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Wisconsin Standards for Mathematics: Algebra 1 Wisconsin Standards for Mathematics: Algebra 1
Wisconsin Standards for Mathematics: Algebra 2 Wisconsin Standards for Mathematics: Algebra 2
Wisconsin Standards for Mathematics: All High School Pathways Wisconsin Standards for Mathematics: All High School Pathways
Wisconsin Standards for Mathematics: Fourth Courses Wisconsin Standards for Mathematics: Fourth Courses
Wisconsin Standards for Mathematics: Geometry Wisconsin Standards for Mathematics: Geometry
Wisconsin Standards for Mathematics: Mathematical Practices Wisconsin Standards for Mathematics: Mathematical Practices
Wisconsin Standards for Mathematics: Mathematics I Wisconsin Standards for Mathematics: Mathematics I
Wisconsin Standards for Mathematics: Mathematics II Wisconsin Standards for Mathematics: Mathematics II
Wisconsin Standards for Mathematics: Mathematics III Wisconsin Standards for Mathematics: Mathematics III
Wisconsin Standards for Mathematics: Precalculus Wisconsin Standards for Mathematics: Precalculus

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N-CN The Complex Number System

N.CN.A Perform arithmetic operations with complex numbers.
- M.N.CN.A.2 (+) Use the relation i² = −1 and the commutative, associative, and distributive properties to add, subtract, and multiply complex numbers.
- M.N.CN.A.3 (+) Find the conjugate of a complex number; use conjugates to find moduli (absolute values) and quotients of complex numbers.
N.CN.B Represent complex numbers and their operations on the complex plane.
- M.N.CN.B.4 (+) Represent complex numbers on the complex plane in rectangular and polar form (including real and imaginary numbers), and explain why the rectangular and polar forms of a given complex number represent the same number.
- M.N.CN.B.5 (+) Represent addition, subtraction, multiplication, and conjugation of complex numbers geometrically on the complex plane; use properties of this representation for computation. For example, (−1 + √3i²)³ = 8 because (−1 + √3i²) has modulus 2 and argument 120°.
- M.N.CN.B.6 (+) 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.
  - Midpoints in the complex plane (PC-W.7)
  - Distance in the complex plane (PC-W.8)
N.CN.C Use complex numbers in polynomial identities and equations.
- M.N.CN.C.8 (+) Extend polynomial identities to the complex numbers. For example, rewrite x2+ 4 as (x + 2i)(x - 2i).
  - Complex conjugate theorem (PC-F.4)
  - Conjugate root theorems (PC-F.5)
- M.N.CN.C.9 (+) Know the Fundamental Theorem of Algebra; show that it is true for quadratic polynomials.
  - Fundamental Theorem of Algebra (PC-F.7)

