Wisconsin Standards for Mathematics: Mathematics III
Skills available for Wisconsin Integrated 3 math standards
Domain
Print skill plan
N
Number and Quantity
N-RN The Real Number System
N.RN.B Use properties of rational and irrational numbers.
N.RN.B Use properties of rational and irrational numbers.
M.N.RN.B.4: Explain why the sum or product of two rational numbers is rational; that the sum of a rational number and an irrational number is irrational; and that the product of a nonzero rational number and an irrational number is irrational.
M.N.RN.B.4: Explain why the sum or product of two rational numbers is rational; that the sum of a rational number and an irrational number is irrational; and that the product of a nonzero rational number and an irrational number is irrational.
N-CN The Complex Number System
N.CN.C Use complex numbers in polynomial identities and equations.
N.CN.C Use complex numbers in polynomial identities and equations.
M.N.CN.C.7: Solve quadratic equations with real coefficients that have complex solutions. Recognize when the quadratic formula gives complex solutions and write them as a ± bi for real numbers a and b.
M.N.CN.C.7: Solve quadratic equations with real coefficients that have complex solutions. Recognize when the quadratic formula gives complex solutions and write them as a ± bi for real numbers a and b.
A
Algebra
A-SSE Seeing Structure in Expressions
A.SSE.A Interpret the structure of expressions. (M)
A.SSE.A Interpret the structure of expressions. (M)
M.A.SSE.A.1: Interpret expressions that represent a quantity in terms of its context.
M.A.SSE.A.1: Interpret expressions that represent a quantity in terms of its context.
M.A.SSE.A.1a: Interpret parts of an expression, such as terms, factors, and coefficients.
M.A.SSE.A.1a: Interpret parts of an expression, such as terms, factors, and coefficients.
M.A.SSE.A.1b: Interpret complicated expressions by viewing one or more of their parts as a single entity.
M.A.SSE.A.1b: Interpret complicated expressions by viewing one or more of their parts as a single entity.
M.A.SSE.A.2: Use the structure of an expression to identify ways to rewrite it.
Polynomial expressions
Binomial Theorem
Radical expressions
Rational expressions
M.A.SSE.A.2: Use the structure of an expression to identify ways to rewrite it.
Polynomial expressions
Binomial Theorem
Radical expressions
Rational expressions
A.SSE.B Write expressions in equivalent forms to solve problems. (M)
A.SSE.B Write expressions in equivalent forms to solve problems. (M)
M.A.SSE.B.4: Derive the formula for the sum of a finite geometric series (when the common ratio is not 1), and use the formula to solve problems.
M.A.SSE.B.4: Derive the formula for the sum of a finite geometric series (when the common ratio is not 1), and use the formula to solve problems.
A-APR Arithmetic with Polynomials and Rational Expressions
A.APR.A Perform arithmetic operations on polynomials.
A.APR.A Perform arithmetic operations on polynomials.
M.A.APR.A.1: Understand that polynomials form a system analogous to the integers, namely, they are closed under the operations of addition, subtraction, and multiplication; add, subtract, and multiply polynomials.
M.A.APR.A.1: Understand that polynomials form a system analogous to the integers, namely, they are closed under the operations of addition, subtraction, and multiplication; add, subtract, and multiply polynomials.
A.APR.B Understand the relationship between zeros and factors of polynomials.
A.APR.B Understand the relationship between zeros and factors of polynomials.
M.A.APR.B.2: Know and apply the Remainder Theorem: For a polynomial p(x) and a number a, the remainder on division by x – a is p(a), so p(a) = 0 if and only if (x – a) is a factor of p(x).
M.A.APR.B.2: Know and apply the Remainder Theorem: For a polynomial p(x) and a number a, the remainder on division by x – a is p(a), so p(a) = 0 if and only if (x – a) is a factor of p(x).
M.A.APR.B.3: Identify zeros of polynomials when suitable factorizations are available, and use the zeros to construct a rough graph of the function defined by the polynomial.
Quadratics
Polynomials
M.A.APR.B.3: Identify zeros of polynomials when suitable factorizations are available, and use the zeros to construct a rough graph of the function defined by the polynomial.
Quadratics
Polynomials
A.APR.C Use polynomial identities to solve problems.
A.APR.C Use polynomial identities to solve problems.
M.A.APR.C.4: Prove polynomial identities and use them to describe numerical relationships.
