High School Mathematics 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.

Explain contextual, mathematical problems using a mathematical model.

Create mathematical models to explain phenomena that exist in the natural sciences, social sciences, liberal arts, fine and performing arts, and/or humanities contexts.

Using abstract and quantitative reasoning, make decisions about information and data from a contextual situation.

Use various mathematical representations and structures with this information to represent and solve real-life problems.

Solve and use first order differential equations.

Solve and use second and higher order differential equations.

Find and use series solutions.

Determine whether a contextual situation results in a system of differential equations and apply the basic existence and uniqueness results for the corresponding initial value problems.

Solve constant coefficient homogeneous systems using eigenvalues and eigenvectors. Solve systems with real, distinct eigenvalues, as well as those with repeated and imaginary eigenvalues.

Draw phase portraits for solutions of homogeneous systems with constant coefficients.

Solve non-homogeneous systems of ordinary differential equations using the method of undetermined coefficients and variation of parameters.

Determine which non-linear systems are locally linear and identify the behavior of the system about each critical point.

Plot locally linear systems.

Use population models derived from locally linear systems.

Use the integral definition to perform Laplace transforms for functions.

Use a Laplace table to accurately and efficiently identify Laplace transforms.

Perform inverse Laplace transforms using a variety of techniques.

Solve first- and second-order differential equations using Laplace transforms that apply to fields such as electrical and mechanical engineering.

Write piecewise functions as compositions of step (Heaviside) functions.

Find the general uniqueness and existence of solutions for step functions, and use Laplace transforms to find solutions to step functions.

Find the Laplace transform of the Dirac delta function.

Solve linear systems of differential equations using Laplace transforms

Express the relationships between points, lines, and planes in three dimensions.

Explore functions of two independent variables of the form $z = f(x, y)$ and implicit functions of the form $f(x, y, z) = 0$.

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.

Explain contextual, mathematical problems using a mathematical model.

Create mathematical models to explain phenomena that exist in the natural sciences, social sciences, liberal arts, fine and performing arts, and/or humanities contexts.

Using abstract and quantitative reasoning, make decisions about information and data from a contextual situation.

Use various mathematical representations and structures with this information to represent and solve real-life problems.

Approximate the partial derivatives at a point of a function defined by a table of data.

Find expressions for the first and second partial derivatives of a function.

Use the total differential to approximate mathematical models.

Represent the partial derivatives of a system of two functions in two variables using the Jacobian.

Find the partial derivatives of the composition of functions using the general chain rule.

Apply partial differentiation to problems of optimization, including problems requiring the use of the Lagrange multiplier.

Find the family of solutions and the envelope of the family of solutions to differential equations, including Clairaut equations.

Define and apply the gradient, the divergence, and the curl in terms of differential vector operations.

Evaluate and apply double and triple integrals.

Evaluate and interpret line and surface integrals.

Apply Integration techniques to solve problems.

Evaluate and interpret definite integrals.

Find the antiderivative of indefinite integrals.

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.

Determine limits graphically, numerically, and analytically.

Apply the definition of the derivative.

Apply rules of differentiation.

Analyze function behavior using the derivative.

Apply the derivative to real-world problems.

Explain contextual, mathematical problems using a mathematical model.

Create mathematical models to explain phenomena that exist in the natural sciences, social sciences, liberal arts, fine and performing arts, and/or humanities contexts.

Using abstract and quantitative reasoning, make decisions about information and data from a contextual situation.

Use various mathematical representations and structures with this information to represent and solve real-life problems.

Use number-theoretic operations.

Prove statements in number theory.

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.

Explain contextual, mathematical problems using a mathematical model.

Create mathematical models to explain phenomena that exist in the natural sciences, social sciences, liberal arts, fine and performing arts, and/or humanities contexts.

Using abstract and quantitative reasoning, make decisions about information and data from a contextual situation.

Use various mathematical representations and structures with this information to represent and solve real-life problems.

Use a counterexample to disprove a statement.

Prove statements directly from definitions and previously proved statements.

Prove statements indirectly by proving the contrapositive of the statement.

Apply the method of reductio ad absurdum (proof by contradiction) to prove statements.

Use the method of mathematical induction to prove statements involving the positive integers.

Represent and interpret statements using logical symbolism.

Use binary to represent and interpret logical statements.

Use set theoretic operations.

Use and interpret Boolean algebra.

Calculate the probability of events.

Use methods of counting to solve problems involving permutations and combinations.

Prove statements involving combinatorics.

Use, apply, and prove graph properties.

Apply graph theory in context.

Distinguish between rational and irrational numbers using decimal expansion. Convert a decimal expansion which repeats eventually into a rational number.

Approximate irrational numbers to compare the size of irrational numbers, locate them approximately on a number line, and estimate the value of expressions.

Apply the properties of integer exponents to generate equivalent numerical expressions.

Use square root and cube root symbols to represent solutions to equations. Recognize that $x^2 = p$ (where p is a positive rational number and |x| ≤ 25) has two solutions and $x^3 = p$ (where p is a negative or positive rational number and |x| ≤ 10) has one solution. Evaluate square roots of perfect squares ≤ 625 and cube roots of perfect cubes ≥ -1000 and ≤ 1000.

Use numbers expressed in scientific notation to estimate very large or very small quantities, and to express how many times as much one is than the other.

Add, subtract, multiply and divide numbers expressed in scientific notation, including problems where both decimal and scientific notation are used. Interpret scientific notation that has been generated by technology (e.g., calculators or online technology tools).

Rewrite algebraic and numeric expressions involving radicals.

Using numerical reasoning, show and explain that the sum or product of rational numbers is rational, the sum of a rational number and an irrational number is irrational, and the product of a nonzero rational number and an irrational number is irrational.

Interpret expressions and parts of an expression, in context, by utilizing formulas or expressions with multiple terms and/or factors.

Describe and solve linear equations in one variable with one solution (x = a), infinitely many solutions (a = a), or no solutions (a = b). Show which of these possibilities is the case by successively transforming the given equation into simpler forms, until an equivalent equation of the form x = a, a = a, or a = b results (where a and b are different numbers).

Create and solve linear equations and inequalities in one variable within a relevant application.

Using algebraic properties and the properties of real numbers, justify the steps of a one-solution equation or inequality.

Solve linear equations and inequalities in one variable with coefficients represented by letters and explain the solution based on the contextual, mathematical situation.

