High School Math IAS Standards

Interpret statistics as a process for making inferences about a population based on a random sample from that population. Recognize the purposes of and differences among sample surveys, experiments, and observational studies; explain how randomization relates to each. (E)

Given a statistical situation, identify it as a sample survey, experiment, or an observational study. Recognize the purpose of using a random sample.

Understand that statistics and data are non-neutral and designed to serve a particular interest. Analyze the possibilities for whose interest might be served and how the representations might be misleading. (E)

Select the model that represents biased or unbiased statistics and data and therefore might be misleading. (E)

Use technology to find a linear function that models a relationship between two quantitative variables to make predictions and interpret the slope and y-intercept. Using technology, compute and interpret the correlation coefficient. (E)

Use the line of best fit to find points that can be used to answer questions about data. (E)

Determine if the correlation coefficient, measured from 0 to 1, is a strong or weak correlation. (E)

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

Given a two-way frequency table, calculate relative frequencies for rows or columns.

Represent real-world problems using linear equations and inequalities in one variable, including those with rational number coefficients and variables on both sides of the equal sign. Solve them fluently, explaining the process used and justify the choice of a solution method. (E)

Write a one-variable equation or inequality given a word-problem. (E)

Solve multi-step linear equations and inequalities in one variable. (E)

Represent linear functions as graphs from equations (with emphasis on technology), equations from graphs, and equations from tables and other given information (e.g., from a given point on a line and the slope of the line). Find the equations of a line in a slope-intercept, point-slope, and standard forms. Recognize that different forms reveal more or less information about a given situation based on the form used.

Represent linear functions using multiple representations. Recognize equations of lines in multiple forms (slope-intercept form and standard form). Describe the attributes revealed by various forms of linear functions.

Represent real-world problems that can be modeled with a linear function using equations, graphs, and tables, including with technology. Translate fluently among these representations and interpret the slope and intercepts. (E)

Represent real-world problems that can be modeled with a linear function (in slope-intercept form) using equations, graphs, and tables. (E)

Solve linear and quadratic equations and formulas for a specified variable to highlight a quantity of interest, using the same reasoning as in solving equations. (E)

Solve linear and quadratic equations, including formulas, for a specified variable limited to simple quadratic formulas (e.g., A = pi*r 2 ).

Simplify square roots of monomial algebraic expressions, including non-perfect squares.

Simplify square roots of non-perfect square whole numbers.

Add, subtract, and multiply polynomials. Divide polynomials by monomials. Use these operations to rewrite algebraic expressions in equivalent forms, and justify them with algebraic properties. (E)

Add, subtract, and multiply monomials and binomials. (E)

Extend understanding of independent/dependent variables to encompass domain/range, as applied to relations using tables, graphs, verbal descriptions, and equations. (E)

Identify the domain and range of a table or graph. Understand the domain is the independent variable and the range is the dependent variable. (E)

Evaluate functions for given elements of the domain, and interpret statements in function notation in terms of a context.

Evaluate functions for given elements of the domain.

Describe, qualitatively, the functional relationship between two quantities by analyzing key features of a graph. Sketch a graph that exhibits given key features of a function that has been verbally described, including intercepts, where the function is increasing or decreasing, where the function is positive or negative, and any relative maximum or minimum values. Identify the independent and dependent variables. (E)

Given the graph of a function, identify its key features (domain, range, intercept(s), increasing, and decreasing). Given the key features (domain, range, increasing, and decreasing) of a function, determine the graph that indicates the key feature. Identify independent and dependent variables of a function. (E)

Distinguish between situations that can be modeled with linear functions and exponential functions. Understand that linear functions grow by equal differences over equal intervals, and that exponential functions grow by equal factors over equal intervals. Compare linear functions and exponential functions that model real-world situations using tables, graphs, and equations. (E)

Identify a function as linear or exponential given a graph, table, or real-world problem. Understand that linear functions grow by equal differences and exponential functions grow by equal factors. (E)

Represent real-world and other mathematical problems that can be modeled with simple exponential functions using tables, graphs, and equations of the form y = ab x (for integer values of x > 1, rational values of b > 0 and b ≠ 1 ) with and without technology; interpret the values of a and b.

Represent real-world or mathematical problems of an exponential equation of the form y = ab x , and identify the a-value as the initial value (or y-intercept) and the b-value as the growth or decay factor.

Solve quadratic equations in one variable by inspection (e.g., for x 2 = 49), finding square roots, using the quadratic formula, and factoring, as appropriate to the initial form of the equation.

Solve quadratic equations using square roots, factoring, and the quadratic formula.

Represent real-world problems using quadratic equations in one or two variables and solve such problems with technology. Interpret the solution(s), and determine whether they are reasonable. (E)

Using technology, determine if a point lies on a quadratic function.

Graph exponential and quadratic functions with and without technology. Identify and describe key features, such as zeros, lines of symmetry, and extreme values in real-world and other mathematical problems involving quadratic functions with and without technology; interpret the results in the real-world contexts.

Graph exponential and quadratic functions with the aid of technology.

Given a graph of an exponential and quadratic function identify key features such as zeros and extreme values.

Describe the relationships among a solution of a quadratic equation, a zero of the function, an x-intercept of the graph, and the factors of the expression. Explain that every quadratic has two complex solutions, which may or may not be real solutions.

Identify zeros of quadratic functions. Understand that quadratic equations have 2, 1, or no real solutions.

Represent real-world problems using linear inequalities in two variables and solve such problems; interpret the solution set, and determine whether it is reasonable. Graph the solutions to a linear inequality in two variables as a half-plane. (E)

Identify a two-variable linear inequality that represents a real-world problem. (E)

Identify solutions to real-world linear inequalities given the graph. (E)

Write and graph a system of two linear equations in two variables that represents a real-world problem and solve the problem graphically and algebraically with and without technology. Interpret the solution, and determine whether the solution is reasonable. (E)

Graph a system of linear equations that represents a real-world problem and determine the reasonableness of the solution. Identify a solution to a system of linear equations. (E)

Represent real-world problems using a system of two linear inequalities in two variables. Graph the solution set to a system of linear inequalities in two variables as the intersection of the corresponding half-planes with and without technology. Interpret the solution set, and determine whether it is reasonable.

