Math 501 Math Content PRAXIS Review

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1 Math 501 Math Content PRAXIS Review Catalog Description: A review of the mathematical concepts included in the ETS PRAXIS (Professional Assessments for Beginning Teachers) Mathematics Content Knowledge secondary teachers examination. (1 credit, PASS/FAIL) Prerequisites: An undergraduate degree in mathematics or at least 24 (graduate and/or undergraduate) credits in mathematics, including courses in precalculus, differential and integral calculus, linear algebra, elementary or mathematical statistics, and geometry. Course objectives: Students will: i) demonstrate strategies for using current technology to solve math problems. ii) compare problem-solving techniques acquired from different areas of mathematics. iii) relate mathematical definitions to new situations. iv) synthesize knowledge from various areas of mathematics. v) assess appropriate strategies for solving mathematics problems. Instructional Procedures: Week-by-week breakdown: This course is best taught in an intensive format (i.e., nine consecutive one and one-half hour summer school days or four consecutive three and one-half hour Saturdays). It is also best taught with students being exposed to a variety of different problems in each class (and as homework) and attempting to first solve them on their own. After that attempt, the class and instructor can discuss similar problems and discuss strategies for solution. Problem solving strategies, test taking strategies, and use of the graphing calculator will be integrated into each class. Content of Course: I. A review of problems from various areas of mathematics A. Arithmetic and basic algebra i. Understand the structure of the natural, integer, rational, real, and complex number systems; have the ability to perform the basic operations

2 (,,, ) on numbers in these systems; identify properties (e.g., closure, commutativity, associativity, distributivity) of the basic operations. ii. Given newly defined operations on a number system, determine whether the closure, commutative, associative, or distributive properties hold. iii. Demonstrate an understanding of the properties of counting numbers (e.g., prime or composite, even or odd, factors, multiples). Solve ratio, proportion, percent, and average (including arithmetic mean and weighted average) problems. iv. Work with algebraic expressions, formulas, and equations. v. Solve and graph systems of equations and inequalities, including those involving absolute value. vi. Use the results of the binomial theorem. vii. Present geometric interpretations of algebraic principles. B. Geometry i. Solve problems using relationships of parts of geometric figures (e.g., medians of triangles, inscribed angles in circles) and among geometric figures (e.g., congruence, similarity), 2-dimensional and 3-dimensional. ii. Describe relationships among sets of special quadrilaterals, such as the square, rectangle, parallelogram, rhombus, and trapezoid. iii. Solve problems using the properties of triangles, quadrilaterals, polygons, circles, parallel and perpendicular lines. iv. Apply the Pythagorean theorem to solve problems. v. Compute perimeter, area/surface area, and volume of 2-dimensional and 3-dimensional figures. vi. Solve problems involving reflections, rotations, and translations of geometric figures in the plane. C. Trigonometry i. Define and use the six basic trigonometric relations using degree or radian measure of angles; know their graphs and be able to identify their period, amplitude, phase displacement or shift, and asymptotes. ii. Know and use the trigonometric functions of special angles (e.g., 2 9,,, ) iii. Apply the law of sines and the law of cosines. x iv. Apply the formulas for the trigonometric functions of, 2x, x y, x y 2 in terms of the trigonometric functions of x and y. v. Solve trigonometric equations and inequalities. vi. Convert between the rectangular and polar coordinate systems. vii. Find the trigonometric form of complex numbers and apply DeMoivre s theorem. D. Functions and their graphs i. Understand and be able to work with functions and their graphs, including functions given as mappings. ii. Given an equation, graph it; given a graph, determine an equation for it.

