Massachusetts Precalculus Math Standards

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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-CN.C Use complex numbers in polynomial identities and equations.
    • 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-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.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.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.a Represent scalar multiplication graphically by scaling vectors and possibly reversing their direction; perform scalar multiplication component-wise, e.g., as c(v_x, v subscript y) = (cv_x, 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.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 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.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.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.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.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.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.B Build new functions from existing functions.
    • 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.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.5 Understand the inverse relationship between exponents and logarithms and use this relationship 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, 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.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-SRT Similarity, Right Triangles, and Trigonometry
  • 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.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-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-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-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.2 Give an informal argument using Cavalieri’s principle for the formulas for the volume of a sphere and other solid figures.