Mathematics 5061

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The Praxis Study Companion
TM

Mathematics:
Content Knowledge
0061/5061

www.ets.org/praxis

Welcome to The Praxis™ Study Companion

Welcome to The Praxis™ Study Companion
Prepare to Show What You Know
You have gained the knowledge and skills you need for your teaching career. Now you are ready to demonstrate
your abilities by taking a PraxisT M test.
Using The Praxis Study Companion is a smart way to prepare for the test so you can do your best on test day. This
guide can help keep you on track and make the most efficient use of your study time.
The Study Companion contains practical information and helpful tools, including:
• An overview of the tests
• Specific information on the Praxis test you are taking
• A template study plan
• Practice questions and explanations of correct answers
• Test-taking tips and strategies
• Frequently asked questions
• Links to more detailed information
So where should you start? Begin by reviewing this guide in its entirety and note those sections that you need
to revisit. Then you can create your own personalized study plan and schedule based on your individual needs
and how much time you have before test day.
Keep in mind that study habits are individual. There are many different ways to successfully prepare for your
test. Some people study better on their own, while others prefer a group dynamic. You may have more energy
early in the day, but another test taker may concentrate better in the evening. So use this guide to develop the
approach that works best for you.
Your teaching career begins with preparation. Good luck!

The PraxisTM Study Companion

Table of Contents

Table of Contents
The PraxisTM Study Companion guides you through the 10 steps to success
1. Know What to Expect......................................................................................................4
Familiarize yourself with the Praxis tests so you know what to expect
2. F
amiliarize Yourself with Test Questions.......................................................................5
Become comfortable with the types of questions you’ll find on the Praxis tests
3. Understand Your Scores..................................................................................................9
Understand how tests are scored and how to interpret your test scores
4. Learn About Your Test.................................................................................................. 11
Learn about the specific test you will be taking
5. Determine Your Strategy for Success.......................................................................... 18
Set clear goals and deadlines so your test preparation is focused and efficient
6. Develop Your Study Plan.............................................................................................. 21
Develop a personalized study plan and schedule
7. Review Smart Tips for Success..................................................................................... 25
Follow test-taking tips developed by experts
8. Practice with Sample Test Questions.......................................................................... 27
Answer practice questions and find explanations for correct answers
9. Check on Testing Accommodations............................................................................ 46
See if you qualify for accommodations that may make it easier to take the Praxis test
10. Do Your Best on Test Day............................................................................................ 47
Get ready for test day so you will be calm and confident
Appendix: Other Questions You May Have .................................................................... 49

The PraxisTM Study Companion

Step 1: Know What to Expect

1. Know What to Expect
Familiarize yourself with the Praxis tests so you know what to expect
Which test should I take?
Each state or agency that uses the Praxis tests sets its own requirements for which test or tests you must take for
the teaching area you wish to pursue.
Before you register for a test, confirm your state or agency’s testing requirements at www.ets.org/praxis/states.

How are the Praxis tests given?
Praxis tests are given in both computer and paper formats. Note: Not all Praxis tests are offered in both formats.

Should I take the computer- or paper-delivered test?
You should take the test in whichever format you are most comfortable. Some test takers prefer taking a paperand-pencil test, while others are more comfortable on a computer. Please note that not all tests are available in
both formats. To help you decide, watch the What to Expect on Test Day video for computer-delivered tests.

If I’m taking more than one Praxis test, do I have to take them all in the same format?
No. You can take each test in the format in which you are most comfortable.

Is there a difference between the subject matter covered on the computer-delivered test
and the paper-delivered test?
No. The computer-delivered test and paper-delivered test cover the same content.

Where and when are the Praxis tests offered?
You can select the test center that is most convenient for you. The Praxis tests are administered through an
international network of test centers, which includes some universities, high schools, Prometric® Testing Centers,
and other locations throughout the world.
Testing schedules depend on whether you are taking computer-delivered tests or paper-delivered tests. See the
Praxis Web site for more detailed test registration information at www.ets.org/praxis/register.

The PraxisTM Study Companion

Step 2: Familiarize Yourself with Test Questions

2. Familiarize Yourself with Test Questions
Become comfortable with the types of questions you’ll find on the Praxis tests
The Praxis tests include two types of questions — multiple-choice (for which you select your answers from a
list of choices) and constructed-response (for which you write a response of your own). You may be familiar
with these question formats from taking other standardized tests. If not, familiarize yourself with them so you
don’t spend time during the test figuring out how to answer them.

Understanding Multiple-Choice Questions
Many multiple-choice questions begin with the phrase “which of the following.” Take a look at this example:
Which of the following is a flavor made from beans?
(A) Strawberry
(B) Cherry
(C) Vanilla
(D) Mint

How would you answer this question?
All of the answer choices are flavors. Your job is to decide which of the flavors is the one made from beans.
Try following these steps to select the correct answer.
1) L imit your answer to one of the choices given. You may know that chocolate and coffee are also flavors
made from beans, but they are not listed. Rather than thinking of other possible answers, focus only on the
choices given (“which of the following”).
2) E
liminate incorrect answers. You may know that strawberry and cherry flavors are made from fruit and that
mint flavor is made from a plant. That leaves vanilla as the only possible answer.
3) V
erify your answer. You can substitute “vanilla” for the phrase “which of the following” and turn the question
into this statement: “Vanilla is a flavor made from beans.” This will help you be sure that your answer is
correct. If you’re still uncertain, try substituting the other choices to see if they make sense. You may want to
use this technique as you answer multiple-choice questions on the practice tests.

The PraxisTM Study Companion

Step 2: Familiarize Yourself with Test Questions

Try a more challenging example
The vanilla bean question is pretty straightforward, but you’ll find that more challenging questions have a
similar structure. For example:
Entries in outlines are generally arranged according
to which of the following relationships of ideas?
(A) Literal and inferential
(B) Concrete and abstract
(C) Linear and recursive
(D) Main and subordinate
You’ll notice that this example also contains the phrase “which of the following.” This phrase helps you
determine that your answer will be a “relationship of ideas” from the choices provided. You are supposed to find
the choice that describes how entries, or ideas, in outlines are related.
Sometimes it helps to put the question in your own words. Here, you could paraphrase the question in this way:
“How are outlines usually organized?” Since the ideas in outlines usually appear as main ideas and subordinate
ideas, the answer is (D).

QUICK TIP: Don’t be intimidated by words you may not understand. It might be easy to be thrown by words
like “recursive” or “inferential.” Read carefully to understand the question and look for an answer that fits. An
outline is something you are probably familiar with and expect to teach to your students. So slow down, and
use what you know.

Watch out for multiple-choice questions containing “NOT,” “LEAST,” and “EXCEPT”
This type of question asks you to select the choice that does not fit. You must be very careful because it is easy
to forget that you are selecting the negative. This question type is used in situations in which there are several
good solutions or ways to approach something, but also a clearly wrong way.

How to approach questions about graphs, tables, or reading passages
When answering questions about graphs, tables, or reading passages, provide only the information that the
questions ask for. In the case of a map or graph, you might want to read the questions first, and then look at the
map or graph. In the case of a long reading passage, you might want to go ahead and read the passage first,
marking places you think are important, and then answer the questions. Again, the important thing is to be sure
you answer the questions as they refer to the material presented. So read the questions carefully.

How to approach unfamiliar formats
From time to time, new question formats are developed to find new ways of assessing knowledge. The latest
tests may include audio and video components, such as a movie clip or animation, instead of the more
traditional map or reading passage. Other tests may allow you to zoom in on details of a graphic or picture.
Tests may also include interactive questions that take advantage of technology to assess knowledge and skills.
They can assess knowledge more than standard multiple-choice questions can. If you see a format you are not
familiar with, read the directions carefully. They always give clear instructions on how you are expected to
respond.
For most questions, you will respond by clicking an oval to select a single answer from a list of options. Other
questions may ask you to respond in the following ways:

The PraxisTM Study Companion

Step 2: Familiarize Yourself with Test Questions

• Typing in an entry box. When the answer is a number, you may be asked to enter a numerical answer or, if the
test has an on-screen calculator, you may need to transfer the calculated result from the calculator to the entry
box. Some questions may have more than one place to enter a response.
• Clicking check boxes. You may be asked to click check boxes instead of an oval when more than one choice
within a set of answers can be selected.
• Clicking parts of a graphic. In some questions, you will select your answers by clicking on a location (or
locations) on a graphic such as a map or chart, as opposed to choosing your answer from a list.
• Clicking on sentences. In questions with reading passages, you may be asked to choose your answers by
clicking on a sentence (or sentences) within the reading passage.
• Dragging and dropping answer choices into targets on the screen. You may be asked to select answers
from a list of options and drag your answers to the appropriate location in a table, paragraph of text or graphic.
• Selecting options from a drop-down menu. You may be asked to choose answers by selecting options from
a drop-down menu (e.g., to complete a sentence).

Remember that with every question you will get clear instructions on how to respond. See the Praxis
Computer-delivered Testing Demonstration on the Praxis website to learn more about Praxis tests and to
see examples of some of the types of questions you may encounter.

QUICK TIP: Don’t make the questions more difficult than they are. Don’t read for hidden meanings or tricks.
There are no trick questions on Praxis tests. They are intended to be serious, straightforward tests that accurately
assess your knowledge.

Understanding Constructed-Response Questions
Constructed-response questions require you to demonstrate your knowledge in a subject area by providing
in-depth explanations on particular topics. Essay and problem solving are types of constructed-response
questions.
For example, an essay question might present you with a topic and ask you to discuss the extent to which you
agree or disagree with the opinion stated. You must support your position with specific reasons and examples
from your own experience, observations, or reading.
Take a look at a few sample essay topics:
• “ Celebrities have a tremendous influence on the young, and for that reason, they have a responsibility to
act as role models.”
• “ We are constantly bombarded by advertisements—on television and radio, in newspapers and
magazines, on highway signs, and the sides of buses. They have become too pervasive. It’s time to put
limits on advertising.”
• “Advances in computer technology have made the classroom unnecessary, since students and teachers
are able to communicate with one another from computer terminals at home or at work.”
A problem-solving question might ask you to solve a mathematics problem such as the one below and show
how you arrived at your solution:
a) In how many different ways can 700 be expressed as the product of two positive integers? Show how
you arrived at your answer.
b) A
mong all pairs of positive integers whose product is 700, which pair has the maximum greatest
common divisor? Explain how you arrived at your answer.

The PraxisTM Study Companion

Step 2: Familiarize Yourself with Test Questions

Keep these things in mind when you respond to a constructed-response question
1) A
nswer the question accurately. Analyze what each part of the question is asking you to do. If the
question asks you to describe or discuss, you should provide more than just a list.
2) A
nswer the question completely. If a question asks you to do three distinct things in your response,
you should cover all three things for the best score. Otherwise, no matter how well you write, you will
not be awarded full credit.
3) A
nswer the question that is asked. Do not change the question or challenge the basis of the
question. You will receive no credit or a low score if you answer another question or if you state, for
example, that there is no possible answer.
4) G
ive a thorough and detailed response. You must demonstrate that you have a thorough
understanding of the subject matter. However, your response should be straightforward and not filled
with unnecessary information.
5) R
eread your response. Check that you have written what you thought you wrote. Be sure not to leave
sentences unfinished or omit clarifying information.

