Praxis Mathematics: Content Knowledge (5161) Book + Online

Praxis Mathematics: Content Knowledge (5161) Book + Online

by Sandra Rush

Paperback(Third Edition, Revised)

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Praxis Mathematics: Content Knowledge (5161) Test Prep with Online Practice Tests

3rd Edition - Completely Aligned with the Current Exam

Scoring well on the Praxis Mathematics Content Knowledge (5061) exam doesn't just help you get certified to teach, it helps you build your career! REA's latest Praxis test prep is perfect for college students, teachers, and career-changing professionals who are looking to become Middle School math teachers.

Written by a math expert and fully aligned with the latest test specifications, our in-depth review guides you through all the topics tested on the Content Knowledge exam: number and quantity, algebra, functions, and calculus, and geometry, probability and statistics, and discrete mathematics. Our author hasn't just researched the test, but has actually taken it herself.

REA's prep package features a diagnostic exam and three full-length practice exams based on actual Praxis questions. Each practice test replicates the Praxis question format and is balanced to include every type of question, subject area, and skill tested on the exam. Detailed explanations of answers help you identify your strengths and weaknesses while building your skills.

Our online tests come complete with instant scoring and diagnostic feedback to help you zero in on the topics and types of questions that give you trouble now, so you'll succeed when it counts.

This test prep is a must-have for anyone who wants to become a math teacher!

REA books and software have proven to be the extra support teacher candidates need to pass their challenging tests for licensure. Our comprehensive test preps are teacher-recommended and written by experts in the field.

Product Details

ISBN-13: 9780738612126
Publisher: Research & Education Association
Publication date: 04/03/2017
Series: PRAXIS Teacher Certification Test Prep
Edition description: Third Edition, Revised
Pages: 468
Sales rank: 248,227
Product dimensions: 8.30(w) x 10.90(h) x 1.30(d)
Age Range: 18 Years

About the Author

Sandra Rush is a bestselling author acclaimed for making mathematics easier to grasp. She has taught math and physics at both the secondary and college levels. As an undergraduate majoring in math at Temple University, Ms. Rush tutored members of the men’s basketball team as well as students at Philadelphia public schools, which sparked her interest in teaching.

After contemplating a career in the space sciences and receiving her master’s degree in ionospheric physics from UCLA, she found fulfillment in teaching in the classroom environment as well as in individual tutoring.

Ms. Rush has pursued her academic career in Massachusetts, Colorado, and Arizona. She has written test-preparation guides for all ages, with her approach to books continually informed by her one-on-one tutoring and coaching.

Read an Excerpt


Used in more than 30 states and the District of Columbia, the Praxis Mathematics Content Knowledge (5161) test is by far the most common teacher-certification test in the United States for prospective secondary math teachers.

The 150-minute test contains 60 selected-response, numeric-entry, drag-and-drop and text completion questions covering two wide-ranging content areas:

Content Category I: Number and Quantity, Algebra, Functions, and Calculus (approximately 41 items)

Content Category II: Geometry, Probability and Statistics, and Discrete Mathematics (approximately 19 items)

There’s no doubt about it: The test is a mental workout, requiring you to make educated guesses, distinguish patterns, employ logic, construct simple proofs, pull together strands from various areas of mathematics, distinguish between viable and nonviable solutions, and develop and use mathematical models to solve real-world problems. Beyond that, you may encounter some questions that the test developer, Educational Testing Service, is trying out for future tests and will therefore not count toward your score.

How to Use This Book + Online Prep

This Book + Online prep will equip you to address the full array of question types and the complete range of difficulty you’ll face on the Praxis Math Content Knowledge test. Developed with the needs of teacher candidates like you in mind, this book, along with REA’s online diagnostic tools, helps you customize your prep so you can make the most of your study time. Use all of it or just what you need to brush up in select areas.

Our Praxis Mathematics Content Knowledge Book + Online Prep package includes:

• Expert test tips and strategies

• Focused content review developed just for the Praxis Mathematics Content Knowledge (5161) test

• An in-depth tutorial on how to get the most out of the Infinity Softworks on-screen calculator

• Practice exercises on every selected-response topic

• Advice on how to solve math problems quickly

• An online diagnostic test to help you pinpoint where you need to spend your valuable study time

• Three full-length practice tests (two in the book and one online)

Test Yourself at the Online REA Study Center

The REA Study Center ( is where you’ll find the online material that accompanies this book—the diagnostic test and a third full-length practice test.

