The Common Core in Wisconsin

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Skills available for Wisconsin high school math standards

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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: 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
Common Core State Standards: Algebra 1 Common Core State Standards: Algebra 1
Common Core State Standards: Algebra 2 Common Core State Standards: Algebra 2
Common Core State Standards: All High School Pathways Common Core State Standards: All High School Pathways
Common Core State Standards: Fourth Courses Common Core State Standards: Fourth Courses
Common Core State Standards: Geometry Common Core State Standards: Geometry
Common Core State Standards: Mathematical Practices Common Core State Standards: Mathematical Practices
Common Core State Standards: Mathematics I Common Core State Standards: Mathematics I
Common Core State Standards: Mathematics II Common Core State Standards: Mathematics II
Common Core State Standards: Mathematics III Common Core State Standards: Mathematics III
Common Core State Standards: Precalculus Common Core State Standards: Precalculus

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N-RN The Real Number System

N.RN.A Extend the properties of exponents to rational exponents.
- M.N.RN.A.1 Explain how the definition of the meaning of rational exponents follows from extending the properties of integer exponents.
- M.N.RN.A.2 Rewrite expressions involving radicals and rational exponents using the properties of exponents.
- M.N.RN.A.3 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.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.

N-Q Quantities

N.Q.A Reason quantitatively and use units to solve problems. (M)
- M.N.Q.A.1 Use units as a way to understand problems and to guide the solution of multi-step problems; choose and interpret units consistently in formulas; choose and interpret the scale and the origin in graphs and data displays.
- M.N.Q.A.2 Define appropriate quantities for the purpose of descriptive modeling.
- M.N.Q.A.3 Choose a level of accuracy appropriate to limitations on measurement when reporting quantities.

N-CN The Complex Number System

N.CN.A Perform arithmetic operations with complex numbers.
- M.N.CN.A.1 Know there is a complex number i such that i² = -1, and every complex number has the form a + bi with a and b real. Understand why complex numbers exist.
  - Introduction to complex numbers (A2-I.1)
- 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 + v3i²)³ = 8 because (-1 + v3i²) 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-S.7)
  - Distance in the complex plane (PC-S.8)
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.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.
  - Fundamental Theorem of Algebra (A2-L.17)
  - Fundamental Theorem of Algebra (PC-D.10)

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.
- 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.
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-U.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.
- 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-U.10)
    - Scalar multiples of three-dimensional vectors (PC-V.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-U.11)
    - Determine the direction of a vector scalar multiple (PC-U.12)
N.VM.C Perform operations on matrices and use matrices in applications. (M)

G-CO Congruence

G.CO.A Experiment with transformations in the plane.
- M.G.CO.A.1 Know precise definitions of angle, circle, perpendicular line, parallel line, and line segment, based on the undefined notions of point, line, distance along a line, and distance around a circular arc.
  - Angle vocabulary (G-C.1)
- M.G.CO.A.2 Represent transformations in the plane using, e.g., transparencies and geometry software; describe transformations as functions that take points in the plane as inputs and give other points as outputs. Compare transformations that preserve distance and angle to those that do not (e.g., translation versus horizontal stretch).
  - Classify congruence transformations (G-L.1)

The Common Core in Wisconsin

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N-RN The Real Number System

N.RN.A Extend the properties of exponents to rational exponents.

M.N.RN.A.1 Explain how the definition of the meaning of rational exponents follows from extending the properties of integer exponents.

M.N.RN.A.2 Rewrite expressions involving radicals and rational exponents using the properties of exponents.

M.N.RN.A.3 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.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.

N-Q Quantities

N.Q.A Reason quantitatively and use units to solve problems. (M)

M.N.Q.A.1 Use units as a way to understand problems and to guide the solution of multi-step problems; choose and interpret units consistently in formulas; choose and interpret the scale and the origin in graphs and data displays.

M.N.Q.A.2 Define appropriate quantities for the purpose of descriptive modeling.

M.N.Q.A.3 Choose a level of accuracy appropriate to limitations on measurement when reporting quantities.

N-CN The Complex Number System

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

M.N.CN.A.1 Know there is a complex number i such that i² = -1, and every complex number has the form a + bi with a and b real. Understand why complex numbers exist.

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 + v3i²)3 = 8 because (-1 + v3i²) 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.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.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.

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.

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)

A-SSE Seeing Structure in Expressions

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.2 Use the structure of an expression to identify ways to rewrite it. For example, see x4-y4 as (x²)² - (y²)², thus recognizing it as a difference of squares that can be factored as (x²- y²)(x²+y²).

A.SSE.B Write expressions in equivalent forms to solve problems. (M)

M.A.SSE.B.3 Choose and produce an equivalent form of an expression to reveal and explain properties of the quantity represented by the expression.

M.A.SSE.B.3a Factor a quadratic expression to reveal the zeros of the function it defines.

M.A.SSE.B.3b Complete the square in a quadratic expression to reveal the maximum or minimum value of the function it defines.

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. For example, calculating mortgage payments or tracking the amount of an antibiotic in the human body when prescribed for an infection.

A-APR Arithmetic with Polynomials and Rational Expressions

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.

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.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.

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.APR.D Rewrite 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)

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.

M.A.CED.A.4 Rearrange formulas to highlight a quantity of interest, using the same reasoning as in solving equations. For example, rearrange the formula C = 5/9 (F-32) so you solve for F.

A-REI Reasoning with Equations and Inequalities

A.REI.A Understand solving equations as a process of reasoning and explain the reasoning.

M.A.REI.A.1 Explain each step in solving a simple equation as following from the equality of numbers asserted at the previous step, starting from the assumption that the original equation has a solution. Construct a viable argument to justify a solution method.

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.B Solve equations and inequalities in one variable.

M.A.REI.B.3 Solve linear equations and inequalities in one variable, including equations with coefficients represented by letters.

A.REI.C Solve systems of equations.

M.A.REI.C.5 Justify that, given a system of two equations in two variables, replacing one equation by the sum of that equation and a multiple of the other produces a system with the same solutions.

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 + v3i²)³ = 8 because (-1 + v3i²) has modulus 2 and argument 120°.

M.A.SSE.A.2 Use the structure of an expression to identify ways to rewrite it. For example, see x⁴-y⁴ as (x²)² - (y²)², thus recognizing it as a difference of squares that can be factored as (x²- y²)(x²+y²).

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.

M.F.LE.A.4 For exponential models, express as a logarithm the solution to abc^t = d where a, c, and d are numbers and the base b is 2,10, or e; evaluate the logarithm using technology.