QUICK TIP: You may find that it helps to circle each of the details of the question in your test book or take
notes on scratch paper so that you don’t miss any of them. Then you’ll be sure to have all the information you
need to answer the question.
For tests that have constructed-response questions, more detailed information can be found in "4. Learn About
Your Test" on page 11.

Understanding Computer-Delivered Questions
Questions on computer-delivered tests are interactive in the sense that you answer by selecting an option
or entering text on the screen. If you see a format you are not familiar with, read the directions carefully. The
directions always give clear instructions on how you are expected to respond.
Interactive question types may ask you to respond by:
• Typing in an entry box, particularly for a constructed-response question.
• Clicking an oval answer option for a multiple-choice question.
• Clicking on sentences. In questions with reading passages, you may be asked to choose your answer by
clicking on a sentence or sentences within the reading passage.
Perhaps the best way to understand computer-delivered questions is to view the Computer-delivered Testing
Demonstration on the Praxis Web site to learn how a computer-delivered test works and see examples of
some types of questions you may encounter.

The PraxisTM Study Companion

Step 3: Understand Your Scores

3. Understand Your Scores
Understand how tests are scored and how to interpret your test scores
Of course, passing the Praxis test is important to you so you need to understand what your scores mean and
what your state requirements are.

What are the score requirements for my state?
States, institutions, and associations that require the tests set their own passing scores. Visit
www.ets.org/praxis/states for the most up-to-date information.

If I move to another state, will my new state accept my scores?
The Praxis Series tests are part of a national testing program, meaning that they are required in more than one
state for licensure. The advantage of a national program is that if you move to another state that also requires
Praxis tests, you can transfer your scores. Each state has specific test requirements and passing scores, which you
can find at www.ets.org/praxis/states.

How do I know whether I passed the test?
Your score report will include information on passing scores for the states you identified as recipients of your
test results. If you test in a state with automatic score reporting, you will receive passing score information for
that state.
A list of states and their passing scores for each test are available online at www.ets.org/praxis/states.

What your Praxis scores mean
You received your score report. Now what does it mean? It’s important to interpret your score report correctly
and to know what to do if you have questions about your scores.
Visit http://www.ets.org/s/praxis/pdf/sample_score_report.pdf to see a sample score report.
To access Understanding Your Praxis Scores, a document that provides additional information on how to read
your score report, visit www.ets.org/praxis/scores/understand.

Put your scores in perspective
Your score report indicates:
• Your score and whether you passed
• The range of possible scores
• The raw points available in each content category
• The range of the middle 50 percent of scores on the test
•Y
our Recognition of Excellence (ROE) Award status, if applicable
(found at www.ets.org/praxis/scores/understand/roe)
If you have taken the same test or other tests in The Praxis Series over the last 10 years, your score report also lists
the highest score you earned on each test taken.

The PraxisTM Study Companion

Step 3: Understand Your Scores

Content category scores and score interpretation
On many of the Praxis tests, questions are grouped into content categories. To help you in future study or
in preparing to retake the test, your score report shows how many “raw points” you earned in each content
category. Compare your “raw points earned” with the maximum points you could have earned (“raw points
available”). The greater the difference, the greater the opportunity to improve your score by further study.

Score scale changes
E T S updates Praxis tests on a regular basis to ensure they accurately measure the knowledge and skills that are
required for licensure. Updated tests cover the same content as the previous tests. However, scores might be
reported on a different scale, so requirements may vary between the new and previous versions. All scores for
previous, discontinued tests are valid and reportable for 10 years.
These resources may also help you interpret your scores:
• Understanding Your Praxis Scores (PDF), found at www.ets.org/praxis/scores/understand
• T he Praxis Series Passing Scores (PDF), found at www.ets.org/praxis/scores/understand
• State requirements, found at www.ets.org/praxis/states

The PraxisTM Study Companion

Step 4: Learn About Your Test

4. Learn About Your Test
Learn about the specific test you will be taking

Mathematics: Content Knowledge (0061/5061)

Test at a Glance
Test Name

Mathematics: Content Knowledge

Test Code

0061 5061

Time

2 hours

Number of Questions

50 50

2 hours

Format

Multiple-choice questions;
Graphing calculator required

Multiple-choice questions; On-screen
graphing calculator available or bring
your own; the test may include nonscored
research sections that consist of the
following question types: multiple-choice
questions–select one or more answers,
numeric entry questions, drag-and-drop
questions, and text completion questions

Test Delivery

Computer delivered
Approximate
Number of
Questions

Paper delivered

Content Categories

Approximate
Percentage of
Examination

I. Algebra and Number Theory

16%

II. Measurement
II
Geometry
III
Trigonometry

3
5
4

6%
10%
8%

III. Functions
Calculus

8
6

16%
12%

IV. Data Analysis and Statistics
Probability

5–6
2–3

10–12%
4–6%

V. Matrix Algebra
Discrete Mathematics

4–5
3–4

8–10%
6–8%

Process Categories
Mathematical Problem Solving
Distributed Across Content Categories
Mathematical Reasoning and Proof
Mathematical Connections
Mathematical Representation
Use of Technology

The PraxisTM Study Companion

Step 4: Learn About Your Test

About This Test
The Praxis Mathematics: Content Knowledge test
is designed to assess the mathematical knowledge
and competencies necessary for a beginning teacher
of secondary school mathematics. Examinees have
typically completed a bachelor’s program with an
emphasis in mathematics or mathematics education.
The examinee will be required to understand
and work with mathematical concepts, to reason
mathematically, to make conjectures, to see patterns,
to justify statements using informal logical arguments,
and to construct simple proofs. Additionally, the
examinee will be expected to solve problems by
integrating knowledge from different areas of
mathematics, to use various representations of
concepts, to solve problems that have several solution
paths, and to develop mathematical models and use
them to solve real-world problems.
This test may contain some questions that will not
count toward your score.
The test is not designed to be aligned with any
particular school mathematics curriculum, but it is
intended to be consistent with the recommendations
of national studies on mathematics education, such
as the National Council of Teachers of Mathematics
(NCTM) Principles and Standards for School
Mathematics (2000) and the National Council for
Accreditation of Teacher Education (NCATE) Program
Standards for Initial Preparation of Mathematics Teachers
(2003).
Graphing calculators without QWERTY
(typewriter) keyboards are required for this test.
Some questions require the use of a calculator. The
minimum capabilities required of the calculator are
described in the section on graphing calculators.
Because many test questions may be solved in more
than one way, examinees should decide first how to
solve each problem and then decide whether to use a
calculator.

For computer-delivered tests, selected notations,
formulas, and definitions are in the Math Reference
tab and available to you on the screen throughout
the test. For paper-delivered tests, selected notations,
formulas, and definitions are printed in the test book.
They are also provided in chapter 8 of this Study
Companion.

Graphing Calculators
If you are taking the paper-delivered test, you must
bring to the examination a graphing calculator with
the built-in capability to
1. produce the graph of a function within an
arbitrary viewing window;
2. find the zeros of a function;
3. compute the derivative of a function
numerically;
4. compute definite integrals numerically.
If you are taking the computer-delivered test, an
on-screen graphing calulator is provided or you may
bring your own graphing calculator as described
above.
Computers, calculators with QWERTY (typewriter)
keyboards, and electronic writing pads are NOT
allowed when taking the test.
Unacceptable machines also include the following:
• Powerbooks and portable/handheld computers
• Pocket organizers
• Electronic writing pads or pen-input/stylusdriven devices (e.g., Palm, PDAs, Casio Class Pad
300, etc.)
• Devices with QWERTY keyboards (e.g., TI-92
PLUS, Voyage 200, etc.)
• Cell-phone calculators

On the test day, examinees taking the paper-delivered
test should bring a calculator they are comfortable
using. Examinees taking the computer-delivered test
may bring a graphing calculator or use the on-screen
graphing calculator provided.

The PraxisTM Study Companion

Step 4: Learn About Your Test

Praxis Graphing Calculator Policy
Test administration staff will clear the memory of all
graphing calculators both before and after the test
administration.
We recommend that you
• back up any important information in your
calculator’s memory, including applications,
before arriving at the test site;
• know how to clear the memory on the approved
calculator that you plan to use during the test.
Note: Instructions on how to back up and clear
the memory of calculators can be found on various
calculator websites.

On-Screen Graphing Calculator
An on-screen graphing calculator is provided for the
computer-delivered test. Please consult the Praxis
Calculator Use web page for further information.
You are expected to know how and when to use the
graphing calculator since it will be helpful for some
questions. You are expected to become familiar with
its functionality before taking the test. To practice
using the calculator, download the 30-day trial
version and view tutorials on how to use it. The
calculator may be used to perform calculations (e.g.,
exponents, roots, trigonometric values, logarithms),
to graph and analyze functions, to find numerical
solutions to equations, and to generate a table of
values for a function.

Using Your Calculator
Take time to download the 30-day trial version
of the calculator. View the tutorials on the
website. Practice with the calculator so that you are
comfortable using it on the test.

Sometimes answer choices are rounded, so the
answer that you get might not match the answer
choices in the question. Since the answer choices
are rounded, plugging the choices into the question
might not produce an exact answer.
Don’t round any intermediate calculations. For
example, if the calculator produces a result for the first
step of a solution, keep the result in the calculator and
use it for the second step. If you round the result from
the first step and the answer choices are close to each
other, you might choose the incorrect answer.
Read the question carefully so that you know what
you are being asked to do. Sometimes a result from
the calculator is NOT the final answer. If an answer you
get is not one of the choices in the question, it may be
that you didn’t answer the question being asked. Read
the question again. It might also be that you rounded
at an intermediate step in solving the problem.
Think about how you are going to solve the question
before using the calculator. You may only need the
calculator in the final step or two. Don’t use it more
than necessary.
Check the calculator modes (degree versus radian,
floating decimal versus scientific notation) to see that
these are correct for the question being asked.
Make sure that you know how to perform the
basic arithmetic operations and calculations (e.g.,
exponents, roots, trigonometric values, logarithms).
Your test may involve questions that require you
to do some of the following: graph functions and
analyze the graphs, find zeros of functions, find points
of intersection of graphs of functions, find minima/
maxima of functions, find numerical solutions to
equations, and generate a table of values for a
function.

There are only some questions on the test for which
a calculator is helpful or necessary. First, decide how
you will solve a problem, then determine if you need a
calculator. For many questions, there is more than one
way to solve the problem. Don’t use the calculator if
you don’t need to; you may waste time.

The PraxisTM Study Companion

Step 4: Learn About Your Test

Mathematics Content
Descriptions
Representative descriptions of the topics covered in
the content categories for the Mathematics: Content
Knowledge test follow. Because the assessment
is designed to measure the ability to integrate
knowledge of mathematics, answering any question
may involve more than one competency and may
involve competencies from more than one content
area.