Use algebraic reasoning to fluently manipulate linear and literal equations expressed in various forms to solve relevant, mathematical problems.

Use the equation y = mx (proportional) for a line through the origin to derive the equation y = mx + b (non-proportional) for a line intersecting the vertical axis at b.

Show and explain that the graph of an equation representing an applicable situation in two variables is the set of all its solutions plotted in the coordinate plane.

Create and solve linear inequalities in two variables to represent relationships between quantities including mathematically applicable situations; graph inequalities on coordinate axes with labels and scales.

Represent constraints of linear inequalities and interpret data points as possible or not possible.

Solve systems of linear inequalities by graphing, including systems representing a mathematically applicable situation.

Interpret quadratic expressions and parts of a quadratic expression that represent a quantity in terms of its context.

Fluently choose and produce an equivalent form of a quadratic expression to reveal and explain properties of the quantity represented by the expression.

Create and solve quadratic equations in one variable and explain the solution in the framework of applicable phenomena.

Represent constraints by quadratic equations and interpret data points as possible or not possible in a modeling framework.

Interpret exponential expressions and parts of an exponential expression that represent a quantity in terms of its framework.

Create exponential equations in one variable and use them to solve problems, including mathematically applicable situations.

Create exponential equations in two variables to represent relationships between quantities, including in mathematically applicable situations; graph equations on coordinate axes with labels and scales.

Represent constraints by exponential equations and interpret data points as possible or not possible in a modeling environment.

Explain a proof of the Pythagorean Theorem and its converse using visual models.

Apply the Pythagorean Theorem to determine unknown side lengths in right triangles within authentic, mathematical problems in two and three dimensions.

Apply the Pythagorean Theorem to find the distance between two points in a coordinate system in practical, mathematical problems.

Apply the formulas for the volume of cones, cylinders, and spheres and use them to solve in relevant problems.

Solve real-life problems involving slope, parallel lines, perpendicular lines, area, and perimeter.

Apply the distance formula, midpoint formula, and slope of line segments to solve real-world 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.

Show and explain that a function is a rule that assigns to each input exactly one output.

Within realistic situations, identify and describe examples of functions that are linear or nonlinear. Sketch a graph that exhibits the qualitative features of a function that has been described verbally.

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

Compare properties (rate of change and initial value) of two functions used to model an authentic situation each represented in a different way (algebraically, graphically, numerically in tables, or by verbal descriptions).

Write and explain the equations $y = mx + b$ (slope-intercept form), $Ax + By = C$ (standard form), and $(y - y_1) = m(x - x_1)$ (point-slope form) as defining a linear function whose graph is a straight line to reveal and explain different properties of the function.

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

Construct a function to model a linear relationship between two quantities. Determine the rate of change and initial value of the function from a description of a relationship or from two $(x, y)$ values, including reading these from a table or from a graph.

Explain the meaning of the rate of change and initial value of a linear function in terms of the situation it models, and in terms of its graph or a table of values.

Graph and analyze linear functions expressed in various algebraic forms and show key characteristics of the graph to describe applicable situations.

Show that straight lines are widely used to model relationships between two quantitative variables. For scatter plots that suggest a linear association, visually fit a straight line, and informally assess the model fit by judging the closeness of the data points to the line of best fit.

Use the equation of a linear model to solve problems in the context of bivariate measurement data, interpreting the slope and intercepts.

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

Use appropriate graphical displays from data distributions involving lines of best fit to draw informal inferences and answer the statistical investigative question posed in an unbiased statistical study.

Interpret and solve relevant mathematical problems leading to two linear equations in two variables.

Show and explain that solutions to a system of two linear equations in two variables correspond to points of intersection of their graphs, because the points of intersection satisfy both equations simultaneously.

Analyze and solve systems of two linear equations in two variables algebraically to find exact solutions.

Create and compare the equations of two lines that are either parallel to each other, perpendicular to each other, or neither parallel nor perpendicular.

Use mathematically applicable situations algebraically and graphically to build and interpret arithmetic sequences as functions whose domain is a subset of the integers.

Construct and interpret the graph of a linear function that models real-life phenomena and represent key characteristics of the graph using formal notation.

Relate the domain and range of a linear function to its graph and, where applicable, to the quantitative relationship it describes. Use formal interval and set notation to describe the domain and range of linear functions.

Use function notation to build and evaluate linear functions for inputs in their domains and interpret statements that use function notation in terms of a mathematical framework.

Analyze the difference between linear functions and nonlinear functions by informally analyzing the graphs of various parent functions (linear, quadratic, exponential, absolute value, square root, and cube root parent curves).

Use function notation to build and evaluate quadratic functions for inputs in their domains and interpret statements that use function notation in terms of a given framework.

Identify the effect on the graph generated by a quadratic function when replacing $f(x)$ with $f(x) + k$, $kf(x)$, $f(kx)$, and $f(x + k)$ for specific values of $k$ (both positive and negative); find the value of $k$ given the graphs.

Graph and analyze the key characteristics of quadratic functions.

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

Rewrite a quadratic function representing a mathematically applicable situation to reveal the maximum or minimum value of the function it defines. Explain what the value describes in context.

Create quadratic functions in two variables to represent relationships between quantities; graph quadratic functions on the coordinate axes with labels and scales.

Estimate, calculate, and interpret the average rate of change of a quadratic function and make comparisons to the average rate of change of linear functions.

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

Compare characteristics of two functions each represented in a different way.

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

Graph and analyze the key characteristics of simple exponential functions based on mathematically applicable situations.

Identify the effect on the graph generated by an exponential function when replacing $f(x)$ with $f(x)$ + $k$, and $k f(x)$, for specific values of $k$ (both positive and negative); find the value of $k$ given the graphs.

Use mathematically applicable situations algebraically and graphically to build and interpret geometric sequences as functions whose domain is a subset of the integers.

Compare characteristics of two functions each represented in a different way.

Use statistics appropriate to the shape of the data distribution to compare and represent center (median and mean) and variability (interquartile range, standard deviation) of two or more distributions by hand and using technology.

Interpret differences in shape, center, and variability of the distributions based on the investigation, accounting for possible effects of extreme data points (outliers).

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

Interpret the slope (predicted rate of change) and the intercept (constant term) of a linear model based on the investigation of the data.