M.A.APR.C.4: Prove polynomial identities and use them to describe numerical relationships.
A.APR.D Rewrite rational expressions.
A.APR.D Rewrite rational expressions.
M.A.APR.D.6: Rewrite simple rational expressions in different forms; write a(x)/b(x) in the form q(x) + r(x)/b(x), where a(x), b(x), q(x), and r(x) are polynomials with the degree of r(x) less than the degree of b(x), using inspection, long division, or, for the more complicated examples, a computer algebra system.
M.A.APR.D.6: Rewrite simple rational expressions in different forms; write a(x)/b(x) in the form q(x) + r(x)/b(x), where a(x), b(x), q(x), and r(x) are polynomials with the degree of r(x) less than the degree of b(x), using inspection, long division, or, for the more complicated examples, a computer algebra system.
M.A.APR.D.7: Understand that rational expressions form a system analogous to the rational numbers, closed under addition, subtraction, multiplication, and division by a nonzero rational expression; add, subtract, multiply, and divide rational expressions.
M.A.APR.D.7: Understand that rational expressions form a system analogous to the rational numbers, closed under addition, subtraction, multiplication, and division by a nonzero rational expression; add, subtract, multiply, and divide rational expressions.
A-CED Creating Equations
A.CED.A Create equations that describe numbers or relationships. (M)
A.CED.A Create equations that describe numbers or relationships. (M)
M.A.CED.A.1: Create equations and inequalities in one variable and use them to solve problems. Include equations arising from linear and quadratic functions, and simple rational and exponential functions.
M.A.CED.A.1: Create equations and inequalities in one variable and use them to solve problems. Include equations arising from linear and quadratic functions, and simple rational and exponential functions.
M.A.CED.A.2: Create equations in two or more variables to represent relationships between quantities; graph equations on coordinate axes with labels and scales.
Linear equations
Absolute value equations
Quadratic equations
Radical equations
Exponential and logarithmic equations
Trigonometric equations
M.A.CED.A.2: Create equations in two or more variables to represent relationships between quantities; graph equations on coordinate axes with labels and scales.
Linear equations
Absolute value equations
Quadratic equations
Radical equations
Exponential and logarithmic equations
Trigonometric equations
M.A.CED.A.3: Represent constraints by equations or inequalities, and by systems of equations and/or inequalities, and interpret solutions as viable or nonviable options in a modeling context. For example, represent inequalities describing nutritional and cost constraints on combinations of different foods.
Linear inequalities
Systems of equations
M.A.CED.A.3: Represent constraints by equations or inequalities, and by systems of equations and/or inequalities, and interpret solutions as viable or nonviable options in a modeling context. For example, represent inequalities describing nutritional and cost constraints on combinations of different foods.
Linear inequalities
Systems of equations
M.A.CED.A.4: Rearrange formulas to highlight a quantity of interest, using the same reasoning as in solving equations.
M.A.CED.A.4: Rearrange formulas to highlight a quantity of interest, using the same reasoning as in solving equations.
A-REI Reasoning with Equations and Inequalities
A.REI.A Understand solving equations as a process of reasoning and explain the reasoning.
A.REI.A Understand solving equations as a process of reasoning and explain the reasoning.
M.A.REI.A.2: Solve simple rational and radical equations in one variable, and give examples showing how extraneous solutions may arise.
M.A.REI.A.2: Solve simple rational and radical equations in one variable, and give examples showing how extraneous solutions may arise.
A.REI.D Represent and solve equations and inequalities graphically.
A.REI.D Represent and solve equations and inequalities graphically.
M.A.REI.D.11: Explain why the x-coordinates of the points where the graphs of the equations y = f(x) and y = g(x) intersect are the solutions of the equation f(x) = g(x); find the solutions approximately, e.g., using technology to graph the functions, make tables of values, or find successive approximations.
Tables
Graphs
M.A.REI.D.11: Explain why the x-coordinates of the points where the graphs of the equations y = f(x) and y = g(x) intersect are the solutions of the equation f(x) = g(x); find the solutions approximately, e.g., using technology to graph the functions, make tables of values, or find successive approximations.