N-VM Vector and Matrix Quantities

N.VM.A Represent and model with vector quantities. (M)
- M.N.VM.A.1 (+) 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||, v).
- M.N.VM.A.2 (+) Find the components of a vector by subtracting the coordinates of an initial point from the coordinates of a terminal point.
  - Find the component form of a vector (PC-Y.2)
  - Find the component form of a three-dimensional vector (PC-Z.2)
- M.N.VM.A.3 (+) Solve problems involving velocity and other quantities that can be represented by vectors.
N.VM.B Perform operations on vectors.
- M.N.VM.B.4 (+) Add and subtract vectors.
  - M.N.VM.B.4a 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.
  - M.N.VM.B.4b Given two vectors in magnitude and direction form, determine the magnitude and direction of their sum.
    - Find the magnitude and direction of a vector sum (PC-Y.9)
  - M.N.VM.B.4c Understand vector subtraction v - w as v + (-w), where -w is the additive inverse of w, with the same magnitude as w and pointing in the opposite direction. Represent vector subtraction graphically by connecting the tips in the appropriate order, and perform vector subtraction component-wise.
    - Subtract vectors (PC-Y.8)
    - Add and subtract three-dimensional vectors (PC-Z.3)
- M.N.VM.B.5 (+) Multiply a vector by a scalar.
  - M.N.VM.B.5a 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).
    - Multiply a vector by a scalar (PC-Y.10)
    - Scalar multiples of three-dimensional vectors (PC-Z.4)
  - M.N.VM.B.5b 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).
    - Find the magnitude of a vector scalar multiple (PC-Y.11)
    - Determine the direction of a vector scalar multiple (PC-Y.12)
N.VM.C Perform operations on matrices and use matrices in applications. (M)
- M.N.VM.C.6 (+) Use matrices to represent and manipulate data, e.g., to represent payoffs or incidence relationships in a network.
- M.N.VM.C.7 (+) Multiply matrices by scalars to produce new matrices, e.g., as when all of the payoffs in a game are doubled.
  - Multiply a matrix by a scalar (PC-Q.4)
- M.N.VM.C.8 (+) Add, subtract, and multiply matrices of appropriate dimensions.
- M.N.VM.C.9 (+) Understand that, unlike multiplication of numbers, matrix multiplication for square matrices is not a commutative operation, but still satisfies the associative and distributive properties.
  - Properties of matrices (PC-Q.8)
- M.N.VM.C.10 (+) Understand that the zero and identity matrices play a role in matrix addition and multiplication similar to the role of 0 and 1 in the real numbers. The determinant of a square matrix is nonzero if and only if the matrix has a multiplicative inverse.
- M.N.VM.C.11 (+) Multiply a vector (regarded as a matrix with one column) by a matrix of suitable dimensions to produce another vector. Work with matrices as transformations of vectors.
- M.N.VM.C.12 (+) Work with 2 x 2 matrices as transformations of the plane, and interpret the absolute value of the determinant in terms of area.

A-APR Arithmetic with Polynomials and Rational Expressions

A.APR.C Use polynomial identities to solve problems.
- M.A.APR.C.5 (+) Know and apply the Binomial Theorem for the expansion of (x + y)ⁿ in powers of x and y for a positive integer n, where x and y are any numbers, with coefficients determined for example by Pascal's Triangle.

A-REI Reasoning with Equations and Inequalities

A.REI.C Solve systems of equations.
- M.A.REI.C.8 (+) Represent a system of linear equations as a single matrix equation in a vector variable.
  - Solve a system of equations using augmented matrices (PC-N.8)
  - Solve a system of equations using augmented matrices: word problems (PC-N.9)
- M.A.REI.C.9 (+) Find the inverse of a matrix if it exists and use it to solve systems of linear equations (using technology for matrices of dimension 3 × 3 or greater).

F-IF Interpreting Functions

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.7e Graph logarithmic functions, showing intercepts and end behavior, and trigonometric functions, showing period, midline, and amplitude.

F-TF Trigonometric Functions

F.TF.A Extend the domain of the trigonometric functions of the unit circle.
- M.F.TF.A.3 3. (+) Use special triangles to determine geometrically the values of sine, cosine, tangent for π/3, π/4 and π/6, and use the unit circle to express the values of sine, cosine, and tangent for π–x, π+x, and 2π–x in terms of their values for x, where x is any real number.
  - Find trigonometric ratios using right triangles (PC-R.5)
  - Find trigonometric ratios of special angles (PC-R.7)
- M.F.TF.A.4 4. (+) Use the unit circle to explain symmetry (odd and even) and periodicity of trigonometric functions.
  - Symmetry and periodicity of trigonometric functions (PC-T.2)
F.TF.B Model periodic phenomena with trigonometric functions. (M)
- M.F.TF.B.6 6. (+) Understand that restricting a trigonometric function to a domain on which it is always increasing or always decreasing allows its inverse to be constructed.
  - Inverses of trigonometric functions (PC-R.9)
- M.F.TF.B.7 7. (+) 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.
  - Solve trigonometric equations (PC-R.11)
F.TF.C Prove and apply trigonometric identities.
- M.F.TF.C.9 9. (+) Prove the addition and subtraction formulas for sine, cosine, and tangent and use them to solve problems.
  - Trigonometric sum and difference identities (PC-T.5)
  - Solve trigonometric equations using sum and difference identities (PC-T.6)