Calculate the line of best fit and interpret the correlation coefficient, $r$, of a linear fit using technology. Use $r$ to describe the strength of the goodness of fit of the regression. Use the linear function to make predictions and assess how reasonable the prediction is in context.

Decide which type of function is most appropriate by observing graphed data.

Distinguish between correlation and causation.

Explain applicable, mathematical problems using a mathematical model.

Create mathematical models to explain phenomena that exist in the natural sciences, social sciences, liberal arts, fine and performing arts, and/or humanities domains.

Use units of measure (linear, area, capacity, rates, and time) as a way to make sense of conceptual problems; identify, use, and record appropriate units of measure within the given framework, within data displays, and on graphs; convert units and rates using proportional reasoning given a conversion factor; use units within multi-step problems and formulas; interpret units of input and resulting units of output.

Use various mathematical representations and structures with this information to represent and solve real-life problems.

Define appropriate quantities for the purpose of descriptive modeling.

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.

Explain contextual, mathematical problems using a mathematical model.

Create mathematical models to explain phenomena that exist in the natural sciences, social sciences, liberal arts, fine and performing arts, and/or humanities contexts.

Using abstract and quantitative reasoning, make decisions about information and data from a contextual situation.

Use various mathematical representations and structures with this information to represent and solve real-life problems.

Solve and explain engineering-based calculus problems; use mathematical and engineering models to explain real-life phenomena, using appropriate terminology and technology.

Describe the impact of engineering and technological advancement on mathematics and society

Express the relationships between points, lines, and planes in three dimensions.

Investigate functions of two and three independent variables to model engineering systems.

Apply partial differentiation of functions of two or more independent variables.

Manipulate integrals by changing the order of integration, introducing variable substitutions, or changing to curvilinear coordinates.

Evaluate and apply line integrals that are independent of path.

Apply properties of integrals to calculate and represent area, volume, or mass.

Use integrals of vectors to define and apply the gradient, divergence, and the curl.

Interpret the theorems of Green, Stokes, and Gauss and apply them to the study of real-world phenomena.

Rewrite algebraic and numeric expressions involving radicals.

Using numerical reasoning, show and explain that the sum or product of rational numbers is rational, the sum of a rational number and an irrational number is irrational, and the product of a nonzero rational number and an irrational number is irrational.

Create and solve linear inequalities in two variables to represent relationships between quantities including mathematically applicable situations; graph inequalities on coordinate axes with labels and scales.

Represent constraints of linear inequalities and interpret data points as possible or not possible.

Solve systems of linear inequalities by graphing, including systems representing a mathematically applicable situation.

Interpret quadratic expressions and parts of a quadratic expression that represent a quantity in terms of its context.

Fluently choose and produce an equivalent form of a quadratic expression to reveal and explain properties of the quantity represented by the expression.

Create and solve quadratic equations in one variable and explain the solution in the framework of applicable phenomena.

Represent constraints by quadratic equations and interpret data points as possible or not possible in a modeling framework.

Interpret exponential expressions and parts of an exponential expression that represent a quantity in terms of its framework.

Create exponential equations in one variable and use them to solve problems, including mathematically applicable situations.

Create exponential equations in two variables to represent relationships between quantities, including in mathematically applicable situations; graph equations on coordinate axes with labels and scales.

Represent constraints by exponential equations and interpret data points as possible or not possible in a modeling environment.

Solve real-life problems involving slope, parallel lines, perpendicular lines, area, and perimeter.

Apply the distance formula, midpoint formula, and slope of line segments to solve real-world 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.

Use mathematically applicable situations algebraically and graphically to build and interpret arithmetic sequences as functions whose domain is a subset of the integers.

Construct and interpret the graph of a linear function that models real-life phenomena and represent key characteristics of the graph using formal notation.

Relate the domain and range of a linear function to its graph and, where applicable, to the quantitative relationship it describes. Use formal interval and set notation to describe the domain and range of linear functions.

Use function notation to build and evaluate linear functions for inputs in their domains and interpret statements that use function notation in terms of a mathematical framework.

Analyze the difference between linear functions and nonlinear functions by informally analyzing the graphs of various parent functions (linear, quadratic, exponential, absolute value, square root, and cube root parent curves).

Use function notation to build and evaluate quadratic functions for inputs in their domains and interpret statements that use function notation in terms of a given framework.

Identify the effect on the graph generated by a quadratic function when replacing $f(x)$ with $f(x) + k$, $kf(x)$, $f(kx)$, and $f(x + k)$ for specific values of $k$ (both positive and negative); find the value of $k$ given the graphs.

Graph and analyze the key characteristics of quadratic functions.

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

Rewrite a quadratic function representing a mathematically applicable situation to reveal the maximum or minimum value of the function it defines. Explain what the value describes in context.

Create quadratic functions in two variables to represent relationships between quantities; graph quadratic functions on the coordinate axes with labels and scales.

Estimate, calculate, and interpret the average rate of change of a quadratic function and make comparisons to the average rate of change of linear functions.

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

Compare characteristics of two functions each represented in a different way.

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

Graph and analyze the key characteristics of simple exponential functions based on mathematically applicable situations.

Identify the effect on the graph generated by an exponential function when replacing $f(x)$ with $f(x)$ + $k$, and $k f(x)$, for specific values of $k$ (both positive and negative); find the value of $k$ given the graphs.

Use mathematically applicable situations algebraically and graphically to build and interpret geometric sequences as functions whose domain is a subset of the integers.

Compare characteristics of two functions each represented in a different way.

Use statistics appropriate to the shape of the data distribution to compare and represent center (median and mean) and variability (interquartile range, standard deviation) of two or more distributions by hand and using technology.

Interpret differences in shape, center, and variability of the distributions based on the investigation, accounting for possible effects of extreme data points (outliers).

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

Interpret the slope (predicted rate of change) and the intercept (constant term) of a linear model based on the investigation of the data.

Calculate the line of best fit and interpret the correlation coefficient, $r$, of a linear fit using technology. Use $r$ to describe the strength of the goodness of fit of the regression. Use the linear function to make predictions and assess how reasonable the prediction is in context.

Decide which type of function is most appropriate by observing graphed data.

Distinguish between correlation and causation.

Explain applicable, mathematical problems using a mathematical model.