We know your time is precious and you want an efficient study experience. Our online content gives you feedback on where you stand right from the start.

Automatic Scoring—Find out how you did on your test, instantly.

Diagnostic Score Reports—Pinpoint the areas that challenge you the most, so you can study effectively.

Detailed Answer Explanations—Learn not just why a response option is correct, but also why the other answer choices are incorrect.

Timed Testing—Manage your time as you practice, so you’ll feel confident on test day.

The Praxis Mathematics Content Knowledge test is computer-based, so practicing online at the REA Study Center will simulate test-day conditions and help you become comfortable with the exam format.

Five Steps to Success

We suggest five simple steps to help ensure your success on test day.

Step One: Take REA’s Praxis Math Diagnostic Test

To get you started, we’ve included a diagnostic test at the online REA Study Center. Our customized diagnostic tools will show you how to spend your time and focus your study, highlighting the areas where you need the most help.

Step Two: Evaluate Your Strengths and Weaknesses

Check your diagnostic score report. Wherever you discover weak spots, you’ll know where you need to extend your review. But even where you’re already strong, be sure to read our answer explanations. This is a prime opportunity to reinforce key concepts and hone your skills.

Step Three: Create Your Customized Study Plan

Based on your diagnostic feedback, you’ll know where to focus your prep. We suggest you start your preparation at least five weeks before test day. This will give you plenty of time to study the review materials and take the practice tests.

Step Four: Review the Content

Improve your weak areas by reviewing the appropriate content areas in this book.

Step Five: Take the Practice Tests

With three full-length REA practice tests, this Book + Online prep will thoroughly test your subject knowledge and build your test-day confidence.

To be assured you’re on the path to success on the Praxis Math Content Knowledge test, you should aim to earn a score of at least 70% correct on your practice tests. Think of this as a good approximation of a passing score. We also encourage you to take the interactive practice tests available from ETS. For details visit

Now let’s turn to the nuts and bolts of what you need to do now through test day.


Setting Up Your Praxis Online Account

To register for the Praxis 5161, you must first create a Praxis account by setting up a user name and password. Go to must provide a valid mailing address, a valid email address, and an accepted form of payment (credit card, debit card or PayPal® account). In addition, for security reasons, you will be asked to select a security question. Remember the answer that you provide to that question.

If you forget your user name and/or password, click the “Forgot your user name or password?” link under “Returning Users” on the Praxis sign-in page ( You will be prompted to enter information about yourself, and to answer the security question you selected when creating your account. For security reasons, user names or passwords are not sent via email, so you will have to create a new password.

You must log into your Praxis account online for almost all interactions regarding registering, ordering test materials, making any necessary changes to or updating the personal information you provided, and viewing or ordering your scores.

Options for Registration

Once you have your Praxis account, you can register by mail, phone, or online. There is a $35 surcharge to register by phone, but there is no extra fee for registration by mail or online. You need to determine whether your state requires the Praxis Math Content Knowledge test (go to Then find out where and when the Praxis 5161 test will be given in your area (see Choose the date and time that is compatible with your schedule. ETS provides real-time seat assignments. You will be advised of seat availability when you select a test center. You must complete the registration process, including payment, and check out to guarantee your seat assignment.

NOTE: Do not wait until the last minute to register for the test. You need to register well ahead of the test date to guarantee your seat. Go to

If you register by mail (ETS-Praxis, PO Box 382065, Pittsburgh, PA 15251-8065) or by phone (800-772-9476), ETS will mail you an admission ticket. If you register online through your Praxis account (, your admission ticket will not be mailed to you. You can always print your admission ticket through your Praxis account. Pay for the test on your Praxis account. Your test scores will not be released until your account is paid in full

Tickets are not required for admission—they are provided only for the test-taker’s information. However, the information on the ticket is what is in the ETS system for your account, so if there is an error on your admission ticket, such as a simple misspelling, corrections may be made online. After any change, including a change in location made by ETS, print an updated admission ticket.

If you find it necessary to change your test center, test date, or test time for a current test registration or to cancel your registration, you can do this up to three days prior to the test date not including the day of your test for a $40 fee online by logging in to your Praxis account or by telephone (800-772-9476).