Identify a system of linear inequalities that represents a given real-world problem.

Identify the solution set to a system of inequalities.

Data Analysis & Statistics

Linear Equations, Inequalities, and Functions

Number Systems, Expressions, and Functions

Quadratic and Exponential Equations and Functions

Systems of Linear Equations and Inequalities

Data Analysis & Statistics

Quadratic and Exponential Equations and Functions

Systems of Linear Equations and Inequalities

Linear Equations, Inequalities, and Functions

Number Systems, Expressions, and Functions

Mathematics Process Standards

Students utilize and interpret statistical claims.

Interpret statistics as a process for making inferences about a population based on a random sample from that population. Recognize the purposes of and differences among sample surveys, experiments, and observational studies; explain how randomization relates to each.

Understand that statistics and data are non-neutral and designed to serve a particular interest. Analyze the possibilities for whose interest might be served and how the representations might be misleading.

Use technology to find a linear function that models a relationship between two quantitative variables to make predictions and interpret the slope and <em>y</em>-intercept. Using technology, compute and interpret the correlation coefficient.

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

Students represent real-world situations with linear functions and use these equations to solve problems.

Represent real-world problems using linear equations and inequalities in one variable, including those with rational number coefficients and variables on both sides of the equal sign. Solve them fluently, explaining the process used and justify the choice of a solution method.

Represent linear functions as graphs from equations (with emphasis on technology), equations from graphs, and equations from tables and other given information (e.g., from a given point on a line and the slope of the line). Find the equations of a line in a slope-intercept, point-slope, and standard forms. Reveal more or less information about a given situation based on the form used.

Represent real-world problems that can be modeled with a linear function using equations, graphs, and tables, including with technology. Translate fluently among these representations and interpret the slope and intercepts.

Solve linear and quadratic equations and formulas for a specified variable to highlight a quantity of interest, using the same reasoning as in solving equations.

Students simplify and manipulate algebraic expressions, equations, and functions in a variety of forms.

Simplify square roots of monomial algebraic expressions, including non-perfect squares.

Add, subtract, and multiply polynomials. Divide polynomials by monomials. Use these operations to rewrite algebraic expressions in equivalent forms, and justify them with algebraic properties.

Extend understanding of independent/dependent variables to encompass domain/range, as applied to relations using tables, graphs, verbal descriptions, and equations.

Evaluate functions for given elements of the domain, and interpret statements in function notation in terms of a context.

Describe, qualitatively, the functional relationship between two quantities by analyzing key features of a graph. Sketch a graph that exhibits given key features of a function that has been verbally described, including intercepts, where the function is increasing or decreasing, where the function is positive or negative, and any relative maximum or minimum values. Identify the independent and dependent variables.

Students represent real-world situations using quadratic and exponential equations and use these equations to solve problems.

Distinguish between situations that can be modeled with linear functions and exponential functions. Understand that linear functions grow by equal differences over equal intervals, and that exponential functions grow by equal factors over equal intervals. Compare linear functions and exponential functions that model real-world situations using tables, graphs, and equations.

Represent real-world and other mathematical problems that can be modeled with simple exponential functions using tables, graphs, and equations of the form <em>y = ab^x</em> (for integer values of <em>x > 1</em>, rational values of <em>b > 0</em> and <em>b ≠ 1</em>) with and without technology; interpret the values of <em>a</em> and <em>b</em>.

Solve quadratic equations in one variable by inspection (e.g., for <em>x² = 49</em>), finding square roots, using the quadratic formula, and factoring, as appropriate to the initial form of the equation.

Represent real-world problems using quadratic equations in one or two variables and solve such problems with technology. Interpret the solution(s), and determine whether they are reasonable.

Graph exponential and quadratic functions with and without technology. Identify and describe key features, such as zeros, lines of symmetry, and extreme values in real-world and other mathematical problems involving quadratic functions with and without technology; interpret the results in the real-world contexts.

Describe the relationships among a solution of a quadratic equation, a zero of the function, an <em>x</em>-intercept of the graph, and the factors of the expression. Explain that every quadratic has two complex solutions, which may or may not be real solutions.

Students represent real-world situations as systems of linear equations and inequalities, using those systems to solve problems.

Represent real-world problems using linear inequalities in two variables and solve such problems; interpret the solution set, and determine whether it is reasonable. Graph the solutions to a linear inequality in two variables as a half-plane.

Write and graph a system of two linear equations in two variables that represents a real-world problem and solve the problem graphically and algebraically with and without technology. Interpret the solution, and determine whether the solution is reasonable.

Represent real-world problems using a system of two linear inequalities in two variables. Graph the solution set to a system of linear inequalities in two variables as the intersection of the corresponding half-planes with and without technology. Interpret the solution set, and determine whether it is reasonable.

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.

Modeling with Quantities

Modeling with Data and Statistics

Modeling with Advanced Algebra

Modeling with Functions and Data

Function Families

Arithmetic and Structure of Expressions, Equations, and Functions

Mathematics Process Standards

Students simplify, manipulate, and solve nonlinear expressions, equations, and functions in a variety of forms.

Explain how extending the properties of integer exponents to rational numbers allows for a notation for radicals in terms of rational exponents (e.g., 5^1/3) and explain how this is defined.

Rewrite algebraic rational expressions in equivalent forms (e.g., using properties of exponents and factoring techniques) and describe how rewriting those expressions reveals mathematical structure. Add, subtract, multiply, and divide algebraic rational expressions.

Solve systems of equations consisting of linear and nonlinear equations or functions in two variables algebraically and graphically.

Solve exponential and logarithmic equations in one variable.

Students represent nonlinear functions in a variety of forms, recognizing and applying key features based on the type of function.

Using technology, identify, create, and connect algebraic and graphical representations of each of the function families listed:

Quadratic

Polynomial

Square root

Rational

Exponential

Logarithmic

Piecewise-defined and absolute value functions

Graph each of the families of function with and without technology. Identify and describe key features, such as intercepts, domain and range, asymptotes, symmetry, and end behavior. Create inverse functions algebraically and/or graphically based on a given function. Model real-world situations with each function family.

Use graphical and algebraic structures and techniques to transform functions into equivalent forms to expose different information and identify key features. Connect the meaning of the key features to contextual situations.

