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BCSS Mathematics Pacing Guide 2012 - 2013 Precalculus Domain Interpreting Functions Content Standard # and Identifier 18c Content Standard Description Graph rational functions, identifying zeros and asymptotes when suitable factorizations are available, and showing end behavior. AHSGE 1st 9 Weeks ACT/Quality Core Standards Description Standards for Mathematical Practice 5, 6 Quality Core Prerequisites Skill Textbook Resources McGraw Hill Precalculus 1.2, 1.3 [F-IF7d] Limits 4 Interpreting Functions 16 Interpreting Functions 17 1|Page Determine numerically, algebraically, and graphically the limits of functions at specific values and at infinity. For a function that models a relationship between two quantities, interpret key features of graphs and tables in terms of the quantities, and sketch graphs showing key features given a verbal description of the relationship. (Key features include intercepts; intervals where the function is increasing, decreasing, positive, or negative; relative maximums and minimums; symmetries; end behavior; and periodicity.)* [F-IF4] Calculate and interpret the average rate of change of a function (presented symbolically or as a table) over a specified interval. Estimate the rate of change from a graph.* [F-IF6] 1.3 2, 4, 5, 6, 7, 8 1.4 2, 4, 5 1.4 BCSS Mathematics Pacing Guide 2012 - 2013 Precalculus Domain Interpreting Functions Content Standard # and Identifier 18, 18a Content Standard Description Graph functions expressed symbolically, and show key features of the graph, by hand in simple cases and using technology for more complicated cases.* [F-IF7] AHSGE 1st 9 Weeks ACT/Quality Core Standards Description Standards for Mathematical Practice 5, 6 Quality Core Prerequisites Skill Textbook Resources McGraw Hill Precalculus 1.2, 1.3, 1.4, 1.5 a. Graph square root, cube root, and piecewise-defined functions, including step functions and absolute value functions. [F-IF7b] Building Functions 19 Building Functions 20 Determine the inverse of a function and a relation. Building Functions 21 Verify by composition that one function is the inverse of another. Building Functions 22 Building Functions 23 Interpreting Functions 18c 2|Page Compose functions. [F-BF1c] 1, 2, 3, 4, 5, 6, 7, 8 1.6 1.7 2, 4, 5, 7 1.7 Read values of an inverse function from a graph or a table, given that the function has an inverse. [F-BF4c] 2, 4, 5, 7 1.7 Produce an invertible function from a non-invertible function by restricting the domain. [F-BF4d] Graph rational functions, identifying zeros and asymptotes when suitable factorizations are available, and showing end behavior. [F-IF7d] 2, 4, 5, 7 1.7 [F-BF4b] 5, 6 2.1, 2.2, 2.4, 2.5 (examples 1-4) BCSS Mathematics Pacing Guide 2012 - 2013 Precalculus Domain Interpreting Functions Content Standard # and Identifier 18b Content Standard Description Graph polynomial functions, identifying zeros when suitable factorizations are available, and showing end behavior. [FIF7c] Building Functions 25 Compare effects of parameter changes on graphs of transcendental functions. Interpreting Functions 18d Graph exponential and logarithmic functions, showing intercepts and end behavior, and trigonometric functions, showing period, midline, and amplitude. [F-IF7e] Building Functions 24 Understand the inverse relationship between exponents and logarithms, and use this relationship to solve problems involving logarithms and exponents. [F- BF5] 3|Page AHSGE 1st 9 Weeks ACT/Quality Core Standards Description Standards for Mathematical Practice 5, 6 Quality Core Prerequisites Skill Textbook Resources McGraw Hill Precalculus 2.2 3.1 (examples 1-3) 5, 6 3.1 (examples 1-3), 3.2 3.2, 3.3, 3.4 BCSS Mathematics Pacing Guide 2012 - 2013 Precalculus Domain Trigonometric Functions Content Standard # and Identifier 29 Content Standard Description AHSGE 2nd 9 Weeks ACT/Quality Core Standards Description Standards for Mathematical Practice Quality Core Prerequisites Skill Textbook Resources McGraw Hill Precalculus 4.3 Use special triangles to determine geometrically the values of sine, cosine, and tangent for 3π, 4π, and 6π, 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 of x, where x is any real number. [F-TF3] Trigonometric Functions 30 Interpreting Functions 18d Trigonometric Functions 26 Trigonometric Functions 31 Trigonometric Functions 32 4|Page Use the unit circle to explain symmetry (odd and even) and periodicity of trigonometric functions. [F-TF4] Graph exponential and logarithmic functions, showing intercepts and end behavior, and trigonometric functions, showing period, midline, and amplitude. [F-IF7e] Determine the amplitude, period, phase shift, domain, and range of trigonometric functions and their inverses. 