Mathematics Praxis II Topics

Mathematics: Content Knowledge (0061)

270

271

272

300

321

340

357

370

451

490

Other

Arithmetic and Basic Algebra

? Understand the structure of the natural, integer, rational, real, and complex number systems

 X

171

471

? Given newly defined operations on a number system, determine whether the closure, commutative, associative, or distributive properties hold

 X

171

471

? Demonstrate an understanding of the properties of counting numbers (e.g. prime or composite, even or odd, factors, multiples)

 X

171

? Solve ratio, proportion, percent, and average problems

171

? Work with algebraic expressions, formulas, and equations

X

X

X

? Solve and graph systems of equations and inequalities, including those involving absolute value

171

? Use the results of the binomial theorem

X

X

171

? Present geometric interpretations of algebraic principles

X

X

X

171

Geometry

? 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

 X

? Describe relationships among sets of special quadrilaterals, such as the square, rectangle, parallelogram, rhombus, and trapezoid

 X

? Solve problems using the properties of triangles, quadrilaterals, polygons, circles, parallel and perpendicular lines

 X

? Apply the Pythagorean Theorem to solve problems

 X

? Compute perimeter, area/surface area, and volume of 2-dimensional and 3-dimensional figures

X

X

 X

? Solve problems involving reflections, rotations, and translations of geometric figures in the plane

X

X

 X

Mathematics: Content Knowledge (0061)

270

271

272

300

321

340

357

370

451

490

Other

?Perform geometric constructions using straight-edge and compass

X

Trigonometry

? 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

171

? Know and use the trigonometric functions of special angles

171

? Apply the law of sines and the law of cosines

X

X

X

? Apply the formulas for the trigonometric functions of x/2, 2x, x+y, and x-y in terms of the trigonometric functions of x and y

X

X

X

? Prove identities using the basic trigonometric identities

171

? Solve trigonometric equations and inequalities

171

? Convert between the rectangular and polar coordinate systems

X

X

? Find the trigonometric form of complex numbers and apply DeMoivre?s theorem

171

Analytic Geometry

? Determine the equations of lines and planes

X

X

? 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)

X

? Translate between the geometric definition of a conic section and its equation

171

Functions and Their Graphs

? Understand and be able to work with functions and their graphs, including functions given as mappings

X

171

? Given and equation, graph it; given a graph, determine an equation for it

X

171

Mathematics: Content Knowledge (0061)

270

271

272

300

321

340

357

370

451

490

Other

? Determine properties of a function, such as domain, range, intercepts, symmetries, intervals of increase or decrease, discontinuities, asymptotes

X

171

? Use the properties of algebraic, trigonometric, logarithmic, and exponential functions to solve problems (e.g., finding composite functions and inverse functions)

X

171

? Find the inverse of a one-to-one function n simple cases and know why one-to-one functions have inverses

X

171

Calculus

? 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

? Know when and how to use L?Hospital?s rule

X

X

? Show that a particular function is continuous; understand the difference between continuity and differentiability

X

? Relate the derivative of a function to a limit and to the slope of a curve

X

? Use standard differentiation and integration techniques

X

X

? Make numerical approximations of derivatives and integrals

X

X

? 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 minima-maxima problems

X

? Understand and be able to use the Mean Value Theorem and the Fundamental Theorem of Calculus

X

? Demonstrate an intuitive understanding of the process of integration

X

X

? Evaluate improper integrals

X

? Calculate the area of regions in the plane; calculate the volume of solids formed by rotating plane figures about a line

X

X

Mathematics: Content Knowledge (0061)

270

271

272

300

321

340

357

370

451

490

Other

? Determine the limits of sequences and simple infinite series

X

471

? Use standard tests to show convergence (either conditional or absolute) or divergence of series (e.g., comparison, ratio)

X

471

Probability and Statistics

? Organize data into a suitable form (e.g., construct a histogram and use it in the calculation of probabilities)

X

? Solve discrete and joint probability problems; know when events are independent and how to calculate the probability of independent events

X

? Solve problems using the binomial distribution and be able to determine when the use of the binomial distribution is appropriate

X

? Find and know the appropriate uses of common measures of central tendency (population mean, sample mean, median, mode) and disersion (range, population standard deviation, sample standard deviation, population variance, sample variance)

X

? Model problems using mathematical expectation of a random variable (e.g., fair coins, expected winnings, expected profit)

X

? Solve problems using the normal, uniform, and chi-square distributions

X

? Recognize a valid test to determine whether to accept or reject a given null hypothesis

X

Discrete Mathematics

? 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)

X

Mathematics: Content Knowledge (0061)

270

271

272

300

321

340

357

370

451

490

Other

? Perform elementary operation on sets

X

X

? Solve basic problems involving permutations and combinations

X

X

? Use the Euclidean Algorithm to find the greatest common divisor of two numbers

X

X

? Work with numbers expressed in bases other than base ten

X

? Find values of functions defined recursively; ?translate? between recursive and closed form

X

? Determine if a binary relation on a set is reflexive, symmetric, antisymmetric, transitive, or an equivalence relation

X

X

? Solve simple linear programming problems

X

Linear Algebra

? Scalar multiply, add, subtract, and multiply vectors and matrices

X

? Find inverses of matrices; understand and use the properties of inverses of matrices

X

? Determine and apply the matrix representation of a linear transformation

X

? Use matrix techniques to solve systems of linear equations

X

Computer Science

 ? Demonstrate an understanding of the roles of the hardware and software components of a computer system

CIS163

? Know basic computer terminology

CIS163

? Develop and debug computer algorithms (written in pseudocode)

CIS163

Mathematics Reasoning and Modeling

? Develop a mathematical model; determine if one model will describe two different situations

285

350

? Determine appropriate problem-solving strategies and consider alternatives. Strategies might include conjectures, counterexamples, inductive reasoning, deductive reasoning, and deciding which tools are appropriate

285

Mathematics: Content Knowledge (0061)

270

271

272

300

321

340

357

370

451

490

Other

? Recognize the reasonableness of results

? 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

 X

  360

? 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 of yet?; ?No one will ever be able to do it.?

 X

? Use the axiomatic method

 X

 X

 X