Accelerated Pre-calculus

MAPA: Accelerated Geometry

MAPA.A: Algebra

MAPA.B: Functions

MAPA.C: Geometry

MAPA.D: Number and Quantity

MAPA.E: Statistics and Probability

MAPA.A.1: use mathematical induction to find and prove formulae for sums of finite series

MAPA.A.2: describe parametric representations of plane curves

MAPA.A.3: convert between Cartesian and parametric form

MAPA.A.4: graph equations in parametric form showing direction and endpoints where appropriate

MAPA.A.5: express coordinates of points in rectangular and polar form

MAPA.A.6: graph and identify characteristics of simple polar equations including lines, circles, cardioids, limaçons and roses

MAPA.A.7: establish and utilize trigonometric identities to simplify expressions and verify equivalence statements (e.g., double angle, half angle, reciprocal, quotient, pythagorean, even, and odd)

MAPA.A.8: represent a system of linear equations as a single matrix equation in a vector variable

MAPA.A.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 3x3 or greater)

MAPA.A.10: solve a simple system consisting of a linear equation and a quadratic equation in two variables algebraically and graphically

MAPA.B.11: find inverse functions

MAPA.B.12: produce an invertible function from a non-invertible function by restricting the domain

MAPA.B.13: use special triangles to determine geometrically the values of sine, cosine, tangent, cosecant, secant, cotangent for pi/3, pi/4 and pi/6, and use the unit circle to express the values of sine, cosine, tangent, cosecant, secant, and cotangent for pi - x, pi + x, and 2pi - x in terms of their values for x, where x is any real number

MAPA.B.14: use the unit circle to explain symmetry (odd and even) and periodicity of trigonometric functions

MAPA.B.15: understand that restricting a trigonometric function to a domain on which it is always increasing or always decreasing allows its inverse to be constructed

MAPA.B.16: 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

MAPA.B.17: prove the addition, subtraction, and double angle formulas for sine, cosine, and tangent and use them to solve problems

MAPA.B.18: explore the continuity of functions of two independent variables in terms of the limits of such functions as (x,y) approaches a given point in the plane

MAPA.B.19: understand radian measure of an angle as the length of the arc on the unit circle subtended by the angle

MAPA.B.20: 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

MAPA.B.21: choose trigonometric functions to model periodic phenomena with specified amplitude, frequency, and midline

MAPA.B.22: 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

MAPA.B.23: 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

MAPA.B.24: graph functions expressed algebraically and show key features of the graph both by hand and by using technology

MAPA.B.25: graph trigonometric functions, showing period, midline, and amplitude

MAPA.C.26: 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

MAPA.C.27: prove the Laws of Sines and Cosines and use them to solve problems

MAPA.C.28: 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)

MAPA.C.29: 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

MAPA.C.30: derive the equation of a parabola given a focus and directrix

MAPA.D.31: 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

MAPA.D.32: work with 2x2 matrices as transformations of the plane, and interpret the absolute value of the determinant in terms of area

MAPA.D.33: use matrices to represent and manipulate data, (e.g., to represent payoffs or incidence relationships in a network)

MAPA.D.34: multiply matrices by scalars to produce new matrices, (e.g., as when all of the payoffs in a game are doubled)

MAPA.D.35: add, subtract, and multiply matrices of appropriate dimensions

MAPA.D.36: understand that, unlike multiplication of numbers, matrix multiplication for square matrices is not a commutative operation, but still satisfies the associative and distributive properties

MAPA.D.37: find the conjugate of a complex number; use conjugates to find moduli and quotients of complex numbers

MAPA.D.38: 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

MAPA.D.39: represent addition, subtraction, multiplication, and conjugation of complex numbers geometrically on the complex plane; use properties of this representation for computation

MAPA.D.40: 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

MAPA.D.41: 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)

MAPA.D.42: find the components of a vector by subtracting the coordinates of an initial point from the coordinates of a terminal point

MAPA.D.43: solve problems involving velocity and other quantities that can be represented by vectors

MAPA.D.44: add and subtract vectors

MAPA.D.45: 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

MAPA.D.46: given two vectors in magnitude and direction form, determine the magnitude and direction of their sum

MAPA.D.47: 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

MAPA.D.48: multiply a vector by a scalar

MAPA.D.49: 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))

MAPA.D.50: 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)

MAPA.D.51: 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

MAPA.E.52: 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

MAPA.E.53: use permutations and combinations to compute probabilities of compound events and solve problems

MAPA.E.54: 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

MAPA.E.55: calculate the expected value of a random variable; interpret it as the mean of the probability distribution

MAPA.E.56: 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)

MAPA.E.57: 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?)

MAPA.E.58: weigh the possible outcomes of a decision by assigning probabilities to payoff values and finding expected values

MAPA.E.59: 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)

MAPA.E.60: 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)

MAPA.E.61: use probabilities to make fair decisions (e.g., drawing by lots, using a random number generator)

MAPA.E.62: analyze decisions and strategies using probability concepts (e.g., product testing, medical testing, pulling a hockey goalie at the end of a game)

MAPA.E.63: use statistics appropriate to the shape of the data distribution to compare center (median, mean) and spread(interquartile, range, standard deviation) of two or more different data sets

MAPA.E.64: use the mean and standard deviation of a data set to fit it to a normal distribution and to estimate population percentages; recognize that there are data sets for which such a procedure is not appropriate; use calculators, spreadsheets, and tables to estimate areas under the normal curve

MAPA.E.65: understand statistics as a process for making inferences about population parameters based on a random sample from that population

MAPA.E.66: decide if a specified model is consistent with results from a given data-generating process, (e.g., using simulation; for example, a model says a spinning coin falls heads up with probability 0.5; would a result of 5 tails in a row cause you to question the model?)

MAPA.E.67: recognize the purposes of and differences among sample surveys, experiments, and observational studies; explain how randomization relates to each

MAPA.E.68: use data from a sample survey to estimate a population mean or proportion; develop a margin of error through the use of simulation models for random sampling

MAPA.E.69: use data from a randomized experiment to compare two treatments; use simulations to decide if differences between parameters are significant

MAPA.E.70: evaluate reports based on data