real-world context.

Apply concepts of density based on area and volume in modeling situations utilizing real-world context.

Apply geometric methods to solve design problems utilizing real-world context.

Understand similarity in terms of similarity transformations.

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

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, SAS, and SSS criterion for two triangles to be similar.

Prove theorems involving similarity.

Prove theorems about triangles. Theorems include: an interior line parallel to one side of a triangle divides the other two proportionally, and conversely; the Pythagorean Theorem proved using triangle similarity.

Use congruence and similarity criteria to prove relationships in geometric figures and solve problems utilizing real-world context.

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 (including inverse trigonometric ratios) and the Pythagorean Theorem to find unknown measurements in right triangles utilizing real-world context.

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.

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, include utilizing real-world context.

Define appropriate quantities for the purpose of descriptive modeling. Include problem-solving opportunities utilizing real-world context.

Choose a level of accuracy appropriate to limitations on measurement when reporting quantities utilizing real-world context.

Reasoning with Matrices

Reasoning with Vectors

Reasoning with Trigonometry

Reasoning with Functions and Relations (RFR)

Analyze Functions (Standards within this strand encompass P.F-IF)

Interpret parameters of a function defined by an expression in the context of the situation.

Sketch the graph of a function that models a relationship between two quantities, identifying key features.

Interpret key features of graphs and tables for a function that models a relationship between two quantities in terms of the quantities.

Use limits to describe long-range behavior, asymptotic behavior, and points of discontinuity.

Sketch the graph of all six trigonometric functions, identifying key features.

Building Functions (Standards within this strand encompass P.F-BF, P.F-TF)

Model relationships between quantities that require adding, subtracting, multiplying, and/or dividing functions

Model relationships through composition and attend to the restrictions of the domain.

Rewrite a function as a composition of functions.

Determine if a function has an inverse. If so, find the inverse. If not, define a restriction on the domain that meets the requirement for invertibility and find the inverse on the restricted domain.

Interpret the meanings of quantities involving functions and their inverses.

Verify by analytical methods that one function is the inverse of another.

Model real-world situations involving trigonometry.

Apply the Law of Sines and Law of Cosines to solve problems.

Use trigonometry to find the area of triangles.

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.

Use inverse functions to solve trigonometric equations utilizing real world context; evaluate the solution and interpret them in terms of context.

Interpreting Conics (Standards within this strand encompass P.G-GPE)

Model real-world situations which involve conic sections.

Identify key features of conic sections (foci, directrix, radii, axes, asymptotes, center) graphically and algebraically.

Sketch a graph of a conic section using its key features.

Use the key features of a conic section to write its equation.

Given a quadratic equation of the form ax2+ by2 + cx + dy + e = 0, determine if the equation is a circle, ellipse, parabola, or hyperbola.

Interpreting Sequences and Series (Standards within this strand encompass P.F-BF.A.2, P.A-SSE.B.4))

Model real-world situations involving sequences or series using recursive and/or explicit definitions.

Use covariational reasoning to describe sequences and series.

Represent finite or infinite series using sigma notation.

Find the sums of finite or infinite series, if they exist.

Using Matrices (Standards within this strand encompass P.N-VM.C)

Use matrices to represent and manipulate data.

Use matrix operations to solve problems. Add, subtract, and multiply matrices of appropriate dimensions. Multiply matrices by scalars to produce new matrices.

Find the inverse and determinant of a matrix.

Use matrices to solve systems of linear equations.

Exploring Polar Equations

Graph polar equations.

Analyze and interpret the graphs of polar equations.

Use polar equations to solve problems.

Extended Triangle Trigonometry (Standards within this strand encompass P.G-SRT, P.F-TF.B)

Reasoning with Trigonometric Structure (Standards within this strand encompass P.F-TF.C)

Use the structure of a trigonometric expression to identify ways to rewrite it.

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

Solve trigonometric equations.

Exploring Vectors

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.

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

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

Add and subtract vectors, and multiply a vector by a scalar.

Modeling with Parametrics

Model real-world contexts with parametric equations.

Use parametric equations to solve problems.

Graph parametric equations and identify orientation.

Analyze and interpret the graphs of parametric equations.