Solve real-world problems with each function family, including situations in the context of science and economic phenomena.

Students use advanced algebra concepts to model real-world function situations and use specific algebraic techniques to reveal and make use of structure with families of functions.

Use algebraic and graphical strategies to make use of structure with quadratic, polynomial, and rational functions to solve real-world problems, including but not limited to:

Completing the square to rewrite contextual quadratic functions in vertex form and interpret the outcome;

Determining the number of solutions to a function using graphical and algebraic forms (including the discriminant and complex numbers as appropriate);

Factoring, chunking, and rewriting functions using properties of exponents; and

Identifying and explaining extraneous roots.

Represent and solve real-world systems of linear equations and inequalities in two or three variables algebraically and using technology. Interpret the solution, and determine whether it is reasonable.

Model real-world phenomena using linear programming and matrices.

Students use statistics and probability techniques to collect and interpret complex data that can be modeled using functions.

Distinguish between random and non-random sampling methods, identify possible sources of bias in sampling, describe how such bias can be controlled and reduced, evaluate the characteristics of a good survey and well-designed experiment, design simple experiments or investigations to collect data to answer questions of interest, and make inferences from sample results.

Using the results of a simulation, decide if a specified model is consistent with the results. Construct a theoretical model, and apply the law of large numbers to show the relationship between the two models.

Use data science techniques such as predictive modeling, linear algebra, and conditional probability to analyze data sets and make and evaluate claims.

Students use families of functions to model real-world situations using multiple mathematical representations.

Define functions and their inverses and illustrate examples algebraically and graphically. Identify real-world situations that can be modeled using functions.

Represent real-world problems that can be modeled by linear, quadratic, exponential, and rational functions using tables, graphs, and equations. Use technology to represent the functional relationships and translate and interpret different forms (e.g., vertex form of a quadratic, intercepts, end behavior) with respect to the context.

Use technology to find a linear, quadratic, or exponential function that models a relationship for a bivariate data set to make predictions; interpret the correlation coefficient for linear models. Compare and evaluate model fit using different function families.

Explore the effects of function transformations using graphing technology. Explain the effects of transformations of functions such as <em>f(x) + k, kf(x), f(kx),</em> or <em>f(x + k)</em> for different functions and values of <em>k</em>.

Students use combinatorics to quantify and model real-world situations.

Model real-world probability situations using permutations, combinations, and the Fundamental Counting Principle.

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.

Modeling with Quantities

Modeling with Data and Statistics

Modeling with Advanced Algebra

Modeling with Functions and Data

Function Families

Arithmetic and Structure of Expressions, Equations, and Functions

Mathematics Process Standards

Students simplify, manipulate, and solve nonlinear expressions, equations, and functions in a variety of forms.

Explain how extending the properties of integer exponents to rational numbers allows for a notation for radicals in terms of rational exponents (e.g., <em>5 ^1/3</em>) and explain how this is defined.

Rewrite algebraic rational expressions in equivalent forms (e.g., using properties of exponents and factoring techniques) and describe how rewriting those expressions reveals mathematical structure. Add, subtract, multiply, and divide algebraic rational expressions.

Solve systems of equations consisting of linear and nonlinear equations or functions in two variables algebraically and graphically.

Students represent nonlinear functions in a variety of forms, recognizing and applying key features based on the type of function.

Using technology, identify, create, and connect algebraic and graphical representations of each of the function families listed. Model real-world situations with each function family:

Quadratic

Polynomial

Square root

Rational

Exponential

Piecewise-defined and absolute value functions

Graph each of the families of function with and without technology. Identify and describe key features, such as intercepts, domain and range, asymptotes, symmetry, and end behavior. Create inverse functions algebraically and/or graphically based on a given function.

Use graphical and algebraic structures and techniques to transform functions into equivalent forms to expose different information and identify key features. Connect the meaning of the key features to contextual situations.

Solve real-world problems with each function family, including situations in the context of science and economic phenomena.

Students use advanced algebra concepts to model real-world function situations and use specific algebraic techniques to reveal and make use of structure with families of functions.

Use algebraic and graphical strategies to make use of structure with quadratic, polynomial, and rational functions to solve real-world problems, including but not limited to:

Completing the square to rewrite contextual quadratic functions in vertex form and interpret the outcome;

Determining the number of solutions to a function using graphical and algebraic forms (including the discriminant and complex numbers as appropriate);

Factoring, chunking, and rewriting functions using properties of exponents; and

Identifying and explaining extraneous roots.

Represent and solve real-world systems of linear equations and inequalities in two or three variables algebraically and using technology. Interpret the solution, and determine whether it is reasonable.

Model real-world phenomena using linear programming and matrices.

Students use statistics and probability techniques to collect and interpret complex data that can be modeled using functions.

Distinguish between random and non-random sampling methods, identify possible sources of bias in sampling, describe how such bias can be controlled and reduced, evaluate the characteristics of a good survey and well-designed experiment, design simple experiments or investigations to collect data to answer questions of interest, and make inferences from sample results.

Using the results of a simulation, decide if a specified model is consistent with the results. Construct a theoretical model, and apply the law of large numbers to show the relationship between the two models.

Use data science techniques such as predictive modeling, linear algebra, and conditional probability to analyze data sets and make and evaluate claims.

Students represent real-world situations with linear and nonlinear functions, and use these equations to solve problems.

Define functions and their inverses and illustrate examples algebraically and graphically. Identify real-world situations that can be modeled using functions.

Represent real-world problems that can be modeled by linear, quadratic, exponential, and rational functions using tables, graphs, and equations. Use technology to represent the functional relationships and translate and interpret different forms (e.g., vertex form of a quadratic, intercepts, end behavior) with respect to the context.

Use technology to find a linear, quadratic, or exponential function that models a relationship for a bivariate data set to make predictions; interpret the correlation coefficient for linear models. Compare and evaluate model fit using different function families.

Explore the effects of function transformations using graphing technology. Explain the effects of transformations of functions such as <em>f(x) + k, kf(x), f(kx),</em> or <em>f(x + k)</em> for different functions and values of <em>k</em>.

Students use combinatorics to quantify and model real-world situations.

Using technology, model real-world probability situations using permutations, combinations, and the Fundamental Counting Principle.

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.