Create mathematical models to explain phenomena that exist in the natural sciences, social sciences, liberal arts, fine and performing arts, and/or humanities domains.

Use units of measure (linear, area, capacity, rates, and time) as a way to make sense of conceptual problems; identify, use, and record appropriate units of measure within the given framework, within data displays, and on graphs; convert units and rates using proportional reasoning given a conversion factor; use units within multi-step problems and formulas; interpret units of input and resulting units of output.

Use various mathematical representations and structures with this information to represent and solve real-life problems.

Define appropriate quantities for the purpose of descriptive modeling.

Interpret polynomial expressions of varying degrees that represent a quantity in terms of its given geometric framework.

Perform operations with polynomials and prove that polynomials form a system analogous to the integers in that they are closed under these operations.

Using algebraic reasoning, add, subtract, and multiply single variable polynomials.

Use geometric reasoning and symmetries of regular polygons to develop definitions of rotations, reflections, and translations.

Verify experimentally the congruence properties of rotations, reflections, and translations: lines are taken to lines and line segments to line segments of the same length; angles are taken to angles of the same measure; parallel lines are taken to parallel lines.

Use geometric descriptions of rigid motions to draw the transformed figures and to predict the effect on a given figure. Describe a sequence of transformations from one figure to another and use transformation properties to determine congruence.

Explain how the criteria for triangle congruence follow from the definition of congruence in terms of rigid motions. Use congruency criteria for triangles to solve problems and to prove relationships in geometric figures.

Use the undefined notions of point, line, line segment, plane, distance along a line segment, and distance around a circular arc to develop and use precise definitions and symbolic notations to prove theorems and solve geometric problems.

Classify quadrilaterals in the coordinate plane by proving simple geometric theorems algebraically.

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

Prove and apply theorems about lines and angles to solve problems.

Use geometric reasoning to establish facts about the angle sum and exterior angle of triangles, about the angles created when parallel lines are cut by a transversal, and the angle-angle criterion for similarity of triangles.

Verify experimentally the properties of dilations.

Given two figures, use and apply the definition of similarity in terms of similarity transformations.

Use the properties of similarity transformations to establish criterion for two triangles to be similar. Use similarity criteria for triangles to solve problems and to prove relationships in geometric figures.

Construct formal proofs to justify and apply theorems about triangles.

Explain 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 for sides and angles of right triangles in applied problems.

Explore and interpret a radian as the ratio of the arc length to the radius of a circle.

Explore and explain the relationship between radian measures and degree measures and convert fluently between degree and radian measures.

Use special right triangles on the unit circle to determine the values of sine, cosine, and tangent for 30° ($\frac{\pi}{6}$), 45° ($\frac{\pi}{4}$) and 60° ($\frac{\pi}{3}$) angle measures. Use reflections of triangles to determine reference angles and identify coordinate values in all four quadrants of the coordinate plane.

Identify and apply angle relationships formed by chords, tangents, secants and radii with circles.

Using similarity, derive the fact that the length of the arc (arc length) intercepted by an angle is proportional to the radius; derive the formula for the area of a sector. Solve mathematically applicable problems involving applications of arc length and area of sector.

Write and graph the equation of circles in standard form.

Use volume formulas for prisms, cylinders, pyramids, cones, and spheres to solve problems including right and oblique solids.

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

Apply concepts of density based on area and volume in modeling situations.

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.

Construct and summarize categorical data for two categories in two-way frequency tables.

Use categorical data in two-way frequency tables to calculate and interpret probabilities based on the investigation.

Describe categories of events as subsets of a sample space using unions, intersections, or complements of other events. Apply the Addition Rule conceptually, P(A or B)= P(A) + P(B)-P(A and B), and interpret the answers in context.

Apply and interpret the general Multiplication Rule conceptually to independent events of a sample space, P(A and B) = [P(A)]x[P(B|A)] =[P(B)]x[P(A|B)] using contingency tables or tree diagrams.

Use conditional probability to interpret risk in terms of decision-making and investigate questions such as those involving false positives or false negatives from screening tests.

Define permutations and combinations and apply this understanding to compute probabilities of compound events and solve meaningful problems.

Interpret the probability distribution for a given random variable and interpret the expected value.

Develop a probability distribution for variables of interest using theoretical and empirical (observed) probabilities and calculate and interpret the expected value.

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

Compare the payoff values associated with the probability distribution for a random variable and make informed decisions based on expected value and measures of variability.

Explain mathematically applicable problems using a mathematical model.

Create mathematical models to explain phenomena that exist in the natural sciences, social sciences, liberal arts, fine and performing arts, and/or humanities contexts.

Using abstract and quantitative reasoning, make decisions about information and data from a mathematically applicable situation.

Use various mathematical representations and structures with this information to represent and solve real-life problems.

Use matrices to represent data, and perform mathematical operations with matrices and scalars, demonstrating that some properties of real numbers hold for matrices, but that others do not.

Rewrite a system of linear equations using a matrix representation.

Use the inverse of an invertible matrix to solve systems of linear equations.

Utilize linear programming to represent constraints by equations or inequalities, and by systems of equations and/or inequalities, and interpret data points as solutions or non-solutions under the established constraints in real-world problems.

Define the three basic trigonometric ratios in terms of x, y, and r using the unit circle centered at the origin of the coordinate plane.

Apply understanding of the angle measures and coordinates of the unit circle to solve practical, real-life problems involving trigonometric equations.

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.

Find the inverse of exponential and logarithmic functions using equations, tables, and graphs, limiting the domain of inverses where necessary to maintain functionality, and prove by composition or verify by inspection that one function is the inverse of another.

Analyze, graph, and compare exponential and logarithmic functions.

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

Create exponential equations and use logarithms to solve mathematical, applicable problems for which only one variable is unknown.

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

Create, interpret, and solve exponential equations to represent relationships between quantities and analyze the relationships numerically with tables, algebraically, and graphically.

Create, interpret, and solve logarithmic equations in two or more variables to represent relationships between quantities.

Rewrite radical expressions as expressions with rational exponents. Extend the properties of integer exponents to rational exponents.

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

Analyze and graph radical functions.

Create, interpret and solve radical equations with one unknown value and use them to solve problems that model real-world situations.

Create, interpret, and solve radical equations in two or more variables to represent relationships between quantities.

Graph and analyze quadratic functions in contextual situations and include analysis of data sets with regressions.