Algebra and Number Theory
A. Demonstrate an understanding of the
structure of the natural, integer, rational, real,
and complex number systems and the ability
to perform basic operations (+, –, ×, and ÷)
on numbers in these systems
B. Compare and contrast properties (e.g.,
closure, commutativity, associativity,
distributivity) of number systems under
various operations
C. Demonstrate an understanding of the
properties of counting numbers (e.g., prime,
composite, prime factorization, even, odd,
factors, multiples)
D. Solve ratio, proportion, percent, and average
(including arithmetic mean and weighted
average) problems
E. Work with algebraic expressions, formulas,
and equations; add, subtract, and multiply
polynomials; divide polynomials; add,
subtract, multiply, and divide algebraic
fractions; perform standard algebraic
operations involving complex numbers,
radicals, and exponents, including fractional
and negative exponents
F. Solve and graph systems of equations and
inequalities, including those involving
absolute value
G. Interpret algebraic principles geometrically
H. Recognize and use algebraic representations
of lines, planes, conic sections, and spheres
I. Solve problems in two and three dimensions
(e.g., distance between two points, the
coordinates of the midpoint of a line
segment)

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II. Measurement
A. Make decisions about units and scales that
are appropriate for problem situations
involving measurement; use unit analysis
B. Analyze precision, accuracy, and approximate
error in measurement situations
C. Apply informal concepts of successive
approximation, upper and lower bounds, and
limit in measurement situations
Geometry
D. 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) in two and three dimensions
E.

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

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

Solve problems using the properties of
circles, including those involving inscribed
angles, central angles, chords, radii, tangents,
secants, arcs, and sectors

Understand and apply the Pythagorean
theorem and its converse

Compute and reason about perimeter, area/
surface area, or volume of two- or threedimensional figures or of regions or solids
that are combinations of these figures

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

Step 4: Learn About Your Test

Trigonometry
K. Define and use the six basic trigonometric
functions using degree or radian measure of
angles; know their graphs and be able to
identify their periods, amplitudes, phase
displacements or shifts, and asymptotes
L. Apply the law of sines and the law of cosines
M. Apply the formulas for the trigonometric
x
functions of , 2x, x, x + y , and x − y;
2
prove trigonometric identities
N. Solve trigonometric equations and
inequalities
O. Convert between rectangular and polar
coordinate systems
III. Functions
A. Demonstrate understanding of and ability to
work with functions in various
representations (e.g., graphs, tables, symbolic
expressions, and verbal narratives) and to
convert flexibly among them
B. Find an appropriate family of functions to
model particular phenomena (e.g.,
population growth, cooling, simple harmonic
motion)
C. Determine properties of functions and their
graphs, such as domain, range, intercepts,
symmetries, intervals of increase or decrease,
discontinuities, and asymptotes
D. Use the properties of trigonometric,
exponential, logarithmic, polynomial, and
rational functions to solve problems
E. Determine the composition of two functions;
find the inverse of a one-to-one function in
simple cases and know why only one-to-one
functions have inverses
F. Interpret representations of functions of two
variables, such as three-dimensional graphs,
level curves, and tables

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Calculus
G. Demonstrate understanding of 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
H. Understand the derivative of a function as a
limit, as the slope of a curve, and as a rate of
change (e.g., velocity, acceleration, growth,
decay)
I. Show that a particular function is continuous;
understand the relationship between
continuity and differentiability
J. Numerically approximate derivatives and
integrals
K. Use standard differentiation and integration
techniques
L. Analyze the behavior of a function (e.g., find
relative maxima and minima, concavity); solve
problems involving related rates; solve
applied minima-maxima problems
M. Demonstrate understanding of and ability to
use the Mean Value Theorem and the
Fundamental Theorem of Calculus
N. Demonstrate understanding of integration as
a limiting sum that can be used to compute
area, volume, distance, or other accumulation
processes
O. Determine the limits of sequences and simple
infinite series

Step 4: Learn About Your Test

IV. Data Analysis and Statistics
A. Organize data into a suitable form (e.g.,
construct a histogram and use it in the
calculation of probabilities)
B. Choose and apply appropriate measures of
central tendency (e.g., population mean,
sample mean, median, mode) and dispersion
(e.g., range, population standard deviation,
sample standard deviation, population
variance, sample variance) to describe and
compare data sets; recognize when to use
sample statistics or population parameters
C. Analyze data from specific situations to
determine what type of function (e.g., linear,
quadratic, exponential) would most likely
model that particular phenomenon; use the
regression feature of the calculator to
determine curve of best fit; interpret the
regression coefficients, correlation, and
residuals in context
D. Understand and apply normal distributions
and their characteristics (e.g., mean, standard
deviation)
E. Understand how sample statistics reflect the
values of population parameters and use
sampling distributions as the basis for
informal inference
F. Understand the differences among various
kinds of studies and which types of inferences
can legitimately be drawn from each
G. Know the characteristics of well-designed
studies, including the role of randomization in
surveys and experiments
Probability
H. Understand the concepts of sample space
and probability distribution and construct
sample spaces and distributions in simple
cases
I. Understand the concepts of conditional
probability and independent events;
understand how to compute the probability
of a compound event
J. Compute and interpret the expected value of
random variables in simple cases (e.g., fair
coins, expected winnings, expected profit)
K. Use simulations to construct empirical
probability distributions and to make informal
inferences about the theoretical probability
distribution

The PraxisTM Study Companion

XII. Matrix Algebra
A. Understand vectors and matrices as systems
that have some of the same properties as the
real number system (e.g., identity, inverse, and
commutativity under addition and
multiplication)
B. Scalar multiply, add, subtract, and multiply
vectors and matrices; find inverses of matrices
C. Use matrix techniques to solve systems of
linear equations
D. Use determinants to reason about inverses of
matrices and solutions to systems of
equations
E. Understand and represent translations,
reflections, rotations, and dilations of objects
in the plane by using sketches, coordinates,
vectors, and matrices
Discrete Mathematics
F. Solve basic problems that involve counting
techniques, including the multiplication
principle, permutations, and combinations;
use counting techniques to understand
various situations (e.g., number of ways to
order a set of objects, to choose a
subcommittee from a committee, to visit n
cities)
G. Find values of functions defined recursively
and understand how recursion can be used
to model various phenomena; translate
between recursive and closed-form
expressions for a function
H. Determine whether a binary relation on a set
is reflexive, symmetric, or transitive; determine
whether a relation is an equivalence relation
I. Use finite and infinite arithmetic and
geometric sequences and series to model
simple phenomena (e.g., compound interest,
annuity, growth, decay)
J. Understand the relationship between discrete
and continuous representations and how
they can be used to model various
phenomena
K. Use difference equations, vertex-edge graphs,
trees, and networks to model and solve
problems

Step 4: Learn About Your Test

Mathematical Process Categories
In addition to knowing and understanding the
mathematics content explicitly described in the
Mathematics Content Descriptions section, entrylevel mathematics teachers must also be able to
think mathematically; that is, they must have an
understanding of the ways in which mathematical
content knowledge is acquired and used. Answering
questions on this assessment may involve one or
more of the processes described below, and all of the
processes may be applied to any of the content topics.

Mathematical Problem Solving
A. Solve problems that arise in mathematics and
those involving mathematics in other
contexts
B. Build new mathematical knowledge through
problem solving
C. Apply and adapt a variety of appropriate
strategies to solve problems
Mathematical Reasoning and Proof
A. Select and use various types of reasoning and
methods of proof
B. Make and investigate mathematical
conjectures
C. Develop and evaluate mathematical
arguments and proofs

The PraxisTM Study Companion

Mathematical Connections
A. Recognize and use connections among
mathematical ideas
B. Apply mathematics in context outside of
mathematics
C. Demonstrate an understanding of how
mathematical ideas interconnect and build
on one another
Mathematical Representation
A. Select, apply, and translate among
mathematical representations to solve
problems
B. Use representations to model and interpret
physical, social, and mathematical
phenomena
C. Create and use representations to organize,
record, and communicate mathematical ideas

Use of Technology
A. Use technology appropriately as a tool for
problem solving and analysis
B. Use technology as an aid to understanding
mathematical ideas

Step 5: Determine Your Strategy for Success

5. Determine Your Strategy for Success
Set clear goals and deadlines so your test preparation is focused and efficient
Effective Praxis test preparation doesn’t just happen. You’ll want to set clear goals and deadlines for yourself
along the way. Otherwise, you may not feel ready and confident on test day. A helpful resource is the Strategies
for Success video, which includes tips for preparing and studying, along with tips for reducing test anxiety.

1) Learn what the test covers.
You may have heard that there are several different versions of the same test. It’s true. You may take one
version of the test and your friend may take a different version a few months later. Each test has different
questions covering the same subject area, but both versions of the test measure the same skills and
content knowledge.
You’ll find specific information on the test you’re taking in "4. Learn About Your Test" on page 11, which
outlines the content categories that the test measures and what percentage of the test covers each topic.
Visit www.ets.org/praxis/testprep for information on other Praxis tests.

2) Assess how well you know the content.
Research shows that test takers tend to overestimate their preparedness—this is why some test takers
assume they did well and then find out they did not pass.
The Praxis tests are demanding enough to require serious review of likely content, and the longer you’ve
been away from the content, the more preparation you will most likely need. If it has been longer than a few
months since you’ve studied your content area, make a concerted effort to prepare.

3) Collect study materials.
Gathering and organizing your materials for review are critical steps in preparing for the Praxis tests. Consider
the following reference sources as you plan your study:
• D
id you take a course in which the content area was covered? If yes, do you still have your books or
your notes?
• Does your college library have a good introductory college-level textbook in this area?
• Does your local library have a high school-level textbook?
Study guides are available for purchase for many Praxis tests at www.ets.org/praxis/testprep. Each guide
provides a combination of test preparation and practice, including sample questions and answers
with explanations.

4) Plan and organize your time.
You can begin to plan and organize your time while you are still collecting materials. Allow yourself plenty of
review time to avoid cramming new material at the end. Here are a few tips:
• C
hoose a test date far enough in the future to leave you plenty of preparation time at
www.ets.org/praxis/register/centers_dates.
• Work backward from that date to figure out how much time you will need for review.
• Set a realistic schedule—and stick to it.

The PraxisTM Study Companion

Step 5: Determine Your Strategy for Success

5) Practice explaining the key concepts.
Praxis tests with constructed-response questions assess your ability to explain material effectively. As a
teacher, you’ll need to be able to explain concepts and processes to students in a clear, understandable
way. What are the major concepts you will be required to teach? Can you explain them in your own words
accurately, completely, and clearly? Practice explaining these concepts to test your ability to effectively
explain what you know.

6) Understand how questions will be scored.
Scoring information can be found in "3. Understand Your Scores" on page 9.

7) Develop a study plan.
A study plan provides a road map to prepare for the Praxis tests. It can help you understand what skills and
knowledge are covered on the test and where to focus your attention. Use the study plan template on page
23 to organize your efforts.
And most important—get started!