If you make a change, however, ETS cannot guarantee that your requested testing time or test center will be available. If you completely cancel your original test by the appropriate deadline, however, you may be eligible for a refund of 50 percent of your test fees. If you are absent from the test (without canceling), your fees are automatically forfeited.

Ordering Materials
You can order test prep materials, such as the ETS Interactive Practice Test or Praxis webinars described in the section “Preparation for the Test” in this chapter, and view your ordering history through your Praxis account.

Changing Your Personal Information
Use your Praxis account to provide demographic information about yourself, to update your personal information or profile, or to make corrections.

Viewing Your Scores
You can view your scores, order additional score reports, or select recipients for your test scores (if applicable) through your Praxis account.


You are expected to know the math to get to the point of calculating the answer, for which the Infinity Softworks on-screen graphing calculator is invaluable to find exponents, roots, trigonometric values, and logarithms, and to solve equations. In addition, the graphing feature of the online calculator provides a means to easily graph and analyze functions as well as generate a table of values for a function. We’ll run through all the features of the calculator in Chapter 2.

Review the math topics presented in the chapters of this book. The Praxis 5161 test is computer-based, but other formats are possible for test-takers approved for accommodations. You can access the online calculator to help answer any question, but the majority of questions can be answered without using this calculator. It is a good idea to get the 90-day trial online graphing calculator from Infinity ( before you start these chapters so you can determine what types of questions require using the calculator and how to do that quickly and efficiently.

Infinity’s URL also provides a link to the manufacturer’s manual and four online video tutorials on its usage. Read the calculator chapter in this book for an overview of the Infinity Softworks graphing calculator and easy-to-follow instructions for its use, including many screenshots to help you follow the instructions. It is up to you to allow yourself plenty of time to become familiar with the online calculator. The 90 days that Infinity Softworks gives you to learn how to use the calculator is not excessive.

With the wealth of material presented in this book, it’s a good idea to make your first stop our online diagnostic test at Once you discover your weak points, you’ll be in a good position to brush them up one by one. To aid you in this process, this book is fully indexed.

You can obtain free or paid books from ETS at The ETS Interactive Practice Test for Content Knowledge is a full-length interactive practice test that consists of one set of authentic test questions in an environment that simulates the computer-delivered test, including a timer. You can move easily from question to question just like on the real test. After completing the practice test, you can see the correct answers as well as their explanations and you can view your results by content category. The ETS Interactive Practice

Test is available for either 90 days from initial use or up to 10 total uses. There is only one version available for the Praxis 5161, so purchasing more than one Praxis 5161 practice test will not give you new or additional practice questions.

ETS offers free webinars ( that deliver in-depth information and demonstrations about ETS test preparation products and services. Each live webinar is interactive, so you have an opportunity to ask questions. View the schedule for more information regarding the live webinars.


You should plan to arrive at the test site at the time noted on your admission ticket. If you drive to the site, be sure you understand where to park—you cannot leave the test, once started, to “feed the meter.” You do not have to have your ticket to be admitted to the test site, but you must have a photo ID that contains your name, signature, and photograph. You must present this as soon as you arrive at the test site. Requirements may vary from site to site; see

You cannot bring food, drinks (including water bottles), scratch paper, study materials, or cell phones into the testing center—only your ID. Your pockets must be empty; leave your cell phone, calculator, and watch at home or store them in a locker at the site or in your car.

Phones, watches, or other electronic, photographic, scanning, recording, and listening devices are strictly prohibited in the test center. Possession of any of these devices at the testing center will result in your being dismissed from the test, your test scores from all tests taken that day will be canceled, and you will forfeit your test fees. In addition, the device is subject to confiscation. For more information, see

Many testing sites are Prometric testing centers, which typically have lockers. The designation STN in front of a testing center name indicates an educational institution or testing center similar to Prometric, but it may not have lockers or storage facilities.

After you have identified yourself, you will be escorted to a test center administrator (TCA) for the check-in procedure. The administrator will check your photo ID, and you should be prepared for the ID procedure to also include fingerprinting, photographing, or ome other form of ID verification, although this secondary verification is not always deemed necessary. To ensure security, prior to entering the testing room, you must show that your pockets are empty, and you may even be subjected to a handheld metal detector. Then you must write and sign a confidentiality statement and a test-taker roster.