3 iii. Determine properties of a function, such as domain, range, intercepts, symmetries, intervals of increase or decrease, discontinuities, asymptotes. iv. Use the properties of algebraic, trigonometric, logarithmic, and exponential functions to solve problems (e.g., finding composite functions and inverse functions). v. Find the inverse of a one-to-one function in simple cases and know why one-to-one functions have inverses. E. Probability and statistics i. Organize data into a suitable form (e.g., construct a histogram and use it in the calculation of probabilities). ii. Solve discrete and joint probability problems; know when events are independent and how to calculate the probability of independent events. iii. Solve problems using the binomial distribution and be able to determine when the use of the binomial distribution is appropriate. iv. Find and know the appropriate uses of common measures of central tendency (population mean, sample mean, median, mode) and dispersion (range, population standard deviation, sample standard deviation, population variance, sample variance). v. Model problems using mathematical expectation of a random variable (e.g., fair coins, expected winnings, expected profit). vi. Solve problems using the normal, uniform, and chi-square distributions. vii. Recognize a valid test to determine whether to accept or reject a given null hypothesis (H 0 ). F. Analytic geometry i. Determine the equations of lines and planes. ii. Make calculations in 2-space or 3-space (e.g., distance between two points, coordinates of a midpoint of a line segment, distance between a point and a plane). iii. Translate between the geometric definition of a conic section and its equation. G. Calculus i. Know what it means for a function to have a limit at a point; calculate limits of functions or determine that the limit does not exist; solve problems using the properties of limits. ii. Know when and how to use L Hopital s rule. iii. Show that a particular function is continuous; understand the difference between continuity and differentiability. iv. Relate the derivative of a function to a limit and to the slope of a curve. v. Use standard differentiation and integration techniques. vi. Make numerical approximations of derivatives and integrals. vii. Analyze the behavior of a function (e.g., find relative maxima and minima, concavity); solve problems involving related rates, rates of change, approximation of roots of a function; solve applied minimamaxima problems. viii. Understand and be able to use the Mean Value Theorem and the Fundamental Theorem of Calculus.

4 ix. Demonstrate an intuitive understanding of the process of integration. x. Evaluate improper integrals. xi. Calculate the area of regions in the plane; calculate the volume of solids formed by rotating plane figures about a line. xii. Determine the limits of sequences and simple infinite series. xiii. Use standard tests to show convergence (either conditional or absolute)or divergence of series (e.g., comparison, ratio). H. Discrete mathematics i. Know the basic terminology of symbolic logic; use truth tables to verify statements; apply laws of Algebra of Propositions to evaluate equivalence of complex logical expressions (e.g., De Morgan s laws). ii. Perform elementary operations on sets. iii. Solve basic problems involving permutations and combinations. iv. Use the Euclidean Algorithm to find the greatest common divisor of two numbers. v. Work with numbers expressed in bases other than base ten. vi. Find values of functions defined recursively; translate between recursive and closed form expressions for a function. vii. Determine if a binary relation on a set is reflexive, symmetric, antisymmetric, transitive, or an equivalence relation. viii. Solve simple linear programming problems. I. Linear algebra i. Scalar multiply, add, subtract, and multiply vectors and matrices. ii. Find inverses of matrices; understand and use the properties of inverses of matrices. iii. Determine and apply the matrix representation of a linear transformation. iv. Use matrix techniques to solve systems of linear equations. J. Computer science i. Demonstrate an understanding of the roles of the hardware and software components of a computer system. ii. Know basic computer terminology. iii. Develop and debug computer algorithms (written in pseudocode). K. Mathematical reasoning and modeling i. Develop a mathematical model; determine if one model will describe two different situations. ii. Determine appropriate problem-solving strategies and consider alternatives. Strategies might include conjectures, counterexamples, inductive reasoning, deductive reasoning (mathematical induction, proof by contradiction, direct proof, other types of proof) and deciding which tools are appropriate (e.g., discussion with others, mental math, pencil and paper, calculator, computer, trees and graphs, fingers). iii. Recognize the reasonableness of results. iv. Estimate answers; determine the accuracy of an estimate by analyzing the effects of roundoff and truncation errors introduced in the course of solving a problem.