Would a Study Group Work for You?
Using this guide as part of a study group
People who have a lot of studying to do sometimes find it helpful to form a study group with others who are
working toward the same goal. Study groups give members opportunities to ask questions and get detailed
answers. In a group, some members usually have a better understanding of certain topics, while others in the
group may be better at other topics. As members take turns explaining concepts to one another, everyone
builds self-confidence.
If the group encounters a question that none of the members can answer well, the group can go to a teacher or
other expert and get answers efficiently. Because study groups schedule regular meetings, members study in a
more disciplined fashion. They also gain emotional support. The group should be large enough so that multiple
people can contribute different kinds of knowledge, but small enough so that it stays focused. Often, three to
six members is a good size.
Here are some ways to use this guide as part of a study group:

• Plan the group’s study program. Parts of the study plan template, beginning on page 21 can help
to structure your group’s study program. By filling out the first five columns and sharing the worksheets,
everyone will learn more about your group’s mix of abilities and about the resources, such as textbooks, that
members can share with the group. In the sixth column (“Dates I will study the content”), you can create an
overall schedule for your group’s study program.
• Plan individual group sessions. At the end of each session, the group should decide what specific
topics will be covered at the next meeting and who will present each topic. Use the topic headings and
subheadings in the Test at a Glance table on page 11 to select topics, and then select practice questions,
beginning on page 31.
• Prepare your presentation for the group. When it’s your to turn present, prepare something that is more
than a lecture. Write two or three original questions to pose to the group. Practicing writing actual questions
can help you better understand the topics covered on the test as well as the types of questions you will
encounter on the test. It will also give other members of the group extra practice at answering questions.

The PraxisTM Study Companion

Step 5: Determine Your Strategy for Success

• T ake the practice test together. The idea of the practice test is to simulate an actual administration of the
test, so scheduling a test session with the group will add to the realism and may also help boost everyone’s
confidence. Remember, complete the practice test using only the time that will be allotted for that test on
your administration day.
• Learn from the results of the practice test. Score one another’s answer sheets. For tests that contain
constructed-response questions, look at the Sample Test Questions section, which also contain sample
responses to those questions and shows how they were scored. Then try to follow the same guidelines that
the test scorers use.
• B
e as critical as you can. You’re not doing your study partner(s) any favors by letting them get away with
an answer that does not cover all parts of the question adequately.
• B
e specific. Write comments that are as detailed as the comments about the sample responses. Indicate
where and how your study partner(s) are doing an inadequate job of answering the question. Writing notes
in the margins of the answer sheet may also help.
• B
e supportive. Include comments that point out what your study partner(s) got right.
Then plan one or more study sessions based on aspects of the questions on which group members performed
poorly. For example, each group member might be responsible for rewriting one paragraph of a response in
which someone else did an inadequate job.
Whether you decide to study alone or with a group, remember that the best way to prepare is to have an
organized plan. The plan should set goals based on specific topics and skills that you need to learn, and it
should commit you to a realistic set of deadlines for meeting those goals. Then you need to discipline yourself
to stick with your plan and accomplish your goals on schedule.

The PraxisTM Study Companion

Step 6: Develop Your Study Plan

6. Develop Your Study Plan
Develop a personalized study plan and schedule
Planning your study time is important because it will help ensure that you review all content areas covered on the
test. Use the sample study plan below as a guide. It shows a plan for the Praxis I® Pre-Professional Skills Test: Reading
test. Following that is a study plan template that you can fill out to create your own plan. Use the “Learn about Your
Test” and “Topics Covered” information beginning on page 11 to help complete it.
Use this worksheet to:
1. Define Content Areas: List the most important content areas for your test as defined in the Topics Covered section.
2. Determine Strengths and Weaknesses: Identify your strengths and weaknesses in each content area.
3. Identify Resources: Identify the books, courses, and other resources you plan to use for each content area.
4. Study: Create and commit to a schedule that provides for regular study periods.
Praxis Test Name:
Praxis Test Code(s):
Test Date:

Praxis I Pre-Professional Skills Test: Reading
0710
11/15/12

Description
of content

How well do
I know the
content?
(scale 1–5)

Main Ideas

Identify summaries
or paraphrases of
main idea or primary
purpose of reading
selection

Middle school
English text
book

College library,
middle school
teacher

9/15/12

Supporting Ideas

Identify summaries
or paraphrases of
supporting ideas and
specific details in
reading selection

Middle school
English text
book

College library,
middle school
teacher

9/17/12

Organization

Identify how reading
selection is organized
in terms of cause/
effect and compare/
contrast

Middle and
high school
English text
book

College library,
middle and
high school
teachers

9/20/12

9/21/12

Organization

Identify key transition
words/phrases in
reading selection and
how used

Middle and
high school
English text
book

College library,
middle and
high school
teachers

9/25/12

9/26/12

Vocabulary in
Context

Identify meanings
of words as used in
context of reading
selection

Middle and
high school
English
text book,
dictionary

College library,
middle and
high school
teachers

9/25/12

9/27/12

Content covered

What
resources do I
have/need for
the content?

Where can I
find the
resources I
need?

Dates I will
study the
content

Date
completed

Literal Comprehension

(continued on next page)

The PraxisTM Study Companion

Step 6: Develop Your Study Plan

Content covered

Description
of content

How well do
I know the
content?
(scale 1–5)

What
resources do I
have/need for
the content?

Where can I
find the
resources I
need?

High school
text book,
college course
notes

College library,
course notes,
high school
teacher, college
professor

10/1/12

High school
text book,
college course
notes

College library,
course notes,
high school
teacher, college
professor

10/1/12

High school
text book,
college course
notes

College library,
course notes,
high school
teacher, college
professor

10/1/12

High school
text book,
college course
notes

College library,
course notes,
high school
teacher, college
professor

10/1/12

High school
text book,
college course
notes

College library,
course notes,
high school
teacher, college
professor

10/8/12

High school
text book,
college course
notes

College library,
course notes,
high school
teacher, college
professor

10/8/12

High school
text book,
college course
notes

College library,
course notes,
high school
teacher, college
professor

10/15/12

10/17/12

High school
text book,
college course
notes

College library,
course notes,
high school
teacher, college
professor

10/22/12

10/24/12

High school
text book,
college course
notes

College library,
course notes,
high school
teacher, college
professor

10/24/12

High school
text book,
college course
notes

College library,
course notes,
high school
teacher, college
professor

10/27/12

Dates I will
study the
content

Date
completed

Critical and Inferential Comprehension

Evaluation

Determine
whether evidence
strengthens,
weakens, or
is relevant to
arguments in reading
selection

Evaluation

Determine role that
an idea, reference, or
piece of information
plays in author’s
discussion/argument

Evaluation

Determine if
information
presented is fact or
opinion

Evaluation

Identify relationship
among ideas
presented in reading
selection

Inferential
Reasoning

Draw inferences/
implications from
directly stated
content of reading
selection

Inferential
Reasoning

Determine logical
assumptions on
which argument or
conclusion is based

Inferential
Reasoning

Determine author’s
attitude toward
materials discussed
in reading selection

Generalization

Recognize or predict
ideas/situations that
are extensions of, or
similar to, what has
been presented in
reading selection

Generalization

Draw conclusions
from materials
presented in reading
selection

Generalization

Apply ideas
presented in a
reading selection to
other situations

The PraxisTM Study Companion

Step 6: Develop Your Study Plan

My Study Plan
Use this worksheet to:
1. Define Content Areas: List the most important content areas for your test as defined in the Learn about Your Test
and Topics Covered sections.
2. Determine Strengths and Weaknesses: Identify your strengths and weaknesses in each content area.
3. Identify Resources: Identify the books, courses, and other resources you plan to use for each content area.
4. Study: Create and commit to a schedule that provides for regular study periods.
Praxis Test Name:
Praxis Test Code:
Test Date:

Content covered

____________________________________________________________
_____________
_____________

Description
of content

How well do
I know the
content?
(scale 1–5)

What
resources do I
have/need for
the content?

Where can I
find the
resources I
need?

Dates I will
study the
content

Date
completed

(continued on next page)

The PraxisTM Study Companion

Step 6: Develop Your Study Plan

Content covered

The PraxisTM Study Companion

Description
of content

How well do
I know the
content?
(scale 1–5)

What
resources do I
have/need for
the content?

Where can I
find the
resources I
need?

Dates I will
study the
content

Date
completed

Step 7: Review Smart Tips for Success

7. Review Smart Tips for Success
Follow test-taking tips developed by experts
Learn from the experts. Take advantage of the following answers to questions you may have and practical tips
to help you navigate the Praxis test and make the best use of your time.

Should I Guess?
Yes. Your score is based on the number of questions you answer correctly, with no penalty or subtraction for an
incorrect answer. When you don’t know the answer to a question, try to eliminate any obviously wrong answers
and then guess at the correct one. Try to pace yourself so that you have enough time to carefully consider
every question.

Can I answer the questions in any order?
Yes. You can go through the questions from beginning to end, as many test takers do, or you can create your
own path. Perhaps you will want to answer questions in your strongest area of knowledge first and then move
from your strengths to your weaker areas. On computer-delivered tests, you can use the “Skip” function to skip a
question and come back to it later. There is no right or wrong way. Use the approach that works best for you.

Are there trick questions on the test?
No. There are no hidden meanings or trick wording. All of the questions on the test ask about subject matter
knowledge in a straightforward manner.

Are there answer patterns on the test?
No. You might have heard this myth: the answers on multiple-choice tests follow patterns. Another myth is that
there will never be more than two questions with the same lettered answer following each other. Neither myth
is true. Select the answer you think is correct based on your knowledge of the subject.

Can I write in the test booklet or, for a computer-delivered test, on the scratch paper I
am given?
Yes. You can work out problems right on the pages of the booklet or scratch paper, make notes to yourself, mark
questions you want to review later or write anything at all. Your test booklet or scratch paper will be destroyed
after you are finished with it, so use it in any way that is helpful to you. But make sure to mark your answers on
the answer sheet or enter them on the computer.

Smart Tips for Taking the Test
1. For a paper-delivered test, put your answers in the right bubbles. It seems obvious, but be sure that
you fill in the answer bubble that corresponds to the question you are answering. A significant number of
test takers fill in a bubble without checking to see that the number matches the question they
are answering.
2. Skip the questions you find extremely difficult. Rather than trying to answer these on your first pass
through the test, leave them blank and mark them in your test booklet. Pay attention to the time as you
answer the rest of the questions on the test, and try to finish with 10 or 15 minutes remaining so that you

The PraxisTM Study Companion

Step 7: Review Smart Tips for Success

can go back over the questions you left blank. Even if you don’t know the answer the second time you read
the questions, see if you can narrow down the possible answers, and then guess.

3. Keep track of the time. Bring a watch to the test, just in case the clock in the test room is difficult for you
to see. Keep the watch as simple as possible—alarms and other functions may distract others or may violate
test security. If the test center supervisor suspects there could be an issue with your watch, they will ask you
to remove it, so simpler is better! You will probably have plenty of time to answer all of the questions, but if
you find yourself becoming bogged down in one section, you might decide to move on and come back to
that section later.
4. Read all of the possible answers before selecting one. Then reread the question to be sure the answer
you have selected really answers the question. Remember, a question that contains a phrase such as “Which
of the following does NOT …” is asking for the one answer that is NOT a correct statement or conclusion.
5. C
heck your answers. If you have extra time left over at the end of the test, look over each question and
make sure that you have answered it as you intended. Many test takers make careless mistakes that they
could have corrected if they had checked their answers.
6. Don’t worry about your score when you are taking the test. No one is expected to answer all of the
questions correctly. Your score on this test is not analogous to your score on the GRE® or other similar-looking
(but in fact very different) tests. It doesn’t matter on the Praxis tests whether you score very high or barely
pass. If you meet the minimum passing scores for your state and you meet the state’s other requirements for
obtaining a teaching license, you will receive a license. In other words, what matters is meeting the minimum
passing score. You can find passing scores for all states that use The Praxis Series tests at
http://www.ets.org/s/praxis/pdf/passing_scores.pdf or on the Web site of the state for which you are
seeking certification/licensure.
7. Use your energy to take the test, not to get angry at it. Getting angry at the test only increases stress
and decreases the likelihood that you will do your best. Highly qualified educators and test development
professionals, all with backgrounds in teaching, worked diligently to make the test a fair and valid measure
of your knowledge and skills. Your state painstakingly reviewed the test before adopting it as a licensure
requirement. The best thing to do is concentrate on answering the questions.