Tables
Graphs
F
Functions
F-IF Interpreting Functions
F.IF.B Interpret functions that arise in applications in terms of context. (M)
F.IF.B Interpret functions that arise in applications in terms of context. (M)
M.F.IF.B.4: 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
Linear functions
Absolute value functions
Quadratic functions
Polynomial functions
Radical functions
Rational functions
Exponential and logarithmic functions
Word problems
M.F.IF.B.4: 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
Linear functions
Absolute value functions
Quadratic functions
Polynomial functions
Radical functions
Rational functions
Exponential and logarithmic functions
Word problems
M.F.IF.B.5: Relate the domain of a function to its graph and find an appropriate domain (discrete or continuous) in the context of the given problem.
Absolute value functions
Quadratic functions
Polynomial functions
Radical functions
Exponential and logarithmic functions
M.F.IF.B.5: Relate the domain of a function to its graph and find an appropriate domain (discrete or continuous) in the context of the given problem.
Absolute value functions
Quadratic functions
Polynomial functions
Radical functions
Exponential and logarithmic functions
M.F.IF.B.6: Calculate and interpret the average rate of change of a linear or nonlinear function (presented symbolically or as a table) over a specified interval. Estimate the rate of change from a graph.
M.F.IF.B.6: Calculate and interpret the average rate of change of a linear or nonlinear function (presented symbolically or as a table) over a specified interval. Estimate the rate of change from a graph.
F.IF.C Analyze functions using different representations. (M)
F.IF.C Analyze functions using different representations. (M)
M.F.IF.C.7: Graph functions expressed symbolically and show key features of the graph using an efficient method.
M.F.IF.C.7: Graph functions expressed symbolically and show key features of the graph using an efficient method.
M.F.IF.C.7b: Graph square root, cube root, and piecewise-defined functions, including step functions and absolute value functions.
M.F.IF.C.7b: Graph square root, cube root, and piecewise-defined functions, including step functions and absolute value functions.
M.F.IF.C.7c: Graph polynomial functions, identifying zeros when suitable factorizations are available, and showing end behavior.
M.F.IF.C.7c: Graph polynomial functions, identifying zeros when suitable factorizations are available, and showing end behavior.
M.F.IF.C.7d: Graph rational functions, identifying zeros and asymptotes when suitable factorizations are available, and showing end behavior.
M.F.IF.C.7d: Graph rational functions, identifying zeros and asymptotes when suitable factorizations are available, and showing end behavior.
M.F.IF.C.7e: Graph logarithmic functions, showing intercepts and end behavior, and trigonometric functions, showing period, midline, and amplitude.
Logarithmic functions
Sine functions
Cosine functions
Sine and cosine functions
M.F.IF.C.7e: Graph logarithmic functions, showing intercepts and end behavior, and trigonometric functions, showing period, midline, and amplitude.
Logarithmic functions
Sine functions
Cosine functions
Sine and cosine functions
M.F.IF.C.8: Write a function defined by an expression in equivalent forms to reveal and explain different properties of the function.
M.F.IF.C.8: Write a function defined by an expression in equivalent forms to reveal and explain different properties of the function.
M.F.IF.C.8a: Use an efficient process to rewrite f(x) = ax2 + bx + c as f(x) = a(x - h)2 + k or f(x) = a(x - p)(x - q) to determine the characteristics of the function and interpret these in terms of a context.
M.F.IF.C.8a: Use an efficient process to rewrite f(x) = ax2 + bx + c as f(x) = a(x - h)2 + k or f(x) = a(x - p)(x - q) to determine the characteristics of the function and interpret these in terms of a context.
M.F.IF.C.8b: Use the properties of exponents to interpret expressions for exponential functions.
M.F.IF.C.8b: Use the properties of exponents to interpret expressions for exponential functions.
M.F.IF.C.9: Compare properties of two functions each represented in a different way (algebraically, graphically, numerically in tables, or by verbal descriptions).
M.F.IF.C.9: Compare properties of two functions each represented in a different way (algebraically, graphically, numerically in tables, or by verbal descriptions).
F-BF Building Functions
F.BF.A Build a function that models a relationship between two quantities. (M)
F.BF.A Build a function that models a relationship between two quantities. (M)
M.F.BF.A.1: Write a function that describes a relationship between two quantities.
M.F.BF.A.1: Write a function that describes a relationship between two quantities.
