The Praxis™ Exam - The Story
What exactly is the Praxis™ Exam and why should I bother taking it?
The Praxis™ Subject Assessments are created by Educational Testing Services® (ETS®) to assess your knowledge of the subject areas you plan to teach, and they are part of the licensing procedure in many states. Your state has adopted the Praxis™ Exam Series tests because it wants to be sure that you have achieved a specified level of mastery of your subject area before it grants you a license to teach in a classroom.
What kind of grade do I need to pass the Praxis™ Exam?
The Praxis™ Exam is used nationally, meaning that the test is used in more than one state. If you move to another state, you can transfer your Praxis™ Exam score to that state. Passing scores are set by state. You can find passing scores for all states that use the Praxis™ Exams in the Understanding Your Praxis™ Scores pamphlet, by calling 609-771-7395. The information is also available on the Praxis™ Exam website: www.ets.org/praxis.
For what exam does the material offered in this website prepare you?
The exam that the material offered is for secondary mathematics. The specific exam is Mathematics: Content Knowledge (# 10061)
How long is the exam and how is the exam graded?
The exam contains 50 multiple-choice problems (a-b-c-d) and lasts 120 minutes. The number of raw points awarded is the number of correct answers. You are not penalized for guessing. Your scaled score is computed from your total number of raw points in a way that adjusts for the difficulty of the questions.
Here is a typical conversion table. They vary slightly from exam to exam.
| Raw score | Scaled Score | Raw score | Scaled Score | Raw score | Scaled Score |
| 0 | 100 | 17 | 108 | 34 | 151 |
| 1 | 100 | 18 | 110 | 35 | 154 |
| 2 | 100 | 19 | 114 | 36 | 156 |
| 3 | 100 | 20 | 116 | 37 | 158 |
| 4 | 100 | 21 | 118 | 38 | 161 |
| 5 | 100 | 22 | 121 | 39 | 164 |
| 6 | 100 | 23 | 124 | 40 | 167 |
| 7 | 100 | 24 | 127 | 41 | 170 |
| 8 | 100 | 25 | 130 | 42 | 173 |
| 9 | 100 | 26 | 133 | 43 | 177 |
| 10 | 100 | 27 | 134 | 44 | 183 |
| 11 | 100 | 28 | 136 | 45 | 186 |
| 12 | 100 | 29 | 138 | 46 | 190 |
| 13 | 100 | 30 | 141 | 47 | 193 |
| 14 | 102 | 31 | 144 | 48 | 197 |
| 15 | 104 | 32 | 147 | 49 | 199 |
| 16 | 105 | 33 | 149 | 50 | 200 |
Do I have to memorize all those formulas?
Not really. You are given many formulas and definitions. The best way to see exactly what you are given is to go to the Praxis™ Exam website: www.ets.org/praxis. Follow the links to 'Tests at a Glance' and then to exams by test title or code. Navigate to test 10061 and download the PDF file with information about the exam, formulas and definitions, sample problems, and solutions.
What are the content categories and how many questions of each can I expect?
| Content Category and approximate number of questions |
Approximate total number of questions | Approximate total percentage of questions |
I. Arithmetic and Basic Algebra (6-8)
Geometry (4-6)
Trigonometry (2-4)
Analytic Geometry (2-4) | 17 Total | 34% |
II. Functions and their Graphs (5-7)
Calculus (5-7) | 12 Total | 24% |
III. Probability and Statistics (3-5)
Discrete Mathematics (3-5)
Linear Algebra (3-5)
Mathematical Reasoning and Modeling (5-7)
Computer Science (5-7) | 21 Total | 42% |
What about using a calculator?
Exam 10061 does not just permit a calculator. It requires a calculator, specifically a graphing calculator. The calculator must have these capabilities:
- Plot the graph of a function within an arbitrary viewing window
- Find the zeros of a function
- Compute the derivative of a function numerically
- Compute definite integrals numerically
While this website does not endorse any specific graphing calculator, the machine which is used in most classrooms and texts is the Texas Instruments TI-83, TI-83+, or TI-84. Laptops are not allowed nor are pocket organizers or calculators with QWERTY keyboards. The material that is being offered to you in this site assumes that students are using TI-83+ calculators or the equivalent and specific instructions are given for their use within each content area.