Define complex numbers $i$ such that $i^2 = –1$ and show that every complex number has the form $a + bi$ where $a$ and $b$ are real numbers and that the complex conjugate is $a - bi$.

Use the relation $i^2 = –1$ and the commutative, associative, and distributive properties to add, subtract, and multiply complex numbers.

Use the structure of an expression to factor quadratics.

Write and solve quadratic equations and inequalities with real coefficients and use the solution to explain a mathematical, applicable situation.

Solve systems of quadratic and linear functions to determine points of intersection.

Create and analyze quadratic equations to represent relationships between quantities as a model for contextual situations.

Identify the number of zeros that exist for any polynomial based upon the greatest degree of the polynomial and the end behavior of the polynomial by observing the sign of the leading coefficient.

Identify zeros of polynomial functions using technology or pre-factored polynomials and use the zeros to construct a graph of the function defined by the polynomial function. Analyze identify key features of these polynomial functions.

Use the structure of an expression to factor polynomials, including the sum of cubes, the difference of cubes, and higher-order polynomials that may be expressed as a quadratic within a quadratic.

Using all the zeros of a polynomial function, list all the factors and multiply to write a multiple of the polynomial function in standard form.

Rewrite simple rational expressions in equivalent forms.

Add, subtract, multiply and divide rational expressions, including problems in context and express rational expressions in irreducible form.

Graph rational functions, identifying key characteristics.

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

Recognize the purposes of and differences among sample surveys, experiments, and observational studies; explain how randomization relates to each. Distinguish between primary and secondary data and how it affects the types of conclusions that can be drawn.

When collecting and considering data, critically evaluate ethics, privacy, potential bias, and confounding variables along with their implications for interpretation in answering a statistical investigative question. Implement strategies for organizing and preparing big data sets.

Distinguish between population distributions, sample data distributions, and sampling distributions. Use sample statistics to make inferences about population parameters based on a random sample from that population and to communicate conclusions using appropriate statistical language.

Calculate and interpret z-scores as a measure of relative standing and as a method of standardizing units.

Given a normally distributed population, estimate percentages using the Empirical Rule, z-scores, and technology.

Model sample-to-sample variability in sampling distributions of a statistic using simulations taken from a given population.

Given a margin of error, develop and compare confidence intervals of different models to make conclusions about reliability.

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

Explain applicable, mathematical problems using a mathematical model.

Create mathematical models to explain phenomena that exist in the natural sciences, social sciences, liberal arts, fine and performing arts, and/or humanities contexts.

Using abstract and quantitative reasoning, make decisions about information and data from a mathematical, applicable situation.

Use various mathematical representations and structures to represent and solve real-life problems.

Analyze sequences using multiple representations.

Analyze series using multiple representations.

Analyze conic sections using different representations.

Extend trigonometry to the polar plane.

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.

Analyze piecewise-defined functions using different representations.

Analyze rational functions using different representations.

Define and analyze trigonometric relationships.

Analyze trigonometric functions and their inverses.

Explain contextual, mathematical problems using a mathematical model.

Create mathematical models to explain phenomena that exist in the natural sciences, social sciences, liberal arts, fine and performing arts, and/or humanities contexts.

Using abstract and quantitative reasoning, make decisions about information and data from a contextual situation.

Use various mathematical representations and structures with this information to represent and solve real-life problems.

Verify trigonometric identities and solve trigonometric equations.

Apply trigonometry to general triangles.

Perform operations with vectors in context.

Model situations with parametric equations.

Use fractions, decimals, percents, and ratios to solve problems related to budgets, income tax rates, payroll deductions, pie charts, percent yield, sales tax, percent populations, rent increase, cost savings, debt-to-income ratios, stock splits, floor plans and scale models, trigonometric calculations, banking services, and other business and financial applications.

Convert numerical quantities of one form (fractions, decimals, percents) to another within financial applications.

Calculate and interpret percent of increase and decrease.

Construct, solve, and interpret algebraic ratios and proportions.

Use and rearrange formulas applicable to real-world contexts.

Investigate the impact of changing the value of the different variables in financial formulas to compare the resulting financial outcomes.

Write algebraic formulas for use in spreadsheets and utilize technology to perform both iterate and formulaic calculations.

Use the simple interest formula, I = Prt, and inverse operations to solve for specified variables in banking services applications and other interest problems.

Demonstrate by iteration (both with technology and without) that the compounding process pays “interest on your interest.”

Derive the compound interest formula, $A = P(1 + \frac{r}{t})^{nt}$, by using patterns and inductive reasoning, then compute compound interest with and without the formula.

Explore the concept of limits of rational functions in discovering the compound continuous formula. Use technology to investigate and verify what happens as the number of compounds approaches infinity.

Apply the natural base e in the continuous compounding formula, $A = Pe^{rt}$.

Use the monthly payment formula to calculate payment amounts in a variety of circumstances.

Utilize the monthly payment formula to assist in calculating the total interest paid (finance charge) when using credit. Compare the total of monthly payments to the original (cash) price.

Interpret and use sigma notation.

Explore and identify how the elements of the present value of a single deposit formula and the periodic deposit investment formula relate to the compound interest formula.

Utilize the present and future value of a periodic investment formulas to make calculations regarding long-term investments and retirement planning.

Write, graph, solve, and interpret systems of linear equations given an applicable financial situation.

Write, graph, solve, and interpret systems of equations containing one linear and one quadratic equation, given an applicable financial situation.

Write, graph, and interpret systems of equations containing one linear and one exponential equation, given an applicable financial situation.

Write, graph, and interpret systems of a linear and a piecewise function.

Solve linear systems of equations and inequalities to identify points of intersection and define domains in the context of the problem situation.

Apply concepts of area, volume, and scale factors to a variety of real-world financial applications.

Use factors of dilations to draw to scale in contextual situations.

Use sectors and central angles of a circle to depict proportional categories on a pie chart when given categorical information.

Solve problems using the Pythagorean Theorem and trigonometric functions and their inverses in 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.

Examine and identify the key characteristics of functions that model financial situations given the parameters of the context.

Solve financial problems given the parameters of the applicable context using a variety of functions.

Describe the meaning of functions and how to determine if a relation is a function or not.

Utilize function notation to represent a functional relation and to evaluate functions.

Create, apply, and interpret linear functions to model real-world financial problems.

