# Praxis II Various Subjects of Mathematics Exam by ETS

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## PRAXIS II Mathematics: Content Knowledge Exam

The exam is aimed for those individuals who wants to teach mathematics at the secondary school level. Two hours will be provided, to complete 50 multiple choice questions. Use of a graphing calculator is allowed. You need to see patterns, make conjecture՚s, construct simple proofs, justify statements with informational logic, reason mathematically and utilize mathematical concepts. Questions will be asked to solve problems by utilizing knowledge from different mathematical areas, solve problems that have potential solutions, develop mathematical prep models and use them to solve real world problems and use different concepts. There will be 3 − 4 questions based on discrete mathematics, 4 − 5 questions based on matrix algebra, 2 − 3 questions based on probability, 5 − 6 questions based on statistics and data analysis, 6 questions based on calculus, 8 questions based on mathematical functions, 4 questions based on trigonometry, 5 questions based on geometry, 3 questions based on measurement and 8 questions based on number theory and algebra.

Questions depending on the use of technology, mathematical representation, mathematical connections, mathematical proofs as well as reasoning and mathematical problem-solving will be distributed across all the section.

You need to use a graphing calculator which will help you to calculate definite integrals numerically, find the zeros of a function, calculate the derivative of a function numerically and make a graph of a function in an arbitrary viewing window.

### Mathematical Process Categories

• Use of Technology: This section will focus on the utilization of technology, problem solving and understanding mathematical ideas.
• Mathematical Representation: This section will focus on the use of representations to organize, communicate and record mathematical ideas and helps in solving problems by application and translation of mathematical representations. Utilizing representations to interpret social, physical and mathematical phenomena will also be covered.
• Mathematical Connections: This section will focus on the interconnection of mathematical ideas and the application of mathematics to different subject areas.
• Mathematical Reasoning and Proof: This section will focus on the development and evaluation of mathematical proofs and arguments, the investigation of mathematical conjectures and the utilization of different methods of proof and types of reasoning.
• Mathematical Problem Solving: This section will focus on application of various strategies to solve problems, acquiring mathematical knowledge through problem-solving and solving problems using mathematics in various subject areas.