Is the Praxis™ Exam really all that difficult?
The Praxis™ Exam is extremely broad. There are eleven content categories and each represent one or two semesters of classroom time - from middle school through college. While each content area is not extremely deep, it is a disservice to tell you that the material is easy. Calculus, for example, is not easy. Still, each content area delves into basic knowledge of that area and the problems can be solved with basic techniques taught in these courses. Students just out of college will have fewer problems than those who have been away from the field of math for awhile.
Exactly what am I expected to know within each content area.
Following is a list of each content areas and specific skills and abilities within each area.
I. Arithmetic and Basic Algebra
- Understand and work with rational, irrational, real and complex numbers and use them correctly in the context of the problem.
- Demonstrate understanding of the properties of whole numbers (prime, composite, even or odd, factors and multiples).
- Apply the correct order of operations to problems involving the 4 operations, roots, and powers, with and without grouping symbols.
- Identify the properties (closure, commutativity, associativity, distributivity) of the basic operations.
- Given newly defined operations on a number system, determine whether operation is closed, commutative, associative, and/or distributive.
- Identify the additive and multiplicative inverses of a number.
- Interpret and apply the concepts of ratio, proportion, and percent in appropriate situations.
- Solve problems that involve measurement in the metric or traditional system.
- Solve problems involving arithmetic mean and weighted average.
- Work with algebraic expressions formulas.
- Add, subtract, multiply, and divide polynomials, as well as algebraic fractions.
- Translate verbal expressions and relationships into algebraic expressions or equations.
- Solve and graph linear equations and inequalities in one or two variables; solve and graph systems of linear equations, and graph inequalities in two variables; solve and graph non-linear algebraic equations, and graph inequalities.
- Determine any term of a binomial expansion using Pascal's triangle or some other method.
- Solve equations and inequalities involving absolute values.
- Interpret and present geometric interpretations of algebraic principles.
II. Geometry
- Solve problems involving the properties of parallel and perpendicular lines.
- Solve problems using special triangles, such as isosceles and equilateral.
- Solve problems using the relationships of the parts of the triangles, such as sides, angles, medians, midpoints, and altitudes.
- Apply the Pythagorean theorem to solve problems.
- Solve problems using the properties of special quadrilaterals (square, rectangle, parallelogram, rhombus, and trapezoid).
- Solve problems using the properties (angles, sum of angles, number of diagonals, and vertices) of polygons with more than four sides.
- Solve problems using the properties of circles, including those involving inscribed angles, central angles, chords, radii, tangents, secants, arcs, and sectors.
- Compute the perimeter and area of triangles, quadrilaterals, and circles, and of regions that are combinations of these figures.
- Use relationships (congruency, similarity) among two-dimensional geometric figures and among three-dimensional figures to solve problems.
- Compute the surface area and volume of right prisms, pyramids, cones, cylinders, and spheres, and of solids that are combinations of these figures.
- Solve problems involving reflections, rotations, and translations or points, lines, or polygons in the plane.
III. Trigonometry
- Perform conversions between radians, degrees, and revolutions.
- Define and use the six basic trigonometric relationships in the context of a right triangle and using radian measure, in the context of the unit circle.
- Solve problems involving right angles and problems involving trigonometric functions evaluated at such numbers as...
 .
- Apply the law of sines and the law of cosines in the solutions of problems.
- Recognize the graphs of the six basic trigonometric functions and identify their period, amplitude, phase shift, vertical shift, and asymptotes.
- Prove identities using the basic trigonometric functions.
- Given a point in the rectangular coordinate system, identify the corresponding point in the polar coordinate system.
- Find the trigonometric form of complex numbers and apply DeMoivre's Theorem.
IV. Analytic Geometry
- Determine equations of lines and planes given appropriate information.
- Determine the length between two points in 2-space or 3-space as well as midpoint and distance between a point and a line.
- Given a geometric definition of a conic section, derive the equation of the conic section.
- Determine the type of conic section represented by an equation (only those whose axis of symmetry is parallel to a coordinate axis).