The PraxisTM Study Companion

Step 8: Practice with Sample Test Questions

8. Practice with Sample Test Questions
Answer practice questions and find explanations for correct answers

Sample Test Questions
This test is available via computer delivery and paper delivery. Other than the delivery method, there is no
difference between the tests. The scope of the test content is the same for both test codes.
To illustrate what the computer-delivered test looks like, the following sample question shows an actual screen
used in a computer-delivered test. For the purposes of this guide, sample questions are provided as they would
appear in a paper-delivered test.

The PraxisTM Study Companion

Step 8: Practice with Sample Test Questions

Selected Notations, Definitions, and Formulas (as provided with the test)

NOTATIONS

( a, b)
{ x : a < x < b}
{ x : a ≤ x < b}
[ a, b)
( a, b]
{ x : a < x ≤ b}
{ x : a ≤ x ≤ b}
[ a, b]
gcd ( m, n ) greatest common divisor of two integers m and n
lcm( m, n ) least common multiple of two integers m and n

[ x]
m ≡ k ( mod n )

greatest integer m such that m ≤ x

f −1

inverse of an invertible function f; (not to be read as 1 )

m and k are congruent modulo n (m and k have the same remainder
when divided by n, or equivalently, m − k is a multiple of n)
f
lim+ f ( x ) right-hand limit of f ( x ) ; limit of f ( x ) as x approaches a from the right
x→a
lim f ( x ) left-hand limit of f ( x ) ; limit of f ( x ) as x approaches a from the left

x → a−

∅

the empty set

x ∈ S

x is an element of set S

S ⊂ T

set S is a proper subset of set T

S ⊆ T

either set S is a proper subset of set T or S = T

complement of set S; the set of all elements not in S that are in some
specified universal set
T \ S
relative complement of set S in set T, i.e., the set of all elements of T that
are not elements of S
S ∪ T

union of sets S and T

S ∩ T

intersection of sets S and T

DEFINITIONS
A relation ℜ on a set S
reflexive if x ℜ x for all x ∈ S
symmetric if x ℜ y ⇒ y ℜ x for all x, y ∈ S
transitive if ( x ℜ y and y ℜ z ) ⇒ x ℜ z for all x, y, z ∈ S
antisymmetric if ( x ℜ y and y ℜ x ) ⇒ x = y for all x, y ∈ S
An equivalence relation is a reflexive, symmetric, and transitive relation.

The PraxisTM Study Companion

Step 8: Practice with Sample Test Questions

FORMULAS
Sum
sin ( x ± y ) = sin x cos y ± cos x sin y
cos ( x ± y ) = cos x cos y  sin x sin y
tan ( x ± y ) =

tan x ± tan y
1  tan x tan y

Half-angle (sign depends on the quadrant of q )
2

sin

1 − cos q
q
=±
2
2

cos

q
1 + cos q
=±
2
2

Range of Inverse Trigonometric Functions
sin −1 x

 p p
 − 2 , 2 

cos −1 x

[0, p ]

tan −1 x

− p , p
 2 2

Law of Sines
sin A sin B sin C
=
=
a
b
c

Law of Cosines
c 2 = a 2 + b 2 − 2ab (cos C )

DeMoivre’s Theorem

(cos q + i sin q )k = cos ( k q ) + i sin ( k q )

The PraxisTM Study Companion

Step 8: Practice with Sample Test Questions

Coordinate Transformation
Rectangular ( x, y ) to polar ( r, q ) : r 2 = x 2 + y 2 ; tan q =

y
x

if x ≠ 0

Polar ( r, q ) to rectangular ( x, y ) : x = r cos q; y = r sin q
Distance from point ( x1 , y1 ) to line Ax + By + C = 0
d=

Ax1 + By1 + C
A2 + B 2

Volume
Sphere with radius r:

4 3
pr
3

Right circular cone with height h and base of radius r: V = 1 pr 2 h
3

Right circular cylinder with height h and base of radius r:

V = pr 2 h

1
Pyramid with height h and base of area B: V = 3 Bh

Right prism with height h and base of area B: V = Bh
Surface Area
Sphere with radius r: A = 4p r 2
Right circular cone with radius r and slant height s:

A = p rs + p r 2

Differentiation
′

( f ( x) g ( x))
′

( f ( g ( x)))

= f ′( x) g ( x) + f ( x) g ′( x)

= f ′ ( g ( x )) g ′ ( x )

 f ( x)  ′ f ′( x) g ( x) − f ( x) g ′( x)
if g ( x ) ≠ 0
2
 g ( x )  =
( g ( x))

Integration by Parts

∫ u dv = uv − ∫ v du

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Step 8: Practice with Sample Test Questions

The sample questions that follow illustrate the kinds of questions
on the test. They are not, however, representative of the entire
scope of the test in either content or difficulty. Answers with
explanations follow the questions.

Directions: Each of the questions or statements below is
followed by four suggested answers or completions. Select
the one that is best in each case.

Algebra and Number Theory
1. Jerry is 50 inches tall and is growing at the
1
rate of
inch per month. Adam is 47 inches
24
1
tall and is growing at the rate of
inch per
8
month. If they each continue to grow at these
rates for the next four years, in how many
months will they be the same height?
(A) 24
(B) 30

4. A taxicab driver charges a fare of $2.00 for the
first quarter-mile or less and $0.75 for each
quarter-mile after that. Which of the following
equations models the fare, f , in dollars, for a
ride m miles long, where m is a positive
integer?
(A) f ( m ) = 2.00 + 0.75 ( m − 1)

m 
− 1
4 

(B) f ( m ) = 2.00 + 0.75 

(

(D) f ( m ) = 2.00 + 0.75 4 ( m − 1)

)

5. For which of the following values of k does the
4
2
equation x − 4 x + x + k = 0 have four
distinct real roots?
I. −2

II. 1

(D) 42

III. 3
408

2. What is the units digit of 33

(A) II only

(A) 1

(B) III only

(B) 3

(D) I, II, and III

(D) 9
2

3. If x and y are even numbers and z = 2x + 4y ,
then the greatest even number that must be a
divisor of z is
(A) 2
(B) 4
(C) 8
(D) 16

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Step 8: Practice with Sample Test Questions

Measurement

Geometry

7. For how many angles θ , where 0 < θ ≤ 2π ,
will rotation about the origin by angle θ map
the octagon in the figure above onto itself?
6. The inside of a rectangular picture frame
measures 36 inches long and 24 inches wide.
The width of the frame is x inches, as shown
in the figure above. When hung, the frame and
its contents cover 1,408 square inches of wall
space. What is the length, y, of the frame, in
inches?

(A) One
(B) Two
(C) Four
(D) Eight

(A) 44
(B) 40
(C) 38
(D) 34

8. In the circle above with center O and radius
2, AP has length 3 and is tangent to the circle
at P. If CP is a diameter of the circle, what is
the length of BC ?
(A) 1.25
(B) 2
(C) 3.2
(D) 5

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Step 8: Practice with Sample Test Questions

Trigonometry

Functions

9. If y = 5 sin x − 6 , what is the maximum value
of y?

12. At how many points in the xy-plane do the
5
2
graphs of y = 4x − 3x −1 and
y = −0.4 − 0.11x intersect?

(A) −6
(B) −1

(A) One

(B) Two

(D) 5

10. In ∆ABC (not shown), the length of side AB is
12, the length of side BC is 9, and the
measure of angle BAC is 30°. What is the
length of side AC ?
(A) 17.10
(B) 4.73
(C) 3.68
(D) It cannot be determined from the
information given
11. In the xy-plane, an acute angle with vertex at
the origin is formed by the positive x-axis and
the line with equation y = 3x. What is the
slope of the line that contains the bisector of
this angle?

(i)

the graph of the function f ( x ) is the line
with slope 2 and y-intercept 1
and

(ii) the graph of the function g ( x ) is the line
with slope -2 and y-intercept -1,
which of the following is an algebraic
representation of the function y = f (g(x)) ?
(A) y = 0
(B) y = −4 x − 3
(C) y = −4 x − 1
(D) y = − ( 2 x + 1)

(A) 3
(B)

13. If

3
2
10 + 1
3
10 − 1
3

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Step 8: Practice with Sample Test Questions

Calculus

P ( t ) = 250 ⋅ ( 3.04 )1.98

14. At the beginning of 1990, the population of
rabbits in a wooded area was 250. The
function above was used to model the
approximate population, P, of rabbits in the
area t years after January 1, 1990. According
to this model, which of the following best
describes how the rabbit population changed
in the area?
(A) The rabbit population doubled every 4
months.
(B) The rabbit population tripled every 6
months.
(C) The rabbit population doubled every 36
months.

16. The figure above is a graph of a differentiable
function f . Which of the following could be
the graph of the first derivative of this
function?

(D) The rabbit population tripled every 24
months.

(A) (B)

15. If f ( x ) = 3 x , what are all real values of a
2

and b for which the graph of g ( x ) = ax + b
2

is below the graph of f ( x ) for all values
of x ?
(A) a ≥ 3 and b is positive.
(B) a ≤ 3 and b is negative.

17. If

lim f ( x ) = 0 and lim g ( x ) = 0 what can
x →c

x →c

be concluded about the value of lim
x →c

f ( x)

g ( x)

(A) The value is not finite.
(B) The value is 0.
(C) The value is 1.
(D) The value cannot be determined from the
information given.