Applications of Integrals

Integrals

Applications of Derivatives

Differentiation

Limits and Continuity

Process Standards For Mathematics

Find the slope of a curve at a point, including points at which there are vertical tangents and no tangents.

Find average and instantaneous rates of change. Understand the instantaneous rate of change as the limit of the average rate of change. Interpret a derivative as a rate of change in applications, including distance, velocity, and acceleration.

Find the velocity and acceleration of a particle moving in a straight line.

Model rates of change, including related rates problems.

Find a tangent line to a curve at a point and a local linear approximation.

Decide where functions are decreasing and increasing. Understand the relationship between the increasing and decreasing behavior of f and the sign of f'.

Solve real-world and other mathematical problems finding local and absolute maximum and minimum points with and without technology.

Analyze real-world problems modeled by curves, including the notions of monotonicity and concavity with and without technology.

Find points of inflection of functions. Understand the relationship between the concavity of f and the sign of f". Understand points of inflection as places where concavity changes.

Use first and second derivatives to help sketch graphs modeling real-world and other mathematical problems with and without technology. Compare the corresponding characteristics of the graphs of f, f', and f".

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

Solve optimization real-world problems with and without technology.

Find specific antiderivatives using initial conditions, including finding velocity functions from acceleration functions, finding position functions from velocity functions, and applications to motion along a line.

Solve separable differential equations and use them in modeling real-world problems with and without technology.

Solve differential equations of the form y' = ky as applied to growth and decay problems.

Use definite integrals to find the area between a curve and the x-axis, or between two curves.

Use definite integrals to find the average value of a function over a closed interval.

Use definite integrals to find the volume of a solid with known cross-sectional area.

Apply integration to model and solve (with and without technology) real-world problems in physics, biology, economics, etc., using the integral as a rate of change to give accumulated change and using the method of setting up an approximating Riemann Sum and representing its limit as a definite integral.

Understand the concept of derivative geometrically, numerically, and analytically, and interpret the derivative as a rate of change.

Understand and apply the relationship between differentiability and continuity.

Understand and apply the Mean Value Theorem.

State, understand, and apply the definition of derivative.

Find the derivatives of functions, including algebraic, trigonometric, logarithmic, and exponential functions.

Find the derivatives of sums, products, and quotients.

Find the derivatives of composite functions, using the chain rule.

Find the derivatives of implicitly-defined functions.

Find the derivatives of inverse functions.

Find second derivatives and derivatives of higher order.

Find derivatives using logarithmic differentiation.

Use rectangle approximations to find approximate values of integrals.

Calculate the values of Riemann Sums over equal subdivisions using left, right, and midpoint evaluation points.

Interpret a definite integral as a limit of Riemann Sums.

Understand the Fundamental Theorem of Calculus: Interpret a definite integral of the rate of change of a quantity over an interval as the change of the quantity over the interval.

Use the Fundamental Theorem of Calculus to evaluate definite and indefinite integrals and to represent particular antiderivatives. Perform analytical and graphical analysis of functions so defined.

Understand and use these properties of definite integrals.

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Understand and use integration by substitution (or change of variable) to find values of integrals.

Understand and use Riemann Sums, the Trapezoidal Rule, and technology to approximate definite integrals of functions represented algebraically, geometrically, and by tables of values.

Understand the concept of limit and estimate limits from graphs and tables of values.

Find the types of discontinuities of a function.

Understand and use the Intermediate Value Theorem on a function over a closed interval.

Understand and apply the Extreme Value Theorem: If f(x) is continuous over a closed interval, then f has a maximum and a minimum on the interval.

Find limits by substitution.

Find limits of sums, differences, products, and quotients.

Find limits of rational functions that are undefined at a point.

Find limits at infinity.

Decide when a limit is infinite and use limits involving infinity to describe asymptotic behavior. Find special limits.

Find one-sided limits.

Understand continuity in terms of limits.

Decide if a function is continuous at a point.

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.

Probability

Optimization

Networks

Matrices

Sets

Process Standards For Mathematics

Add, subtract, and multiply matrices of appropriate dimensions (i.e. up to 3x3 matrices). Multiply matrices by scalars. Calculate row and column sums for matrix equations.

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.

Understand the determinant of a square matrix is nonzero if and only if the matrix has a multiplicative inverse.

Solve problems represented by matrices using row-reduction techniques and properties of matrix multiplication, including identity and inverse matrices.

Use matrices to solve real-world problems that can be modeled by a system of equations (i.e. up to 3 linear equations) in two or three variables using technology.

Build and use matrix representations to model polygons, transformations, and computer animations.

Use networks, traceable paths, tree diagrams, Venn diagrams, and other pictorial representations to find the number of outcomes in a problem situation.

Optimize networks in different ways and in different contexts by finding minimal spanning trees, shortest paths, and Hamiltonian paths including real-world problems.

Use critical-path analysis in the context of scheduling problems and interpret the results.

Construct and interpret directed and undirected graphs, decision trees, networks, and flow charts that model real-world contexts and problems.

Use graph-coloring techniques to solve problems.

Construct vertex-edge graph models involving relationships among a finite number of elements. Describe a vertex-edge graph using an adjacency matrix. Use vertex-edge graph models to solve problems in a variety of real-world settings.

Use bin-packing techniques to solve problems of optimizing resource usage.

Use geometric and algebraic techniques to solve optimization problems with and without technology.

Use the Simplex method to solve optimization problems with and without technology.

Use Markov chains to solve problems with and without technology.

Use the relative frequency of a specified outcome of an event to estimate the probability of the outcome and apply the law of large numbers in simple examples.

Understand and use the addition rule to calculate probabilities for mutually exclusive and non mutually exclusive events.

Understand and use the multiplication rule to calculate probabilities for independent and dependent events. Understand that two events A and B are independent if the probability of A and B occurring together is the product of their probabilities, and use this characterization to determine if they are independent.

Understand the multiplication counting principle, permutations, and combinations; use them to solve real-world problems. Use simulations with and without technology to solve counting and probability problems.

Calculate the probabilities of complementary events.

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

Analyze decisions and strategies using probability concepts. Analyze probabilities to interpret odds and risk of events.

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

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

Know and use the concepts of sets, elements, and subsets.

Perform operations on sets (union, intersection, complement, cross product) and illustrate using Venn diagrams.