V. Functions and their Graphs
- Understand function notation and be able to determine whether a graph in a plane is a function.
- Use multiple representations of a function such as an equation, a graph, a table, or a verbal statement.
- Use the definition of mapping to define a function.
- Find the domain and range of a function given in as a graph or as an equation.
- Use the properties of algebraic, trigonometric, logarithmic, and exponential functions to solve problems.
- Be able to find composite functions.
- Find the inverse of a relation and determine whether the relation is a function.
- Determine the graphical properties and sketch the graph of linear, step, absolute-value, or quadratic function (slope, intercept, intervals of increase or decrease, axis of symmetry).
- Understand how altering an equation will produce vertical shifts, horizontal shifts, stretching or translation across an axis.
VI. Discrete Math
- Use the basic terminology and given the definitions, use the symbols of logic.
- Use truth tables to verify statements.
- Use laws of Propositions to evaluate equivalence of complex logical expressions.
- Solve functions involving union and intersections of sets, subsets, and disjoint sets.
- Solve basic problems involving permutations and combinations.
- Use the Euclidean algorithm to find the greatest common divisor of two numbers.
- Work with numbers expressed in bases other than base 10.
- Find values of functions defined recursively and translate between recursive and closed form expressions for a function.
- Determine whether a binary relation on a set is reflexive, symmetric, antisymmetric, transitive, or an equivalence relations.
- Solve simple linear programming problems.
VII. Calculus
- Discuss informally 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.
- Use limits to show that a particular function is continuous.
- Use L'Hopital's Rule, where applicable, to calculate the limits of functions.
- Relate the derivative of a function to a limit or to the slope of the curve.
- Explain conditions under which a continuous function does not have a derivative.
- Differentiate algebraic expressions, trigonometric functions, and exponential and logarithmic functions using the sum, product, quotient, and chain rules.
- Perform implicit differentiation.
- Make numerical approximations of derivatives and integrals.
- Use differential calculus to analyze the behavior of a function (points of relative maxima and minima and intervals of concavity).
- Use differential calculus to solve problems involving related rates and rates of change.
- Approximate the roots of a function (Newton's method).
- Use differential calculus to solve applied maxima-minima problems.
- Solve problems using the Mean Value Theorem of differential calculus.
- Solve problems using the Fundamental Theorem of calculus.
- Demonstrate an understanding of integration by use of a limiting process of finding areas in the plane.
- Integrate functions using standard integration techniques.
- Evaluate improper integrals.
- Use integral calculus to calculate the area of regions in the plane and the volumes of solids formed by rotating plane figures about a line.
- Determine the limits of sequences and simple infinite series.
- Use standard tests to show convergence or divergence of series (comparison, ratio, root, etc).
VIII. Probability and Statistics
- Organize data into a presentation that is appropriate for solving a problem (histograms, boxplots, etc).
- Solve probability problems involving finite sample spaces by actually counting outcomes appropriately.
- Solve probability problems by using counting techniques (permutations, combinations).
- Solve probability problems involving independent trials.
- Solve problems using the binomial distribution and determine when such use is appropriate.
- Solve problems involving joint probability.
- Find and interpret measures of central tendency (population mean, sample mean, median, mode) and know which one is most meaningful in a particular situation.
- Find and interpret measures of spread (range, population variance and standard deviation, sample variance and standard deviation).
- Perform regression to predict data.
- Determine the mathematical expectation of a discrete random variable.
- Solve problems using the normal distributions.
- Solve basic intuitive problems using concepts of uniform and chi-square distributions.
- Recognize a valid test to determine whether to accept or reject a given null hypothesis.
IX. Linear Algebra
- Add, subtract, and scalar multiply vectors using geometric interpretations of these operations and use in real-world applications.
- Scalar multiply, add, subtract, and multiply matrices.
- Demonstrate an understanding and the basic properties of inverses of matrices.
- Determine and apply the matrix representation of a linear transformation.
- Use matrix techniques to solve systems of linear equations.
X. Mathematical Reasoning and Modeling
- Demonstrate an understanding of a physical situation or a verbal description of a situation, develop a mathematical model of it, and determine whether one mathematical will describe two apparently different situations.