The administrator will explain the testing procedures and rules for the exam and answer all of your questions. The administrator will provide everything you need during the exam, including scratch paper and pencil.

During the Test

The TCA will then escort you to your assigned test station and computer. Each test station has a video camera that is continually monitored by a TCA, so be aware that your every move is recorded.

Get familiar and comfortable with your computer and mouse. Put the scratch paper and pencil that the TCA gave to you in a convenient location. When the session begins, you will be shown several screens. Click continue to go from one screen to the next. The first screen contains your personal data (name, ID, title of test) and your photo. Be sure that the information is correct and then log in. (If it is in error, raise your hand and the TCA will come to you.)

The next set of screens present general directions and testing tool information, including test navigation and how to answer various types of questions, as well as how to mark specific questions so you can come back to them during the test. Also included are directions for accessing the specific test and math resource information, such as the math reference sheets and the online graphing calculator. The last screen is a confidentiality agreement. When you click continue after this last screen, the test (and the clock) will begin.

A TCA will walk around the room during test time. If you have any questions during the test, raise your hand and the TCA will come to you to help you. Most centers do not offer a break during the test, even though it is 150 minutes long. If you need to take a break, the clock will not stop. You will need to sign out and back in again, so be sure to take your photo ID with you. You may even need to go through another security procedure (empty pockets, metal detector) before reentering the testing room.

Table of Contents

About the Author
About REA

Chapter 1: Getting Started
Devising Your Plan of Attack
How to Use This Book + Online Prep
Registering for the Test
Setting Up Your Praxis Online Account
Preparing for the Test
What to Expect on Test Day
During the Test
Diagnostic Test

Chapter 2: All About the Online Calculator
The Basics
Standard Keypad
Storing and Recalling
Mode Choices
Other Choices
Navigate and Data
Entering Data
My Data
My Graphs
Graphing More than One Function
How the Online Calculator Can Get Solutions Quickly

Chapter 3: Refresher on Numbers and Quantity
3.1 The Number Line
3.2 The Complex Number System
3.3 Closure of Rational and Irrational Numbers
3.4 Properties of Real Numbers
3.5 Order of Operations
3.6 Types of Numbers
3.7 Fractions, Decimals, and Percentages
3.8 Powers
3.9 Radical Expressions
3.10 Quantity
3.11 Measurement
3.12 Vectors and Matrices
Practice Exercises
Practice Exercise Solutions

Chapter 4: Algebra
4.1 Expressions
4.2 Linear Equations
4.3 Linear Inequalities
4.4 Multiplying Binomials
4.5 Quadratic Equations
4.6 Polynomials
Practice Exercises
Practice Solutions

Chapter 5: Geometry
5.1 Definitions
5.2 Parallel and Perpendicular Lines
5.3 Segmenting Lines
5.4 Triangles
5.5 Quadrilaterals
5.6 Polygons
5.7 Circles
5.8 Conic Sections
5.9 Transformations in the Coordinate Plane
5.10 Similarity, Dilation, Congruence, Mapping
5.11 Constructions
5.12 Three-Dimensional Figures
5.13 Trigonometry
Practice Exercises
Practice Solutions

Chapter 6: Functions
6.1 Definitions and Criteria
6.2 Types of Functions
6.3 Transformations of Functions
6.4 Composition of Functions
6.5 The Inverse of a Function
6.6 Functions of Two Variables
Practice Exercises
Practice Solutions

Chapter 7: Calculus
7.1 Limits
7.2 Derivatives and Differentiation
7.3 Integrals
Practice Exercises
Practice Solutions

Chapter 8: Probability and Statistics
8.1 Probability
8.2 Expected Value
8.3 Counting Principle
8.4 Statistics
8.5 Comparing Data Sets
8.6 Normal Distributions
Practice Exercises
Practice Solutions

Chapter 9: Discrete Mathematics
9.1 Sequences and Series
9.2 Set Theory
9.3 Binary Relations
9.4 Logic
9.5 Counting Techniques
Practice Exercises
Practice Solutions

Praxis Math Content Knowledge Practice Test 1
Practice Test 1 Answer Sheet
Practice Test 1
Answer Key
Answers and Explanations

Praxis Math Content Knowledge Practice Test 2
Practice Test 2 Answer Sheet
Practice Test 2
Answer Key
Answers and Explanations

Self-Assessment Guides for Practice Tests 1 and 2

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