The Praxis Series ebooks The Official Study Guide Mathematics: Content Knowledge Test Code: 0061 Revised 009 Study Topics Practice Questions Directly From the Test Makers Test-Taking Strategies www.ets.org/praxis
Study Guide for the Mathematics: Content Knowledge Test Revised 009 A PUBLICATION OF ETS
Copyright 009 by Educational Testing Service. All rights reserved. ETS, the ETS logo, and LISTENING. LEARNING. LEADING. are registered trademarks of Educational Testing Service (ETS) in the United States and other countries. THE PRAXIS SERIES is a trademark of ETS.
Table of Contents Study Guide for the Mathematics: Content Knowledge Test
TABLE OF CONTENTS Chapter 1 Background Information on The Praxis Series Assessments............................ 1 Chapter Introduction to the Mathematics: Content Knowledge Test and Suggestions for Using this Study Guide......................................... 5 Chapter 3 Succeeding on Multiple-Choice Questions......................................... 10 Chapter 4 Study Topics................................................................ 19 Chapter 5 Mathematics: Content Knowledge Test Practice Questions............................... 5 Chapter 6 Right Answers and Explanations for the Practice Questions............................. 45 Chapter 7 Are You Ready? Last-Minute Tips................................................ 81 Appendix A Study Plan Sheet............................................................. 84 Appendix B For More Information......................................................... 86
Chapter 1 Background Information on The Praxis Series Assessments
CHAPTER 1 What Are The Praxis Series Subject Assessments? The Praxis Series Subject Assessments are designed by Educational Testing Service (ETS) to assess your knowledge of specific subject areas. They are a part of the licensing procedure in many states. This study guide covers an assessment that tests the knowledge your state licensing agency expects you to have in order to begin a career as a professional mathematics educator. Your state has adopted The Praxis Series tests because it wants to confirm that you have achieved a specified level of mastery in your subject area before it grants you a license to teach in a classroom. The Praxis Series tests are part of a national testing program, meaning that the test covered in this study guide is required in more than one state for licensure. The advantage of a national program is that if you want to move to another state, you can transfer your scores from one state to another. However, each state has specific test requirements and passing scores. If you are applying for a license in another state, you will want to verify the appropriate test and passing score requirements. This information is available online at www.ets.org/praxis/prxstate.html or by calling ETS at 800-77-9476 or 609-771-7395. What Is Licensure? 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. Why Does My State Require The Praxis Series Assessments? Your state chose The Praxis Series Assessments because the tests 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. Praxis Study Guide for the Mathematics: Content Knowledge Test
CHAPTER 1 What Do the Tests Measure? The Praxis Series Subject Assessments are tests of content knowledge. They measure your understanding and skills in a particular subject area. 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 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. They measure your knowledge of a subject and of how to teach it. The teachers in your field who help us design and write these tests, and the states that require them, do so in the belief that knowledge of your subject area is the first requirement for licensing. Teaching combines many complex skills, only some of which can be measured by a single test. While the tests covered in this study guide measure content knowledge, your teaching ability is a skill that is typically measured in other ways; for example, through observation, videotaped practice, or portfolios. How Were These Tests Developed? ETS began the development of The Praxis Series Subject Assessments with a survey. For each subject, teachers around the country in various teaching situations were asked to judge which knowledge and skills a beginning teacher in that subject needs to possess. Professors in schools of education who prepare teachers were asked the same questions. These responses were ranked in order of importance and sent out to hundreds of teachers for review. All of the responses to these surveys (called job analysis surveys ) were analyzed to summarize the judgments of these professionals. From their consensus, we developed guidelines, or specifications, for the multiplechoice and constructed-response tests. Each subject area had a committee of practicing teachers and teacher educators who wrote the specifications, which were reviewed and eventually approved by teachers. From the test specifications, groups of teachers and professional test developers created test questions that met content requirements and satisfied the ETS Standards for Quality and Fairness.