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-TF6] 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-TF7] 4.3 5, 6 4.4, 4.5 (examples 1-4) 4.4, 4.5 (ex. 1-4), 4.6 (ex. 1-5) 4.6 (ex. 1-5) 4.3-4.6 BCSS Mathematics Pacing Guide 2012 - 2013 Precalculus Domain Similarity, Right Triangles, and Trigonometry Content Standard # and Identifier 35 Content Standard Description (+) 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. (Apply formulas previously derived in Geometry.) [G-SRT9] Prove the Pythagorean identity sin2(θ) + cos2(θ) = 1, and use it to find sin(θ), cos(θ), or tan(θ) given sin(θ), cos(θ), or tan(θ) and the quadrant of the angle. [FTF8] Prove the addition and subtraction formulas for sine, cosine, and tangent, and use them to solve problems. [F-TF9] AHSGE 2nd 9 Weeks ACT/Quality Core Standards Description Standards for Mathematical Practice Quality Core Prerequisites Skill Textbook Resources McGraw Hill Precalculus 4.7 example 8 Trigonometric Functions 33 Trigonometric Functions 34 Trigonometric Functions 27 Use the sum, difference, and half-angle identities to find the exact value of a trigonometric function. 5.4, 5.5 (example 5) Vector and Matrix Quantities 11 (+) Work with 2 x 2 matrices as transformations of the plane, and interpret the absolute value of the determinant in terms of area. [N-VM12] 6.2 (page 384 #45-48), page 387, page 393 # 37 Reasoning with Equations and Inequalities 14 (+) Represent a system of linear equations as a single matrix equation in a vector variable. [A-REI8] 6.3, Enrichment: ex. 3 & 4 5|Page 3, 7, 8 5.1 5.4 BCSS Mathematics Pacing Guide 2012 - 2013 Precalculus Domain Conic Sections Content Standard # and Identifier 15, 15a Content Standard Description AHSGE 3rd 9 Weeks ACT/Quality Core Standards Description Standards for Mathematical Practice Quality Core Prerequisites Skill Textbook Resources McGraw Hill Precalculus 7.1 (skip ex. 5), 7.2, 7.3 Create graphs of conic sections, including parabolas, hyperbolas, ellipses, circles, and degenerate conics from second degree equations. Ex. Graph x2 – 6x + y2 – 12y + 41 = 0 or y2 – 4x + 2y + 5 = 0. a.Formulate equations of conic sections from their determining characteristics. Ex. Write the equation of an ellipse with center (5, -3), a horizontal major axis of length 10, and a minor axis of length 4. Expressing Geometric Properties with Equations Expressing Geometric Properties with Equations 6|Page 36 Derive the equations of a parabola given a focus and a directrix. 2, 3, 7, 8 7.1 (skip ex. 5) [G-GPE2] 37 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-GPE3] 7.2, 7.3 BCSS Mathematics Pacing Guide 2012 - 2013 Precalculus Domain Trigonometric Functions Content Standard # and Identifier 28, 28a, 28b Content Standard Description Utilize parametric equations by graphing and by converting to rectangular form. AHSGE 3rd 9 Weeks ACT/Quality Core Standards Description Standards for Mathematical Practice Quality Core Prerequisites Skill Textbook Resources McGraw Hill Precalculus 7.5 a. Solve application-based problems involving parametric equations. b. Solve applied problems that include sequences with recurrence relations. Vector and Matrix Operations 5 Vector and Matrix Operations 6 Vector and Matrix Operations 7 7|Page 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-VM1] Find the components of a vector by subtracting the coordinates of an initial point from the coordinates of a terminal point. [N-VM2] Solve problems involving velocity and other quantities that can be represented by vectors. [N-VM3] 8.1, 8.2 8.1, 8.2 8.1, 8.2 BCSS Mathematics Pacing Guide 2012 - 2013 Precalculus Domain Vector and Matrix Operations Content Standard # and Identifier 8, 8a, 8b, 8c Content Standard Description (+) Add and subtract vectors. [N-VM4] 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-VM4a] b. (+) Given two vectors in magnitude and direction form, determine the magnitude and direction of their sum. [NVM4b] 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. [N-VM4c] 8|Page AHSGE 3rd 9 Weeks ACT/Quality Core Standards Description Standards for Mathematical Practice Quality Core Prerequisites Skill Textbook Resources McGraw Hill Precalculus 8.1, 8.2 BCSS Mathematics Pacing Guide 2012 - 2013 Precalculus Domain Vector and Matrix Operations Content Standard # and Identifier 9, 9a, 9b Content Standard Description (+) Multiply a vector by a scalar. [NVM5] AHSGE 3rd 9 Weeks ACT/Quality Core Standards Description Standards for Mathematical Practice Quality Core Prerequisites Skill Textbook Resources McGraw Hill Precalculus 8.1 a. (+) Represent scalar multiplication graphically by scaling vectors and possibly reversing their direction; perform scalar multiplication componentwise, e.g., as c(vx, vy) = (cvx, cvy). [NVM5a] 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). [NVM5b] Vector and Matrix Operations 10 The Complex Number System 1 9|Page 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-VM11] 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-CN4] Page 517 9.5 (ex. 1-6) BCSS Mathematics Pacing Guide 2012 - 2013 Precalculus Domain The Complex Number System Content Standard # and Identifier 2 Content Standard Description AHSGE 3rd 9 Weeks ACT/Quality Core Standards Description Standards for Mathematical Practice Quality Core Prerequisites Skill Textbook Resources McGraw Hill Precalculus Pages P6-P7 (examples 1-3), 9.5 (ex. 1-6) Represent addition, subtraction, multiplication, and conjugation of complex numbers geometrically on the complex plane; use properties of this representation for computation. [N-CN5] The Complex Number System 3 Seeing Structure in Expressions 12 Ex. (-1 + i)3 = 8 because (-1 + i) has modulus 2 and argument 120°. 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. 9.5 (ex. 1-6) [N-CN6] Derive the formula for the sum of a finite geometric series (when the common ratio is not 1), and use the formula to solve problems.* [A-SSE4] 3, 4, 7, 8 10.3 (examples 6-8) Example: Calculate mortgage payments. Arithmetic with Polynomials and Rational Expressions 10 | P a g e 13 (+) 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. (The Binomial Theorem can be proved by mathematical induction or by a combinatorial argument.) [A-APR5] 10.5 BCSS Mathematics Pacing Guide 2012 - 2013 Precalculus Domain Limits Content Standard # and Identifier 4, 4a Content Standard Description AHSGE 4th 9 Weeks ACT/Quality Core Standards Description Standards for Mathematical Practice Quality Core Prerequisites Skill Textbook Resources McGraw Hill Precalculus Review 1.3; 12.1, 12.2 (skip 12.2 ex. 7) Determine numerically, algebraically, and graphically the limits of functions at specific values and at infinity. a. Apply limits in problems involving convergence and divergence. Expressing Geometric Properties with Equations 38 Interpreting Categorical and Quantitative Data Interpreting Categorical and Quantitative Data 39 11 | P a g e 40 Give an informal argument using Cavalieri’s principle for the formulas for the volume of a sphere and other solid figures. [G-GMD2] 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. (Focus on increasing rigor using standard deviation). [S-ID2] Interpret differences in shape, center, and spread in the context of the data sets, accounting for possible effects of extreme data points (outliers). (Identify unifrom, skewed, and normal distridutions in a set of data. Determine the quartiles and interquartile range for a set of data.) [SID3] http://www.cut-theknot.org/Curriculum/Calculus/Cavalieri.shtml http://www.answers.com/topic/method-ofindivisibles 1, 2, 3, 4, 5, 7 11.1, 11.2 1, 2, 3, 4, 5, 7 11.1 BCSS Mathematics Pacing Guide 2012 - 2013 Precalculus Domain Interpreting Categorical and Quantitative Data Content Standard # and Identifier 41 Content Standard Description 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. [S-ID4] AHSGE 4th 9 Weeks ACT/Quality