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Study Resources for the Praxis? Mathematics: Content Knowledge Test (5161)The links below allow you to connect content topics on this Praxis? test directly to free Khan Academy study resources.Praxis Mathematics: Content Knowledge (5161) Content TopicsStudy ResourcesI. Number and Quality, Algebra, Functions, and CalculusLessonNumber and QualityUnderstands the properties of exponentsUnderstands the role of phonics and word analysis in literacy developmentAlgebra I / Exponent properties reviewAlgebra 2 / Rational exponentsDemonstrate an understanding of the properties of exponential expressionsAlgebra I / Exponent properties reviewAlgebra 2 / Properties of exponents (rational exponents)Use the properties of exponents to rewrite expressions that have radicals or rational exponentsAlgebra I / RadicalsAlgebra I / Simplifying square rootsAlgebra 2 / Evaluating exponents & radicalsAlgebra 2 / Equivalent forms of exponential expressionsUnderstands the properties of rational and irrational numbers, and the interactions between those sets of numbersAlgebra I / Irrational numbersAlgebra I / Proofs concerning irrational numbersRecognize that the sum or product of two rational numbers is rationalAlgebra I / Sums and products of rational and irrational numbersRecognize that the sum of a rational number and an irrational number is irrationalAlgebra I / Sums and products of rational and irrational numbersRecognize that the product of a nonzero rational number and an irrational number is irrationalAlgebra I / Sums and products of rational and irrational numbersRecognize that the sum or product of two irrational numbers can be rational or irrationalAlgebra I / Sums and products of rational and irrational numbersUnderstands how to solve problems by reasoning quantitatively (e.g., dimensional analysis, reasonableness of solutions)Use units as a way to understand problems and to guide the solution of multistep problemsAlgebra I / Rate conversionChoose and interpret units consistently in formulasAlgebra I / Appropriate unitsChoose and interpret the scale and the origin in graphs and data displays Recognize the reasonableness of results within the context of a given problemAlgebra I / Appropriate unitsAlgebra I / Word problems with multiple unitsUnderstands the structure of the natural, integer, rational, real, and complex number systems and how the basic operations (+,-, ×, and ÷) on numbers in these systems are performedSolve problems using addition, subtraction, multiplication, and division of rational, irrational, and complex numbersAlgebra I / Substitution and evaluating expressionsAlgebra I / Division by zeroAlgebra 2 / The imaginary unit iAlgebra 2 / Complex numbers introductionAlgebra 2 / Adding and subtracting complex numbersAlgebra 2 / Multiplying complex numbersPrecalculus / What are the imaginary numbers?Precalculus / What are the complex numbers?Precalculus / Adding and subtracting complex numbersPrecalculus / Multiplying complex numbersPrecalculus / Complex conjugates and dividing complex numbersPrecalculus / Identities with complex numbersApply the order of operationsPre-algebra / Order of operationsGiven operations on a number system, determine whether the properties (e.g., commutative, associative, distributive) holdCompare, classify, and order real numbersSimplify and approximate radicalsAlgebra I / Simplifying square rootsAlgebra 2 / The imaginary unit iFind conjugates of complex numbersPrecalculus / Complex conjugates and dividing complex numbersDemonstrate an understanding of the properties of counting numbers (e.g., prime, composite, prime factorization, even, odd, factors, multiples)Understands how to work with complex numbers when solving polynomial equations and rewriting polynomial expressionsSolve quadratic equations with real coefficients that have complex solutionsAlgebra I / The quadratic formulaAlgebra 2 / Quadratic equations with complex solutionsExtend polynomial identities to the complex numbers (e.g., Algebra I / Special products of binomialsAlgebra 2 / Special products of polynomialsAlgebra 2 / Polynomial identitiesPrecalculus / Identities with complex numbersVerify the fundamental theorem of algebra for quadratic polynomialsAlgebra I / The quadratic formulaAlgebra 2 / Quadratic equations with complex solutionsUnderstands how to perform operations on matrices and how to use matrices in applicationsUse matrices to represent and manipulate dataPrecalculus / Introduction to matricesPrecalculus / Model real-world situations with matricesMultiply matrices by scalars to produce new matricesPrecalculus / Multiplying matrices by scalarsPrecalculus / Properties of matrix addition & scalar multiplicationAdd, subtract, and multiply matrices of appropriate dimensionsPrecalculus / Adding and subtracting matricesPrecalculus / Properties of matrix addition & scalar multiplicationPrecalculus / Multiplying matrices by matricesUnderstand that matrix multiplication for square matrices is not a commutative operation but still satisfies the associative and distributive propertiesPrecalculus / Properties of matrix multiplicationUnderstand the role played by zero, and identity matrices in matrix addition and multiplicationPrecalculus / Properties of matrix addition & scalar multiplicationPrecalculus / Properties of matrix multiplicationUnderstand that the determinant of a square matrix is nonzero if and only if the matrix has a multiplicative inversePrecalculus / The determinant of a 2x2 matrixPrecalculus / Introduction to matrix inversesPrecalculus / Finding the inverse of a matrix using its determinantPrecalculus / Practice finding the inverses of 2x2 matricesWork with matrices as transformations of the plane, and interpret the absolute value of the determinant in terms of areaPrecalculus / Matrices as transformationsUnderstands how to solve problems involving ratios, proportions, averages, percents, and metric and traditional unit conversionsApply the concept of a ratio and use ratio language and notation to describe a relationship between two quantitiesPre-algebra / Intro to ratiosCompute unit rates.Pre-algebra / Intro to ratesUse ratio reasoning to convert ratesPre-algebra / Visualize ratiosPre-algebra / Ratio applicationSolve problems involving scale factorsRecognize and represent proportional and inversely proportional relationships between two quantitiesPre-algebra / Constant of proportionality Pre-algebra / Identifying proportional relationshipsPre-algebra / Writing & solving proportionsAlgebra (all content) / Direct and inverse variationUse proportional relationships to solve multistep ratio, average, and percent problemsPre-algebra / Intro to percentsPre-algebra / Percent, fraction, decimal conversionsPre-algebra / Percent problemsPre-algebra / Percent word problemsSolve measurement and estimation problems involving time, length, temperature, volume, and mass in both the U.S. customary system and the metric system, where appropriateAlgebra I / Rate conversionAlgebra I / Appropriate unitsAlgebra I / Word problems with multiple unitsConvert units within the metric and customary systemsAlgebra I / Rate conversionAlgebra I / Appropriate unitsAlgebra I / Word problems with multiple unitsKnows how to analyze both precision and accuracy in measurement situationsChoose a level of accuracy appropriate to limitations on measurement when reporting quantities.Calculate or estimate absolute and relative error in the numerical answer to a problemUnderstands various ways to represent and compare very large and very small numbers (e.g., scientific notation, orders of magnitude)Represent and compare very large and very small numbersPre-algebra / Scientific notationUnderstands how to both estimate and