G-SRT Similarity, Right Triangles, and Trigonometry

G.SRT.D Apply trigonometry to general triangles.
- M.G.SRT.D.9 (+) Derive the formula A = 1/2 ab sin(C) for the area of a triangle by drawing an auxiliary line from a vertex perpendicular to the opposite side.
  - Area of a triangle: sine formula (PC-R.18)
- M.G.SRT.D.10 (+) Prove the Laws of Sines and Cosines and use them to solve problems.
- M.G.SRT.D.11 (+) Understand and apply the Law of Sines and the Law of Cosines to find unknown measurements in right and non-right triangles (e.g., surveying problems, resultant forces).
  - Solve a triangle (PC-R.17)

G-GPE Expressing Geometric Properties

G.GPE.A Translate between the geometric description and the equation for a conic section.
- M.G.GPE.A.2 (+) Derive the equation of a parabola given a focus and directrix.
- M.G.GPE.A.3 (+) Derive the equations of ellipses and hyperbolas given the foci, using the fact that the sum or difference of distances from the foci is constant.
  - Write equations of ellipses in standard form (PC-U.9)
  - Write equations of hyperbolas in standard form (PC-U.12)

S-CP Conditional Probability and the Rules of Probability

SP.CP.B Use the rules of probability to compute probabilities of compound events in a uniform probability model.
- M.SP.CP.B.8 (+) Use a representation such as a tree diagram to apply the general Multiplication Rule in a uniform probability model, P(A and B) = P(A)P(B|A) = P(B)P(A|B), and interpret the answer in terms of the model.
  - Find conditional probabilities (PC-CC.6)
- M.SP.CP.B.9 (+) Use permutations and combinations to compute probabilities of compound events and solve problems.
  - Combinations and permutations (PC-CC.2)
  - Find probabilities using combinations and permutations (PC-CC.3)

S-MD Using Probability to Make Decisions

SP.MD.A Calculate expected values and use them to solve problems. (M)
- M.SP.MD.A.1 (+) 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.
  - Write a discrete probability distribution (PC-DD.3)
  - Graph a discrete probability distribution (PC-DD.4)
- M.SP.MD.A.2 (+) Calculate the expected value of a random variable; interpret it as the mean of the probability distribution.
  - Expected values of random variables (PC-DD.5)
- M.SP.MD.A.3 (+) Develop a probability distribution for a random variable defined for a sample space in which theoretical probabilities can be calculated; find the expected value. For example, find the theoretical probability distribution for the number of correct answers obtained by guessing on all five questions of a multiple-choice test where each question has four choices, and find the expected grade under various grading schemes.
  - Write the probability distribution for a game of chance (PC-DD.8)
  - Expected values for a game of chance (PC-DD.9)
- M.SP.MD.A.4 (+) Develop a probability distribution for a random variable defined for a sample space in which probabilities are assigned empirically; find the expected value. For example, find a current data distribution on the number of TV sets per household in the United States, and calculate the expected number of sets per household. How many TV sets would you expect to find in 100 randomly selected households?
SP.MD.B Use probability to evaluate outcomes of decisions. (M)
- M.SP.MD.B.5 (+) Weigh the possible outcomes of a decision by assigning probabilities to payoff values and finding expected values.
  - M.SP.MD.B.5a Find the expected payoff for a game of chance. For example, find the expected winnings from a state lottery ticket or a game at a fast-food restaurant.
    - Expected values for a game of chance (PC-DD.9)
  - M.SP.MD.B.5b Evaluate and compare strategies on the basis of expected values. For example, compare a high-deductible versus a low-deductible automobile insurance policy using various, but reasonable, chances of having a minor or a major accident.
    - Choose the better bet (PC-DD.10)

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N-CN The Complex Number System

N.CN.A Perform arithmetic operations with complex numbers.