Create, apply, and interpret exponential functions of the form y = $ab^x$ and classify them as exponential decay (when 0 < b < 1) or as exponential growth (when b > 1).

Create, apply, and interpret quadratic functions to model real-world financial applications.

Create, apply, and interpret the greatest integer function in real-world financial applications.

Create, apply, and interpret piecewise functions in real-world financial applications.

Recognize real-world situations where square root, cubic, or rational functions apply.

Create and use inequalities to define domains when creating algebraic expressions and functions.

Interpret measures of central tendency (mean, median, mode) and spread (range, interquartile range, variance, standard deviation) to analyze contextualized data sets.

Construct and interpret common data displays (bar graphs, line graphs, stock bar charts, candlestick charts, box and whisker plots, stem and leaf plots, and circle graphs) to recognize and interpret trends.

Construct and interpret scatterplots to recognize and interpret trends.

Use technology to find, interpret, and graph linear, quadratic, and exponential regression equations to make predictions about the corresponding context.

Use technology to determine the correlation coefficient of linear, quadratic, and exponential regression curves.

Distinguish between causation and correlation for bivariate data.

Create and analyze discrete probability distributions.

Apply the Arithmetic Average Formula to calculate and interpret a d-day simple moving average given a set of n data points, $p_1$, $p_2$, $p_3$, ..., $p_{n -1}$, $p_n$.

Identify a contextual, real-life problem that can be answered using investigative research.

Develop statistical questions that can help solve a real-life problem involved in business and financial decision-making.

Create a statistical study using sound methodology to answer statistical questions and to solve the real-life problem.

Explain how the sample size impacts the precision with which estimates of the population parameters can be made.

Recognize that random selection from a population plays a different role than random assignment in an experiment.

Incorporate random designs in data collection.

Describe ways in which “big data” can be used to make decisions in various business enterprises and in the context of business and financial decision-making.

Use distributions to identify the key features of the data collected.

Interpret results and make connections to the original research question.

Explain contextual, mathematical problems using a mathematical model.

Create mathematical models to explain phenomena that exist in the natural sciences, social sciences, liberal arts, fine and performing arts, and/or humanities contexts.

Using abstract and quantitative reasoning, make decisions about information and data from a contextual situation.

Use various mathematical representations and structures with this information to represent and solve real-life problems.

Create and verify identification numbers.

Analyze and evaluate the mathematics behind various methods of voting and selection.

Evaluate various voting and selection processes to determine an appropriate method for a given situation.

Apply various ranking algorithms to determine an appropriate method for a given situation.

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

Determine, represent, and analyze mathematical models for income, expenditures, and various types of loans and investments.

Represent situations and solve problems using vectors, in areas such as transportation, computer graphics, and the physics of force and motion.

Represent geometric transformations and solve problems using matrices.

Solve problems represented by a vertex-edge graphs.

Construct, analyze, and interpret flow charts to develop an algorithm to describe processes such as quality control procedures.

Investigate the scheduling of projects using Program Evaluation Review Technique (PERT).

Consider problems that can be resolved by coloring graphs.

Create and use two-dimensional and three-dimensional representations to model authentic situations.

Solve problems involving inaccessible distances using basic trigonometric principles including extensions of right triangle trigonometry.

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.

Determine whether a problem situation involving two quantities is best modeled by a discrete or continuous relationship.

Use linear, exponential, logistic, and piecewise functions to construct a model.

Apply statistical methods to design, conduct, and analyze statistical studies. Identify a contextual, real-life problem that can be answered using investigative research.

Build the skills and vocabulary necessary to analyze and critique reported statistical information, summaries, and graphical displays. Develop statistical investigative questions that can help solve a real-life problem involved in business and financial decision-making.

Create a statistical study using sound methodology to answer statistical investigative questions and to solve the real-life problem.

Explain how the sample size impacts the precision with which estimates of the population parameters can be made (i.e., the larger the sample size the more precision).

Recognize that random selection from a population plays a different role than random assignment in an experiment.

Incorporate random designs in data collection.

Describe ways in which big data can be used to make decisions in various business enterprises and in the context of business and financial decision-making.

Use distributions to identify the key features of the data collected.

Interpret results and make connections to the original research question.

Determine conditional probabilities and probabilities of compound events to make decisions in problem situations.

Use probabilities to make and justify decisions about risks in everyday life.

Calculate expected value to analyze mathematical fairness, payoff, and risk.

Analyze real-life situations involving strategic interactions using the mathematics of zero-sum games.

Construct a mathematical model of probabilistic situations to make mathematical assumptions.

Explain contextual, mathematical problems using a mathematical model.

Create mathematical models to explain phenomena that exist in the natural sciences, social sciences, liberal arts, fine and performing arts, and/or humanities contexts.

Using abstract and quantitative reasoning, make decisions about information and data from a contextual situation.

Use relevant information to create various mathematical representations and structures to solve real-life problems.

Apply proportions, ratios, rates, and percentages to various settings, including business, media, and consumerism.

Solve problems involving ratios in mechanical and agricultural contexts.

Use proportions to solve problems involving large quantities that are not easily measured.

Use averages and weighted averages to make decisions.

Calculate and interpret indices.

Use matrices to solve systems of linear equations.

Operations on and with matrices.

Solve applied problems involving matrices using programming.

Use properties of vector spaces to solve problems.

Find and use vectors as a basis.

Apply vector spaces and vectors as a basis in programming to solve contextual problems.

Use and apply determinants.

Find and apply eigenvalues and eigenvectors.

Use vectors to find and interpret geometric relationships.

Use the dot product and the cross product of two vectors to find and interpret geometric relationships.

Solve applied problems involving vectors using programming.

Find and use matrices which represent geometric transformations.

Use matrices as functions.

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.

Model a finite random process using transition matrices in a Markov chain.

Simulate the different stages of a Markov chain using random numbers.

Use matrix algebra to calculate the probability of future states of a Markov chain.

Determine the attractor for a regular Markov chain.

Use transition matrices to identify absorbing states of a Markov chain.

Apply Markov chains in context.

Write a program to model the probabilities of real-life phenomena using a Markov chain.

Explain contextual, mathematical problems using a mathematical model.

Create mathematical models to explain phenomena that exist in the natural sciences, social sciences, liberal arts, fine and performing arts, and/or humanities contexts.