M.F.BF.A.1a: Determine an explicit expression, a recursive process, or steps for calculation from a context
M.F.BF.A.1a: Determine an explicit expression, a recursive process, or steps for calculation from a context
M.F.BF.A.1b: Combine standard function types using arithmetic operations.
Polynomial expressions
Rational expressions
Function operations
M.F.BF.A.1b: Combine standard function types using arithmetic operations.
Polynomial expressions
Rational expressions
Function operations
M.F.BF.A.1c: Work with composition of functions using tables, graphs and symbols.
M.F.BF.A.1c: Work with composition of functions using tables, graphs and symbols.
F.BF.B Build new functions from existing functions.
F.BF.B Build new functions from existing functions.
M.F.BF.B.3: Identify the effect on the graph of replacing f(x) by f(x) + k, k f(x), f(kx), and f(x + k) using transformations for specific values of k (both positive and negative); find the value of k given the graphs. Experiment with cases and illustrate an explanation of the effects on the graph using technology. Include recognizing even and odd functions from their graphs and algebraic expressions for them.
Translations
Reflections
Dilations
Mixed transformations
M.F.BF.B.3: Identify the effect on the graph of replacing f(x) by f(x) + k, k f(x), f(kx), and f(x + k) using transformations for specific values of k (both positive and negative); find the value of k given the graphs. Experiment with cases and illustrate an explanation of the effects on the graph using technology. Include recognizing even and odd functions from their graphs and algebraic expressions for them.
Translations
Reflections
Dilations
Mixed transformations
M.F.BF.B.4: Identify and create inverse functions, using tables, graphs, and symbolic methods to solve for the other variable.
M.F.BF.B.4: Identify and create inverse functions, using tables, graphs, and symbolic methods to solve for the other variable.
M.F.BF.B.5: Understand the inverse relationship between exponents and logarithms and use this relationship to solve problems involving logarithms and exponents.
Exponential and logarithmic form
Exponential equations
Logarithmic equations
M.F.BF.B.5: Understand the inverse relationship between exponents and logarithms and use this relationship to solve problems involving logarithms and exponents.
Exponential and logarithmic form
Exponential equations
Logarithmic equations
F-LE Linear, Quadratic, and Exponential Models
F.LE.A Construct and compare linear, quadratic, and exponential models and solve problems. (M)
F.LE.A Construct and compare linear, quadratic, and exponential models and solve problems. (M)
M.F.LE.A.4: For exponential models, express as a logarithm the solution to abct = d where a, c, and d are numbers and the base b is 2,10, or e; evaluate the logarithm using technology.
Exponential and logarithmic form
Solve exponential equations
M.F.LE.A.4: For exponential models, express as a logarithm the solution to abct = d where a, c, and d are numbers and the base b is 2,10, or e; evaluate the logarithm using technology.
Exponential and logarithmic form
Solve exponential equations
F-TF Trigonometric Functions
F.TF.A Extend the domain of the trigonometric functions of the unit circle.
F.TF.A Extend the domain of the trigonometric functions of the unit circle.
M.F.TF.A.1: Understand radian measure of an angle as the length of the arc on the unit circle subtended by the angle.
M.F.TF.A.1: Understand radian measure of an angle as the length of the arc on the unit circle subtended by the angle.
M.F.TF.A.2: 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.
M.F.TF.A.2: 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.
F.TF.B Model periodic phenomena with trigonometric functions. (M)
F.TF.B Model periodic phenomena with trigonometric functions. (M)
M.F.TF.B.5: Choose trigonometric functions to model periodic phenomena with specified amplitude, frequency, and midline.
Sine functions
Cosine functions
M.F.TF.B.5: Choose trigonometric functions to model periodic phenomena with specified amplitude, frequency, and midline.
Sine functions
Cosine functions
F.TF.C Prove and apply trigonometric identities.
F.TF.C Prove and apply trigonometric identities.
M.F.TF.C.8: Prove the Pythagorean identity sin2(θ) + cos2(θ) = 1 and use it to find sin(θ), cos(θ), or tan(θ) given sin(θ), cos(θ), or tan(θ) and the quadrant of the angle.
M.F.TF.C.8: Prove the Pythagorean identity sin2(θ) + cos2(θ) = 1 and use it to find sin(θ), cos(θ), or tan(θ) given sin(θ), cos(θ), or tan(θ) and the quadrant of the angle.
G
Geometry
G-GPE Expressing Geometric Properties
G.GPE.A Translate between the geometric description and the equation for a conic section.