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Step 8: Practice with Sample Test Questions

18. In a certain chemical reaction, the number of
grams, N, of a substance produced t hours
after the reaction begins is given by
N ( t ) = 16t − 4t 2 , where 0 < t < 2. At what
rate, in grams per hour, is the substance being
produced 30 minutes after the reaction
begins?
(A) 7
(B) 12
(C) 16
(D) 20

Data Analysis and Statistics
19. The measures of the hand spans of ninthgrade students at Tyler High School are
approximately normally distributed, with a
mean of 7 inches and a standard deviation of
1 inch. Of the following groups of
measurements of hand span, which is
expected to contain the largest number of
ninth-graders?
(A) Less than 6 inches

Stem
9
8
7
6

Leaf
1
2
0
1

3
2
2
3

4
5
4
7

5
6
5
9

7
6
8

8
8

20. The stem plot above shows the course grades
that each of 22 students received in a history
course. The course grade is represented by
using the tens digit of each grade as a stem
and the corresponding units digit as a leaf.
For example, the stem 9 and the leaf 1 in the
first row of the table represent a grade of 91.
What was the median course grade of the 22
students?
(A) 78
(B) 80
(C) 80.7
(D) 82

Probability

(B) Greater than 7 inches
(C) Between 6 and 8 inches
(D) Between 5 and 7 inches

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21. A two-sided coin is unfairly weighted so that
when it is tossed, the probability that heads
will result is twice the probability that tails will
result. If the coin is to be tossed 3 separate
times, what is the probability that tails will
result on exactly 2 of the tosses?
(A)

2
9

(B)

3
8

(C)

4
9

(D)

2
3

Step 8: Practice with Sample Test Questions

Matrix Algebra

Discrete Mathematics

22. The orthogonal projection of 3-space onto the
xy-plane takes the point ( x, y , z ) onto the
point ( x, y , 0 ) . This transformation can be
represented by the matrix equation

24. Given the recursive function defined by

 x  x
M  y  =  y  , where M is which of the
   
 z  0
following matrices?

f (1) = −3,

f ( n ) = f ( n − 1) − 6 for n ≥ 2,

what is the value of f ( 4 ) ?
(A) −2
(B) −9
(C) −10

(D) − 21

0 0
0 0 0


0 0 1

(A)  0

25. For lines in the plane, the relation “is
perpendicular to” is
(A) reflexive but not transitive

(B)  1

0 0
0 1 0


0 0 0

(B) symmetric but not transitive
(C) transitive but not symmetric
(D) both symmetric and transitive

0 0
0 0 0


0 0 1

0 0
0 1 0


0
0
1



(D)  0

( )

1 4
23. For what value of x is the matrix x 6 NOT
invertible?
(A) −

3
2

(B) 0
(C)

3
2

(D) 2

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Step 8: Practice with Sample Test Questions

Answers to Sample Questions
Algebra and Number Theory
1. The heights in this question can be expressed as
two linear equations. Jerry’s height in inches, J, can be
expressed as J = 50 + 1 m, where m is the number of

months from now. Adam’s height in inches, A, can be

1
expressed as A = 47 + m. The question asks, “in how
8
many months will they be the same height?” This is the
same as asking, “for what value of m will J = A ?” The
solution can be found by solving

50 + 1 m = 47 + 1 m for m.
24
8
50 + 1 m = 47 + 1 m
24
8


50 − 47 =  1 − 1  m
 8 24 
 3
1
3=  24 − 24  m

3= 1 m
12
m= 36

The correct answer is (C), 36 months.

2. To find the units digit of 33408 , it is helpful to find the
first few integer powers of 33 and look for a pattern. For
example,

331 = 33
332 = 1089
333 = 35,937
334 = 1185
, ,921
335 = 39,135,393
You can see that the pattern in the units digits is
3, 9, 7, 1, 3, … and that it will continue to repeat with
every four integers of the exponent. Dividing 408 by 4
yields 102 with no remainder. So the units digit of 33408
will be the same as the units digit of 334 , which is 1.
The correct answer is (A).
3. Since 2 is a divisor of both 2x 2 and 4 y 2 , it follows
that 2 is a divisor of z. To find out if there is a greater
even number that must be a divisor of z, you need to
consider the additional information given, which is that
x and y are both even numbers. Since x and y are even
numbers, they can be expressed as x = 2m and y = 2n ,
respectively, where m and n can be either odd or even
integers. Substituting these values for x and y in the
2
2
expression for z yields z = 2(2m) + 4 (2n) . It follows
then that z = 8m2 + 16n2 and that 8 is a divisor of z. The
number 16 would also be a divisor of z if m is even, but
not if m is odd. Since m and n can be either even or odd
and the question asks for the largest even number that
must be a divisor of z, the correct answer is (C), 8.
4. This question asks you to determine which of the
four equations given as choices models the fare for a
taxi ride of m miles, where m is a positive integer. The
question states that the fare is $2.00 for the first quartermile or less and $0.75 for each quarter mile after that.
You will notice by examining the answer choices that all
of the choices include a constant term of 2.00 (for the
first quarter-mile). Thus, the task is to model the fare for
the remaining distance beyond the first quarter-mile.
Since the question states that $0.75 is charged for each
quarter-mile after the first, you must determine how
many quarter-miles the trip is. Since the trip is given as
m miles (where m is an integer), the number of quartermiles in the trip would be 4m. The charge for the first
quarter mile is $2.00, so that would leave 4m –1 quarter
miles to be charged at a rate of $0.75 each. The total fare
for the trip would thus be modeled by the equation
f = 2.00 + 0.75(4m − 1). By comparing this with the
choices given, you will see that the correct answer is (C).

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Step 8: Practice with Sample Test Questions

5. You may recall from your study of solutions to
polynomial equations that a fourth-degree polynomial
has at most four distinct real roots and that the roots of
the equation are the x-intercepts of the graph of the
equation. One way to determine for which of the three
given values of k the equation will have four distinct
real roots is to graph the equations using your
calculator.
I. x 4 − 4 x 2 + x − 2 = 0

II. x 4 − 4 x 2 + x + 1= 0

Measurement
6. This question requires you to use your knowledge of
the area of a rectangle in order to find the outer length,
y, of the picture frame described. You should recall that
the area of a rectangle can be found by multiplying the
length of the rectangle by the width of the rectangle.
The inside dimensions of the frame are given as 36
inches long and 24 inches wide. The width of the frame
is given as x inches, so that the outside dimensions of
the frame would be 36 + 2x inches long and 24 + 2x
inches wide. The area of the rectangle with the outside
dimensions of the frame is given as 1,408 square
inches. This area can then be represented as
(36 + 2x) (24 + 2x) = 1,408). Solving this for x yields

,
(36 + 2x )(24 + 2x ) = 1408
864 + 48 x + 72x + 4 x 2 = 1408
,
2
4 x + 120 x − 544 = 0
x 2 + 30 x − 136 = 0

III. x 4 − 4 x 2 + x + 3 = 0

Using an appropriate viewing window to see the
behavior of the graphs for the three values of k clearly,
you can see that the values of k given in I and III each
result in the equation having only two distinct real
roots. The value of k given in II results in the equation
having four distinct real roots. The correct answer is (A),
II only.

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( x + 34)( x − 4) = 0
x = −34 or x = 4 .
Only x = 4 makes sense in the context of this question,
so the width of the frame is 4 inches and, therefore, the
outer length, y, of the frame is 36 + 2x = 36 + 2( 4) = 44.
The correct answer is (A), 44 inches.

Step 8: Practice with Sample Test Questions

Geometry

Trigonometry

7. The question asks you to consider rotation about
the origin of the octagon in the figure and to
determine for how many angles q , where 0 < θ ≤ 2π,
would rotation of the octagon result in the octagon
being mapped onto itself. One way to begin is to
consider a single point on the octagon, such as the
point (0, 4), at the “top” of the octagon in the figure. This
point is 4 units from the origin, so any rotation that
maps the octagon onto itself would need to map this
point onto a point that is also 4 units from the origin.
The only other point on the octagon that is 4 units
from the origin is the point (0, −4). A rotation of angle
θ = π would map the point (0, 4) onto the point (0, −4).
You can see that the octagon is symmetric about both
the x- and y-axes, so a rotation of angle θ = π would
map all of the points of the octagon onto
corresponding points of the octagon. Likewise, a
rotation of angle θ = 2π would map the point (0, 4)
onto itself (and map all other points of the octagon
onto themselves). No other values of q such that
0 < θ ≤ 2π would map the octagon onto itself.
Therefore, the correct answer is two, choice (B).
8. To determine the length of BC, it would be helpful to
first label the figure with the information given. Since
the circle has radius 2, then both OC and OP have
length 2 and CP has length 4. AP is tangent to the circle
at P, so angle APC is a right angle. The length of AP is
given as 3. This means that triangle ACP is a 3-4-5 right
triangle and AC has length 5. You should also notice
that since CP is a diameter of the circle, angle CBP is
also a right angle. Angle BCP is in both triangle ACP and
triangle PCB and, therefore, the two triangles are similar.
You can then find the length of BC by setting up a
proportion between the corresponding parts of the
similar triangles as follows:

CP = BC
AC PC

4 BC

4
BC = 16 = 3.2
5

The correct answer, 3.2, is (C).

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9. There are two ways to answer this question. You
should be able to use either method. The first solution
is based on reasoning about the function f ( x ) = sin x .
First, you need to recall that the maximum value of
sin x is 1 and, therefore, the maximum value of 5sin x is
5. The maximum value of y = 5sin x − 6 is then
5 − 6 = −1. Alternatively, you could graph the function
y = 5sin x − 6 and find the maximum value of y from
the graph.

The maximum value is –1, and the correct answer is (B).
10. In this question, you are given the length of two
sides of a triangle and the measure of the angle
opposite one of those two sides of the triangle. You are
asked to find the length of the third side of the triangle.
You should recall that the law of sines relates the
lengths of two sides of a triangle and the sines of the
angles opposite the sides. (The law of sines is included
in the Notations, Definitions, and Formulas pages that
are included in this document and at the beginning of
each of the Content Knowledge tests.) Using the law of
sines yields
sin( ∠BAC ) BC
sin30° = 9 .
=
and
sin( ∠BCA) BA
sin( ∠BCA) 12

4
3

2
3

Therefore, sin( ∠BCA) = sin30° = .
You should recall that this is an example of the
ambiguous case of the law of sines—that since the
value of the sine is between 0 and 1, there are two
angles between 0 and 180 degrees, one acute and one
obtuse, associated with this sine and therefore there
are two possible triangles with the given sides and
angle measure.
The two values of the measure of BCA are
approximately 41.8° and 138.2°. Using either the law of
sines again (with BAC and ABC, or with BCA and
ABC ) or the law of cosines, you can determine that
the length of side AC is either approximately 3.68 or
17.10. Since the length of side AC cannot be uniquely
determined, the correct answer is (D), "It cannot be
determined from the information given."

Step 8: Practice with Sample Test Questions

11. To answer this question, it might be helpful to first
draw a figure such as the one shown below.

Simplifying yields
1− cos ( ∠AOB )
2

1+ cos ( ∠AOB )

1+ cos ( ∠AOB )
.

1+ cos ( ∠AOB )

1− cos2 ( ∠AO
OB )

1+ cos ( ∠AOB )
2

sin( ∠AOB )
1+ cos( ∠AOB )
3
10
=
1+ 1
10

Consider the triangle OAB, where O is the origin, A is
the point (1, 3), and B is the point (1, 0). Point A lies on
the line y = 3x. The acute angle described in the
question is the angle AOB. The question asks you to find
the slope of the line that contains the angle bisector of
angle AOB. Let α be the angle between the x-axis and
the angle bisector of angle AOB. Then the slope of the
line that contains the angle bisector of angle AOB will
be equal to tan α. You can use the half-angle formulas
in the Notations, Definitions, and Formulas that begin
page 28, and also are provided when you take the
Mathematics: Content Knowledge test, to find tan α in
terms of the sine and cosine of angle AOB. From your
figure, you can see that OB = 1, AB = 3, and OA = 10 .

3
10 + 1

(

)

3 10 − 1
10 − 1

= 10 − 1
3
.
The correct answer is (D).

Functions
12. To answer this question, you should graph the
equations on your calculator using an appropriate
viewing window and then see how many points of
intersection are shown. The figure below shows one
view of the intersections of the two graphs.

∠AOB 
1− cos ( ∠AOB )
2 
2
tana =
=
 ∠AOB 
1
+
cos
( ∠AOB )
cos 
 2 
2
sin


You should also convince yourself that there are no
additional points of intersection that are not visible in
this viewing window. One way to do that is to verify
that y = 4 x 5 − 3x 2 − 1 has only two relative extrema,
both of which are shown. (Find where y′ = 0.) Only one
point of intersection is shown in the figure above, so
the correct answer is (A).