perform calculations on very large and very small quantitiesUse orders of magnitude to estimate very large and very small numbersPerform calculations on numbers in scientific notationPre-algebra / Computing with scientific notationAlgebraUnderstands how to write algebraic expressions in equivalent formsUse the structure of an expression to identify ways to rewrite itAlgebra I / Combining like termsAlgebra I / Introduction to equivalent expressionsAlgebra I / Factoring quadratics with difference of squares Algebra 2 / Factoring using structureAlgebra 2 / Factoring monomialsAlgebra 2 / Factoring higher degree polynomialsUnderstand how to rewrite quadratic expressions for specific purposes (e.g., factoring/finding zeros, completing the square/finding maxima or minima).Algebra I / Factoring quadratics introAlgebra I / Factoring quadratics by groupingAlgebra I / Factoring quadratics with difference of squaresAlgebra I / Factoring quadratics with perfect squaresAlgebra I / Strategy in factoring quadraticsAlgebra I / Vertex formAlgebra I / Completing the squareAlgebra I / Quadratic standard formAlgebra I / Features & forms of quadratic functionsUse the properties of exponents to rewrite expressions for exponential functions.Algebra I / Exponent properties reviewUnderstands how to perform arithmetic operations on polynomialsAdd, subtract, and multiply polynomialsAlgebra I / Combining like termsAlgebra I / Multiplying monomials by polynomialsAlgebra I / Multiplying binomialsAlgebra 2 / Intro to polynomialsAlgebra 2 / Adding and subtracting polynomialsAlgebra 2 / Multiplying monomials by polynomialsAlgebra 2 / Multiplying binomials by polynomialsPrecalculus / Adding and subtracting polynomials: two variablesUnderstands the relationship between zeros of polynomial functions (including their graphical representation) and factors of the related polynomial expressionsKnow and apply the remainder theorem: for a polynomial and a number a, the remainder on division by is , so if and only if is a factor of .Algebra 2 / Polynomial Remainder TheoremPrecalculus / Polynomial divisionUse factorization to identify zeros of polynomialsAlgebra I / Introduction to factoringAlgebra 2 / Dividing quadratics by linear factorsAlgebra 2 / Dividing polynomials by linear factorsAlgebra 2 / Zeros of polynomialsUse zeros of a polynomial to construct a rough graph of the function defined by the polynomial.Algebra I / Solving and graphing with factored formAlgebra 2 / Zeros of polynomialsAlgebra 2 / Positive and negative intervals of polynomialsAlgebra 2 / End behavior of polynomialsAlgebra 2 / Putting it all togetherPrecalculus / The fundamental theorem of algebraUnderstands how to use polynomial identities (e.g., difference of squares, sum and difference of cubes) to solve problems.Apply the binomial theorem for the expansion of in powers of x and y for a positive integer n.Precalculus / Adding and subtracting polynomials: two variablesPrecalculus / The binomial theoremUnderstands how to rewrite rational expressions and perform arithmetic operations on rational expressionsRewrite simple rational expressions in different formsAlgebra 2 / Cancelling common factorsUnderstand that rational expressions form a system analogous to the rational numbers, closed under addition, subtraction, multiplication, and division by a nonzero rational expression.Algebra 2 / Taking common factors Algebra 2 / Dividing quadratics by linear factorsAlgebra 2 / Dividing polynomials by linear factorsAdd, subtract, multiply, and divide rational expressionsAlgebra 2 / Dividing polynomials by xPrecalculus / Polynomial divisionAlgebra 2 / Multiplying and dividing rational expressionsAlgebra 2 / Adding and subtracting rational expressions introAlgebra 2 / Adding and subtracting rational expressions (factored)Algebra 2 / Adding and subtracting rational expressions (not factored)Understands how to create equations and inequalities that describe relationshipsCreate equations and inequalities in one variable and use them to solve problems and graph solutions on the number lineAlgebra I / Linear equations with variables on both sidesAlgebra I / Linear equations with parenthesesAlgebra I / Linear equations with unknown coefficientsAlgebra I / Multi-step inequalitiesAlgebra 2 / Modeling with equations and inequalitiesCreate equations and inequalities in two or more variables to represent relationships between quantities, solve problems, and graph them on the coordinate plane with labels and scalesAlgebra I / Two-variable linear equations introAlgebra I / Graphing two-variable inequalitiesAlgebra I / SlopeRepresent constraints by equations, inequalities, or systems of equations and/or inequalities, and interpret solutions as viable or nonviable options in a modeling contextAlgebra I / Systems of equations word problemsAlgebra I / Modeling with linear inequalitiesAlgebra 2 / Modeling with equations and inequalitiesRearrange formulas to highlight a quantity of interest (e.g., solve )Algebra 2 / Manipulating formulasUnderstands how to justify the reasoning process used to solve equations, including analysis of potential extraneous solutions.Explain each step in solving a simple equationAlgebra I / Linear equations with variables on both sidesAlgebra I / Linear equations with parenthesesSolve simple rational and radical equations in one variable, incorporating analysis of possible extraneous solutionsAlgebra 2 / Rational equationsAlgebra 2 / Square-root equationsAlgebra 2 / Extraneous solutionsAlgebra 2 / Cube-root equationsUnderstands how varied techniques (e.g., graphical, algebraic) are used to solve equations and inequalities in one variableSolve linear equations and inequalities in one variable, including equations with coefficients represented by lettersAlgebra I / Linear equations with variables on both sidesAlgebra I / Linear equations with parenthesesAlgebra I / Linear equations with unknown coefficientsAlgebra I / Multi-step inequalitiesAlgebra I / Compound inequalitiesAlgebra I / Summary: Forms of two-variable linear equationsUse the method of completing the square to transform any quadratic equation in x into the equivalent form Algebra I / Completing the squareSolve equations using a variety of methods (e.g., using graphs, using the quadratic formula, or factoring)Algebra I / Factoring quadratics introAlgebra I / Factoring quadratics by groupingAlgebra I / Factoring quadratics with difference of squaresAlgebra I / Factoring quadratics with perfect squaresAlgebra I / Strategy in factoring quadraticsAlgebra I / Solving quadratics by factoringAlgebra I / The quadratic formulaPrecalculus / The fundamental theorem of algebraPrecalculus / Solving equations by graphingUse different methods (e.g., discriminant analysis, graphical analysis) to determine the nature of the solutions of a quadratic equationAlgebra I / Intro to parabolasAlgebra I / Solving and graphing with factored formAlgebra I / Solving by taking the square rootAlgebra I / Vertex formAlgebra I / The quadratic formulaAlgebra I / Quadratic standard formAlgebra I / Features & forms of quadratic functionsWrite complex solutions in the form .Algebra I / The quadratic formulaAlgebra 2 / The imaginary unit iUnderstands how varied techniques (e.g., graphical, algebraic, matrix) are used to solve systems of equations and inequalitiesExplain why, when solving a system of two equations using the elimination method, replacing one or both equations with a scalar multiple produces a system with the same solutions as the solutions of the original systemAlgebra I / Equivalent systems of equations and the elimination methodSolve a system consisting of two linear equations in two variables algebraically and graphicallyAlgebra I / Introduction to systems of