Standards for Mathematical Practice

Discrete Mathematics

Contemporary Mathematics

Using Probability to Make Decisions

Conditional Probability and the Rules of Probability

Making Inferences and Justifying Conclusions

Statistics and Probability

Geometric Measurement and Dimension

Expressing Geometric Properties with Equations

Circles

Similarity, Right Triangles, and Trigonometry

Geometry

Trigonometric Functions

Building Functions

Interpreting Functions

Functions

Reasoning with Equations and Inequalities

Arithmetic with Polynomials and Rational Expressions

Algebra

Vector and Matrix Quantities

The Complex Number System

Number and Quantity

Use polynomial identities to solve problems.

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. The Binomial Theorem can be proved by mathematical induction or by a combinatorial argument.

Rewrite rational expressions.

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.

Solve systems of equations.

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).

Understand and apply vertex-edge graph topics

Study the following topics related to vertex-edge graph: Euler circuits, Hamilton circuits, shortest path, vertex coloring, and adjacency matrices.

Understand, analyze, and apply vertex-edge graphs to model and solve problems related to paths, circuits, networks, and relationships among a finite number of elements, in real-world and abstract settings.

Devise, analyze, and apply algorithms for solving vertex-edge graph problems.

Extend work with adjacency matrices for graphs, such as interpreting row sums and using the nth power of the adjacency matrix to count paths of length n in a graph.

Build a function that models a relationship between two quantities.

Write a function that describes a relationship between two quantities.

Compose functions.

Build new functions from existing functions.

Find inverse functions.

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.

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

Extend the domain of trigonometric functions using 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.

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 utilizing real world context; evaluate the solution and interpret them in terms of context.

Apply trigonometric identities.

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

Understand and apply theorems about circles.

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

Explain volume formulas and use them to solve problems.

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

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

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.

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 = ½ 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.

Perform arithmetic operations with 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. For example, (-1 + √3i)³ = 8 because (-1 + √3i) has modulus 2 and argument 120°.

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.

Extend polynomial identities to the complex numbers.

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

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.

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

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

Add and subtract vectors.

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.

Multiply matrices by scalars to produce new matrices.

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.

Perform operations on vectors.

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

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

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 random sample 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.

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. For example, find the expected winnings from a state lottery ticket or a game at a fast-food restaurant.

Evaluate and compare strategies on the basis of expected values.

Use randomization to make fair decisions based on probabilities.

Analyze decisions and strategies using probability concepts.

Covariational Reasoning (CR)

Discrete Mathematical Reasoning (DMR)

Financial Reasoning (FR)

Standards for Mathematical Practice (MP)

Numerical Reasoning (NR)

Analyze and compare growth and decay using absolute and relative change utilizing real-world contexts.

Compare, reason and communicate about proportional and non-proportional models utilizing real-world contexts.

Identify, create, and use appropriate models for bivariate data sets (i.e. linear, exponential) to estimate solutions for contextual questions, identify patterns and identify how changing parameters affect the models.

Understand, analyze, and apply vertex-edge graphs to model and make informed decisions related to paths, circuits, networks, and relationships in real-world settings. Encompasses P.CM-DM.A.1, P.CM-DM.A.2

Devise, analyze, and apply algorithms for solving vertex-edge graph problems. P.CM-DM.A.3

Extend work with adjacency matrices for graphs, such as interpreting row sums and using the nth power of the adjacency matrix to count paths of length n in a graph. P.CM-DM.A.4

Identify and research a career goal. Develop a plan and time table for achieving it including educational/training requirements, costs, and other factors (e.g. cost versus savings, income and debt).

Understand and apply strategies to monitor income and expenses, plan for spending, implement a diversified investment strategy, and save for future goals.

Use models to solve and communicate about contextual financial questions such as credit card debt, installment savings, amortization schedules, mortgage and other loan scenarios.

Identify and explain personal and societal consequences of financial decisions.

Make sense of problems and persevere in solving them. Mathematically proficient students explain to themselves the meaning of a problem, look for entry points to begin work on the problem, and plan and choose a solution pathway. While engaging in productive struggle to solve a problem, they continually ask themselves, “Does this make sense?" to monitor and evaluate their progress and change course if necessary. Once they have a solution, they look back at the problem to determine if the solution is reasonable and accurate. Mathematically proficient students check their solutions to problems using different methods, approaches, or representations. They also compare and understand different representations of problems and different solution pathways, both their own and those of others.

Reason abstractly and quantitatively. Mathematically proficient students make sense of quantities and their relationships in problem situations. Students can contextualize and decontextualize problems involving quantitative relationships. They contextualize quantities, operations, and expressions by describing a corresponding situation. They decontextualize a situation by representing it symbolically. As they manipulate the symbols, they can pause as needed to access the meaning of the numbers, the units, and the operations that the symbols represent. Mathematically proficient students know and flexibly use different properties of operations, numbers, and geometric objects and when appropriate they interpret their solution in terms of the context.