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Step 8: Practice with Sample Test Questions

13. This question asks you to find an algebraic
representation of the composition of the functions
f ( x ) and g ( x ) . First, you should write algebraic
representations of the individual functions. You are
given the slopes and y-intercepts of the lines that are
the graphs of f ( x ) and g ( x ) . Using the slope-intercept
form of the equation of a line (y = mx + b, where m is
the slope and b is the y-intercept) and the information
given in parts (i) and (ii) yields the following functions,
which have the graphs described in the question:
f (x) = 2x + 1 and g(x) = −2x − 1. These functions imply
that f ( g ( x )) = 2( −2x − 1) + 1= − 4 x − 1. So y = −4 x − 1,
and (C) is the correct answer.
14. In this question, a model is given for the growth of
the rabbit population as a function of time, t, in years.
The question asks for a verbal description of the
change in the rabbit population, based on the function
given. You should recall the meaning of the base
(growth factor) and the exponent in an exponential
growth model. You should note that the function given
t

P (t ) = 250 ⋅ (3.04)1.98 ≈ 250 ⋅ 32. You can observe from
this approximation (with base 3, and exponent

t
) that the population tripled every two years. Thus,
2
the correct answer is (D), “The rabbit population tripled
every 24 months.”

The PraxisTM Study Companion

15. This question is asking about your understanding
of how changing the values of the coefficient a and
y-intercept b in a quadratic function f ( x ) = ax 2 + b
affects the graph of the function. You should recall that
for a > 0, as a decreases, the width of the parabola that
is the graph of y = ax 2 increases, and for a < 0, the
graph opens downward. You should also recall that as
the value of b decreases, the vertex of the graph of

y = ax 2 + b moves in a negative direction along the
y-axis. So for the graph of g ( x ) = ax 2 + b to be below
the graph of f ( x ) = 3x 2 for all values of x, a must be
less than or equal to 3 and b must be negative (the
vertex will be below the vertex of f ( x ), which is at the
origin). The correct answer, therefore, is (B).

Calculus
16. This question asks you to determine the possible
shape of the graph of the first derivative of a
differentiable function from the shape of the graph of
the function. You should recall that the first derivative
of the function at a point is equal to the slope of the
graph of the function at that point. By inspection, you
will see that, starting near x = 0, the slope of the graph
of f ( x ) is negative and becomes less negative as x
approaches a and that the slope is 0 at x = a (at the
minimum value of f ) and then becomes increasingly
positive as x increases. Only (B) is consistent with this
behavior. Therefore, (B) is the correct answer.

Step 8: Practice with Sample Test Questions

17. In a problem such as this, which contains the
answer choice “It cannot be determined from the
information given,” you should be careful to base your
answer on correct reasoning. If you conclude that the
value can be determined, you should base your
conclusion on known mathematical facts or principles;
however, if you conclude that the value cannot be
determined, you should support your conclusion by
producing two different possible values for the limit.
You should recall that the quotient property of limits
states that if lim f ( x ) = L and lim
g ( x ) = M, and if
x →c
x →c

f ( x ) = L . However, this property
M ≠ 0, then lim
x →c g ( x )
M
f (x)
cannot be used to determine lim
for the problem
x →c g ( x )
at hand since the value of lim
g ( x ) is 0 and the quotient
x →c
property is inconclusive in this case. In fact, for this
problem,

f (x) 0
f ( x ) = lim
x →c
lim
= . Note that the expression 0
x →c g ( x ) lim g ( x ) 0
0
x →c

18. In this question, you are given a function, N, that
models the production of a certain chemical reaction
in grams as a function of time, t, in hours. You are asked
to find the rate of production at 30 minutes after the
reaction begins. The rate of production will be equal to
the first derivative of N evaluated at 30 minutes. You
should recognize that you first need to convert 30
minutes into hours and then evaluate the first
derivative of N at that value of t. Since 30 minutes
 1
equals 1 hour, you will need to evaluate N ′  2 .
2

First, find N ′ (t ).

N ′ (t ) = 16 − 8t.
 1
 1
Therefore, N ′  2 = 16 − 8  2 = 12. The answer is 12

grams per hour, so the correct answer is (B).

does not represent a real number; in particular, it is not

f (x)
equal to either 0 or 1. Thus, the value of lim
x →c g ( x )
cannot be determined by using the basic properties of
limits. As a result, you should suspect that, in fact,

f (x)
lim
cannot be determined and verify your hunch
x →c g ( x )
by producing examples to show that the value of the
limit depends on the particular functions f and g.
In the remaining discussion, it will be assumed that
c = 0 . (It is always possible to apply a translation of c
units to the two functions.) You should be aware that,
although both f and g have the limit 0 as x →0, one
function might be approaching 0 more quickly than
the other, which would affect the value of the limit of
the quotient. Thus, if one of the functions is x and the
other x 2 , then the quotient is either x or 1 , and so the

limit of the quotient is either 0 or nonexistent,
respectively. The value of the limit can, in fact, be any
nonzero real number b, as the functions bx and x show.
Thus, answer choices (A), (B), and (C) are incorrect, and
the correct answer is (D).

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Step 8: Practice with Sample Test Questions

Data Analysis and Statistics
19. In this question, you will need to use your
knowledge of a normally distributed set of data. In
particular, you should know that approximately 68
percent of a normally distributed set of data lie within
±1 standard deviation of the mean and that
approximately 95 percent of the data lie within ±2
standard deviations of the mean. The question asks you
to identify which of the groups given in the answer
choices is expected to correspond to the greatest
number of ninth-graders if the hand spans of ninthgraders are approximately normally distributed with a
mean of 7 inches and a standard deviation of 1 inch.
You will need to evaluate each answer choice in order
to determine which of the groups is largest.
(A) is the group of hand spans less than 6 inches. Since
the mean hand span is 7 inches and the standard
deviation is 1 inch, the group of hand spans that is less
than 6 inches is the group that is more than 1 standard
deviation less than the mean. The group of hand spans
that is less than 7 inches includes 50 percent of the
measurements. Approximately 34 percent ( 1 of 68
2

percent) of the measurements are between 6 inches
and 7 inches (within 1 standard deviation less than the
mean). So the group with hand spans less than 6
inches would be approximately equal to 50 − 34, or 16
percent of the measurements.
(B) is the group of hand spans greater than 7 inches.
Since 7 inches is the mean, approximately 50 percent
of the measurements are greater than the mean.
(C) is the group of hand spans between 6 and 8 inches.
This is the group that is within ±1 standard deviation of
the mean. This group contains approximately 68
percent of the measurements.
(D) is the group of hand spans between 5 and 7 inches.
This group is between the mean and 2 standard
deviations less than the mean. Approximately 47.5

1
percent ( of 95 percent) of the measurements are
2
between 5 inches and 7 inches.

20. A stem plot such as the one shown in this question
is a very useful way to display data such as these when
you are interested in determining the median value of
the data. The data in a stem plot is ordered, so finding
the median, the middle number when the data are
ordered from least to greatest or greatest to least, is
straightforward. You are given the course grades
received by 22 students. The median course grade
would be the average of the course grades of the 11th
and 12th students. You can start at either the least or
greatest data entry and count in increasing (or
decreasing) order along the leaves until you reach the
11th and 12th entries. In this case, both the 11th and
12th entries have a value of 82 (i.e., a stem value of 8
and a leaf value of 2). Therefore, the median course
grade received by the 22 students is 82. The correct
answer is (D).

Probability
21. In this question, you are asked to apply your
knowledge of independent events to find the
probability of tossing tails exactly 2 out of 3 times when
using an unfairly weighted coin. Because each toss of
the coin is an independent event, the probability of
tossing heads then 2 tails, P (HTT ), is equal to

P (H ) ⋅ P (T ) ⋅ P (T ), where P (H ) is the probability of
tossing heads and P (T ) is the probability of tossing

tails. In this case, you are given that the probability of
tossing heads is twice the probability of tossing tails.
So, P (H ) = 2 and P (T ) = 1. (Out of 3 tosses, 2 would be
3
3
expected to be heads and 1 would expected to be
   
tails.) Therefore, P (HTT ) =  2  1  1 = 2 . There are 3
 3  3  3 27
ways in which exactly 2 of 3 tosses would be tails and
each of them has an equal probability of occurring:
P (THT ) = P (TTH ) = P (HTT ) = 2 . Therefore, the total
27
probability that tails will result exactly 2 times in 3
tosses is 3 2  = 2 . The correct answer is (A).
 27  9

Of the answer choices given, the group described in (C)
is expected to contain the greatest percent of the
measurements, approximately 68 percent, and would
correspond to the largest number of ninth-graders, so
(C) is the correct answer.

The PraxisTM Study Companion

Step 8: Practice with Sample Test Questions

Matrix Algebra
22. In order to answer this question, you need to
consider how matrix multiplication is performed. You
are asked to find a matrix, M, that when multiplied by
 x

 x

 
 z 

 
 0 

any matrix of the form  y  , yields the result  y  . You
will notice that all of the answer choices are 3 × 3
matrices. You can either solve this problem for the
general case or reason to the answer. First, the general
solution:
 a b c
 x   a b c   x   ax + by + cz 


  

  
M =  d e f  . Then M  y  =  d e f   y  =  dx + ey + fz  for all






 
 
 z   g h j   z   gx + hy + jz 
 g h j






 x
 
 y .
 
 z 

must all be 0. Therefore,

 1 0 0


M =  0 1 0


 0 0 0

and the correct

answer is (B).
23. This question asks you to find the value of x for
which the given matrix is NOT invertible. A matrix is not
invertible if the determinant of the matrix is equal to
 a b
zero. The determinant of the matrix   is equal to
 c d

ad − bc. For the matrix given in the question, the
determinant is equal to (1)(6) − ( 4)( x ). This equals 0

3
2

when 6 − 4 x = 0, or x = . The correct answer is (C).

 x  x
 x
   
 
Since M  y  =  y  for all  y  , then
 
   
 z 
 z   0 

ax + by + cz = y

dx + ey + fz = y for all x, y, and z. This implies a =1,
 gx + hy + jz = 0


b = 0, c = 0; and d = 0, e =1, f = 0; and

g = h = j = 0; and, therefore,

 1 0 0


M =  0 1 0 .


 0 0 0

The correct answer is (B). You could also reason to the
answer by inspecting the answer choices given. Since
 x
 
multiplying the first row of M by the matrix  y  has to
 
 z 

result in only the x term for all x, y, and z, the first entry
in the first row must be 1 and the others 0. Likewise,
 x
 
multiplying the second row of M by the matrix  y  will
 
 z 

result only in the y term for all x, y, and z, so the entries
in the second row must be 0, 1, 0, in that order.
 x
 

Multiplying the third row of M by the matrix  y  results
 
 z 

in 0 for all x, y, and z, so the entries in the third row

The PraxisTM Study Companion

Step 8: Practice with Sample Test Questions

Discrete Mathematics
24. Given the recursive function defined in the
question, in order to find f ( 4), you first need to find

f (2) and f (3). ( f (1) is given.)
Since f (1) = −3 and f (n) = f (n − 1) − 6 for n≥ 2, then

f (2) = −3 − 6 = −9
f (3) = −9 − 6 = −15

f ( 4) = −15 − 6 = −21
The correct answer is (D).