equationsAlgebra I / Solving systems of equations with substitutionAlgebra I / Number of solutions to systems of equationsSolve a system consisting of a linear equation and a quadratic equation in two variables algebraically and graphicallyAlgebra 2 / Quadratic systemsRepresent a system of linear equations as a single matrix equationPrecalculus / Representing linear systems of equations with augmented matricesFind the inverse of a matrix if it exists, and use it to solve systems of linear equationsPrecalculus / Solving equations with inverse matricesExplain why the x-coordinates of the intersection points of the graphs of and are the solutions of .Precalculus / Solving equations by graphingFind the solutions of approximately (e.g., use technology to graph the functions, make tables of values, find successive approximations). Include cases where and/or are linear, polynomial, rational, absolute value, exponential, or logarithmic functionsPrecalculus / Solving equations by graphingGraph the solutions to a linear inequality in two variables as a half-plane (excluding the boundary in the case of a strict inequality), and graph the solution set to a system of linear inequalities in two variables as the intersection of the corresponding half-planesAlgebra I / Checking solutions of two-variable inequalitiesAlgebra I / Graphing two-variable inequalitiesUnderstands the properties of number systems under various operationsGiven operations on algebraic expressions, determine whether the properties hold (e.g., commutative, associative, distributive)Understands the concept of rate of change of nonlinear functionsAlgebra I / Average rate of changeAlgebra I / Average rate of change word problemsAlgebra 2 / Average rate of change of polynomialsCalculate and interpret the average rate of change of a function presented symbolically, numerically, or graphically over a specified intervalUnderstands the concepts of intercept(s) of a line and slope as a rate of changeCalculate and interpret the intercepts of a lineAlgebra I / x-intercepts and y-interceptsAlgebra I / Applying intercepts and slopeCalculate and interpret the slope of a line presented symbolically, numerically, or graphicallyAlgebra I / SlopeAlgebra I / Horizontal & vertical linesAlgebra I / Applying intercepts and slopeAlgebra I / Intro to slope-intercept formAlgebra I / Graphing slope-intercept equationsAlgebra I / Writing slope-intercept equationsAlgebra I / Point-slope formAlgebra I / Standard formAlgebra I / Summary: Forms of two-variable linear equationsEstimate the rate of change of a linear function from a graphAlgebra I / Applying intercepts and slopeUnderstands how to find the zero(s) of functionsUses a variety of techniques to find and analyze the zero(s) (real and complex) of functionsAlgebra 2 / Zeros of polynomialsAlgebra 2 / Solving equations by graphingPrecalculus / Solving equations by graphingFunctionsUnderstands the function concept and the use of function notationRecognize that functions are sets of ordered pairsAlgebra I / Recognizing functionsUnderstand that a function from one set (called the domain) to another set (called the range) assigns to each element of the domain exactly one element of the rangeAlgebra I / Inputs and outputs of a functionUse function notation, evaluate functions, and interpret statements that use function notation in terms of a contextAlgebra I / Evaluating functionsAlgebra I / Functions and equationsAlgebra I / Interpreting function notationRecognize that sequences are functions, sometimes defined recursively, whose domain is a subset of the integersUnderstands how to find the domain and range of a function and a relationIdentify the domain and range of a function or relationAlgebra I / Introduction to the domain and range of a functionDetermine the domain of a function from a function rule (e.g., , graph, set of ordered pairs, or tableAlgebra I / Determining the domain of a functionUnderstands how function behavior is analyzed using different representations. (e.g., graphs, mappings, tables).For a function that models a relationship between two quantities, interpret key features of graphs and tables (e.g., increasing/decreasing, maximum/minimum, periodicity) in terms of the quantitiesAlgebra I / Maximum and minimum pointsAlgebra I / Intervals where a function is positive, negative, increasing, or decreasingAlgebra 2 / Interpreting features of functionsGiven a verbal description of a relation, sketch graphs that show key features of that relationAlgebra I / Interpreting features of graphsGraph functions (i.e., radical, piecewise, absolute value, polynomial, rational, logarithmic, trigonometric) expressed symbolically and identify key features of the graphAlgebra I / Graphs of absolute value functionsAlgebra I / Piecewise functionsAlgebra I / Graphs of exponential growthAlgebra I / Quadratic standard formAlgebra I / Features & forms of quadratic functionsAlgebra I / Transforming quadratic functionsAlgebra 2 / Putting it all togetherAlgebra 2 / Graphs of square and cube root functionsAlgebra 2 / Graphs of exponential functionsAlgebra 2 / Graphs of logarithmic functionsAlgebra 2 / Graphs of sin(x), cos(x), and tan(x)Algebra 2 / Graphing sinusoidal functionsAlgebra 2 / End behavior of rational functionsAlgebra 2 / Discontinuities of rational functionsAlgebra 2 / Graphs of rational functionsWrite a function that is defined by an expression in different but equivalent forms to reveal different properties of the function (e.g., zeros, extreme values, symmetry of the graph)Algebra I / Solving and graphing with factored formAlgebra I / Vertex formInterpret the behavior of exponential functions (e.g., growth, decay)Algebra I / Exponential growth & decayUnderstand how to determine if a function is odd, even, or neither and any resulting symmetriesAlgebra 2 / Symmetry of functionsUnderstands how functions and relations are used to model relationships between quantities.Write a function that relates two quantitiesAlgebra I / Exponential vs. linear modelsAlgebra 2 / Solving exponential modelsAlgebra 2 / Modeling with rational functionsDetermine an explicit expression or a recursive process that builds a function from a contextAlgebra I / Exponential vs. linear modelsAlgebra 2 / Solving exponential modelsUnderstands how new functions are obtained from existing functions (e.g., compositions, transformations, inverses)Describe how the graph of is related to the graph of , where , , , or for specific values of k (both positive and negative), and find the value of k given the graphsAlgebra I / Transforming quadratic functionsAlgebra 2 / Shifting functionsAlgebra 2 / Reflecting functionsAlgebra 2 / Scaling functionsAlgebra 2 / Putting it all togetherDetermine if a function has an inverse and write an expression for the inverseAlgebra I / Intro to inverse functionsPrecalculus / Invertible functionsVerify by composition if one function is the inverse of anotherAlgebra I / Intro to inverse functionsPrecalculus / Verifying inverse functions by compositionGiven that a function f has an inverse, find values of the inverse function from a graph or a table of fAlgebra I / Intro to inverse functionsPrecalculus / Invertible functionsGiven a noninvertible function, determine a largest possible domain of the function that produces an invertible functionPrecalculus / Invertible functionsUnderstand the inverse relationship between exponential and logarithmic functions and use this relationship to solve problemsAlgebra 2 / Introduction to logarithmsAlgebra 2 / The constant e and the natural logarithmPrecalculus / Invertible functionsCombine standard function types using arithmetic operationsAlgebra 2 / Modeling with function combinationPerform domain analysis on functions resulting from