Construct viable arguments and critique the reasoning of others. Mathematically proficient students construct mathematical arguments (explain the reasoning underlying a strategy, solution, or conjecture) using concrete, pictorial, or symbolic referents. Arguments may also rely on definitions, assumptions, previously established results, properties, or structures. Mathematically proficient students make conjectures and build a logical progression of statements to explore the truth of their conjectures. They are able to analyze situations by breaking them into cases, and can recognize and use counterexamples. Mathematically proficient students present their arguments in the form of representations, actions on those representations, and explanations in words (oral or written). Students critique others by affirming or questioning the reasoning of others. They can listen to or read the reasoning of others, decide whether it makes sense, ask questions to clarify or improve the reasoning, and validate or build on it. Mathematically proficient students can communicate their arguments, compare them to others, and reconsider their own arguments in response to the critiques of others.

Model with mathematics. Mathematically proficient students apply the mathematics they know to solve problems arising in everyday life, society, and the workplace. When given a problem in a contextual situation, they identify the mathematical elements of a situation and create a mathematical model that represents those mathematical elements and the relationships among them. Mathematically proficient students use their model to analyze the relationships and draw conclusions. They interpret their mathematical results in the context of the situation and reflect on whether the results make sense, possibly improving the model if it has not served its purpose.

Use appropriate tools strategically. Mathematically proficient students consider available tools when solving a mathematical problem. They choose tools that are relevant and useful to the problem at hand. Proficient students are sufficiently familiar with tools appropriate for their grade or course to make sound decisions about when each of these tools might be helpful; recognizing both the insight to be gained and their limitations. Students deepen their understanding of mathematical concepts when using tools to visualize, explore, compare, communicate, make and test predictions, and understand the thinking of others.

Attend to precision. Mathematically proficient students clearly communicate to others using appropriate mathematical terminology, and craft explanations that convey their reasoning. When making mathematical arguments about a solution, strategy, or conjecture, they describe mathematical relationships and connect their words clearly to their representations. Mathematically proficient students understand meanings of symbols used in mathematics, calculate accurately and efficiently, label quantities appropriately, and record their work clearly and concisely.

Look for and make use of structure. Mathematically proficient students use structure and patterns to assist in making connections among mathematical ideas or concepts when making sense of mathematics. Students recognize and apply general mathematical rules to complex situations. They are able to compose and decompose mathematical ideas and notations into familiar relationships. Mathematically proficient students manage their own progress, stepping back for an overview and shifting perspective when needed.

Look for and express regularity in repeated reasoning. Mathematically proficient students look for and describe regularities as they solve multiple related problems. They formulate conjectures about what they notice and communicate observations with precision. While solving problems, students maintain oversight of the process and continually evaluate the reasonableness of their results. This informs and strengthens their understanding of the structure of mathematics which leads to fluency.

Represent quantities, using equivalent forms when appropriate, to investigate and describe quantitative and geometric relationships and solve problems in real-world contexts.

Reason, model, and communicate with and about percentages (change, incorrect, deceptive, relative and absolute).

Understand and compare magnitudes of numbers utilizing real-world context. Understand the importance and impact of unit selection.

Use and justify estimation skills, and know why, how, and when to estimate results. Assess and justify the reasonableness of estimations using the context and comparisons to other known values.

Reason and communicate about the validity of claims based on empirical, theoretical, and subjective probabilities. Draw conclusions or make decisions related to risk, pay-off, expected value, and false negatives/positives in various probabilistic contexts. Encompasses P.S-CP.B.9, P.S-MD.A.2, P.S-MD.A.3, P.S-MD.A.4, P.S-MD.B.5, P.S-MD.B.7

Analyze statistical information and identify limitations, strengths, or lack of information in studies including data collection methods (e.g. sampling, experimental, observational) and possible sources of bias. Identify errors or misuses of statistics to justify particular conclusions. Encompasses P.S-IC.B.3

Represent numerical summaries and visual displays of real-world data to make informed decisions. Reason, communicate, and describe strengths, limitations, and fallacies of various displays. Encompasses P.S-IC.B.6

Represent center, shape, and spread of two or more data sets. Reason, communicate, and compare data sets in context.

Statistical and Probabilistic Reasoning (SPR)