M.N.CN.A.2 (+) Use the relation i² = −1 and the commutative, associative, and distributive properties to add, subtract, and multiply complex numbers.

M.N.CN.A.3 (+) Find the conjugate of a complex number; use conjugates to find moduli (absolute values) and quotients of complex numbers.

N.CN.B Represent complex numbers and their operations on the complex plane.

M.N.CN.B.4 (+) Represent complex numbers on the complex plane in rectangular and polar form (including real and imaginary numbers), and explain why the rectangular and polar forms of a given complex number represent the same number.

M.N.CN.B.5 (+) Represent addition, subtraction, multiplication, and conjugation of complex numbers geometrically on the complex plane; use properties of this representation for computation. For example, (−1 + √3i²)3 = 8 because (−1 + √3i²) has modulus 2 and argument 120°.

M.N.CN.B.6 (+) 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.

N.CN.C Use complex numbers in polynomial identities and equations.

M.N.CN.C.8 (+) Extend polynomial identities to the complex numbers. For example, rewrite x2+ 4 as (x + 2i)(x - 2i).

M.N.CN.C.9 (+) Know the Fundamental Theorem of Algebra; show that it is true for quadratic polynomials.

N-VM Vector and Matrix Quantities

N.VM.A Represent and model with vector quantities. (M)

M.N.VM.A.1 (+) 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||, v).

M.N.VM.A.2 (+) Find the components of a vector by subtracting the coordinates of an initial point from the coordinates of a terminal point.

M.N.VM.A.3 (+) Solve problems involving velocity and other quantities that can be represented by vectors.

N.VM.B Perform operations on vectors.

M.N.VM.B.4 (+) Add and subtract vectors.

M.N.VM.B.4a 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.

M.N.VM.B.4b Given two vectors in magnitude and direction form, determine the magnitude and direction of their sum.

M.N.VM.B.5 (+) Multiply a vector by a scalar.

M.N.VM.B.5a 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).

M.N.VM.B.5b 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).

N.VM.C Perform operations on matrices and use matrices in applications. (M)

M.N.VM.C.6 (+) Use matrices to represent and manipulate data, e.g., to represent payoffs or incidence relationships in a network.

M.N.VM.C.7 (+) Multiply matrices by scalars to produce new matrices, e.g., as when all of the payoffs in a game are doubled.

M.N.VM.C.8 (+) Add, subtract, and multiply matrices of appropriate dimensions.

M.N.VM.C.9 (+) Understand that, unlike multiplication of numbers, matrix multiplication for square matrices is not a commutative operation, but still satisfies the associative and distributive properties.

M.N.VM.C.10 (+) Understand that the zero and identity matrices play a role in matrix addition and multiplication similar to the role of 0 and 1 in the real numbers. The determinant of a square matrix is nonzero if and only if the matrix has a multiplicative inverse.

M.N.VM.C.11 (+) Multiply a vector (regarded as a matrix with one column) by a matrix of suitable dimensions to produce another vector. Work with matrices as transformations of vectors.

M.N.VM.C.12 (+) Work with 2 x 2 matrices as transformations of the plane, and interpret the absolute value of the determinant in terms of area.

A-APR Arithmetic with Polynomials and Rational Expressions

A.APR.C Use polynomial identities to solve problems.

M.A.APR.C.5 (+) Know and apply the Binomial Theorem for the expansion of (x + y)n in powers of x and y for a positive integer n, where x and y are any numbers, with coefficients determined for example by Pascal's Triangle.

A-REI Reasoning with Equations and Inequalities

A.REI.C Solve systems of equations.

M.A.REI.C.8 (+) Represent a system of linear equations as a single matrix equation in a vector variable.