arithmetic operationsCompose functions algebraically, numerically, and graphicallyPrecalculus / Composing functionsPrecalculus / Modeling with composite functionsPerform domain analysis on functions resulting from compositionsUnderstands differences between linear, quadratic, and exponential models, including how their equations are created and used to solve problems.Understand that linear functions grow by equal differences over equal intervals, and that exponential functions grow by equal factors over equal intervals.Algebra I / Exponential vs. linear growthRecognize situations in which one quantity changes at a constant rate per unit interval relative to anotherAlgebra I / Exponential vs. linear growthRecognize situations in which a quantity grows or decays by a constant percent rate per unit interval relative to anotherAlgebra I / Exponential expressionsAlgebra 2 / Interpreting the rate of change of exponential modelsAlgebra 2 / Advanced interpretation of exponential modelsConstruct linear and exponential functions, including arithmetic and geometric sequences, given a graph, a description of a relationship, or two ordered pairs (include reading these from a table).Algebra I / Exponential functions from tables & graphsAlgebra 2 / Constructing exponential models according to rate of changeAlgebra 2 / Advanced interpretation of exponential modelsObserve that a quantity increasing exponentially eventually exceeds a quantity increasing linearly, quadratically, or (more generally) as a polynomial function.Algebra I / Exponential vs. linear growth over timeExpress the solution to an exponential equation with base b as a logarithm (e.g., ,)Algebra 2 / Introduction to logarithmsAlgebra 2 / The constant e and the natural logarithmAlgebra 2 / Properties of logarithmsAlgebra 2 / The change of base formula for logarithmsAlgebra 2 / Solving exponential equations with logarithmsUse technology to evaluate logarithms that have any baseAlgebra 2 / Introduction to logarithmsAlgebra 2 / The constant e and the natural logarithmAlgebra 2 / Properties of logarithmsAlgebra 2 / The change of base formula for logarithmsInterpret the parameters in a linear or exponential function in terms of a context (e.g., )Algebra I / Exponential vs. linear modelsAlgebra 2 / Solving exponential modelsUse quantities that are inversely related to model phenomenaUnderstands how to construct the unit circle and how to use it to find values of trigonometric functions for all angle measures in their domains.Understand radian measure (e.g., 1 radian is the measure of a central angle that subtends an arc with length equal to the length of the radiusHigh school geometry / Introduction to radiansAlgebra 2 / Rotation-by-radians-and-quadrantsUnderstand how the domains of trigonometric functions can be extended beyond 0 to 2π using the unit circleUse special triangles (i.e., 30-60-90, 45-45-90) to determine geometrically the values of sine, cosine, and tangent for , , and .Algebra 2 / Trigonometric values of special anglesUse reference angles to find the values of trigonometric functions at angles outside the interval 0 to .Use the unit circle to explain symmetry and periodicity of trigonometric functionsUnderstands how periodic phenomena are modeled using trigonometric functionsChoose trigonometric functions to model periodic phenomena with specified amplitude, frequency, and midlineAlgebra 2 / Amplitude, midline and periodAlgebra 2 / Transforming sinusoidal graphsAlgebra 2 / Sinusoidal modelsPrecalculus / Sinusoidal modelsUnderstand how to restrict the domain of a trigonometric function so that its inverse can be constructedPrecalculus / Invertible functionsPrecalculus / Inverse trigonometric functionsUse inverse functions to solve trigonometric equations that arise in modeling contexts, and interpret them in terms of the contextPrecalculus / Inverse trigonometric functionsUnderstands the application of trigonometric identities (e.g., Pythagorean, double angle, half angle, sum of angles, difference of angles).Use Pythagorean identities (e.g., )Algebra 2 / The Pythagorean identityUse trigonometric identities to rewrite expressions and solve equationsPrecalculus / Trigonometric identitiesPrecalculus / Angle addition identitiesPrecalculus / Using trigonometric identitiesUnderstand trigonometric identities in the context of equivalent graphs of trigonometric functions (e.g., and are equivalent graphs)Algebra 2 / Transforming sinusoidal graphsAlgebra 2 / Graphing sinusoidal functionsProve Pythagorean identities (e.g., .Algebra 2 / The Pythagorean identityKnows how to interpret representations of functions of two variables (e.g., three-dimensional graphs, tables).Interpret representations of functions of two variablesUnderstands how to solve equations (e.g., trigonometric, logarithmic, exponential).Solve trigonometric, logarithmic, and exponential equationsAlgebra 2 / Solving exponential equations using properties of exponentsAlgebra 2 / Solving exponential equations with logarithmsAlgebra 2 / Solving equations by graphingPrecalculus / Sinusoidal equationsCalculusUnderstands the meaning of a limit of a function and how to calculate limits of functions, determine when the limit does not exist, and solve problems using the properties of limits.Graphically analyze the limit of as x approaches a fixed value from both left and rightDifferential calculus / Limits introDifferential calculus / Estimating limits from graphsDifferential calculus / Limits at infinitySolve limit problems (e.g., a constant times a function, the sum of two functions, the product and quotient of two functions) using properties of limits, where all limits of the individual functions exist at the value that x is approachingDifferential calculus / Properties of limitsDifferential calculus / Limits by direct substitutionDifferential calculus / Limits using algebraic manipulationDifferential calculus / Strategy in finding limitsDifferential calculus / L’H?pital’s ruleDifferential calculus / L’H?pital’s rule: composite exponential functionsAnalyze one-sided limits for various functions to see whether or not the limit existsDifferential calculus / Estimating limits from graphsDifferential calculus / Estimating limits from tablesRecognize limits that do not exist, such as and .Differential calculus / Estimating limits from graphsDifferential calculus / Estimating limits from tablesUnderstands the derivative of a function as a limit, as the slope of a line tangent to a curve, and as a rate of change. Construct a function graph for a given function and a given point , and explain what happens to the succession of slopes of secant lines connecting to as x approaches a, from both the right side and the left sideDifferential calculus / Average vs. instantaneous rate of changeDifferential calculus / Secant linesState the limit definition of the derivative, and use it to find the derivative of a given function at a given value of x and to find the derivative functionDifferential calculus / Derivative definitionDifferential calculus / Estimating derivativesUnderstands how to show that a particular function is continuous. Apply the three steps (i.e., exists, exists, and ) that are part of the definition of what it means for a function to be continuous at to verify whether a given function is continuous at a given pointDifferential calculus / Types of discontinuitiesDifferential calculus / Continuity at a pointDifferential calculus / Continuity over an intervalDifferential calculus / Removing discontinuitiesKnows the relationship between continuity and differentiabilityGive examples of functions that are continuous at but not differentiable at , and explain whyDifferential calculus / DifferentiabilityUnderstands how to approximate