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Skills available for New Jersey high school math standards

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Student Learning Standards: Algebra 1 Student Learning Standards: Algebra 1
Student Learning Standards: Algebra 2 Student Learning Standards: Algebra 2
Student Learning Standards: All High School Pathways Student Learning Standards: All High School Pathways
Student Learning Standards: Geometry Student Learning Standards: Geometry
Student Learning Standards: Mathematics I Student Learning Standards: Mathematics I
Student Learning Standards: Mathematics II Student Learning Standards: Mathematics II
Student Learning Standards: Mathematics III Student Learning Standards: Mathematics III
Student Learning Standards: Precalculus Student Learning Standards: Precalculus

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N Number and Quantity

N-CN The Complex Number System
- N-CN.A Perform arithmetic operations with complex numbers.
  - N-CN.A.3 Find the conjugate of a complex number; use conjugates to find moduli and quotients of complex numbers.
- N-CN.B Represent complex numbers and their operations on the complex plane.
  - 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.
  - 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.
  - 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-VM Vector and Matrix Quantities
- N-VM.A Represent and model with vector quantities.
  - 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).
  - 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-U.2)
    - Find the component form of a three-dimensional vector (PC-V.2)
  - N-VM.A.3 Solve problems involving velocity and other quantities that can be represented by vectors.
- N-VM.B Perform operations on vectors.
  - N-VM.B.4 Add and subtract vectors.
    - 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.
    - 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)
    - 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.
  - N-VM.B.5 Multiply a vector by a scalar.
    - 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)
    - 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.
  - N-VM.C.6 Use matrices to represent and manipulate data, e.g., to represent payoffs or incidence relationships in a network.
  - 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-L.4)
  - N-VM.C.8 Add, subtract, and multiply matrices of appropriate dimensions.
  - 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-L.8)
  - 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.
  - 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.
  - N-VM.C.12 Work with 2 × 2 matrices as a transformations of the plane, and interpret the absolute value of the determinant in terms of area.

A Algebra

A-REI Reasoning with Equations and Inequalities
- A-REI.C Solve systems of equations
  - 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-I.8)
    - Solve a system of equations using augmented matrices: word problems (PC-I.9)
  - 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 Functions

F-IF Interpreting Functions
- F-IF.C Analyze functions using different representations
  - F-IF.C.7 Graph functions expressed symbolically and show key features of the graph, by hand in simple cases and using technology for more complicated cases.
    - F-IF.C.7d Graph rational functions, identifying zeros and asymptotes when suitable factorizations are available, and showing end behavior.
      - Rational functions: asymptotes and excluded values (PC-E.1)
F-BF Building Functions
- F-BF.A Build a function that models a relationship between two quantities
  - F-BF.A.1 Write a function that describes a relationship between two quantities.
    - F-BF.A.1c Compose functions.
      - Composition of functions (PC-A.11)
- F-BF.B Build new functions from existing functions
  - F-BF.B.4 Find inverse functions.
    - F-BF.B.4b Verify by composition that one function is the inverse of another.
      - Identify inverse functions (PC-A.12)
      - Check whether two rational functions are inverses (PC-E.3)
    - F-BF.B.4c Read values of an inverse function from a graph or a table, given that the function has an inverse.
      - Find values of inverse functions from tables (PC-A.13)
      - Find values of inverse functions from graphs (PC-A.14)
    - F-BF.B.4d Produce an invertible function from a non-invertible function by restricting the domain.
  - F-BF.B.5 Use the inverse relationship between exponents and logarithms to solve problems involving logarithms and exponents.
F-TF Trigonometric Functions
- F-TF.A Extend the domain of trigonometric functions using the unit circle
  - F-TF.A.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, cosines, 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-M.5)
    - Find trigonometric ratios of special angles (PC-M.7)
  - F-TF.A.4 Use the unit circle to explain symmetry (odd and even) and periodicity of trigonometric functions.
    - Symmetry and periodicity of trigonometric functions (PC-O.2)
- F-TF.B Model periodic phenomena with trigonometric functions
  - F-TF.B.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-M.9)
    - Inverses of trigonometric functions using a calculator (PC-M.10)
  - F-TF.B.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-M.11)
- F-TF.C Prove and apply trigonometric identities
  - F-TF.C.9 Prove the addition and subtraction formulas for sine, cosine, and tangent and use them to solve problems.