* When your state adopted The Praxis Series Subject Assessments, local panels of practicing teachers and teacher educators in each subject area met to examine the tests and to evaluate each question for its relevance to beginning teachers in your state. This is called a validity study because local practicing teachers validate that the test content is relevant to the job. For the test to be adopted in your state, teachers in your state must judge that it is valid. During the validity study, the panel also provides a passing-score recommendation. This process includes a rigorous review to determine how many of the test questions a beginning teacher in that state would be able to answer correctly. Your * ETS Standards for Quality and Fairness (003, Princeton, NJ) are consistent with the Standards for Educational and Psychological Testing, industry standards issued jointly by the American Educational Research Association, the American Psychological Association, and the National Council on Measurement in Education (1999, Washington, DC). Praxis Study Guide for the Mathematics: Content Knowledge Test 3
CHAPTER 1 state s licensing agency then reviewed the panel s recommendations and made a final determination of the passing-score requirement. Throughout the development process, practitioners in the teaching field teachers and teacher educators participated in defining what The Praxis Series Subject Assessments would cover, which test would be used for licensure in your subject area, and what score would be needed to achieve licensure. This practice is consistent with how professional licensure works in most fields: those who are already licensed oversee the licensing of new practitioners. When you pass The Praxis Series Subject Assessments, you and the practitioners in your state will have evidence that you have the knowledge and skills required for beginning teaching practice. 4 Praxis Study Guide for the Mathematics: Content Knowledge Test
Chapter Introduction to the Mathematics: Content Knowledge Test and Suggestions for Using this Study Guide
CHAPTER Introduction to the Mathematics: Content Knowledge Test The Mathematics: Content Knowledge test is designed for prospective secondary school mathematics teachers. The test is designed to reflect current standards for knowledge, skills, and abilities in mathematics. Educational Testing Service (ETS) works in collaboration with teacher educators, higher education content specialists, and accomplished practicing teachers in the field of mathematics to keep the tests updated and representative of current standards. This guide covers the Mathematics: Content Knowledge test. This test is multiple-choice; that is, it presents questions with several possible answers choices, from which you must choose the best answer and indicate your response on an answer sheet. The Mathematics: Content Knowledge test (0061) consists of 50 multiple-choice questions and covers ten content categories, grouped into five reporting categories, in the following proportions: Grouped Content Categories Percentage of Questions Approximate Number of Questions I. Algebra and Number Theory 16 18% 8 II. Measurement, Geometry, and Trigonometry 6% 1 III. Functions and Calculus 6 30% 14 IV. Data Analysis and Statistics; Probability 14 18% 8 V. Matrix Algebra and Discrete Mathematics 14 18% 8 Process Standards Mathematical Problem Solving Mathematical Reasoning and Proof Mathematical Connections Mathematical Representations Use of Technology Test takers have two hours to complete the test. Please note that a graphing calculator is required for the test. Distributed across all content categories The test is not intended to assess your teaching skills but rather your knowledge in the major areas of mathematics. 6 Praxis Study Guide for the Mathematics: Content Knowledge Test
CHAPTER How to Use This Study Guide This study guide gives you instruction, practice, and test-taking tips to help you prepare for taking the Mathematics: Content Knowledge test. You should turn to chapters 3, 4, 5, and 6 to review the topics likely to be covered on the test review some approaches to solving math problems get tips on succeeding at multiple-choice tests answer practice questions see the answers to the practice questions, along with explanations of those answers So where should you start? All users of this study guide will probably want to begin with the following two steps: Become familiar with the test content. Note what the appropriate chapter of the study guide says about the topics covered in the test you plan to take. Consider how well you know the content in each subject area. Perhaps you already know that you need to build up your skills in a particular area. If you re not sure, skim over the chapters that cover test content to see what topics they cover. If you encounter material that is unfamiliar or difficult, mark those pages to remind yourself to spend extra time in these sections. Also, all users of this study guide will probably want to end with these two steps: Familiarize yourself with test taking. Chapter 3 answers frequently asked questions about multiple-choice tests, such as whether it is a good idea to guess on a test. You can simulate the experience of the test by answering practice questions within the specified time limits. Choose a time and place where you will not be interrupted or distracted. For a multiple-choice test, use the appropriate chapter to score