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.

Circles

Define, identify, and use relationships among the following: radius, diameter, arc, measure of an arc, chord, secant, tangent, congruent circles, and concentric circles.

Define and label parts of a circle including: radius, diameter, arc, and chord. (E)

Explore and use relationships among inscribed angles, radii, and chords, including the following: a. The relationship that exists between central, inscribed, and circumscribed angles; b. Inscribed angles on a diameter are right angles; and c. The radius of a circle is perpendicular to a tangent where the radius intersects the circle.

Explore and use relationships among radii, chords, and sectors to include the following: a. Draw/label a central angle and find the relationship between the angle and intercepted arc; and b. Draw/label an inscribed angle and find the relationship between the angle and intercepted arc.

Solve real-world and other mathematical problems that involve finding measures of circumference, areas of circles and sectors, and arc lengths and related angles (central, inscribed, and intersections of secants and tangents). (E)

Solve real-world and mathematical problems involving circumference and area of a circle. (E)

Describe the structure of and relationships within an axiomatic system (undefined terms, definitions, axioms and postulates, methods of reasoning, and theorems) and explain differences among supporting evidence, counterexamples, and actual proofs. (E)

Define geometric terms including point, line, plane, skew, parallel lines, parallel planes, perpendicular lines, and perpendicular planes. Be able to draw and label pictures involving the vocabulary. (E)

State, use, and examine the validity of the converse, inverse, and contrapositive of conditional (“if – then”) and bi-conditional (“if and only if”) statements.

State the converse, inverse, and contrapositive of a real-world conditional (“if-then”) statement. (E)

Develop geometric proofs, including those involving coordinate geometry, using two-column, paragraph, and flow chart formats.

Organize two-column geometric proofs given the statements and reasons to create a logical flow of arguments.

Prove, construct, and apply theorems about parallel and perpendicular lines, parallel lines and transversals, vertical angles, and perpendicular bisectors. (E)

Apply theorems about parallel lines and transversals to name and label angle relationships including: vertical angles, corresponding angles, alternate interior angles, alternate exterior angles, and consecutive angles. Apply theorems about perpendicular bisectors.

Determine if a pair of lines are parallel, perpendicular, or neither by comparing the slopes in coordinate graphs and equations. (E)

Determine if a pair of lines are parallel, perpendicular, or neither by comparing the slopes in coordinate graphs and equations in the form y = mx+b. (E)

Use tools to explain and justify the process to construct congruent segments and angles, angle bisectors, perpendicular bisectors, altitudes, medians, parallel and perpendicular lines, and parallel lines and transversals.

Use tools to construct congruent segments, midpoints, perpendicular bisectors, and congruent angles.

Develop the distance formula using the Pythagorean Theorem. Find the lengths and midpoints of line segments in the two-dimensional coordinate system. (E)

Find lengths and midpoints of line segments in the two-dimensional coordinate system using distance and midpoint formulas. (E)

Prove and apply theorems about parallelograms, including those involving angles, diagonals, and sides. (E)

Apply theorems about parallelograms including those involving angles, diagonals, and sides.

Prove that given quadrilaterals are parallelograms, rhombuses, rectangles, squares, kites, or trapezoids. Include coordinate proofs of quadrilaterals in the coordinate plane.

Prove that given quadrilaterals are parallelograms, rectangles, or squares.

Develop and use formulas to find measures of interior and exterior angles of polygons.

Use formulas to find measures of interior and exterior angles of regular polygons.

Compute perimeters and areas of regular and irregular polygons to solve real-world and other mathematical problems. (E)

Find the perimeter and area of regular polygons given the formula and labeled diagram. (E)

Prove and apply theorems about triangles, including: a. Interior angles of a triangle sum to 180° b. The Isosceles Triangle Theorem and its converse c. The Pythagorean Theorem d. The segment joining midpoints of two sides of a triangle is parallel to the third side and half the length e. A line parallel to one side of a triangle divides the other two proportionally, and its converse f. The Angle Bisector Theorem g. Triangle inequality h. Inequality in one triangle i. Hinge Theorem and its converse (E)

Apply theorems about triangles, including: a. Interior angles of a triangle sum to 180⁰ b. The Isosceles Triangle Theorem c. The Pythagorean Theorem d. The Midsegment Theorem

Prove and apply criteria for triangle congruence (ASA, SAS, AAS, SSS, and HL) from the definition of congruence in terms of rigid motions. (E)

Determine which side or angle is "included" between the given angles or sides of a triangle. Prove triangle congruence using SSS, SAS, and ASA by naming which corresponding sides and angles of a triangle are congruent based on given information.

Use the definition of similarity in terms of similarity transformations to determine if two given triangles are similar. Explore and develop the meaning of similarity for triangles.

Use the definition of similarity to determine if two given triangles are similar. (E)

Use congruent and similar triangles to solve real-world and mathematical problems involving sides, perimeters, and areas of triangles. (E)

Use congruent triangles to solve real-world and mathematical problems involving sides, perimeters, and areas of triangles. (E)

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.

Label the parts of a right triangle identifying the hypotenuse, the opposite, and the adjacent side given an angle of reference. Identify the trigonometric ratio needed to solve problems involving right triangles based on given information. (E)

Use trigonometric ratios (sine, cosine, tangent, and their inverses) and the Pythagorean Theorem to solve real-world and mathematical problems involving right triangles. (E)

Use trigonometric ratios (sine, cosine, and tangent) and the Pythagorean Theorem to solve real-world and mathematical problems involving right triangles.

Use the relationship between the sides of special right triangles (30° - 60° and 45° - 45°) to solve real-world and other mathematical problems. (E)

Find the length of the third side of a special right triangle given the length of the other two sides using the relationship between the sides of 30° - 60° - 90° and 45° - 45° - 90° triangles.

Use geometric descriptions of rigid motions to transform figures and to predict and describe the results of translations, reflections and rotations on a given figure. Describe a motion or series of motions that will show two shapes are congruent. (E)

Describe the results of translations, reflections, and rotations. (E)

Verify experimentally the properties of dilations given by a center and a scale factor. Understand the dilation of a line segment is longer or shorter in the ratio given by the scale factor.

Describe a dilation as an enlargement or reduction, then calculate the scale factor.