Core Standards Description Standards for Mathematical Practice 1, 2, 3, 4, 5, 7 Quality Core Prerequisites Skill Textbook Resources McGraw Hill Precalculus 11.2, 11.3 Using Probability to Make Decisions 50 (+) 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. [S-MD1] 11.2 Using Probability to Make Decisions 51 (+) Calculate the expected value of a random variable; interpret it as the mean of the probability distribution. [S-MD2] 11.2 12 | P a g e BCSS Mathematics Pacing Guide 2012 - 2013 Precalculus Domain Using Probability to Make Decisions Content Standard # and Identifier 54, 54a, 54b Content Standard Description (+) Weigh the possible outcomes of a decision by assigning probabilities to payoff values and finding expected values. [S-MD5] a. Find the expected payoff for a game of chance. [S-MD5a] Examples: Find the expected winnings from a state lottery ticket or a game at a fast-food restaurant. b. Evaluate and compare strategies on the basis of expected values. [S-MD5b] Example: Compare a high-deductible versus a low-deductible automobile insurance policy using various, but reasonable, chances of having a minor or a major accident. 13 | P a g e AHSGE 4th 9 Weeks ACT/Quality Core Standards Description Standards for Mathematical Practice Quality Core Prerequisites Skill Textbook Resources McGraw Hill Precalculus 11.2 BCSS Mathematics Pacing Guide 2012 - 2013 Precalculus Domain Using Probability to Make Decisions Content Standard # and Identifier 52 Content Standard Description AHSGE 4th 9 Weeks ACT/Quality Core Standards Description Standards for Mathematical Practice Quality Core Prerequisites Skill Textbook Resources McGraw Hill Precalculus 11.3 (+) Develop a probability distribution for a random variable defined for a sample space in which theoretical probabilities can be calculated; find the expected value. [S-MD3] Example: 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. Using Probability to Make Decisions 53 11.3 (+) Develop a probability distribution for a random variable defined for a sample space in which probabilities are assigned empirically; find the expected value. [SMD4] Example: Find a current data distribution on the number of television sets per household in the United States, and calculate the expected number of sets per household. How many television sets would you expect to find in 100 randomly selected households? Making Inferences and Justifying Conclusions 14 | P a g e 44 Understand statistics as a process for making inferences about population parameters based on a random sample from that population. [S-IC1] 2, 4, 6 11.4 BCSS Mathematics Pacing Guide 2012 - 2013 Precalculus Domain Making Inferences and Justifying Conclusions Making Inferences and Justifying Conclusions Making Inferences and Justifying Conclusions Making Inferences and Justifying Conclusions Making Inferences and Justifying Conclusions 15 | P a g e Content Standard # and Identifier 45 Content Standard Description Decide if a specified model is consistent with results from a given data-generating process, e.g., using simulation. [S-IC2] AHSGE 4th 9 Weeks ACT/Quality Core Standards Description Standards for Mathematical Practice 1, 2, 3, 4, 5, 6, 7, 8 Quality Core Prerequisites Skill Textbook Resources McGraw Hill Precalculus 11.4, 11.5 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? 46 Recognize the purposes of and differences among sample surveys, experiments, and observational studies; explain how randomization relates to each. [S-IC3] 2, 3, 4, 6 11.4 47 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. [S-IC4] 1, 3, 4, 5, 6 11.4 48 Use data from a randomized experiment to compare two treatments; use simulations to decide if differences between parameters are significant. [S-IC5] 1, 3, 4, 5, 6, 8 11.6 49 Evaluate reports based on data. [S-IC6] 1, 2, 3, 4, 5, 6, 7, 8 11.1-11.7 BCSS Mathematics Pacing Guide 2012 - 2013 Precalculus Domain Interpreting Categorical and Quantitative Data Interpreting Categorical and Quantitative Data 16 | P a g e Content Standard # and Identifier 42 43 Content Standard Description Compute (using technology) and interpret the correlation coefficient of a linear fit. [S-ID8] Distinguish between correlation and