G.GPE.A Translate between the geometric description and the equation for a conic section.
M.G.GPE.A.1: Derive the equation of a circle of given center and radius using the Pythagorean Theorem; complete the square to find the center and radius of a circle given by an equation.
M.G.GPE.A.1: Derive the equation of a circle of given center and radius using the Pythagorean Theorem; complete the square to find the center and radius of a circle given by an equation.
S
Statistics and Probability
S-ID Interpreting Categorical and Quantitative Data
SP.ID.A Summarize, represent, and interpret data on a single count or measurement variable. (M)
SP.ID.A Summarize, represent, and interpret data on a single count or measurement variable. (M)
M.SP.ID.A.4: Use the mean and standard deviation of a data set to fit it to a normal distribution and to estimate population percentages. Recognize that there are data sets for which such a procedure is not appropriate. Use statistical packages calculators, spreadsheets, and tables to estimate areas under the normal curve.
M.SP.ID.A.4: Use the mean and standard deviation of a data set to fit it to a normal distribution and to estimate population percentages. Recognize that there are data sets for which such a procedure is not appropriate. Use statistical packages calculators, spreadsheets, and tables to estimate areas under the normal curve.
S-IC Making Inferences and Justifying Conclusions
SP.IC.A Understand and evaluate random processes underlying statistical experiments. (M)
SP.IC.A Understand and evaluate random processes underlying statistical experiments. (M)
M.SP.IC.A.1: Understand statistics as a process for making inferences about population parameters based on a random sample from that population.
M.SP.IC.A.1: Understand statistics as a process for making inferences about population parameters based on a random sample from that population.
M.SP.IC.A.2: Decide if a specified model is consistent with results from a given data-generating process (e.g., using simulation). For example, a model says a spinning coin falls heads up with probability 0.5. Would a result of 5 tails in a row cause you to question the model?
M.SP.IC.A.2: Decide if a specified model is consistent with results from a given data-generating process (e.g., using simulation). For example, a model says a spinning coin falls heads up with probability 0.5. Would a result of 5 tails in a row cause you to question the model?
SP.IC.B Make inferences and justify conclusions from sample surveys, experiments, and observational studies. (M)
SP.IC.B Make inferences and justify conclusions from sample surveys, experiments, and observational studies. (M)
M.SP.IC.B.3: Recognize the purposes of and differences among sample surveys, experiments, and observational studies; explain how randomization relates to each.
M.SP.IC.B.3: Recognize the purposes of and differences among sample surveys, experiments, and observational studies; explain how randomization relates to each.
M.SP.IC.B.4: 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.
M.SP.IC.B.4: 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.
M.SP.IC.B.5: Use data from a randomized experiment to compare two treatments; use simulations to decide if differences between parameters are significant.
M.SP.IC.B.5: Use data from a randomized experiment to compare two treatments; use simulations to decide if differences between parameters are significant.
M.SP.IC.B.6: Evaluate reports based on data.
M.SP.IC.B.6: Evaluate reports based on data.
S-MD Using Probability to Make Decisions
SP.MD.B Use probability to evaluate outcomes of decisions. (M)
SP.MD.B Use probability to evaluate outcomes of decisions. (M)
M.SP.MD.B.6: Use probabilities to make fair decisions (e.g., drawing for a party door prize where attendees earn one entry to the drawing for each activity they complete, using an electronic spinner to pick a team spokesperson at random from a group, flip a coin to decide which of two friends gets to choose the movie, using a random number generator to select people to include in a sample for an experiment).
M.SP.MD.B.6: Use probabilities to make fair decisions (e.g., drawing for a party door prize where attendees earn one entry to the drawing for each activity they complete, using an electronic spinner to pick a team spokesperson at random from a group, flip a coin to decide which of two friends gets to choose the movie, using a random number generator to select people to include in a sample for an experiment).
M.SP.MD.B.7: Analyze decisions and strategies using probability concepts (e.g., balancing expected gains and risk, medical product testing, choosing an investment option, deciding when to kick an extra point vs. two point conversion after a touchdown in football).
M.SP.MD.B.7: Analyze decisions and strategies using probability concepts (e.g., balancing expected gains and risk, medical product testing, choosing an investment option, deciding when to kick an extra point vs. two point conversion after a touchdown in football).