M.A.REI.C.9 (+) Find the inverse of a matrix if it exists and use it to solve systems of linear equations (using technology for matrices of dimension 3 × 3 or greater).

F-IF Interpreting Functions

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.7e Graph logarithmic functions, showing intercepts and end behavior, and trigonometric functions, showing period, midline, and amplitude.

F-TF Trigonometric Functions

F.TF.A Extend the domain of the trigonometric functions of the unit circle.

M.F.TF.A.3 3. (+) Use special triangles to determine geometrically the values of sine, cosine, tangent for π/3, π/4 and π/6, and use the unit circle to express the values of sine, cosine, and tangent for π–x, π+x, and 2π–x in terms of their values for x, where x is any real number.

M.F.TF.A.4 4. (+) Use the unit circle to explain symmetry (odd and even) and periodicity of trigonometric functions.

F.TF.B Model periodic phenomena with trigonometric functions. (M)

M.F.TF.B.6 6. (+) Understand that restricting a trigonometric function to a domain on which it is always increasing or always decreasing allows its inverse to be constructed.

M.F.TF.B.7 7. (+) 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.

F.TF.C Prove and apply trigonometric identities.

M.F.TF.C.9 9. (+) Prove the addition and subtraction formulas for sine, cosine, and tangent and use them to solve problems.

G-SRT Similarity, Right Triangles, and Trigonometry

G.SRT.D Apply trigonometry to general triangles.

M.G.SRT.D.9 (+) Derive the formula A = 1/2 ab sin(C) for the area of a triangle by drawing an auxiliary line from a vertex perpendicular to the opposite side.

M.G.SRT.D.10 (+) Prove the Laws of Sines and Cosines and use them to solve problems.

M.G.SRT.D.11 (+) Understand and apply the Law of Sines and the Law of Cosines to find unknown measurements in right and non-right triangles (e.g., surveying problems, resultant forces).

G-GPE Expressing Geometric Properties

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

M.G.GPE.A.2 (+) Derive the equation of a parabola given a focus and directrix.

M.G.GPE.A.3 (+) Derive the equations of ellipses and hyperbolas given the foci, using the fact that the sum or difference of distances from the foci is constant.

S-CP Conditional Probability and the Rules of Probability

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

M.SP.CP.B.8 (+) Use a representation such as a tree diagram to apply the general Multiplication Rule in a uniform probability model, P(A and B) = P(A)P(B|A) = P(B)P(A|B), and interpret the answer in terms of the model.

M.SP.CP.B.9 (+) Use permutations and combinations to compute probabilities of compound events and solve problems.

S-MD Using Probability to Make Decisions

SP.MD.A Calculate expected values and use them to solve problems. (M)

M.SP.MD.A.1 (+) 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.

M.SP.MD.A.2 (+) Calculate the expected value of a random variable; interpret it as the mean of the probability distribution.

SP.MD.B Use probability to evaluate outcomes of decisions. (M)

M.SP.MD.B.5 (+) Weigh the possible outcomes of a decision by assigning probabilities to payoff values and finding expected values.

M.SP.MD.B.5a Find the expected payoff for a game of chance. For example, find the expected winnings from a state lottery ticket or a game at a fast-food restaurant.

M.SP.MD.B.5b Evaluate and compare strategies on the basis of expected values. For example, compare a high-deductible versus a low-deductible automobile insurance policy using various, but reasonable, chances of having a minor or a major accident.

M.N.CN.B.5 (+) Represent addition, subtraction, multiplication, and conjugation of complex numbers geometrically on the complex plane; use properties of this representation for computation. For example, (−1 + √3i²)³ = 8 because (−1 + √3i²) has modulus 2 and argument 120°.

M.A.APR.C.5 (+) Know and apply the Binomial Theorem for the expansion of (x + y)ⁿ in powers of x and y for a positive integer n, where x and y are any numbers, with coefficients determined for example by Pascal's Triangle.