your responses. The scoring key identifies which topic each question addresses, so you can see which areas are your strongest and weakest. Look over the explanations of the questions you missed and see whether you understand them and could answer similar questions correctly. Then plan any additional studying according to what you ve learned about your understanding of the topics and your strong and weak areas. Register for the test and consider last-minute tips. Consult http://www.ets.org/praxis/index.html to learn how to register for the test, and review the checklist in Chapter 7 to make sure you are ready for the test. What you do between these first steps and these last steps depends on whether you intend to use this study guide to prepare on your own or as part of a class or study group. Praxis Study Guide for the Mathematics: Content Knowledge Test 7
CHAPTER Using this study guide to prepare on your own If you are working by yourself to prepare for this test, you may find it helpful to fill out the Study Plan Sheet in appendix A. This work sheet will help you to focus on what topics you need to study most, identify materials that will help you study, and set a schedule for doing the studying. Using this study 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 preparing 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 each other, everyone builds self-confidence. If the group encounters a question that none of the members can answer well, the members can go as a group to a teacher or other expert and get answers efficiently. Because study groups schedule regular meetings, group members study in a more disciplined fashion. They also gain emotional support. The group should be large enough so that various people can contribute various kinds of knowledge, but small enough so that it stays focused. Often, three to six people is a good size. Here are some ways to use this study guide as part of a study group: Plan the group s study program. Parts of the Study Plan Sheet in appendix A can help to structure your group s study program. By filling out the first five columns and sharing the work sheets, 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 planned for study of 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 chapter that covers the topics for the test you will take. Prepare your presentation for the group. When it s your turn to be presenter, prepare something that s more than a lecture. Write five to ten 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. Take 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 will also help boost everyone s confidence. 8 Praxis Study Guide for the Mathematics: Content Knowledge Test
CHAPTER Learn from the results of the practice test. For each test, use the corresponding chapter with the correct answers to score each other s answer sheets. Then plan one or more study sessions based on the questions that group members got wrong. For example, each group member might be responsible for a question that he or she got wrong and could use it as a model to create an original question to pose to the group, together with an explanation of the correct answer modeled after the explanations in this study guide. 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 these goals. Then you need to discipline yourself to stick with your plan and accomplish your goals on schedule. Note: Every effort is made to provide the most recent information in this study guide. However, The Praxis Series tests are continually evaluated and updated. You will always find the most recent information about these tests, including the topics covered, number of questions, time allotted, and scoring criteria, in the Test at a Glance materials available online at http://www.ets.org/praxis/prxtest.html. Praxis Study Guide for the Mathematics: Content Knowledge Test 9
Chapter 3 Succeeding on Multiple-Choice Questions
CHAPTER 3 The goal of this chapter is to provide you with background information and advice from experts so that you can improve your skill in answering multiple-choice questions about mathematics. Tools: Calculators and Commonly Used Formulas It is important to remember the tools that will be available to you during the test administration. As you prepare for one or more of the mathematics tests, it is probably a good idea to use these tools as you would at an actual administration. Calculators You will be expected to use a graphing calculator without a QWERTY (typewriter-layout) keyboard. Calculator memories will not be cleared. Your calculator should be able to produce the graph of a function within an arbitrary viewing window, find the zeros of a function, compute the derivative of a function numerically, and compute definite integrals numerically. As you work the practice questions and check your answers in this study guide, use your calculator to answer questions when appropriate. Commonly Used Formulas You will notice that the practice questions contain three pages of commonly used notations, formulas, and definitions. These pages will be exactly the same in the test book at your actual administration. Get to know what is contained in these pages so that during the actual test you will not have to use valuable time becoming familiar with the information contained there. As you take the test, you can refer to these pages as needed to find information to help you answer the questions. Multiple-Choice Questions Multiple-choice questions in mathematics are different from those in most other subject areas in several important ways. Wording Mathematics questions are economically worded that is, every word counts in terms of information you need to get to the right answer. Therefore be sure to read the questions carefully to identify and understand each piece of information that is needed to find a solution. Praxis Study Guide for the Mathematics: Content Knowledge Test 11
