21 ST CENTURY LEARNING CURRICULUM FRAMEWORK PERFORMANCE RUBRICS FOR MATHEMATICS PRE-CALCULUS

Transcription

1 21 ST CENTURY LEARNING CURRICULUM FRAMEWORK PERFORMANCE RUBRICS FOR MATHEMATICS PRE-CALCULUS

2 Table of Contents Functions... 2 Polynomials and Rational Functions... 3 Exponential Functions... 4 Logarithmic Functions... 5 Vectors... 6 Polar Coordinates... 7 Parametric Equations... 8 Conic Sections... 9 Systems and Matrices Sequences and Series Statistics Probability Three Dimensions Analytic Trigonometry

3 Functions Topic: In addition to, in-depth inferences and applications that go beyond what was taught. Level 3.5 In addition to performance, in-depth inferences and applications with partial While engaged in grade appropriate tasks involving functions, the student: determines characteristics of a graph (required: domain, range, inc/dec intervals, continuity, relative extrema, end behavior, asymptotes) decomposes a given function sketches graphs of functions using transformations (required: horizontal reflections, stretches and shrinks) knowledge of the more o identifies basic examples of domain, range, inc/dec intervals, continuity, relative extrema, end behavior, asymptotes o identifies examples of a decomposed function o sketches simple graphs using a basic transformation o domain, range o inc/dec intervals, continuity, relative extrema, end behavior, asymptotes o composition, decomposition o domain restrictions o transformation o relation, function, odd and even functions more 2

4 Topic: Polynomials and Rational Functions In addition to, in-depth inferences and applications that go beyond what was taught. Level 3.5 In addition to performance, in-depth inferences and applications with partial While engaged in grade appropriate tasks involving polynomials and rational functions, the student: analyzes and graphs polynomial functions (required: intermediate value thm) analyzes and graphs rational functions (required: end behavior) solves a polynomial inequality (required: analytically and graphically) solves a rational inequality (required: analytically and graphically) knowledge of the more o domain, range o relative extrema, end behavior, continuity, asymptotes o limits at infinity o increasing/decreasing intervals o intermediate value theorem, rational root theorem o synthetic division o interval and set notation o test intervals o domain restrictions o extraneous solutions more 3

5 Topic: Exponential Functions In addition to, in-depth inferences and applications that go beyond what was taught. Level 3.5 In addition to performance, in-depth inferences and applications with partial While engaged in grade appropriate tasks involving exponential functions, the student: solves exponential equations (required: base e, different bases) analyzes and graphs exponential functions (required: horizontal reflections, stretches and shrinks) solves applications of exponential functions (required: average rate of change, instantaneous rate of change) knowledge of the more o base of exponents o logarithms o change of base o properties of exponents o inverse operations o definition of a logarithm o asymptotes o transformations o natural base o average rate change o instantaneous rate of change o extraneous solution more 4

6 Topic: Logarithmic Functions In addition to, in-depth inferences and applications that go beyond what was taught. Level 3.5 In addition to performance, in-depth inferences and applications with partial While engaged in grade appropriate tasks involving logarithmic functions, the student: solves logarithmic equations (required: natural logarithms) analyzes and graphs logarithmic functions (required: horizontal reflections, stretches and shrinks) solves applications of logarithmic functions knowledge of the more o logarithms o change of base o properties of exponents o inverse operations o definition of a logarithm o exponentiate o asymptotes o transformations o natural base o average rate change o instantaneous rate of change o extraneous solutions o expanding and condensing logarithms o domain restrictions more 5

7 Topic: Vectors In addition to, in-depth inferences and applications that go beyond what was taught. Level 3.5 In addition to performance, in-depth inferences and applications with partial While engaged in grade appropriate tasks involving vectors, the student: writes vectors in component form calculates the angle between two vectors projects one vector onto another vector knowledge of the more performs basic processes: o linear combination form o dot product o orthogonal vectors o projection o unit vector more 6

8 Topic: Polar Coordinates In addition to, in-depth inferences and applications that go beyond what was taught. Level 3.5 In addition to performance, in-depth inferences and applications with partial While engaged in grade appropriate tasks involving polar coordinates, the student: converts between rectangular and polar coordinates analyzes and graphs polar equations writes the polar form of a complex number solves operations of complex numbers in polar form (required: multiply, divide, powers, roots) knowledge of the more o rectangular coordinates o polar coordinates o complex numbers more 7

9 Topic: Parametric Equations In addition to, in-depth inferences and applications that go beyond what was taught. Level 3.5 In addition to performance, in-depth inferences and applications with partial While engaged in grade appropriate tasks involving parametric equations, the student: eliminates the parameter to write a rectangular equation writes a rectangular equation in parametric form (required: conic sections) uses parametric equations to simulate the motion of a projectile knowledge of the more o parameter o rectangular form o parametric form o conic section o projectile more 8

10 Topic: Conic Sections In addition to, in-depth inferences and applications that go beyond what was taught. Level 3.5 In addition to performance, in-depth inferences and applications with partial While engaged in grade appropriate tasks involving conic sections, the student: analyzes and graphs conic sections (required: rotating axes) converts between polar equation and rectangular equation of conic sections graphs polar equations of conic sections analyzes polar equations of conic sections (required: eccentricity, type) knowledge of the more performs basic processes: o focus o directrix vertex o axis o asymptotes o degree of rotation o eccentricity o standard form, general form more 9