G Geometry

G-GPE Expressing Geometric Properties with Equations
- G-GPE.A Translate between the geometric description and the equation for a conic section
  - 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-P.9)
    - Write equations of hyperbolas in standard form (PC-P.12)
G-GMD Geometric Measurement and Dimension
- G-GMD.A Explain volume formulas and use them to solve problems
  - G-GMD.A.2 Give an informal argument using Cavalieri's principle for the formulas for the volume of a sphere and other solid figures.

S Statistics and Probability

S-MD Using Probability to Make Decisions
- S-MD.A Calculate expected values and use them to solve problems
  - S-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-Y.2)
    - Graph a discrete probability distribution (PC-Y.3)
  - S-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-Y.4)
  - S-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.
    - Write the probability distribution for a game of chance (PC-Y.7)
    - Expected values for a game of chance (PC-Y.8)
  - S-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.
- S-MD.B Use probability to evaluate outcomes of decisions
  - S-MD.B.5 Weigh the possible outcomes of a decision by assigning probabilities to payoff values and finding expected values.
    - S-MD.B.5a Find the expected payoff for a game of chance.
      - Expected values for a game of chance (PC-Y.8)
    - S-MD.B.5b Evaluate and compare strategies on the basis of expected values.
      - Choose the better bet (PC-Y.9)

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N Number and Quantity

N-CN The Complex Number System

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

N-CN.A.3 Find the conjugate of a complex number; use conjugates to find moduli and quotients of complex numbers.

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

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.

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.

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-VM Vector and Matrix Quantities

N-VM.A Represent and model with vector quantities.

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).

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.

N-VM.A.3 Solve problems involving velocity and other quantities that can be represented by vectors.

N-VM.B Perform operations on vectors.

N-VM.B.4 Add and subtract vectors.

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.

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

N-VM.B.5 Multiply a vector by a scalar.

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).

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.

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

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.

N-VM.C.8 Add, subtract, and multiply matrices of appropriate dimensions.

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.

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.

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.

N-VM.C.12 Work with 2 × 2 matrices as a transformations of the plane, and interpret the absolute value of the determinant in terms of area.

A Algebra

A-REI Reasoning with Equations and Inequalities

A-REI.C Solve systems of equations

A-REI.C.8 Represent a system of linear equations as a single matrix equation in a vector variable.

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 Functions

F-IF Interpreting Functions

F-IF.C Analyze functions using different representations

F-IF.C.7 Graph functions expressed symbolically and show key features of the graph, by hand in simple cases and using technology for more complicated cases.

F-IF.C.7d Graph rational functions, identifying zeros and asymptotes when suitable factorizations are available, and showing end behavior.

F-BF Building Functions

F-BF.A Build a function that models a relationship between two quantities

F-BF.A.1 Write a function that describes a relationship between two quantities.

F-BF.A.1c Compose functions.

F-BF.B Build new functions from existing functions

F-BF.B.4 Find inverse functions.

F-BF.B.4b Verify by composition that one function is the inverse of another.

F-BF.B.4c Read values of an inverse function from a graph or a table, given that the function has an inverse.

F-BF.B.4d Produce an invertible function from a non-invertible function by restricting the domain.

F-BF.B.5 Use the inverse relationship between exponents and logarithms to solve problems involving logarithms and exponents.

F-TF Trigonometric Functions

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

F-TF.A.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, cosines, and tangent for π–x, π+x, and 2π–x in terms of their values for x, where x is any real number.

F-TF.A.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

F-TF.B.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.

F-TF.B.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

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

G Geometry

G-GPE Expressing Geometric Properties with Equations

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

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.

G-GMD Geometric Measurement and Dimension

G-GMD.A Explain volume formulas and use them to solve problems

G-GMD.A.2 Give an informal argument using Cavalieri's principle for the formulas for the volume of a sphere and other solid figures.

S Statistics and Probability

S-MD Using Probability to Make Decisions

S-MD.A Calculate expected values and use them to solve problems

S-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.

S-MD.A.2 Calculate the expected value of a random variable; interpret it as the mean of the probability distribution.

S-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.

S-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.

S-MD.B Use probability to evaluate outcomes of decisions

S-MD.B.5 Weigh the possible outcomes of a decision by assigning probabilities to payoff values and finding expected values.

S-MD.B.5a Find the expected payoff for a game of chance.

S-MD.B.5b Evaluate and compare strategies on the basis of expected values.