Using abstract and quantitative reasoning, make decisions about information and data from a contextual situation.

Use various mathematical representations and structures with this information to represent and solve real-life problems.

Use an object-oriented programming language to complete computer programming tasks.

Make sense of problems and persevere in solving them.

Reason abstractly and quantitatively.

Construct viable arguments and critique the reasoning of others.

Model with mathematics.

Use appropriate tools strategically.

Attend to precision.

Look for and make use of structure.

Look for and express regularity in repeated reasoning.

Use program evaluation review technique (PERT) to investigate completion times of a project.

Develop and apply transition matrices to make predictions using Markov Chains.

Apply queuing theory

Consider contextual factors and investigate issues within the decision-making process.

Explain contextual, mathematical problems using a mathematical model.

Create mathematical models to explain phenomena that exist in the natural sciences, social sciences, liberal arts, fine and performing arts, and/or humanities contexts.

Using abstract and quantitative reasoning, make decisions about information and data from a contextual situation.

Use various mathematical representations and structures with this information to represent and solve real-life problems.

Use advanced linear programming to make decisions and interpret results in real-life contexts.

Distinguish among continuous, integer, and binary contexts

Model and interpret results of a contextual problem with three or more variables using linear programming.

Solve problems with three or more variables using technology and principles of linear programming.

Examine cause and effect of contextual changes.

Find the optimal median location in a one-dimensional context.

Find the optimal median location in a rectilinear context.

Find the optimal location given three equally weighted, noncollinear points

Find the optimal location in a set covering context.

Relate context to a network representation.

Apply appropriate recursive algorithms.

Examine alternate decisions in response to contextual changes.

Use properties of normal distributions to make decisions about optimization and efficiency.

Calculate, analyze and interpret theoretical and empirical probabilities using standardized and non-standardized data.

Consider contextual factors and investigate issues within the decision-making process.

Apply techniques to quality control settings.

Calculate theoretical and empirical probabilities using standardized and non-standardized data.

Analyze and interpret the probabilities in terms of context.

Consider contextual factors and investigate issues within the decision-making process.

Use technology to simulate a real-world situation.

Analyze, evaluate, and interpret results of simulations.

Examine alternate decisions in response to contextual changes of simulations.

Develop and analyze fair methods for voting.

Develop and analyze fair methods for apportioning representatives.

Develop fair methods for setting voting district boundaries.

Historical computation methods

Use historical methods to solve equations.

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.

Explain contextual, mathematical problems using a mathematical model.

Create mathematical models to explain phenomena that exist in the natural sciences, social sciences, liberal arts, fine and performing arts, and/or humanities contexts.

Using abstract and quantitative reasoning, make decisions about information and data from a contextual situation.

Use various mathematical representations and structures with this information to represent and solve real-life problems.

Greek geometry

Greek algebra and number sense

Greek culture and society

Non-European mathematics in the middle ages

European mathematics emerges from the dark ages.

Use historical multiplication and division algorithms.

Use Cardano’s cubic formula to find a solution to a cubic equation.

Explain the cultural factors that encouraged the development of algebra in 15th century Italy, and how this development influenced mathematical thought throughout Europe.

Identify the works of Galileo, Copernicus, and Kepler as a landmark in scientific thought, describe the conflict between their explanation of the workings of the solar system and then-current perspectives, and contrast their works to those of Aristotle.

Describe the mathematical contributions of Fermat, Pascal, and Descartes.

The origins of calculus

Non-Euclidean geometry

Abstract algebra and number theory

The nature of mathematicians in the 17th, 18th, and 19th centuries.

The modern nature of mathematics

The modern nature of mathematicians

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.

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

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

Formulate multivariable statistical investigative questions.

Formulate inferential statistical investigative questions regarding association and prediction.

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

Distinguish between surveys, observational studies, and experiments.

Design sample surveys, experiments, and observational studies using accepted practices.

Distinguish between random selection and random assignment and identify their impact on conclusions.

Describe potential sources and effects of bias and confounding variables

Describe and adhere to the ethical use of data (e.g., sensitive information, privacy, and living subjects).

Identify when data can be generalized to a target population.

Summarize quantitative or categorical data using tables, graphical displays, and numerical summary statistics.

Summarize and describe relationships among multiple variables.

Use sampling distributions developed through simulation to describe the sample-to-sample variability of sample statistics.

Use sampling distributions to compute simulated p-values.

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

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

Use statistical evidence from analyses to answer the formulated statistical investigative questions.

Interpret the impact of outliers, missing values, or erroneous values on the results.

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

Interpret a given margin of error associated with an estimate of a population characteristic.

Explain the impact of multiple variables on one another.

Explain contextual, mathematical problems using a mathematical model.

Create mathematical models to explain phenomena that exist in the natural sciences, social sciences, liberal arts, fine and performing arts, and/or the humanities.

Using abstract and quantitative reasoning, make decisions about information and data from a real-life situation.

Use various mathematical representations and structures with this information to represent and solve real-life problems.

Through multi-step/multi-operational problems, perform mathematical operations on real numbers demonstrating fluency using the order of operations.

Represent and solve problems using proportional reasoning with ratios, rates, proportions, and scaling.

Apply the rules of exponents to simplify numerical expressions, extending the properties of exponents to rational exponents.

Perform mathematical operations on real numbers to include numerical radical expressions and complex fractions.

Estimate solutions to problems with real numbers and use the estimates to assess the reasonableness of results in the context of the problem.

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

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

Using multiple representations, solve equations and inequalities and use the solutions to draw reasonable conclusions about a situation being modeled, including possible constraints.

Solve quadratic equations using a variety of methods.

Rearrange literal equations to highlight a specified variable using the same reasoning as in solving equations.

Solve inequalities in one variable graphically and algebraically.

Using multiple methods, create and solve systems of linear equations and inequalities.

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

Use the distance formula, midpoint formula or slope to verify simple geometric properties.

Use coordinates to compute perimeters of polygons, circumference of circles and areas of triangles, rectangles and circles.

Informally derive the formulas for the volume and surface area of a cylinder, sphere, prism, pyramid, and cone.

Use formulas for finding the volume and surface area of spheres, right and oblique prisms, cylinders, pyramids, and cones.

Apply the Pythagorean Theorem and trigonometric ratios to solve problems involving right triangles.

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.

Define a function through maps, sets, equations and graphs using function notation.

