Massachusetts Curriculum Frameworks: Precalculus
Skills available for Massachusetts Precalculus standards
Domain
Print skill plan
N
Number and Quantity
N-CN The Complex Number System
N-CN.A Perform arithmetic operations with complex numbers.
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 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-CN.C Use complex numbers in polynomial identities and equations.
N-CN.C Use complex numbers in polynomial identities and equations.
N-CN.C.8: Extend polynomial identities to the complex numbers.
N-CN.C.8: Extend polynomial identities to the complex numbers.
N-CN.C.9: Know the Fundamental Theorem of Algebra; show that it is true for quadratic polynomials.
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.
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 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, 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, 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, v subscript y) = (cvx, cv subscript y).
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, v subscript y) = (cvx, cv subscript y).
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 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 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 transformations of the plane, and interpret the absolute value of the determinant in terms of area.
A
Algebra
A-APR Arithmetic with Polynomials and Rational Expressions
A-APR.C Use polynomial identities to solve problems.
A-APR.C Use polynomial identities to solve problems.
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.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.
A-APR.D Rewrite rational expressions.
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-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-REI Reasoning with Equations and Inequalities
A-REI.C Solve systems of equations.
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.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
Functions
F-IF Interpreting Functions
F-IF.C Analyze functions using different representations.
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-BF Building Functions
F-BF.A Build a function that models a relationship between two quantities.
F-BF.A Build a function that models a relationship between two quantities.
F-BF.A.1: Write a function (linear, quadratic, exponential, simple rational, radical, logarithmic, and trigonometric) that describes a relationship between two quantities.
F-BF.A.1: Write a function (linear, quadratic, exponential, simple rational, radical, logarithmic, and trigonometric) 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 Build new functions from existing functions.
F-BF.B.4: Find inverse functions algebraically and graphically.
F-BF.B.4: Find inverse functions algebraically and graphically.
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: Understand the inverse relationship between exponents and logarithms and use this relationship to solve problems involving logarithms and exponents.
Convert between forms
Solve exponential equations
Solve logarithmic equations
F-BF.B.5: Understand the inverse relationship between exponents and logarithms and use this relationship to solve problems involving logarithms and exponents.
Convert between forms
Solve exponential equations
Solve logarithmic equations
F-TF Trigonometric Functions
F-TF.A Extend the domain of trigonometric functions using the unit circle.
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, cosine, 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, cosine, 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 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 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.
F-TF.C.9: Prove the addition and subtraction formulas for sine, cosine, and tangent and use them to solve problems.
G
Geometry
G-SRT Similarity, Right Triangles, and Trigonometry
G-SRT.D Apply trigonometry to general triangles.
G-SRT.D Apply trigonometry to general triangles.
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.
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.
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).
G-C Circles
G-C.A Understand and apply theorems about 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-GPE Expressing Geometric Properties with Equations
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.
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.A.3.a: Use equations and graphs of conic sections to model real-world problems.
G-GPE.A.3.a: Use equations and graphs of conic sections to model real-world problems.
G-GMD Geometric Measurement and Dimension
G-GMD.A Explain volume formulas and use them to solve problems.
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.