Explore properties of congruent and similar solids, including prisms, regular pyramids, cylinders, cones, and spheres, and use them to solve problems.

Describe solids as congruent and similar solids, including prisms, regular pyramids, cylinders, cones, and spheres.

Solve real-world and other mathematical problems involving volume and surface area of prisms, cylinders, cones, spheres, and pyramids, including problems that involve composite solids and algebraic expressions. (E)

Solve real-world and other mathematical problems involving volume of prisms when provided the formula. Solve problems involving surface area of prisms, cylinders, and pyramids, when given the net of the solid. (E)

Apply geometric methods to create and solve design problems. (E)

Design and draw 3D models of prisms, pyramids, cylinders, and/or cones.

Geometry Foundations

Quadrilaterals & Other Polygons

Triangles

Transformations & Three-Dimensional Solids

Transformations & Three-Dimensional Solids

Circles

Quadrilaterals & Other Polygons

Triangles

Geometry Foundations

Mathematics Process Standards

Students solve real-world and mathematical problems involving circles.

Define, identify, and use relationships among the following: radius, diameter, arc, measure of an arc, chord, secant, tangent, congruent circles, and concentric circles.

Explore and use relationships among inscribed angles, radii, and chords, including the following:

The relationship that exists between central, inscribed, and circumscribed angles;

Inscribed angles on a diameter are right angles; and

The radius of a circle is perpendicular to a tangent where the radius intersects the circle.

Solve real-world and other mathematical problems that involve finding measures of circumference, areas of circles and sectors, and arc lengths and related angles (central, inscribed, and intersections of secants and tangents).

Students apply the logic of geometric proofs, including in core concepts related to angles and lines.

Describe the structure of and relationships within an axiomatic system (undefined terms, definitions, axioms and postulates, methods of reasoning, and theorems) and explain differences among supporting evidence, counterexamples, and actual proofs.

State, use, and examine the validity of the converse, inverse, and contrapositive of conditional ("if – then") and bi-conditional ("if and only if") statements.

Develop geometric proofs, including those involving coordinate geometry, using two-column, paragraph, and flow chart formats.

Prove, construct, and apply theorems about parallel and perpendicular lines, parallel lines and transversals, vertical angles, and perpendicular bisectors.

Determine if a pair of lines are parallel, perpendicular, or neither by comparing the slopes in coordinate graphs and equations.

Use tools to explain and justify the process to construct congruent segments and angles, angle bisectors, perpendicular bisectors, altitudes, medians, parallel and perpendicular lines, and parallel lines and transversals.

Develop the distance formula using the Pythagorean Theorem. Find the lengths and midpoints of line segments in the two-dimensional coordinate system.

Students solve real-world and mathematical problems involving regular and irregular polygons, including proofs of theorems.

Prove and apply theorems about parallelograms, including those involving angles, diagonals, and sides.

Prove that given quadrilaterals are parallelograms, rhombuses, rectangles, squares, kites, or trapezoids. Include coordinate proofs of quadrilaterals in the coordinate plane.

Develop and use formulas to find measures of interior and exterior angles of polygons.

Compute perimeters and areas of regular and irregular polygons to solve real-world and other mathematical problems.

Students solve real-world and mathematical problems involving triangles, including proofs of theorems and definitions of trigonometric ratios.

Prove and apply theorems about triangles, including:

Interior angles of a triangle sum to 180°

The Isosceles Triangle Theorem and its converse

The Pythagorean Theorem

The segment joining midpoints of two sides of a triangle is parallel to the third side and half the length

A line parallel to one side of a triangle divides the other two proportionally, and its converse

The Angle Bisector Theorem

Triangle inequality

Inequality in one triangle

Hinge Theorem and its converse

Prove and apply criteria for triangle congruence (ASA, SAS, AAS, SSS, and HL) from the definition of congruence in terms of rigid motions.

Use the definition of similarity in terms of similarity transformations to determine if two given triangles are similar. Explore and develop the meaning of similarity for triangles.

Use congruent and similar triangles to solve real-world and mathematical problems involving sides, perimeters, and areas of 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.

Use trigonometric ratios (sine, cosine, tangent, and their inverses) and the Pythagorean Theorem to solve real-world and mathematical problems involving right triangles.

Use the relationship between the sides of special right triangles (30° - 60° and 45° - 45°) to solve real-world and other mathematical problems.

Students solve real-world and mathematical problems involving transformations of figures and three-dimensional solids.

Use geometric descriptions of rigid motions to transform figures and to predict and describe the results of translations, reflections and rotations on a given figure. Describe a motion or series of motions that will show two shapes are congruent.

Verify experimentally the properties of dilations given by a center and a scale factor. Understand the dilation of a line segment is longer or shorter in the ratio given by the scale factor.

Explore properties of congruent and similar solids, including prisms, regular pyramids, cylinders, cones, and spheres, and use them to solve problems.

Solve real-world and other mathematical problems involving volume and surface area of prisms, cylinders, cones, spheres, and pyramids, including problems that involve composite solids and algebraic expressions.

Apply geometric methods to create and solve design 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.

Conics

Sequences and Series

Exponential and Logarithmic Functions

Quadratic, Polynomial and Rational Equations and Functions

Functions

Process Standards For Mathematics

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

Construct the equation of a circle of given center and radius. Complete the square to find the center and radius of a circle given by an equation.

Construct the equations of ellipses and hyperbolas given at least 2 of the following: foci, vertices, length of an axis, or point on the curve.

Graph conic sections. Identify and describe features like center, vertex or vertices, focus or foci, directrix, axis of symmetry, major axis, minor axis, and eccentricity.

Use the definition of logarithms to convert logarithms from one base to another and prove simple laws of logarithms.

Use the laws of logarithms to simplify logarithmic expressions, approximate the value of a logarithmic expression, and solve logarithmic equations.

Graph and solve real-world and other mathematical problems that can be modeled using exponential and logarithmic functions; interpret the solution and determine whether it is reasonable. Identify and describe features such as intercepts, domain, range, asymptotes, and end behavior.

Use technology to find a quadratic, exponential, logarithmic, or power function that models a relationship for a bivariate data set to make predictions.