Identify and sketch by hand the parent graph of functions expressed algebraically and show key characteristics of the graph using technology.

Using tables, graphs, and verbal descriptions, interpret the key characteristics of a function.

Calculate and interpret the average rate of change of a function over a specified interval. Estimate the rate of change from a graph.

Compare characteristics of two functions each represented in a different way.

Construct linear and exponential functions, given a graph, a description of a relationship, or two input-output pairs.

Construct arithmetic and geometric sequences recursively and explicitly, use them to model situations, and translate between the two forms. Connect linear functions to arithmetic sequences and exponential functions to geometric sequences.

Identify the effect on the parent graph of replacing $f(x)$ by $f(x) + k$, $kf(x)$, 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 using technology.

Represent univariate data on the real number line.

Calculate, compare, and interpret shape, center, and spread of two or more univariate data sets, accounting for possible effects of extreme data points.

Summarize categorical data for two categories in two-way frequency tables using relative frequencies in the context of the data.

Represent bivariate data on a scatter plot and describe how the variables are related in terms of strength and direction.

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 “r” of a linear fit.

Distinguish between correlation and causation, and interpolation and extrapolation.

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

Use the two-way frequency table to calculate conditional probabilities.

Calculate the conditional probability of A given B.

Explain contextual, mathematical problems using a mathematical model.

Create mathematical models to explain phenomena that exist in the natural sciences, social sciences, liberal arts, fine and performing arts, and/or humanities contexts.

Using abstract and quantitative reasoning, make decisions about information and data from a contextual situation.

Use various mathematical representations and structures with this information to represent and solve real-life problems.

Use matrices to represent data, and perform mathematical operations with matrices and scalars, demonstrating that some properties of real numbers hold for matrices, but that others do not.

Rewrite a system of linear equations using a matrix representation.

Use the inverse of an invertible matrix to solve systems of linear equations.

Utilize linear programming to represent constraints by equations or inequalities, and by systems of equations and/or inequalities, and interpret data points as solutions or non-solutions under the established constraints in real-world problems.

Analyze sequences using multiple representations.

Analyze series using multiple representations.

Define the three basic trigonometric ratios in terms of x, y, and r using the unit circle centered at the origin of the coordinate plane.

Apply understanding of the angle measures and coordinates of the unit circle to solve practical, real-life problems involving trigonometric equations.

Analyze conic sections using different representations.

Extend trigonometry to the polar plane.

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.

Find the inverse of exponential and logarithmic functions using equations, tables, and graphs, limiting the domain of inverses where necessary to maintain functionality, and prove by composition or verify by inspection that one function is the inverse of another.

Analyze, graph, and compare exponential and logarithmic functions.

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

Create exponential equations and use logarithms to solve mathematical, applicable problems for which only one variable is unknown.

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

Create, interpret, and solve exponential equations to represent relationships between quantities and analyze the relationships numerically with tables, algebraically, and graphically.

Create, interpret, and solve logarithmic equations in two or more variables to represent relationships between quantities.

Rewrite radical expressions as expressions with rational exponents. Extend the properties of integer exponents to rational exponents.

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

Analyze and graph radical functions.

Create, interpret and solve radical equations with one unknown value and use them to solve problems that model real-world situations.

Create, interpret, and solve radical equations in two or more variables to represent relationships between quantities.

Graph and analyze quadratic functions in contextual situations and include analysis of data sets with regressions.

Define complex numbers $i$ such that $i^2 = –1$ and show that every complex number has the form $a + bi$ where $a$ and $b$ are real numbers and that the complex conjugate is $a - bi$.

Use the relation $i^2 = –1$ and the commutative, associative, and distributive properties to add, subtract, and multiply complex numbers.

Use the structure of an expression to factor quadratics.

Write and solve quadratic equations and inequalities with real coefficients and use the solution to explain a mathematical, applicable situation.

Solve systems of quadratic and linear functions to determine points of intersection.

Create and analyze quadratic equations to represent relationships between quantities as a model for contextual situations.

Identify the number of zeros that exist for any polynomial based upon the greatest degree of the polynomial and the end behavior of the polynomial by observing the sign of the leading coefficient.

Identify zeros of polynomial functions using technology or pre-factored polynomials and use the zeros to construct a graph of the function defined by the polynomial function. Analyze identify key features of these polynomial functions.

Use the structure of an expression to factor polynomials, including the sum of cubes, the difference of cubes, and higher-order polynomials that may be expressed as a quadratic within a quadratic.

Using all the zeros of a polynomial function, list all the factors and multiply to write a multiple of the polynomial function in standard form.

Rewrite simple rational expressions in equivalent forms.

Add, subtract, multiply and divide rational expressions, including problems in context and express rational expressions in irreducible form.

Graph rational functions, identifying key characteristics.

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

Analyze piecewise-defined functions using different representations.

Analyze rational functions using different representations.

Define and analyze trigonometric relationships.

Analyze trigonometric functions and their inverses.

Recognize the purposes of and differences among sample surveys, experiments, and observational studies; explain how randomization relates to each. Distinguish between primary and secondary data and how it affects the types of conclusions that can be drawn.

When collecting and considering data, critically evaluate ethics, privacy, potential bias, and confounding variables along with their implications for interpretation in answering a statistical investigative question. Implement strategies for organizing and preparing big data sets.

Distinguish between population distributions, sample data distributions, and sampling distributions. Use sample statistics to make inferences about population parameters based on a random sample from that population and to communicate conclusions using appropriate statistical language.

Calculate and interpret z-scores as a measure of relative standing and as a method of standardizing units.

Given a normally distributed population, estimate percentages using the Empirical Rule, z-scores, and technology.

Model sample-to-sample variability in sampling distributions of a statistic using simulations taken from a given population.

Given a margin of error, develop and compare confidence intervals of different models to make conclusions about reliability.

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

Explain contextual, mathematical problems using a mathematical model.

Create mathematical models to explain phenomena that exist in the natural sciences, social sciences, liberal arts, fine and performing arts, and/or humanities contexts.

Using abstract and quantitative reasoning, make decisions about information and data from a contextual situation.

Use various mathematical representations and structures with this information to represent and solve real-life problems.

Verify trigonometric identities and solve trigonometric equations.

Apply trigonometry to general triangles.

Perform operations with vectors in context.

Model situations with parametric equations.