derivatives and integrals numericallyGiven a table of values, use the slope of a secant line to approximate a derivative.Differential calculus / Secant linesDifferential calculus / Estimating derivativesUse the midpoint rule, trapezoid rule, or other Riemann sums to find numerical approximations for integralsUnderstands how and when to use standard differentiation and integration techniquesUse standard differentiation techniquesDifferential calculus / Power ruleDifferential calculus / Derivative rules: constant, sum, difference, and constant multipleDifferential calculus / Combining the power rule with other derivative rulesDifferential calculus / Derivatives of cos(x), sin(x), ??, and ln(x)Differential calculus / Product ruleDifferential calculus / Quotient ruleDifferential calculus / Derivatives of tan(x), cot(x), sec(x), and csc(x)Differential calculus / Chain ruleDifferential calculus / More chain rule practiceDifferential calculus / Implicit differentiationDifferential calculus / Implicit differentiation (advanced examples)Differential calculus / Differentiating inverse functionsDifferential calculus / Derivatives of inverse trigonometric functionsDifferential calculus / Strategy in differentiating functionsDifferential calculus / Differentiation using multiple rulesDifferential calculus / Disguised derivativesDifferential calculus / Logarithmic differentiationUse standard integration techniquesIntegral calculus / Properties of definite integralsIntegral calculus / Reverse power ruleIntegral calculus / Indefinite integrals of common functionsIntegral calculus / Definite integrals of common functionsIntegral calculus / Integrating with u-substitutionIntegral calculus / Integrating using long division and completing the squareIntegral calculus / Integrating using trigonometric identitiesIntegral calculus / Trigonometric substitutionIntegral calculus / Integration by partsIntegral calculus / Integrating using linear partial fractionsIntegral calculus / Improper integralsUnderstand the relationship between position, velocity, and acceleration functions of a particle in motionDifferential calculus / Straight-line motionIntegral calculus / Straight-line motionUnderstands how to analyze the behavior of a function (e.g., extrema, concavity, symmetryUse the first and second derivatives to analyze the graph of a functionDifferential calculus / Second derivativesDifferential calculus / Intervals on which a function is increasing or decreasingDifferential calculus / Relative (local) extremaDifferential calculus / Absolute (global) extremaDifferential calculus / Concavity and inflection points introDifferential calculus / Analyzing concavity and inflection pointsDifferential calculus / Second derivative testDifferential calculus / Sketching curvesDifferential calculus / Connecting f, f', and f''Understands how to apply derivatives to solve problems (e.g., related rates, optimization)Apply derivatives to solve problemsDifferential calculus / Introduction to related ratesDifferential calculus / Solving related rates problemsDifferential calculus / Approximation with local linearityDifferential calculus / Solving optimization problemsDifferential calculus / Analyzing implicit relationsUnderstands the foundational theorems of calculus (e.g., fundamental theorems of calculus, mean value theorem, intermediate value theorem)Solve problems using the foundational theorems of calculusDifferential calculus / Intermediate value theoremDifferential calculus / Mean value theoremDifferential calculus / Extreme value theorem and critical pointsUnderstand the relationship between differentiation and integration, including the role of the fundamental theorems of calculusIntegral calculus / Fundamental theorem of calculus and accumulation functionsIntegral calculus / Interpreting the behavior of accumulation functionsIntegral calculus / Fundamental theorem of calculus and definite integralsMatch graphs of functions with graphs of their derivatives or accumulationsUnderstand how to use differentiation and integration of a function to express rates of change and total changeDifferential calculus / Meaning of the derivative in contextDifferential calculus / Non-motion applications of derivativesUnderstand and calculate the average value of a function over an interval (i.e., mean value theorem of integrals).Integral calculus / Average value of a functionUnderstands integration as a limit of Riemann sumsIntegral calculus / Approximation with Riemann sumsIntegral calculus / Summation notation reviewIntegral calculus / Riemann sums in summation notationIntegral calculus / Defining integrals with Riemann sumsCalculate a definite integral using a limit of Riemann sums.Understands how to use integration to compute area, volume, distance, or other accumulation processesIntegral calculus / Accumulations of change introductionIntegral calculus / Fundamental theorem of calculus and accumulation functionsIntegral calculus / Interpreting the behavior of accumulation functionsIntegral calculus / Non-motion applications of integralsIntegral calculus / Area: vertical area between curvesIntegral calculus / Area: horizontal area between curvesIntegral calculus / Area: curves that intersect at more than two pointsIntegral calculus / Volume: squares and rectangles cross sectionsIntegral calculus / Volume: triangles and semicircles cross sectionsIntegral calculus / Volume: disc method (revolving around x- and y-axes)Integral calculus / Volume: disc method (revolving around other axes)Integral calculus / Volume: washer method (revolving around x- and y-axes)Integral calculus / Volume: washer method (revolving around other axes)Use integration techniques to compute area, volume, distance, or other accumulation processesKnows how to determine the limits of sequences, if they exist.Determine the limits of sequences when they existIs familiar with simple infinite seriesAlgebra 2 / Geometric seriesPrecalculus / Geometric seriesPrecalculus / Geometric series (with summation notation)Precalculus / Arithmetic seriesIntegral calculus / Convergent and divergent infinite seriesIntegral calculus / Infinite geometric seriesIntegral calculus / nth-term testIntegral calculus / Integral testIntegral calculus / Harmonic series and p-seriesIntegral calculus / Comparison testsIntegral calculus / Alternating series testIntegral calculus / Ratio testIntegral calculus / Absolute and conditional convergenceDetermine if simple infinite series converge or divergeAlgebra 2 / Geometric seriesPrecalculus / Geometric seriesPrecalculus / Geometric series (with summation notation)Precalculus / Arithmetic seriesIntegral calculus / Convergent and divergent infinite seriesFind the sum of a simple infinite series if it existsAlgebra 2 / Geometric seriesPrecalculus / Geometric seriesPrecalculus / Geometric series (with summation notation)Precalculus / Arithmetic seriesIntegral calculus / Convergent and divergent infinite seriesFind the partial sum of a simple infinite seriesIntegral calculus / Approximation with Riemann sumsIntegral calculus / Summation notation reviewIntegral calculus / Riemann sums in summation notationIntegral calculus / Defining integrals with Riemann sumsII. Geometry, Probability and Statistics, and Discrete MathematicsLessonGeometryUnderstands transformations in a planeKnow precise definitions of angle, circle, line segment, perpendicular lines, and parallel linesHigh school geometry / Intro to Euclidean geometryHigh school geometry / Circle basicsRepresent transformations in the planeHigh school geometry / Introduction to rigid transformationsDescribe transformations as functions that take points in the plane as inputs, and give other points as outputsHigh school geometry / Introduction to rigid