Massachusetts flagMassachusetts: Model Precalculus [PC] Math Standards

42 standards · 11 domains

ALGEBRA - ARITHMETIC WITH POLYNOMIALS AND RATIONAL EXPRESSIONS

  • PC.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.
  • PC.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.

ALGEBRA - REASONING WITH EQUATIONS AND INEQUALITIES

  • PC.A-REI.C.8 (+) Represent a system of linear equations as a single matrix equation in a vector variable.
  • PC.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 x 3 or greater).

FUNCTIONS - BUILDING FUNCTIONS

  • PC.F-BF.A.1.c (+) Compose functions. For example, if T(y) is the temperature in the atmosphere as a function of height, and h(t) is the height of a weather balloon as a function of time, then T(h(t)) is the temperature at the location of the weather balloon as a function of time.
  • PC.F-BF.B.4.b (+) Verify by composition that one function is the inverse of another.
  • PC.F-BF.B.4.c (+) Read values of an inverse function from a graph or a table, given that the function has an inverse.
  • PC.F-BF.B.4.d (+) Produce an invertible function from a non-invertible function by restricting the domain.
  • PC.F-BF.B.5 (+) Understand the inverse relationship between exponents and logarithms and use this relationship to solve problems involving logarithms and exponents.

FUNCTIONS - INTERPRETING FUNCTIONS

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

FUNCTIONS - TRIGONOMETRIC FUNCTIONS

  • PC.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.
  • PC.F-TF.A.4 (+) Use the unit circle to explain symmetry (odd and even) and periodicity of trigonometric functions.
  • PC.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.
  • PC.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.
  • PC.F-TF.C.9 (+) Prove the addition and subtraction formulas for sine, cosine, and tangent and use them to solve problems.

GEOMETRY - CIRCLES

  • PC.G-C.A.4 (+) Construct a tangent line from a point outside a given circle to the circle.

GEOMETRY - GEOMETRIC MEASUREMENT AND DIMENSION

  • PC.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.

GEOMETRY - EXPRESSING GEOMETRIC PROPERTIES WITH EQUATIONS

  • PC.G-GPE.A.3.a (+) Use equations and graphs of conic sections to model real-world problems.

GEOMETRY - SIMILARITY, RIGHT TRIANGLES, AND TRIGONOMETRY

  • PC.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.
  • PC.G-SRT.D.10 (+) Prove the Laws of Sines and Cosines and use them to solve problems.
  • PC.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).

NUMBER AND QUANTITY - THE COMPLEX NUMBER SYSTEM

  • PC.N-CN.A.3 (+) Find the conjugate of a complex number; use conjugates to find moduli and quotients of complex numbers.
  • PC.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.
  • PC.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+√(3i))^3=8 because (-1+√(3i)) has modulus 2 and argument 120°.
  • PC.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.
  • PC.N-CN.C.8 (+) Extend polynomial identities to the complex numbers. For example, rewrite x^2 + 4 as (x + 2i)(x – 2i).
  • PC.N-CN.C.9 (+) Know the Fundamental Theorem of Algebra; show that it is true for quadratic polynomials.

NUMBER AND QUANTITY - VECTOR AND MATRIX QUANTITIES

  • PC.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).
  • PC.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.
  • PC.N-VM.A.3 (+) Solve problems involving velocity and other quantities that can be represented by vectors.
  • PC.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.
  • PC.N-VM.B.4.b (+) Given two vectors in magnitude and direction form, determine the magnitude and direction of their sum.
  • PC.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.
  • PC.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_y) = (cv_x, cv_y).
  • PC.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).
  • PC.N-VM.C.6 (+) Use matrices to represent and manipulate data, e.g., to represent payoffs or incidence relationships in a network.
  • PC.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.
  • PC.N-VM.C.8 (+) Add, subtract, and multiply matrices of appropriate dimensions.
  • PC.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.
  • PC.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.
  • PC.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.
  • PC.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.