5 v. Demonstrate an understanding of the different levels of mathematical impossibility, such as: I lack the mathematical skills to do it ; No one has been able to do it as yet (e.g., prove Goldbach s conjecture); No one will ever be able to do it (e.g., trisect a general angle with straight edge and compass). vi. Use the axiomatic method. II. III. IV. Problem solving strategies Test-taking strategies Use of the graphing calculator in problem solving to: A. Produce the graph of a function within an arbitrary viewing window. B. Find the zeros of a function. C. Compute the derivative of a function numerically. D. Compute definite integrals numerically. Evaluation Measures: Home study and work on assigned sample test problems (50%) Class discussion of sample test problems and problem solving strategies (25%) Practice tests (25%) The course is PASS/FAIL. Students who satisfactorily complete the above components will PASS. Required Text: Educational Testing Service (ETS) The PRAXIS Series Study Guide. Subject Assessments: Mathematics. Approaches to Solving Math Problems. Book 2. Princeton, NJ. Educational Testing Service (ETS) The PRAXIS Series Study Guide. Subject Assessments: Mathematics. Answers and Explanations for the Mathematics: Content Knowledge Test. Book 3. Princeton, NJ. Educational Testing Service (ETS) The PRAXIS Series 0061 Mathematics: Content Knowledge Test. Princeton, NJ. Bibliography: Angel, Allen R., and Stuart R. Porter. A Survey of Mathematics with Applications. Addison-Wesley Publishing, New York, Aufmann, Richard N., and Vernon C. Barker. College Algebra and Trigonometry, 3rd ed. Houghton Mifflin Company, New York, 1996.

6 Barnett, Raymond A., Karl E. Byleen, and Martin R. Ziegler. Precalculus: Functions and Graphs, 5th ed. McGraw Hill Company, New York, Bittenger, Marvin L., and David J. Ellenbogen. Precalculus: Graphs, Models, & Graphing Calculator Manual, 2nd ed. Addison Wesley Longman, Inc., New York, Bond, Robert J., and Willliam J. Keane. An Introduction to Abstract Mathematics. Brooks/Cole Publishing Co., Pacific Grove, CA, Cohen, David. Precalculus, 5 th ed. West Publishing Company, New York, COMAP, For All Practical Purposes: Mathematical Literacy in Today s World, 5 th ed. W. H. Freeman Company, New York, Demana, Franklin. Graphing Calculator and Computer Graphing Laboratory Manual, 2 nd ed. Addison-Wesley Publishing, New York, Eccles, Peter J. An Introduction to Mathematical Reasoning: Numbers, Sets and Functions. Cambridge University Press, Feller, W. An Introduction to Probability Theory and its Applications, Volume I, 3 rd ed. John Wiley & Sons, New York, Freund, John E. Mathematical Statistics, 6 th ed. Prentice-Hall, Englewood Cliffs, NJ, Ghahramani, Saeed. Fundamentals of Probability. Prentice Hall, Upper Saddle River, NJ, Hungerford, Thomas W. Contemporary Precalculus: A Graphing Approach, 3 rd ed. Saunders College Publishing, Philadelphia, Johnson, Robert. Just the Essentials of Elementary Statistics, 2nd ed. Wadsworth Publishing Co., New York, Larson, Roland E., Robert P. Hostetler, and Bruce Edwards. Calculus with Analytic Geometry, 7 th ed. Houghton Mifflin Company, Boston, Larson, Roland E., and Bruce H. Edwards. Elementary Linear Algebra, 4 th ed. D. C. Heath & Co., New York, Lucas, John F. Introduction to Abstract Mathematics, 2nd ed. Ardsley House, Publishers, Inc., New York, 1990.

7 Moise, Edwin E. Elementary Geometry from an Advanced Standpoint, 3 rd ed. Addison-Wesley Publishing, New York, Posamentier, Alfred S., and Jay Stepelman. Teaching Secondary Mathematics, 5th ed. Prentice Hall. Upper Saddle River, NJ, 1999 Spiegel, Murray R., and Robert E. Moyer. Schaum's Outline of College Algebra, 2nd ed. McGraw Hill Company, New York, Stewart, James. Calculus, 4th ed. International Thomson Publishing, New York, Yoshiwara, Droyan. Modeling Functions and Graphs, 3rd ed. Brooks Cole Publishing Co., Pacific Grove, CA, 2001.