For a function that models a relationship between two quantities, interpret key features of graphs and tables in terms of the quantities, and sketch graphs showing key features given a verbal description of the relationship. Key features include: intercepts; intervals where the function is increasing, decreasing, positive, or negative; relative maximums and minimums; symmetries; end behavior; and periodicity.

Find linear models by using median fit and least squares regression methods, making use of technology. Decide which among several linear models gives a better fit. Interpret the slope and intercept in terms of the original context.

Compose functions and find the domain of composite functions.

Determine if a graph or table has an inverse, and justify if the inverse is a function, relation, or neither. Identify the values of an inverse function/relation from a graph or a table, given that the function has an inverse. Derive the inverse equation from the values of the inverse.

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

Recognize even and odd functions from their graphs and algebraic expressions.

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

Understand and use addition, subtraction, multiplication, and conjugation of complex numbers.

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.

Know and apply the Remainder Theorem and the Factor Theorem.

Understand the Fundamental Theorem of Algebra. Find a polynomial function of lowest degree with real coefficients when given its roots.

Graph rational functions with and without technology. Identify and describe features such as intercepts, domain and range, and asymptotic and end behavior.

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

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

Find partial sums of arithmetic and geometric series and represent them using sigma notation.

Model and solve real-world problems involving applications of sequences and series, interpret the solutions and determine whether the solutions are reasonable.

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.

Vectors

Polar Coordinates and Complex Numbers

Identities

Periodic Functions

Triangles

Unit Circle

Process Standards For Mathematics

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.

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

Verify trigonometric identities and simplify expressions using trigonometric identities.

Prove the addition and subtraction identities for sine, cosine, and tangent. Use the identities to solve problems.

Prove the double- and half-angle identities for sine, cosine, and tangent. Use the identities to solve problems.

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

State, prove, and use DeMoivre's Theorem.

Define polar coordinates and relate polar coordinates to Cartesian coordinates.

Translate equations from rectangular coordinates to polar coordinates and from polar coordinates to rectangular coordinates. Graph equations in the polar coordinate plane.

Graph trigonometric functions with and without technology. Use the graphs to model and analyze periodic phenomena, stating amplitude, period, frequency, phase shift, and midline (vertical shift).

Model a data set with periodicity using a sinusoidal function and explain the parameters of the model.

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

Construct the inverse trigonometric functions of sine, cosine, and tangent by restricting the domain.

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

Define and use the trigonometric ratios (sine, cosine, tangent, cotangent, secant, cosecant) in terms of angles of right triangles and the coordinates on the unit circle.

Solve real-world problems with and without technology that can be modeled using right triangles, including problems that can be modeled using trigonometric ratios. Interpret the solutions and determine whether the solutions are reasonable.

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

Prove the Laws of Sines and Cosines.

Understand and apply the Laws of Sines and Cosines to solve real-world and other mathematical problems involving right and non-right triangles.

Derive the formula A = ½ ab sin(C) for the area of a triangle by drawing an auxiliary line. Use the formula to find areas of triangles.

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

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

Use special triangles to determine the values of sine, cosine, and tangent for π/3, π/4, and π/6. Apply special right triangles to the unit circle and use them to express the values of sine, cosine, and tangent for x, π ± x, and 2π ± x in terms of their values for x, where x is any real number.

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

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

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.

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

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

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

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

Probability

Experimental Design

Data Analysis

Process Standards For Mathematics

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.

Create, compare, and evaluate different graphic displays of the same data, using histograms, frequency polygons, cumulative frequency distribution functions, pie charts, scatterplots, stem-and-leaf plots, and box-and-whisker plots. Draw these with and without technology.

Use data from a sample survey to estimate a population mean or proportion; develop a margin of error through the use of simulation models for random sampling.

Find linear models by using median fit and least squares regression methods to make predictions. Decide which among several linear models gives a better fit. Interpret the slope and intercept in terms of the original context. Informally assess the fit of a function by plotting and analyzing residuals.

Evaluate reports based on data by considering the source of the data, the design of the study, the way the data are analyzed and displayed, and whether the report confuses correlation with causation. Distinguish between correlation and causation.

Compute and use mean, median, mode, weighted mean, geometric mean, harmonic mean, range, quartiles, variance, and standard deviation. Use tables and technology to estimate areas under the normal curve. Fit a data set to a normal distribution and estimate population percentages. Recognize that there are data sets not normally distributed for which such procedures are inappropriate.

Understand the central limit theorem and use it to solve problems.

Understand hypothesis tests of means and differences between means and use them to reach conclusions. Compute and use confidence intervals to make estimates. Construct and interpret margin of error and confidence intervals for population proportions.

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

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.

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

Understand the meaning of measurement data and categorical data, of univariate and bivariate data, and of the term variable.

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

Formulate questions that can be addressed with data. Collect, organize, and display relevant data to answer the questions formulated.

Use election theory techniques to analyze election data. Use weighted voting techniques to decide voting power within a group.

Construct simulated sampling distributions of sample proportions and use sampling distributions to identify which proportions are likely to be found in a sample of a given size.

Use simulations to explore the variability of sample statistics from a known population and to construct sampling distributions.

Model and solve real-world problems using the geometric distribution or waiting-time distribution, with or without technology.

Model and solve real-world problems involving patterns using recursion and iteration, growth and decay, and compound interest.

Understand and apply basic ideas related to the design, analysis, and interpretation of surveys and sampling, such as background information, random sampling, causality and bias.

Understand how basic statistical techniques are used to monitor process characteristics in the workplace.

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

Understand and use the addition rule to calculate probabilities for mutually exclusive and nonmutually exclusive events.

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

Understand and use the multiplication rule to calculate probabilities for independent and dependent events. Understand that two events A and B are independent if the probability of A and B occurring together is the product of their probabilities, and use this characterization to determine if they are independent.

Understand the multiplication counting principle, permutations, and combinations; use them to solve real-world problems. Use simulations with and without technology to solve counting and probability problems.

Calculate the probabilities of complementary events.

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

Analyze decisions and strategies using probability concepts. Analyze probabilities to interpret odds and risk of events.

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.

Develop a probability distribution for a random variable defined for a sample space in which theoretical probabilities can be calculated; Compute and interpret the expected value of random variables.

Derive the binomial theorem by combinatorics. Use combinatorial reasoning to solve problems.