causation. [S-ID9] AHSGE 4th 9 Weeks ACT/Quality Core Standards Description Standards for Mathematical Practice 4, 5, 6 3, 4, 6 Quality Core Prerequisites Skill Textbook Resources McGraw Hill Precalculus 11.7 11.7 BCSS Mathematics Pacing Guide 2012 - 2013 Precalculus Essential Vocabulary 1st 9 Weeks Chapter 1: zeros, roots, line symmetry, point symmetry, even function, odd function, continuous function, limit, discontinuous function, infinite discontinuity, jump discontinuity, removable discontinuity, nonremovable discontinuity, end behavior, increasing, decreasing, constant, critical point, extrema, maximum, minimum, point of inflection, average rate of change, secant line, parent function, constant function, zero function, identity function, quadratic function, cubic function, square root function, reciprocal function, absolute value function, step function, greatest integer function, transformation, translation, reflection, dilation, composition, inverse relation, inverse function, oneto-one Chapter 2: power function, monomial function, radical function, extraneous solution, polynomial function, leading coefficient, quartic function, turning point, repeated zero, multiplicity, Rational Zero Theorem, lower bound, upper bound, Descartes’ Rule of Signs, Fundamental Theorem of Algebra, Linear Factorization Theorem, Conjugate Root Theorem, complex conjugates, rational function, asymptote, vertical asymptote, horizontal asymptote, oblique asymptote Chapter 3: algebraic function, transcendental function, exponential function, natural base, continuous function, interest, logarithmic function, logarithm, common logarithm, natural logarithm 17 | P a g e 2nd 9 Weeks Chapter 4: quadrantal angle, reference angle, unit circle, circular function, periodic function, period, sinusoid, amplitude, frequency, phase shift, vertical shift, midline, arcsine function, arccosine function, arctangent function Chapter 5: identity, trigonometric identity, reduction identity Chapter 6: identity matrix, inverse matrix, inverse, invertible, singular matrix, determinant, square system, Cramer’s Rule 3rd 9 Weeks Chapter 7: conic section, degenerate conic, locus, parabola, focus, directrix, axis of symmetry, vertex, latus rectum, ellipse, foci, major axis, center, minor axis, vertices, co-vertices, eccentricity, hyperbola, transverse axis, conjugate axis, parametric equation, parameter, orientation, parametric curve Chapter 8: vector, initial point, terminal point, standard position, direction, magnitude, quadrant bearing, true bearing, parallel vectors, equivalent vectors, opposite vectors, resultant, triangle method, parallelogram method, zero vector, components, rectangular components, component form, unit vector, linear combination Chapter 9: complex plane, real axis, imaginary axis, Argand plane, absolute value of a complex number, polar form, trigonometric form, modulus, argument Chapter 10: geometric sequence, common ratio, geometric series, binomial coefficients, Pascal’s Triangle, Binomial Theorem 4th 9 Weeks Chapter 11: univariate, negatively skewed distribution, symmetrical distribution, positively skewed distribution, resistant statistic, cluster, bimodal distribution, percentiles, percentile graph, random variable, discrete random variable, continuous random variable, probability distribution, expected value, binomial experiment, binomial distribution, binomial probability distribution function, normal distribution, empirical rule, z-value, standard normal distribution, sampling distribution, standard error of mean, sampling error, continuity correction factor, inferential statistics, parameter, point estimate, interval estimate, confidence level, maximum error of estimate, critical value, confidence interval, t-distribution, degrees of freedom, hypothesis test, null hypothesis, alternative hypothesis, level of significance, left-tailed test, twotailed test, right-tailed test, p-value, correlation, bivariate, explanatory variable, response variable, correlation coefficient, regression line, line of best fit, residual, least-squares regression line, residual plot, influential, interpolation, extrapolation Chapter 12: one-sided limit, two-sided limit, direct substitution, indeterminate form