### Content Categories

• Discrete Mathematics: This section will test your knowledge of finite and infinite arithmetic and geometric sequences and series to model simple phenomena. It tests relationship between discrete and continuous representations, basic problem-solving in counting techniques, multiplication, permutations, recursion, function values and, closed form expressions for a function will also be assessed. The conclusion of weather a binary set՚s relation is reflexive, transitive, symmetric or equivalent and; use of equations, trees, networks, and use of vertex edged graphs to solve problems.
• Matrix Algebra: This section will test your knowledge of reflections, translations, dilations, and rotations of objects in the plane by utilizing matrices, vectors, coordinates and sketches. Using matrix techniques to solve linear equations, reasoning the inverses of matrices with determinants, scalar multiply, subtract, and, multiplying matrices and vectors and finding inverses of matrices will also be covered in this section of the exam. Your knowledge of the system properties of matrices and vectors and properties of the real number system will also be tested in this section.
• Probability: This section will test your knowledge to construct empirical probability distributions and make informal inferences based on the theoretical probability distribution by using simulations. Your ability to interpret, and calculate the expected value of random variables, in simple cases will be covered in this section of the exam. Your knowledge of conditional probability and independent events, computation of the probability of a compound event, and the components of probability distribution and sample space will also be covered.
• Data Analysis and Statistics: This section will focus on characteristics of a well-designed study, understand various types of studies and types of inferences that can be drawn from each, know the values of population parameters, and be able to use sample distributions for informal inference. Your ability to utilize normal distribution characteristics ex. Standard deviation, mean and analyze data to determine what type of function would model the particular phenomenon will also be covered. Utilization of the regression feature on your calculator will help you determine the best curve to use and, interpret the residuals, regression, and coefficients in context. Your knowledge of utilization of the central tendency common measures (ex. Mean, median and node) will be assessed in this section of the exam. Your ability to organize data and construction a histogram will also be covered.
• Calculus: This section will test your knowledge of the following: Simple infinite series, limits of sequences, integration as a limiting sum that can be used to calculate volumes and area; analysis of the behavior of a function when solving problems of related rates and maxima and minima problems. The ability to utilize standard integration and differentiation techniques, understand integrals and derivatives, and understand the relationship between differentiability and continuity will be assessed in this section. Your knowledge of the derivative of a function as a slope of a curve, its rate of change, the limit point of a function, calculation of function limits, determination of function limits and, problem-solving with properties of limits will also be tested in the section.
• Functions: This section will test your knowledge of working with various functions such as verbal narratives, symbolic expressions, tables and graphs. The selection of functions for modeling of a particular phenomena, determination of symmetries, range, domain, intercepts, intervals of decrease or increase, and asymptotes will be covered in this section. Your ability to understand the composition of two functions, find the inverse of a one-to-one function and the knowledge of why only one to one functions have inverses will be assessed. Your knowledge of the utilization of polynomial, logarithmic, exponential, rational and, trigonometric functions to solve problems and interpretation of tables, level curves and three-dimensional graphs will also be covered.
• Measurement: This section will test your ability to solve trigonometric equations and inequalities, apply trigonometric formulas, prove trigonometric identities, apply the law of cosines and seines, understand Pythagorean theorem and its converse, apply the concepts of limits, upper and lower bounds, and successive approximation in measurement situations. Your ability to approximate error, and analyze precision and accuracy in measurement situations will be assessed. Problem-solving by the utilization of reflections, translations, rotations, parallel and perpendicular lines, polygons, triangles, quadrilaterals, arcs, secants, sectors, tangents, radii, cords, central angles, inscribed angles, circles, in two and three dimensions will be covered in this section of the exam. The relationship between the trapezoid, rhombus, parallelogram, rectangle, and square will also be covered in this section of the exam. Your ability to calculate the surface area, volume and perimeter of solids, and two and three dimensional figures will be assessed. Your ability to define and utilize the six basic trigonometric relations using radian measure of angles or degree, identify their asymptotes, shifts, phase displacements, amplitudes, periods and graphs will be covered.
• Algebra Number Theory: This section will test your ability to solve problems in 2 and 3 dimensions, use algebraic representations of spheres, conic sections, planes, lines, interpret algebraic principles geometrically, solve and graph equations and inequalities, and compare and contrast properties of number systems under a variety of operations. It tests fractions, exponents, radicals, complex numbers, negative numbers, polynomials, formulas and equations, algebraic expressions, ratios, percentages, proportion, and averaging in exam. Also test ability to perform addition, subtraction, multiplication, and division on rational, integer, real, and complex numbers along with odd, even, prime, composite, and prime factors of numbers and counting.

## PRAXIS II Mathematics: Pedagogy Exam

The exam is aimed for those individuals who wants to become instructors of mathematics at the secondary school level. One hour will be provided to complete the 3 essay questions that comprise this exam. There will be 1 question based on assessing instruction, 1 question based on implementation of instruction and 1 question based on planning of instruction. Each question counts as 33 % of the final grade. The relationship between mathematical pedagogy and reasoning, communication and problem solving will be the focus of this exam. The question based on assessing instruction will focus on student evaluation and student understanding of the material presented. The question based on implementation of instruction will focus on how the instructor presents the material to the students. The question based on instructional planning will focus on the teacher՚s preparation prior to instructing their students. You will be permitted to use a calculator for this exam. The level of mathematics utilized for this exam will be at a first year level in statistics, probability, informal geometry, informal proof and algebra.

## PRAXIS II Mathematics: Proofs, Models, and Problems. Part 1 Exam

The exam is aimed for those individuals who wants to be instructors of general mathematics at the secondary school level.

One hour will be provided to complete two problems, one model and one proof. Use of a graphing calculator is allowed for this exam. 30 % of the final grade will be scored on the proof question, 30 % on the model question, and 40 % on the two problems you will be asked to solve. When the graphing calculator is used to answer a question, your answer should include a mathematical description of your calculator usage. For this exam, you will need to write a proof, construct a model, and solve two problems. The following content areas will be covered: Discrete mathematics, matrix algebra, probability and statistics, data analysis, functions, trigonometry, measurement, geometry, algebra, and number theory. The focus will be on mathematical representations as well as connections, proof, reasoning and problem-solving.

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