# PRAXIS II Mathematics: Content Knowledge Exam

This exam is designed for individuals who would like to teach mathematics at the secondary school level. You will be given two and a half hours to complete this 60 question multiple-choice exam. The usage of an on-screen graphing calculator is required. You will be required to see patterns, make conjecture’s, construct simple proofs, justify statements with informational logic, reason mathematically, and utilize mathematical concepts. You will also be asked to solve problems by utilizing knowledge from different mathematical areas, solve problems that have several potential solutions, develop mathematical prep models and utilize them to solve real world problems, and utilize various concepts. There will be 41 questions about numbers, algebra, functions, and calculus. There will also be 19 questions regarding geometry, probability, statistics, and discrete mathematics.

Questions regarding the use of technology, mathematical representation, mathematical connections, mathematical proofs and reasoning, and mathematical problem-solving will be distributed across all content areas of this exam.

You will be required to utilize an on-screen graphing calculator that can compute and definite integrals numerically, find the zeros of a function, compute the derivative of a function numerically, and produce the graph of a function in an arbitrary viewing window for this exam.

Mathematical Process Categories

Use of Technology
This section of the exam will cover the utilization of technology, problem solving and understanding mathematical ideas.

Mathematical Representation
This section of the exam will cover the utilization of representations to organize, communicate and record mathematical ideas and, solving problems by application and translation of mathematical representations. Utilizing representations to interpret social, physical and mathematical phenomena will also be included in this section of the exam.

Mathematical Connections
This section of the exam will cover the interconnection of mathematical ideas and, the application of mathematics to different subject areas.

Mathematical Reasoning and Proof
This section of the exam will cover the development and evaluation of mathematical proofs and arguments, the investigation of mathematical conjectures, and the utilization of various methods of proof and types of reasoning.

Mathematical Problem Solving
This section of the exam will cover 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 of the exam will assess your knowledge of finite and infinite arithmetic and geometric sequences and series to model simple phenomena. Your knowledge of the 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; the utilization of equations, trees, networks, and vertex edged graphs to solve problems are also included in this section of the exam.

Matrix Algebra
This section of the exam will assess 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 included 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 assessed.

Probability
This section of the exam will assess your ability to construct empirical probability distributions and make informal inferences regarding the theoretical probability distribution by using simulations. Your ability to interpret, and compute the expected value of random variables, in simple cases will be included 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 in this section of the exam.

Data Analysis and Statistics
For this section of the exam you will be required to recognize the 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 included in this section. 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 included in this section.

Calculus
This section of the exam will assess 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 assessed.

Functions
This section of the exam will assess 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 included 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 included in this section of the exam.

Measurement
This section of the exam will assess 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 included in this section of the exam. The relationship between the trapezoid, rhombus, parallelogram, rectangle, and square will also be included in this section of the exam. Your ability to compute 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 included in this section of the exam.

Algebra Number Theory
This section of the exam will assess your ability to solve problems in two and three 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. Your ability to work with fractions, exponents, radicals, complex numbers, negative numbers, polynomials, formulas and equations, algebraic expressions, ratios, percentages, proportion, and averaging will be included in this section of the exam. Your 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 will be assessed in a section of the exam.

PRAXIS II Mathematics: Content Knowledge Practice Questions

1. Basic information technology concepts include:

A. binary
B. ASCII
C. routing
D. All of the above

2. How does the evolution of information technology affect educators?

A. IT changes slowly.
B. Schools have bottomless budgets.
C. Time is needed to prepare requests and obtain approval.
D. They desire to stay with known software and hardware.

3. What do software developers do?

A. Analyze requirements
B. Determine specifications
C. Design and write the program
D. All of the above

4. Which of the following is not part of a design pattern?

A. Recognizable code
B. Testing required
C. Documentation available
D. List of issues and their solutions

5. Computer programming includes:

A. writing
B. testing and troubleshooting
C. maintenance of source code
D. All of the above

Answer Key For Mathematics: Content Knowledge

There are six basic concepts important to understanding information technology (IT). These concepts should be introduced to students beginning in the first grade and continue through high school:

BINARY: brief representations of long strings of numbers used in computations. Only zeroes and ones are used to represent these number strings.
ASCII (American Standard Code for Information Interchange): letters and punctuations signs combined with strings of binary numbers
HIERARCHY: There is an order to understanding and using information technology; this concept is sometimes referred to as “nesting quality.”
WORLD WIDE WEB: The www is a communication tool that links information via bursts of binary data.
STORED PROGRAM: A CPU (central processing unit) stores multiple sets of instructions and determines the next step in a process.
ROUTING: Sequences of binary numbers move through a connected network.

Information technology changes rapidly. It may be only a few months before the next generation is introduced. Because of this, it can be a challenge to keep current hardware accessible and be knowledgeable about the newest versions of popular software. Educators face especially difficult obstacles because of budget restraints and the time needed to research, prepare requests, and obtain approval for new software and hardware (which in all probability means most school districts are several generations behind).

How can a teacher cope with this real-world situation? The best way is to stay current with advances in information technology through reading and research. Develop lesson plans that show students how to approach new and unfamiliar software and hardware. Most software applications do not change dramatically from one version to the next. Many improvements are slight and based on user requests. There are only so many ways to set up a spreadsheet or create a document. New hardware usually just adds more memory and faster response time along with a few “bells and whistles.” The basics do not change.

Software design is the process of plotting a solution to solve a problem. Developers analyze requirements, determine specifications, and write a program to address the needs. The process may be automated (no discussion with users) or semiautomated, which involves end-user input. A design may be a simple flow chart defining a sequence of events or a complicated series of interrelated instructions. The result may be platform-specific, meaning it can only be used on a particular framework, or platform-independent, meaning it will work on most frameworks.

When designing software, there are several things to consider, depending upon the desired goals:

EXTENSION: ease with which new features can be added without changing the architecture
ROBUST: ability to tolerate unpredictable input
RELIABLE: does what it is supposed to do
SECURE: ability to withstand unwanted intrusions
MAINTAIN: easily accepts updates when needed
COMPATIBLE: operates with other software
REUSE: all components contain essential functions, so they can be used in similar designs

A design pattern is a template with recognizable code that provides a solution to a common design problem. It speeds up the software design process because the pattern has already been tested and proven to work. The documentation describes the context, the issues, and the solution. There is no standard format, but one outlined by the Gang of Four (Erich Gamma, Richard Helm, Ralph Johnson, and John Vlissides) in their book Design Patterns is a good starting point. The structure, participants, and collaboration sections are especially important:

PATTERN NAME: descriptive and unique
INTENT: goals and reasons to use it
ALSO KNOWN AS: other names
MOTIVATION: how it is used
APPLICABILITY: situations in which it can be used
STRUCTURE: graphical representation
PARTICIPANTS: classes, objects, and their roles
COLLABORATION: how classes and objects interact
CONSEQUENCES: results, side effects, and tradeoffs
IMPLEMENTATION: how to set up and integrate
SAMPLE CODE: how to use it in programming language
KNOWN USES: specific examples
RELATED PATTERNS: differences with similar patterns