Pre-Calculus

MAPR: Pre-Calculus

MAPR.A: Algebra

MAPR.B: Functions

MAPR.C: Geometry

MAPR.D: Number and Quantity

MAPR.E: Statistics and Probability

MAPR.A.1: solve a simple system consisting of a linear equation and a quadratic equation in two variables algebraically and graphically

MAPR.A.2: represent a system of linear equations as a single matrix equation in a vector variable

MAPR.A.3: find the inverse of a matrix if it exists and use it to solve systems of linear equations (using technology for matrices of dimension 3x3 or greater)

MAPR.B.4: using tables, graphs, and verbal descriptions, interpret the key characteristics of a function which models the relationship between two quantities; sketch a graph showing key features including: intercepts; interval where the function is increasing, decreasing, positive, or negative; relative maximums and minimums; symmetries; end behavior; and periodicity

MAPR.B.5: graph functions expressed algebraically and show key features of the graph both by hand and by using technology

MAPR.B.6: graph trigonometric functions, showing period, midline, and amplitude

MAPR.B.7: find inverse functions

MAPR.B.8: produce an invertible function from a non-invertible function by restricting the domain

MAPR.B.9: use special triangles to determine geometrically the values of sine, cosine, tangent for pi/3, pi/4 and pi/6, and use the unit circle to express the values of sine, cosine, and tangent for pi - x, pi + x, and 2pi - x in terms of their values for x, where x is any real number

MAPR.B.10: use the unit circle to explain symmetry (odd and even) and periodicity of trigonometric functions

MAPR.B.11: understand that restricting a trigonometric function to a domain on which it is always increasing or always decreasing allows its inverse to be constructed

MAPR.B.12: 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

MAPR.B.13: prove the addition and subtraction formulas for sine, cosine, and tangent and use them to solve problems

MAPR.B.14: understand radian measure of an angle as the length of the arc on the unit circle subtended by the angle

MAPR.B.15: explain how the unit circle in the coordinate plane enables the extension of trigonometric functions to all real numbers, interpreted as radian measures of angles traversed counterclockwise around the unit circle

MAPR.B.16: choose trigonometric functions to model periodic phenomena with specified amplitude, frequency, and midline

MAPR.B.17: prove the Pythagorean identity (sin A)² + (cos A)² = 1 and use it to find sin A, cos A, or tan A, given sin A, cos A, or tan A, and the quadrant of the angle

MAPR.C.18: derive the equation of a parabola given a focus and directrix.

MAPR.C.19: 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

MAPR.C.20: prove the Laws of Sines and Cosines and use them to solve problems

MAPR.C.21: 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)

MAPR.C.22: 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

MAPR.D.23: find the conjugate of a complex number; use conjugates to find moduli and quotients of complex numbers

MAPR.D.24: 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

MAPR.D.25: represent addition, subtraction, multiplication, and conjugation of complex numbers geometrically on the complex plane; use properties of this representation for computation

MAPR.D.26: 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

MAPR.D.27: 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

MAPR.D.28: find the components of a vector by subtracting the coordinates of an initial point from the coordinates of a terminal point

MAPR.D.29: solve problems involving velocity and other quantities that can be represented by vectors

MAPR.D.30: add and subtract vectors

MAPR.D.31: 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

MAPR.D.32: given two vectors in magnitude and direction form, determine the magnitude and direction of their sum

MAPR.D.33: 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

MAPR.D.34: multiply a vector by a scalar

MAPR.D.35: 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))

MAPR.D.36: compute the magnitude of a scalar multiple cv using \\cv\\ = \c\v; compute the direction of cv knowing that when \c\v not equal to 0, the direction of cv is either along v (for c > 0) or against v (for c < 0)

MAPR.D.37: 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

MAPR.D.38: use matrices to represent and manipulate data, (e.g., to represent payoffs or incidence relationships in a network)

MAPR.D.39: multiply matrices by scalars to produce new matrices, (e.g., as when all of the payoffs in a game are doubled)

MAPR.D.40: add, subtract, and multiply matrices of appropriate dimensions

MAPR.D.41: understand that, unlike multiplication of numbers, matrix multiplication for square matrices is not a commutative operation, but still satisfies the associative and distributive properties

MAPR.D.42: 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

MAPR.D.43: work with 2x2 matrices as transformations of the plane, and interpret the absolute value of the determinant in terms of area

MAPR.E.44: apply the general Multiplication Rule in a uniform probability model, P(A and B) = [P(A)]x[P(B\A)] =[P(B)]x[P(A\B)], and interpret the answer in terms of the model

MAPR.E.45: use permutations and combinations to compute probabilities of compound events and solve problems

MAPR.E.46: 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

MAPR.E.47: calculate the expected value of a random variable; interpret it as the mean of the probability distribution

MAPR.E.48: develop a probability distribution for a random variable defined for a sample space in which theoretical probabilities can be calculated; find the expected value (e.g., find the theoretical probability distribution for the number of correct answers obtained by guessing on all five questions of a multiple-choice test where each question has four choices, and find the expected grade under various grading schemes)

MAPR.E.49: develop a probability distribution for a random variable defined for a sample space in which probabilities are assigned empirically; find the expected value (e.g., find a current data distribution on the number of TV sets per household in the United States, and calculate the expected number of sets per household; how many TV sets would you expect to find in 100 randomly selected households?)

MAPR.E.50: weigh the possible outcomes of a decision by assigning probabilities to payoff values and finding expected values

MAPR.E.51: find the expected payoff for a game of chance (e.g., find the expected winnings from a state lottery ticket or a game at a fast-food restaurant)

MAPR.E.52: evaluate and compare strategies on the basis of expected values (e.g., compare a high-deductible versus a low-deductible automobile insurance policy using various, but reasonable, chances of having a minor or a major accident)

MAPR.E.53: use probabilities to make fair (equally likely) decisions (e.g., drawing by lots, using a random number generator)

MAPR.E.54: analyze decisions and strategies using probability concepts (e.g., product testing, medical testing, pulling a hockey goalie at the end of a game)