transformationsRecognize whether a transformation preserves distance and angle measureHigh school geometry / TranslationsHigh school geometry / RotationsHigh school geometry / Reflections Given a rectangle, parallelogram, trapezoid, or regular polygon, describe the rotations and reflections that map it onto itselfHigh school geometry / Rigid transformations overviewHigh school geometry / SymmetryDevelop definitions of rotations, reflections, and translations in terms of angles, circles, perpendicular lines, parallel lines, and line segmentsHigh school geometry / Properties & definitions of transformationsGiven a geometric figure and a rotation, reflection, or translation, draw the transformed figureHigh school geometry / Rigid transformations overviewSpecify a sequence of transformations that will map a given figure onto another figureHigh school geometry / Rigid transformations overviewHigh school geometry / Properties & definitions of transformationsUnderstands how to prove geometric theorems such as those about lines and angles, triangles, and parallelogramsProve theorems about lines and anglesHigh school geometry / AnglesProve theorems about trianglesHigh school geometry / Theorems concerning triangle propertiesHigh school geometry / Proofs of general theorems that use triangle congruenceHigh school geometry / Angle bisector theoremHigh school geometry / Pythagorean theorem proofsHigh school geometry / PolygonsProve theorems about parallelogramsHigh school geometry / Theorems concerning quadrilateral propertiesHigh school geometry / PolygonsUnderstands how geometric constructions are made with a variety of tools and methodsRecognize formal geometric constructionsHigh school geometry / Constructing bisectors of lines & anglesExplain how formal geometric constructions are made (e.g., an equilateral triangle, a square, a regular hexagon inscribed in a circle).High school geometry / Constructing bisectors of lines & anglesHigh school geometry / Inscribed shapes problem solvingHigh school geometry / Constructing regular polygons inscribed in circlesHigh school geometry / Constructing circumcircles & incirclesUnderstands congruence and similarity in terms of transformationsUse geometric descriptions of rigid motions to transform figures and to predict the effect of a given rigid motion on a given figureHigh school geometry / Rigid transformations overviewVerify the properties of dilations given by a center and a scale factorHigh school geometry / DilationsGiven two figures, use the definition of congruence in terms of rigid motions to decide if they are congruentHigh school geometry / Transformations & congruenceGiven two figures, use the definition of similarity in terms of dilations to decide if the figures are similarHigh school geometry / Definitions of similarityExplain how the criteria for triangle congruence (e.g., ASA, SAS, HL) follow from the definition of congruence in terms of rigid motionsHigh school geometry / Triangle congruenceUse the properties of similarity transformations to establish the AA criterion for two triangles to be similarHigh school geometry / Introduction to triangle similarityHigh school geometry / Solving similar trianglesUse congruence and similarity criteria for triangles to solve problems and to prove relationships in geometric figuresHigh school geometry / Theorems concerning triangle propertiesHigh school geometry / Working with trianglesHigh school geometry / Proofs of general theorems that use triangle congruenceHigh school geometry / Solving problems with similar & congruent trianglesHigh school geometry / Solving modeling problems with similar & congruent trianglesUnderstands how trigonometric ratios are defined in right trianglesUnderstand 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.High school geometry / Introduction to the trigonometric ratiosHigh school geometry / Trigonometric ratios & similarityAlgebra 2 / Unit circle introductionExplain and use the relationship between the sine and cosine of complementary anglesHigh school geometry / Introduction to the trigonometric ratiosHigh school geometry / Sine & cosine of complementary anglesUse trigonometric ratios and the Pythagorean theorem to solve right triangles in applied problemsHigh school geometry / Pythagorean theoremHigh school geometry / Solving for a side in a right triangle using the trigonometric ratiosHigh school geometry / Solving for an angle in a right triangle using the trigonometric ratiosHigh school geometry / Modeling with right trianglesUnderstands how trigonometry is applied to general trianglesDerive the formula for the area of a triangle by drawing an auxiliary line from a vertex perpendicular to the opposite side and use it to solve problemsHigh school geometry / Law of sinesHigh school geometry / Solving general trianglesApply the law of sines and the law of cosines to find unknown measurements in triangles.High school geometry / Law of sinesHigh school geometry / Law of cosinesHigh school geometry / Solving general trianglesUnderstands and applies theorems about circles. Identify and describe relationships among inscribed angles, radii, and chordsHigh school geometry / Arc measureHigh school geometry / Inscribed shapes problem solvingProve properties of angles for a quadrilateral inscribed in a circleHigh school geometry / Arc length (from degrees)High school geometry / Arc length (from radians)High school geometry / Inscribed anglesHigh school geometry / Inscribed shapes problem solvingConstruct a tangent line from a point outside a given circle to the circle.High school geometry / Properties of tangentsHigh school geometry / Constructing a line tangent to a circleUnderstands arc length and area measurements of sectors of circlesDerive and use the fact that the length of the arc intercepted by a central angle is proportional to the circumferenceHigh school geometry / Arc measureDerive and use the formula for the area of a sectorHigh school geometry / SectorsKnows how to translate between a geometric description (e.g., focus, asymptotes, directrix) and an equation for a conic sectionDerive and use the equation of a circle of given center and radiusHigh school geometry / Standard equation of a circlePrecalculus / The features of a circlePrecalculus / Standard equation of a circleComplete the square to find the center and radius of a circle given by an equation in standard formHigh school geometry / Expanded equation of a circlePrecalculus / Expanded equation of a circlePrecalculus / Identifying conic sections from their equationsDerive the equation of a parabola given a focus and directrixPrecalculus / Introduction to conic sectionsPrecalculus / Focus and directrix of a parabolaPrecalculus / Identifying conic sections from their equationsDerive and use the equations of ellipses and hyperbolas given the foci, using the fact that the sum or difference of distances from a point on the curve to the foci is constantPrecalculus / Introduction to conic sectionsPrecalculus / Center and radii of an ellipsePrecalculus / Foci of an ellipsePrecalculus / Introduction to hyperbolasPrecalculus / Foci of a hyperbolaPrecalculus / Hyperbolas not centered at the originPrecalculus / Identifying conic sections from their equationsUnderstands how to use coordinate geometry to algebraically prove simple geometric theoremsUse coordinates to prove simple geometric theorems algebraicallyHigh school geometry / Distance and midpointsProve the slope criteria for parallel and perpendicular lines, and use parallel and perpendicular lines to solve geometric problemsHigh school geometry / Parallel & perpendicular lines on the coordinate planeHigh school geometry / Equations of parallel & perpendicular linesFind the point on a directed line segment between