- Determine appropriate mathematical strategies to solve a problem including conjectures, counterexamples, inductive reasoning, deductive reasoning, proof by contradiction, and direct proof) and deciding which tools are appropriate.
- Recognize the reasonableness of solutions given the context of a problem.
- Using estimation, test the reasonableness of solutions.
- Estimate the actual and relative error in the numerical solutions to a problems by analyzing the effects of round-off and truncation error.
- Communicate results in an appropriate form.
XI. Computer Science
- Demonstrate an understanding of the roles of hardware and software components of a computer system (output devices, CPU, disks, operating systems, secondary storage devices).
- Know basic computer terminology (files, I/O, records).
- Develop computer algorithms to solve mathematical problems.
- Trace and debug existing computer algorithms.
HOW TO GET THE ANSWERS TO THE PRAXIS™ EXAM!
So you've seen the websites that promise great success with hardly any studying. They are going to sell you 'the secret' to the exam... they are going to give you the answers you need, effortlessly. Don't be fooled by clever advertising. Think about it for a minute. Would your state want teachers going into your school systems based on who paid the most for the answers? Remember the goal of this exam is to help qualify teachers for their teaching position. Here is the answer... learn the math. If you know the math, you are going to know the answers on the exam. It's that simple. It requires preparation but the test is broader than it is deep and attempts to test your exposure to concepts rather than complex in-depth knowledge of a topic. Read the section below to learn the best way to prepare for the exam. Don't worry... you'll do well if you take some time to brush up on your skills.
Why spend all my time studying? Isn't there a quick way to prepare?
How do you sum up as much as 8 years of mathematics into one tidy little list of things you can do to get through this exam? A number of prospective teachers have taken this exam countless times, wasting time and money, until they finally pass the exam. Not passing can cost you a job.
The best way to prepare is to be acquainted with all eleven sections on the exam so that you have some chance of getting a problem correct in any of the sections.
In most states, you do not need a tremendous number of problems correct. Below is a chart detailing the passing score for each state currently using the Praxis™ test and the approximate number of problems you need to get correct. Please note that this information is accurate as of February 2005, and is subject to change.
| State | Score | Approximate Problems Correct |
| Alabama | 118 | 21 |
| Alaska | 146 | 32 |
| Arkansas | 116 | 20 |
| California | * | * |
| Colorado | 156 | 36 |
| Connecticut | 137 | 29 |
| Delaware | 121 | 21 |
| District of Columbia | 141 | 30 |
| Georgia | 136 | 28 |
| Hawaii | 136 | 28 |
| Idaho | 119 | 22 |
| Indiana | 136 | 28 |
| Kansas | * | * |
| Kentucky | 125 | 24 |
| Louisiana | 125 | 24 |
| Maryland | 141 | 30 |
| Minnesota | 124 | 23 |
| Mississippi | 123 | 23 |
| * under review |
| |
| State | Score | Approximate Problems Correct |
| Missouri | 137 | 29 |
| Nevada | 144 | 31 |
| New Hampshire | 127 | 25 |
| New Jersey | 137 | 29 |
| North Carolina | 136 | 28 |
| North Dakota | 139 | 29 |
| Ohio | 139 | 29 |
| Oregon | 138 | 29 |
| Pennsylvania | 136 | 28 |
| South Carolina | 131 | 26 |
| South Dakota | 124 | 23 |
| Tennessee | 136 | 28 |
| Utah | * | * |
| Vermont | 141 | 30 |
| Virginia | 147 | 33 |
| Washington | 134 | 27 |
| West Virginia | 133 | 27 |
| Wisconsin | 135 | 28 |
| Average | 133.8 | 27.3 |
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Depending on your state, you may need to put more or less study into this exam. You may decide to concentrate on the basics: arithmetic, algebra, geometry, trigonometry, and probability and statistics and forego calculus and the more advanced subjects. But a safer technique is to review each topic so that you have at least some idea what a question is asking.
In short, there is no easy way to prepare for this exam. It will take time. But considering the fact that it could eventually decide whether you get that coveted teaching position, it probably is worth it.
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