Statistics

Probabilistic Reasoning to Assess Risk

Modeling

Ratio and Proportional Reasoning

Numeracy

Process Standards For Mathematics

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 and critique mathematical models and be able to describe their limitations, including distinguishing between correlation and causation and determine whether interpolation and/or extrapolation are appropriate.

Use models, including models created with spreadsheets or other tools, to estimate solutions to contextual questions, such as functional models to estimate future population or spreadsheets to model financial applications (e.g. credit card debt, installment savings, amortization schedules, mortgage and other loan scenarios). Identify patterns and identify how changing parameters affect the results.

Choose and create, with and without technology, linear, exponential, logistic, or periodic models and curves of best fit for bivariate data sets. Use the models to answer questions and draw conclusions or make decisions, addressing limitations and long-term ramifications of chosen models when appropriate. Recognize when a change in model is needed.

Analyze real-world problem situations and use variables to construct and solve equations involving one or more unknown or variable quantities to answer questions about the situations, such as creating spreadsheet formulas to calculate prices based on percentage mark-up or solving formulas for specified values. Demonstrate understanding of the meaning of a solution. Identify when there is insufficient information given to solve a problem.

Apply geometric concepts to model situations and solve problems such as those arising in art, architecture, and other fields.

The student uses a variety of network models represented graphically to organize data in quantitative situations, make informed decisions, and solve problems, such as in scheduling or routing situations that can be modeled using different methods, e.g., vertex-edge graphs using critical paths, Euler paths, or minimal spanning trees.

Represent quantities in equivalent forms (fractions, decimals, and percentages) to investigate and describe quantitative relationships and solve real-world problems in a variety of contexts. Compare the size of numbers in different forms arising in authentic real-world contexts, such as growth expressed as a fraction versus as a percentage. Interpret the meaning of numbers in different forms, such as the meaning of a fraction or the meaning of a percentage greater than 100 and its validity in a given context. Recognize incorrect or deceptive uses of fractions, decimals, or percentages.

Solve problems involving calculations with percentages and interpret the results, such as calculating percentage rates or differentiating between a discount of 30% and two consecutive discounts of 15%. Calculate relative change and explain how it differs from absolute change. Recognize incorrect or deceptive uses of percentages.

Interpret numbers in different forms in terms of authentic contexts to solve real-world problems, such as interpreting a growth rate less than 1%. Compare and precisely communicate with numbers in different forms (including words, fractions, decimals, standard notation, and scientific notation), such as comparing relative and absolute changes in quantities.

Compare magnitudes of numbers in context, such as the population of the US compared to the population of the world. Perform such comparisons when numbers are in different forms (including words, fractions, decimals, standard notation, and scientific notation).

Perform accurate and efficient calculations using large and small numbers in different forms, to an appropriate precision, with and without technology. Include calculations in context, such as ratios representing water use per capita for a large population.

Use estimation skills, and know why, how, and when to estimate results. Identify and use numeric benchmarks for estimating calculations (e.g., using 25% as an estimate for 23%). Identify and use contextual benchmarks for comparison to other numbers (e.g., using the US population as a benchmark to evaluate reasonableness of statistical claims or giving context to numbers). Check for reasonableness using both types of benchmarks.

Use dimensional analysis to convert between units of measurements and to solve problems involving multiple units of measurement, such as converting between currencies, calculating the cost of gasoline to drive a given car a given distance, or calculating dosages of medicine.

Solve real-life problems requiring interpretation and comparison of complex numeric summaries which extend beyond simple measures of center, such as problems requiring interpreting and/or comparing weighted averages, indices, coding, and ranking. Evaluate claims based on complex numeric summaries.

Understand and communicate percentages as rates per 100, and identify uses and misuses of percentages related to a proper understanding of the base in real-world and mathematical problems.

Solve real-life problems requiring interpretation and comparison of various representations of ratios, (i.e. fractions, decimals, rate, and percentages), such as problems that involve non-standard ratios (e.g., media and risk reporting) or part-to-part versus part-to-whole ratios taken from meaningful context.

Analyze growth and decay using absolute and relative change and make comparisons using absolute and relative difference.

Distinguish between proportional and non-proportional situations, and, when appropriate, apply proportional reasoning, such as when solving for an unknown quantity in proportional situations; solving real-life problems requiring conversion of units using dimensional analysis; or applying scale factors to perform indirect measurements (e.g., maps, blueprints, concentrations, dosages, and densities). Recognize when proportional techniques do not apply.

Determine the constant of proportionality in proportional situations (both real-life and mathematical), leading to a symbolic model for the situation (i.e. an equation based upon a rate of change, <em>y = kx</em>).

Determine the nature and number of elements in a finite sample space to model the outcomes of real-world events using counting techniques, and build the sample space by making lists, tables, or tree diagrams.

Determine the number of ways an event may occur using the Fundamental Counting Principle.

Evaluate 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.

Use data displays and models, such as two-way tables, tree diagrams, Venn diagrams, and area models, to determine probabilities (including conditional probabilities) and use these probabilities to make informed decisions.

Analyze statistical information from studies, surveys, and polls (including when reported in condensed form or using summary statistics) to make informed judgments as to the validity of claims or conclusions, such as when interpreting and comparing the results of polls using margin of error.

Identify limitations, strengths, or lack of information in studies, including data collection methods (e.g. sampling, experimental, observational) and possible sources of bias, and identify errors or misuses of statistics to justify particular conclusions.

Create (with and without technology) and use visual displays of real world data, such as charts, tables and graphs.

Interpret and analyze visual representations of data, and describe strengths, limitations, and fallacies of various graphical displays.

Read, interpret, and make decisions about data summarized numerically using measures of center and spread, in tables, and in graphical displays (line graphs, bar graphs, scatterplots, and histograms), e.g., explain why the mean may not represent a typical salary; explain the difference between bar graphs and histograms; critique a graphical display by recognizing that the choice of scale can distort information.

Summarize, represent, and interpret data sets on a single count or measurement variable using plots and statistics appropriate to the shape of the data distribution to represent it.

Compare center, shape, and spread of two or more data sets and interpret the differences in context.

Use properties of distributions, including uniform and normal distributions, to analyze data and answer questions.

Recognize when data are normally distributed and use the mean and standard deviation of the data to fit it to a normal distribution.