two given points that partitions the segment in a given ratioHigh school geometry / Dividing line segmentsUse coordinates to compute perimeters of polygons and areas of triangles and quadrilateralsHigh school geometry / Problem solving with distance on the coordinate planeUnderstands how perimeter, area, surface area, and volume formulas are used to solve problemsGive an informal argument for the formulas for the circumference of a circle, area of a circle, volume of a cylinder, pyramid, and coneHigh school geometry / AreaHigh school geometry / Solid geometry introUse the perimeter and area of geometric shapes to solve problemsHigh school geometry / AreaUse the surface area and volume of prisms, cylinders, pyramids, cones, and spheres to solve problemsHigh school geometry / AreaKnows how to visualize relationships (e.g., cross section, nets, rotations) between two-dimensional and three-dimensional objectsIdentify the shapes of two-dimensional cross sections of three-dimensional objects, and identify three-dimensional objects generated by rotations of two-dimensional objectsHigh school geometry / 2D vs. 3D objectsUse two-dimensional representations of three-dimensional objects to visualize and solve problems.High school geometry / AreaKnows how to apply geometric concepts in real-world situationsUse geometric shapes, their measures, and their properties to describe objectsApply concepts of density based on area and volume in modeling situationsHigh school geometry / DensityApply geometric methods to solve design problemsUnderstands the properties of parallel and perpendicular lines, triangles, quadrilaterals, polygons, and circles and how they can be used in problem solvingSolve problems involving parallel, perpendicular, and intersecting linesHigh school geometry / AnglesApply angle relationships (e.g., supplementary, vertical, alternate interior) to solve problemsHigh school geometry / AnglesSolve problems that involve medians, midpoints, and altitudesHigh school geometry / PolygonsHigh school geometry / Proofs of general theorems that use triangle congruenceSolve problems involving special triangles (e.g., isosceles, equilateral, right)High school geometry / Special right trianglesHigh school geometry / PolygonsHigh school geometry / Working with trianglesKnow geometric properties of various quadrilaterals (e.g., parallelograms, trapezoids)High school geometry / PolygonsHigh school geometry / Theorems concerning quadrilateral propertiesKnow relationships among quadrilateralHigh school geometry / PolygonsSolve problems involving angles and diagonalsHigh school geometry / AnglesHigh school geometry / Theorems concerning quadrilateral propertiesSolve problems involving polygons with more than four sidesHigh school geometry / PolygonsProbability and StatisticsUnderstands how to summarize, represent, and interpret data collected from measurements on a single variable (e.g., box plots, dot plots, normal distributionsRepresent data with plots on the real number line (e.g., dot plots, histograms, and box plots).Statistics and probability / Displaying quantitative data with graphsStatistics and probability / More on data displaysStatistics and probability / Box and whisker plotsHigh school statistics / Displays of distributionsHigh school statistics / Box and whisker plotsUse statistics appropriate to the shape of the data distribution to compare center (e.g., median, mean) and spread (e.g., interquartile range, standard deviation) of two or more different data sets.Statistics and probability / Describing and comparing distributionsStatistics and probability / Measuring center in quantitative dataStatistics and probability / Interquartile range (IQR)Statistics and probability / Variance and standard deviation of a populationStatistics and probability / Other measures of spreadHigh school statistics / Summarizing center of distributions (central tendency)High school statistics / Summarizing spread of distributionsHigh school statistics / Comparing distributionsInterpret differences in shape, center, and spread in the context of the data sets, accounting for possible effects of outliersStatistics and probability / More on mean and medianUse the mean and standard deviation of a data set to fit it to a normal distribution and to estimate population percentages, and recognize that there are data sets for which such a procedure is not appropriateStatistics and probability / Variance and standard deviation of a sampleStatistics and probability / More on standard deviationEstimate areas under the normal curveStatistics and probability / PercentilesStatistics and probability / Z-scoresStatistics and probability / Effects of linear transformationsStatistics and probability / Density curvesStatistics and probability / Normal distributions and the empirical ruleStatistics and probability / Normal distribution calculationsStatistics and probability / More on normal distributionsHigh school statistics / Normal distributionsUnderstands how to summarize, represent, and interpret data collected from measurements on two variables, either categorical or quantitative (e.g., scatterplots, time series)Summarize and interpret categorical data for two categories in two-way frequency tables (e.g., joint, marginal, conditional relative frequenciesStatistics and probability / Two-way tablesStatistics and probability / Distributions in two-way tablesHigh school statistics / Two-way frequency tablesHigh school statistics / Two-way relative frequency tablesRecognize possible associations and trends in the dataStatistics and probability / Introduction to scatterplotsHigh school statistics / Creating and interpreting scatterplotsRepresent data for two quantitative variables on a scatterplot, and describe how the variables are related.Statistics and probability / Introduction to scatterplotsUnderstands how to create and interpret linear regression models (e.g., rate of change, intercepts, correlation coefficientUse technology to fit a function to data (i.e., linear regression)Statistics and probability / Introduction to trend linesHigh school statistics / Estimating with trend linesUse functions fitted to data to solve problems in the context of the dataStatistics and probability / Introduction to trend linesAssess the fit of a function by plotting and analyzing residualsStatistics and probability / Least-squares regression equationsStatistics and probability / Assessing the fit in least-squares regressionStatistics and probability / More on regressionInterpret the slope and the intercept of a regression line in the context of the data.Statistics and probability / Least-squares regression equationsCompute and interpret a correlation coefficientStatistics and probability / Correlation coefficientsDistinguish between correlation and causationStatistics and probability / correlation-and-causalityHigh school statistics / Creating and interpreting scatterplotsUnderstands statistical processes and how to evaluate themUnderstand statistics as a process for making inferences about population parameters based on a random sample from that populationStatistics and probability / Statistical questionsDecide if a specified model is consistent with results from a given data-generating process (e.g., using simulationStatistics and probability / Experimental probabilityStatistics and probability / Randomness, probability, and simulationUnderstands how to make inferences and justify conclusions from samples, experiments, and observational studiesRecognize the purposes of and differences among samples, experiments, and observational studies, and explain how randomization relates to eachStatistics and probability / Sampling and observational studiesStatistics and probability / Sampling