11 Topic: Systems and Matrices In addition to, in-depth inferences and applications that go beyond what was taught. Level 3.5 In addition to performance, in-depth inferences and applications with partial While engaged in grade appropriate tasks involving systems and matrices, the student: solves operations on matrices solves a system (required: Gauss-Jordan elimination, Cramer s Rule) solves a system using an inverse matrix decomposes partial fractions knowledge of the more o matrix o scalar multiple o matrix multiplication o row-reduction o inverse o determinant o Cramer s Rule o decomposition of fractions more 10

12 Topic: Sequences and Series In addition to, in-depth inferences and applications that go beyond what was taught. Level 3.5 In addition to performance, in-depth inferences and applications with partial While engaged in grade appropriate tasks involving sequences and series, the student: writes the formula for the nth term of a sequence given critical information (required: arithmetic, geometric, finite differences) evaluates a finite arithmetic and geometric series evaluates an infinite geometric series writes a series in summation notation proves summation formulas using induction knowledge of the more performs basic operations such: o sequence o series o summation notation o sigma o arithmetic o geometric o induction o nth partial sum Partial knowledge of the simpler details and processes but major errors or omissions Level 0.5 With help, a partial understanding of some of the simpler details and processes but not the more 11

13 Topic: Statistics In addition to, in-depth inferences and applications that go beyond what was taught. Level 3.5 In addition to performance, in-depth inferences and applications with partial While engaged in grade appropriate tasks involving statistics, the student: calculates and analyzes measures of variation (required: range, and standard deviation) determines and analyzes the intervals in which percentages of data lie (required: empirical, Chebyshev) uses measures of position to compare different data (required: z-score, percentile rankings, boxplot) knowledge of the more o frequency distribution (categorical, grouped, relative, cumulative, cumulative relative) o class, class boundaries, class midpoint, class width, class limit (lower, upper) o histogram, ogive, stem-and-leaf, Pareto, pie chart, scatterplot, time-series graph o bell-shaped distribution o uniformity o skew (left, right, symmetric) o modality (unimodal, bimodal, multimodal, nonmodal) o central tendency, mean, median, mode, weighted mean, modal class o variation, range o standard deviation, z-score/standard score o percentile, decile, quartile, Inter-Quartile range o outliers, o Empirical Rule, Chevyshev s Theorem o boxplot more 12

14 Topic: Probability In addition to, in-depth inferences and applications that go beyond what was taught. Level 3.5 In addition to performance, in-depth inferences and applications with partial While engaged in grade appropriate tasks involving probability, the student: calculates and analyzes the probability of an event (required: addition/multiplication rule, permutations/combinations) uses the binomial theorem to expand a binomial expression calculates and analyzes mean and standard deviation for discrete probability distributions (required: binomial) calculates and analyzes probabilities in continuous probability distributions (required: standard normal, non-standard normal, and Central Limit Thm) solves for data values given probabilities in continuous probability distributions and analyze results knowledge of the more performs basic processes: o events (mutually exclusive, independent, dependent, complementary, equally likely, simple, compound) o conditional probability o Law of Large Numbers o sample space o classical probability, empirical probability o factorial o permutation o Fundamental Counting Rule o tree diagram, Venn diagram o expected value o binomial experiment, binomial distribution o discrete probability distribution o random variable o normal distribution, standard normal distribution, sampling distribution of sample means o z-value more 13

15 Topic: Three Dimensions In addition to, in-depth inferences and applications that go beyond what was taught. Level 3.5 In addition to performance, in-depth inferences and applications with partial While engaged in grade appropriate tasks involving probability, the student: finds the equation of a sphere and critical information graphs a sphere finds the magnitude and direction of a vector determines if two vectors are parallel, orthogonal, or neither uses cross product to find an orthogonal vector given two vectors writes a set of parametric and symmetric equations given a point and parallel to a given vector knowledge of the more o sphere o vector o magnitude o direction o orthogonal o cross product o dot product o parallel o parametric o symmetric more 14

16 Topic: Analytic Trigonometry In addition to, in-depth inferences and applications that go beyond what was taught. Level 3.5 In addition to performance, in-depth inferences and applications with partial While engaged in grade appropriate tasks involving analytic trigonometry, the student: simplifies trigonometric expressions (required: half angle, double angle, sum and difference, Pythagorean identities, reciprocal identities, quotient identities, cofunction identities, even and odd identities) evaluates trigonometric expressions (required: half angle, double angle, sum and difference, reciprocal identities, quotient identities, cofunction identities, even and odd identities) verifies trigonometric identities (required: half angle, double angle, sum and difference, Pythagorean identities, reciprocal identities, quotient identities, cofunction identities, even and odd identities, 2 column proof format, common denominators, factoring, the conjugate, multiplying/dividing by an expression) knowledge of the more performs basic operations such as: o simplifies trigonometric expressions (required: sum and difference, Pythagorean identities, reciprocal identities, quotient identities, cofunction identities, even and odd identities) o evaluates trigonometric expressions (required: sum and difference, reciprocal identities, quotient identities, cofunction identities, even and odd identities o verifies trigonometric identities (required: Pythagorean identities, reciprocal identities, quotient identities, cofunction identities, even and odd identities, 2 column proof format, common denominators, factoring) o 2-column proof format o conjugate o factoring more 15