25. To answer this question, you must read each
answer choice and find the statement that correctly
describes the properties of the relation defined as “is
perpendicular to.” You can see that each answer choice
includes two of three properties: reflexivity, symmetry,
or transitivity. It may be most efficient to consider each
of these properties first and then find the statement
that describes these properties correctly for the given
relation. The definition of these properties can be
found in Notation, Definitions, and Formulas page
28. These are also provided when you take the test.
A relation ℜ is reflexive if x ℜy for all x. In this case, a
line cannot be perpendicular to itself, so the relation
given in the question is not reflexive.
A relation ℜ is symmetric if x ℜx ⇒ y ℜx for all x
and y. In this case, if line j is perpendicular to line k, it
follows that line k is perpendicular to line j. This relation
is symmetric.
A relation ℜ is transitive if (x ℜy and y ℜz )⇒ x ℜz for
all x, y, and z. In this case, if line j is perpendicular to
line k and line k is perpendicular to line l, then lines
j and l are either the same line or are parallel to each
other. Thus, line j is not perpendicular to line l. So this
relation is not transitive.
The answer that correctly describes the relation “is
perpendicular to” is (B), “symmetric but not transitive.”

The PraxisTM Study Companion

Step 9: Check on Testing Accommodations

9. Check on Testing Accommodations
See if you qualify for accommodations that may make it easier to take the Praxis test
What if English is not my primary language?
Praxis tests are given only in English. If your primary language is not English (PLNE), you may be eligible for
extended testing time. For more details, visit www.ets.org/praxis/register/accommodations/plne.

What if I cannot take the paper-based test on Saturday?
Monday is the alternate paper-delivered test day for test takers who can’t test on Saturday due to:
• r eligious convictions
• duties as a member of the United States armed forces
Online registration is not available for Monday test takers. You must complete a registration form and provide a
photocopy of your military orders or a letter from your cleric. You’ll find details at
www.ets.org/praxis/register/accommodations/monday_testing.

What if I have a disability or other health-related need?
The following accommodations are available for Praxis test takers who meet the Americans with Disabilities Act
(ADA) Amendments Act disability requirements:
•
•
•
•
•
•
•
•
•
•
•
•
•
•
•

E xtended testing time
Additional rest breaks
Separate testing room
Writer/recorder of answers
Test reader
Sign language interpreter for spoken directions only
Perkins Brailler
Braille slate and stylus
Printed copy of spoken directions
Oral interpreter
Audio test
Braille test
Large print test book (14 pt.)
Large print answer sheet
Listening section omitted

For more information on these accommodations, visit www.ets.org/praxis/register/disabilities.

Note: Test takers who have health-related needs requiring them to bring equipment, beverages, or snacks into
the testing room or to take extra or extended breaks must request these accommodations by following the
procedures described in the Bulletin Supplement for Test Takers with Disabilities or Health-Related Needs (PDF),
which can be found at http://www.ets.org/praxis/register/disabilities.
You can find additional information on available resources for test takers with disabilities or health-related needs
at www.ets.org/disabilities.

The PraxisTM Study Companion

Step 10: Do Your Best on Test Day

10. Do Your Best on Test Day
Get ready for test day so you will be calm and confident
You followed your study plan. You are prepared for the test. Now it’s time to prepare for test day.
Plan to end your review a day or two before the actual test date so you avoid cramming. Take a dry run to the
test center so you’re sure of the route, traffic conditions, and parking. Most of all, you want to eliminate any
unexpected factors that could distract you from your ultimate goal—passing the Praxis test!
On the day of the test, you should:
• be well rested
• wear comfortable clothes and dress in layers
• eat before you take the test and bring food with you to eat during break to keep your energy level up
• bring an acceptable and valid photo identification with you
• bring a supply of well-sharpened No. 2 pencils (at least 3) and a blue or black pen for the essay or
constructed-response questions for a paper-delivered test
• be prepared to stand in line to check in or to wait while other test takers check in
• select a seat away from doors, aisles, and other high-traffic areas
You can’t control the testing situation, but you can control yourself. Stay calm. The supervisors are well trained
and make every effort to provide uniform testing conditions, but don’t let it bother you if the test doesn’t start
exactly on time. You will have the necessary amount of time once it does start.
You can think of preparing for this test as training for an athletic event. Once you’ve trained, prepared, and
rested, give it everything you’ve got.

What items am I restricted from bringing into the test center?
You cannot bring into the test center personal items such as:
• handbags, knapsacks, or briefcases
• water bottles or canned or bottled beverages
• study materials, books, or notes
• scrap paper
• any electronic, photographic, recording, or listening devices

Note: All cell phones, smart phones (e.g., BlackBerry® devices, iPhones®, etc.), PDAs, and other electronic,
photographic, recording, or listening devices are strictly prohibited from the test center. If you are seen with
such a device, you will be dismissed from the test, your test scores will be canceled, and you will forfeit your test
fees. If you are seen USING such a device, the device will be confiscated and inspected. For more information on
what you can bring to the test center, visit www.ets.org/praxis/test_day/bring.

The PraxisTM Study Companion

Step 10: Do Your Best on Test Day

Are You Ready?
Complete this checklist to determine whether you are ready to take your test.
❒ Do you know the testing requirements for the license or certification you are seeking in the state(s) where
you plan to teach?
❒ Have you followed all of the test registration procedures?
❒ Do you know the topics that will be covered in each test you plan to take?
❒ Have you reviewed any textbooks, class notes, and course readings that relate to the topics covered?
❒ Do you know how long the test will take and the number of questions it contains?
❒ Have you considered how you will pace your work?
❒ Are you familiar with the types of questions for your test?
❒ Are you familiar with the recommended test-taking strategies?
❒ Have you practiced by working through the practice questions in this study companion or in a study
guide or practice test?
❒ If constructed-response questions are part of your test, do you understand the scoring criteria for
these items?
❒ If you are repeating a Praxis test, have you analyzed your previous score report to determine areas where
additional study and test preparation could be useful?
If you answered “yes” to the questions above, your preparation has paid off. Now take the Praxis test, do your
best, pass it—and begin your teaching career!

The PraxisTM Study Companion

Appendix: Other Questions You May Have

Appendix: Other Questions You May Have
Here is some supplemental information that can give you a better understanding of the Praxis tests.

What do the Praxis tests measure?
The Praxis tests measure the specific pedagogical skills and knowledge that beginning teachers need. The
tests do not measure an individual’s disposition toward teaching or potential for success. The assessments are
designed to be comprehensive and inclusive, but are limited to what can be covered in a finite number of
questions and question types. Ranging from Agriculture to World Languages, there are more than 100 Praxis
tests, which contain multiple-choice questions, constructed-response questions, or a combination of both.

What is the difference between Praxis multiple-choice and constructed-response tests?
Multiple-choice tests measure a broad range of knowledge across your content area. Constructed-response
tests measure your ability to provide in-depth explanations of a few essential topics in a given subject area.
Content-specific Praxis pedagogy tests, most of which are constructed-response, measure your understanding
of how to teach certain fundamental concepts in a subject area.
The tests do not measure your actual teaching ability, however. Teaching combines many complex skills that
are typically measured in other ways, including classroom observation, videotaped practice, or portfolios not
included in the Praxis test.

Who takes the tests and why?
Some colleges and universities use the Praxis Core Academic Skills for Educators tests (Reading, Writing, and
Mathematics) to evaluate individuals for entry into teacher education programs. The assessments are generally
taken early in your college career. Many states also require Praxis Core Academic Skills for Educators test scores
as part of their teacher licensing process.
Individuals entering the teaching profession take the Praxis tests as part of the teacher licensing and certification
process required by many states. In addition, some professional associations and organizations require Praxis
tests for professional licensing.

Do all states require these tests?
The Praxis Series tests are currently required for teacher licensure in approximately 40 states and United States
territories. These tests are also used by several professional licensing agencies and by several hundred colleges
and universities. Teacher candidates can test in one state and submit their scores in any other state that requires
Praxis testing for licensure. You can find details at www.ets.org/praxis/states.

What is licensure/certification?
Licensure in any area—medicine, law, architecture, accounting, cosmetology—is an assurance to the public that
the person holding the license possesses sufficient knowledge and skills to perform important occupational
activities safely and effectively. In the case of teacher licensing, a license tells the public that the individual has
met predefined competency standards for beginning teaching practice.
Because a license makes such a serious claim about its holder, licensure tests are usually quite demanding. In
some fields, licensure tests have more than one part and last for more than one day. Candidates for licensure
in all fields plan intensive study as part of their professional preparation. Some join study groups, others study
alone. But preparing to take a licensure test is, in all cases, a professional activity. Because it assesses the entire
body of knowledge for the field you are entering, preparing for a licensure exam takes planning, discipline, and
sustained effort.

The PraxisTM Study Companion

Appendix: Other Questions You May Have

Why does my state require The Praxis Series tests?
Your state chose The Praxis Series tests because they assess the breadth and depth of content—called the
“domain”—that your state wants its teachers to possess before they begin to teach. The level of content
knowledge, reflected in the passing score, is based on recommendations of panels of teachers and teacher
educators in each subject area. The state licensing agency and, in some states, the state legislature ratify the
passing scores that have been recommended by panels of teachers.

How are the tests updated to ensure the content remains current?
Praxis tests are reviewed regularly. During the first phase of review, E T S conducts an analysis of relevant state
and association standards and of the current test content. State licensure titles and the results of relevant
job analyses are also considered. Revised test questions are then produced following the standard test
development methodology. National advisory committees may also be convened to review existing test
specifications and to evaluate test forms for alignment with the specifications.

How long will it take to receive my scores?
Scores for computer-delivered tests are available faster than scores for paper-delivered tests. Scores for most
computer-delivered multiple-choice tests are reported on the screen immediately after the test. Scores for tests
that contain constructed-response questions or essays aren’t available immediately after the test because of the
scoring process involved. Official scores for computer-delivered tests are reported to you and your designated
score recipients approximately two to three weeks after the test date. Scores for paper-delivered tests will be
available within four weeks after the test date. See the test dates and deadlines calendar at
www.ets.org/praxis/register/centers_dates for exact score reporting dates.

Can I access my scores on the Web?
All test takers can access their test scores via their Praxis account free of charge for one year from the posting
date. This online access replaces the mailing of a paper score report.
The process is easy—simply log in to your Praxis account at www.ets.org/praxis and click on your score report.
If you do not already have a Praxis account, you must create one to view your scores.

Note: You must create a Praxis account to access your scores, even if you registered by mail or phone.

The PraxisTM Study Companion

Your teaching career is worth preparing for, so start today!
Let the Praxis Study Companion guide you.
TM

To search for the Praxis test prep resources
that meet your specific needs, visit:

www.ets.org/praxis/testprep

To purchase official test prep made by the creators
of the Praxis tests, visit the E T S Store:

www.ets.org/praxis/store

Copyright © 2013 by Educational Testing Service. All rights reserved. E T S, the E T S logo, LISTENING. LEARNING. LEADING., GRE, PRAXIS
I and PRAXIS II are registered trademarks of Educational Testing Service (E T S). PRAXIS and THE PRAXIS SERIES are trademarks of E T S.
All other trademarks are property of their respective owners. 19117

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