methodsStatistics and probability / Types of studies (experimental vs. observational)Statistics and probability / ExperimentsHigh school statistics / Samples and surveysHigh school statistics / Observational studies and experimentsUse data from a sample to estimate a population mean or proportionUse data from a randomized experiment to compare two treatmentsUse results of simulations to decide if differences between parameters are significantEvaluate reports based on dataUnderstands the concepts of independence and conditional probability and how to apply these concepts to dataDescribe events as subsets of a sample space using characteristics of the outcomes, or as unions, intersections, or complements of other eventsPrecalculus / Venn diagrams and the addition ruleStatistics and probability / Basic theoretical probabilityStatistics and probability / Probability using sample spacesStatistics and probability / Basic set operationsHigh school statistics / Probability basicsUnderstand that two events, A and B, are independent if and only if Precalculus / Compound probability of independent events using the multiplication rulePrecalculus / Dependent eventsStatistics and probability / Multiplication rule for independent eventsStatistics and probability / Multiplication rule for dependent eventsHigh school statistics / Multiplication rule for independent eventsHigh school statistics / Multiplication rule for dependent eventsUnderstand the conditional probability of A given B as , and interpret independence of A and B as saying that and Statistics and probability / Conditional probability and independenceRecognize and explain the concepts of conditional probability and independenceStatistics and probability / Conditional probability and independenceUnderstands how to compute probabilities of simple events, probabilities of compound events, and conditional probabilitiesCalculate probabilities of simple and compound eventsPrecalculus / Basic probabilityStatistics and probability / Basic theoretical probabilityConstruct 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 probabilitiesHigh school statistics / Conditional probabilityFind , and interpret it in terms of a given model.High school statistics / Conditional probabilityApply the addition rule, and interpret it in terms of a given modelPrecalculus / Venn diagrams and the addition ruleStatistics and probability / Addition ruleHigh school statistics / Addition rule for probabilityApply the general multiplication rule in a uniform probability model, , and interpret it in terms of a given modelCalculate probabilities using the binomial probability distributionPrecalculus / PermutationsPrecalculus / CombinationsPrecalculus / Probability using combinatoricsStatistics and probability / Counting principle and factorialStatistics and probability / PermutationsStatistics and probability / CombinationsStatistics and probability / Combinatorics and probabilityHigh school statistics / Counting with permutationsHigh school statistics / Counting with combinationsHigh school statistics / Probability with counting, permutations, combinationsKnows how to make informed decisions using probabilities and expected valuesDefine a random variable for a quantity of interest by assigning a numerical value to each event in a sample space, and graph the corresponding probability distribution using the same graphical displays as for data distributionsStatistics and probability / Discrete random variablesCalculate the expected value of a random variable, and interpret it as the mean of the probability distributionStatistics and probability / Discrete random variablesHigh school statistics / Expected valueDevelop a probability distribution for a random variable, defined for a sample space in which theoretical probabilities can be calculated, and find the expected valueHigh school statistics / Expected valueDevelop a probability distribution for a random variable, defined for a sample space in which probabilities are assigned empirically, and find the expected value.High school statistics / Expected valueWeigh the possible outcomes of a decision by assigning probabilities to payoff values and finding expected valuesStatistics and probability / More on expected valueHigh school statistics / Expected valueAnalyze decisions and strategies using probability concepts (e.g., fairness).Understands how to use simulations to construct experimental probability distributions and to make informal inferences about theoreticalGiven the results of simulations, construct experimental probability distributionsHigh school statistics / Simulation and randomnessGiven the results of simulations, make informal inferences about theoretical probability distributions.High school statistics / Simulation and randomnessUnderstands how to find probabilities involving finite sample spaces and independent trialsUse the fundamental counting principle to find probabilities involving finite sample spaces and independent trialsStatistics and probability / Counting principle and factorialStatistics and probability / PermutationsStatistics and probability / CombinationsStatistics and probability / Combinatorics and probabilityDiscrete MathematicsUnderstands sequences (e.g., arithmetic, recursively defined, geometric).Write arithmetic and geometric sequences both recursively and with an explicit formula, use them to model situations, and translate between the two formsAlgebra I / Constructing arithmetic sequencesAlgebra I / Constructing geometric sequencesPrecalculus / Arithmetic sequencesPrecalculus / Geometric sequencesEvaluate, extend, or algebraically represent rules that involve number patternsAlgebra I / Introduction to arithmetic sequencesAlgebra I / Introduction to geometric sequencesAlgebra I / General sequencesExplore patterns in order to make conjectures, predictions, or generalizationsAlgebra I / Modeling with sequencesIs familiar with how recursion can be used to model various phenomenaFind values of functions defined recursively, and understand how recursion can be used to model various phenomenaConvert between recursive and closed-form expressions for a function, where possibleHas knowledge of equivalence relationsDetermine whether a binary relation on a set is reflexive, symmetric, or transitiveDetermine whether a relation is an equivalence relationUnderstands the differences between discrete and continuous representations (e.g., data, functions) and how each can be used to model various phenomenaUnderstand the differences between discrete and continuous representations (e.g., data, functionsUnderstand how discrete and continuous representations can be used to model various phenomena.Understands basic terminology and symbols of logicUnderstand the basic terminology of logicUnderstand the symbols of logicUse logic to evaluate the truth of statementsUse logic to evaluate the equivalence of statements (e.g., statement and contrapositiveUnderstands how to use counting techniques such as the multiplication principle, permutations, and combinationsUse counting techniques to solve problemsPrecalculus / PermutationsPrecalculus / CombinationsStatistics and probability / Counting principle and factorialStatistics and probability / PermutationsStatistics and probability / CombinationsHigh school statistics / Counting with permutationsHigh school statistics / Counting with combinationsUnderstands basic set theory (e.g., unions, differences, Venn diagrams). Solve problems using basic set theory (i.e., union, intersection, complement, differenceStatistics and probability / Basic set operationsUse Venn diagrams to answer questions about sets.Statistics and probability / Basic set operations ................
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