New Jersey Student Learning Standards (adopted in 2023): Precalculus
Skills available for New Jersey Precalculus standards
Section
Print skill plan
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.
Complex conjugates
Operations with complex numbers
Absolute value
N.CN.A.3 Find the conjugate of a complex number; use conjugates to find moduli and quotients of complex numbers.
Complex conjugates
Operations with complex numbers
Absolute value
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.
Complex numbers
Polar and rectangular forms
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.
Complex numbers
Polar and rectangular forms
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.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.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.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.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.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.4 Add and subtract vectors.
N.VM.B.4.a: 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.4.a: 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.4.b: Given two vectors in magnitude and direction form, determine the magnitude and direction of their sum.
N.VM.B.4.b: Given two vectors in magnitude and direction form, determine the magnitude and direction of their sum.
N.VM.B.4.c: 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.4.c: 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.5 Multiply a vector by a scalar.
N.VM.B.5.a: 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.5.a: 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.5.b: 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.B.5.b: 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.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.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.
Addition and subtraction
Multiplication
Mixed operations
N.VM.C.8 Add, subtract, and multiply matrices of appropriate dimensions.
Addition and subtraction
Multiplication
Mixed operations
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.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.
Determinant of a matrix
Inverse of a matrix
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.
Determinant of a matrix
Inverse of a matrix
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.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.
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.REI
Reasoning with Equations and Inequalities
A.REI.C Solve systems of equations.
A.REI.C.5 Prove 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.
A.REI.C.5 Prove 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.
A.REI.C.8 Represent a system of linear equations as a single matrix equation in a vector variable.
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).
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.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.1 Write a function that describes a relationship between two quantities.
F.BF.A.1.c: Compose functions.
F.BF.A.1.c: Compose functions.
F.BF.B Build new functions from existing functions.
F.BF.B.4 Find inverse functions.
F.BF.B.4 Find inverse functions.
F.BF.B.4.b: Verify by composition that one function is the inverse of another.
F.BF.B.4.b: Verify by composition that one function is the inverse of another.
F.BF.B.4.c: Read values of an inverse function from a graph or a table, given that the function has an inverse.
F.BF.B.4.c: Read values of an inverse function from a graph or a table, given that the function has an inverse.
F.BF.B.4.d: Produce an invertible function from a non-invertible function by restricting the domain.
F.BF.B.4.d: 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.
Convert between forms
Solve exponential equations
Solve logarithmic equations
F.BF.B.5 Use the inverse relationship between exponents and logarithms to solve problems involving logarithms and exponents.
Convert between forms
Solve exponential equations
Solve logarithmic equations
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.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.7.d: Graph rational functions, identifying zeros and asymptotes when suitable factorizations are available, and showing end behavior.
F.IF.C.7.d: Graph rational functions, identifying zeros and asymptotes when suitable factorizations are available, and showing end behavior.
F.IF.C.7.f: Graph trigonometric functions, showing period, midline, and amplitude.
F.IF.C.7.f: Graph trigonometric functions, showing period, midline, and amplitude.
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.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.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.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.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.8 Prove the Pythagorean identity sin2 (θ) + cos2θ = 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.
F.TF.C.8 Prove the Pythagorean identity sin2 (θ) + cos2θ = 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.
F.TF.C.9 Prove the addition and subtraction formulas for sine, cosine, and tangent and use them to solve problems.
F.TF.C.9 Prove the addition and subtraction formulas for sine, cosine, and tangent and use them to solve problems.
G.C
Circles
G.C.A Understand and apply theorems about circles.
G.C.A.4 Construct a tangent line from a point outside a given circle to the circle.
G.C.A.4 Construct a tangent line from a point outside a given circle to the circle.
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.
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.
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.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.
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.
G.GPE.A.2 Derive the equation of a parabola given a focus and directrix.
G.GPE.A.2 Derive the equation of a parabola given a focus and directrix.
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.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.GPE.B Use coordinates to prove simple geometric theorems algebraically.
G.GPE.B.6 Find the point on a directed line segment between two given points that partitions the segment in a given ratio.
G.GPE.B.6 Find the point on a directed line segment between two given points that partitions the segment in a given ratio.
G.SRT
Similarity, Right Triangles, and Trigonometry
G.SRT.D Apply trigonometry to general triangles.
G.SRT.D.9 Derive the formula A = ½ab sin(C) for the area of a triangle by drawing an auxiliary line from a vertex perpendicular to the opposite side.
G.SRT.D.9 Derive the formula A = ½ab sin(C) for the area of a triangle by drawing an auxiliary line from a vertex perpendicular to the opposite side.
G.SRT.D.10 Prove the Laws of Sines and Cosines and use them to solve problems.
G.SRT.D.10 Prove the Laws of Sines and Cosines and use them to solve problems.
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.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).
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.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.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.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.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.5 Weigh the possible outcomes of a decision by assigning probabilities to payoff values and finding expected values.
S.MD.B.5.a: Find the expected payoff for a game of chance.
S.MD.B.5.a: Find the expected payoff for a game of chance.
S.MD.B.5.b: Evaluate and compare strategies on the basis of expected values.
S.MD.B.5.b: Evaluate and compare strategies on the basis of expected values.