CHAPTER 3 Here is an example of a typical mathematics question. If (i) the graph of the function f xis the line with slope 3 and y-intercept 1 and (ii) the graph of the function gx is the semicircle in the upper half plane with center at the origin and radius, what is the domain of gf x? [ ] (A) 0, (B) È -1, ÎÍ 1 3 [ ] (C), (D), The structure of the question is simple and concise: If a and b, then what? Given two pieces of information, what is the third piece of information? It is critical to look carefully at each of the three parts of the question. Within each statement there is a wealth of information. The first statement reads: (i) the graph of the function f x is the line with slope 3 and y-intercept 1 From this statement, you need to figure out that the graph of the function f x represented by y 3x 1. The second statement provides another critical piece of information: is the line (ii) the graph of the function gx is the semicircle in the upper half plane with center at the origin and radius is 4 From this statement, you need to figure out that the function g x x, where x. Here is how to do this. Recall that the equation of a circle with center (0, 0) and radius can be written x y 4. Using some elementary Algebra, you can rewrite this as y 4 x or y 4 x. (Remember that this is not a function.) The graph of gx ( ), the semicircle in the upper half of the plane, is represented by y 4 x. By sketching and inspecting the graph of gx ( ), you can infer that the domain of gx ( ) is x. You can confirm this by graphing the semicircle as well as the line on a calculator using the viewing window 4 x 4 and 3 y 3. 1 Praxis Study Guide for the Mathematics: Content Knowledge Test
CHAPTER 3 The third part of the question presents the way you are to put the first two parts together: what is the domain of gf x? You need to understand from this part of the question that the domain of gf x is the set of all x for which gf x is defined. is defined for all x such that f x Since g f x f x 3x 1, the domain of g f x is given by the equation, and since f x is the set of all x satisfying the double inequality 3x 1 3 3x 1 1 1 x 3 It follows that the domain of gf x is the closed interval - You can solve the problem a different way. You can let Y Y Y Y 3 1 x, or Y 3 4 ( 3 1) x. È ÎÍ 1, 1 ; thus the correct answer is (B). 3 3x 1, Y 4 x, and 1 The domain of Y 3 is those values of x for which 4 ( 3x 1) 0. By inspection and comparison 1 with the inequality above, you will find that 1 x. 3 This can be confirmed by graphing Y 3 on your calculator and inspecting the graph, which shows x to have values from 1 and 1 3. (B) is the only answer choice consistent with this graph; therefore the correct answer is (B). Praxis Study Guide for the Mathematics: Content Knowledge Test 13
CHAPTER 3 As this problem shows, the length of the question itself does not necessarily indicate how many steps will be needed to solve it, how a graphing calculator should be used, or how difficult or easy the problem is. In multiple-choice questions in mathematics, you must take your time, read carefully, and use every bit of information from each word, phrase, and sentence. Timing Multiple-choice questions in mathematics are also different from questions in many other subject areas because they can vary greatly in the time it takes to arrive at the correct answer. Some of the problems are more conceptual, are relatively straightforward, and take little time to solve, provided you know the concept. Others require more effort to read, understand, set up, and solve, even when you know your content extremely well. It is fine to spend a little extra time trying to solve the longer problems, provided you can make it up on other problems. For the Content Knowledge test, you have two hours (10 minutes) to complete 50 questions, which means almost two-and-a-half (½) minutes per question. You should expect some questions to take more time than that, and some to take less. During the test, you may want to consider checking your progress in half-hour increments you should complete at least 1 or 13 questions each half hour in order to finish the test in two hours. Multiple Ways of Finding a Solution There is often more than one way of arriving at a correct answer in a mathematics question, and you should rely on your own ways of solving problems when you take the test. If you tend to draw pictures or make graphs, then use the blank spaces in the test book to do so. If you typically use your calculator for virtually all problem solving, then use it in that way during the test. Whatever ways you use to work math problems, however, be sure to read the question carefully, use every bit of information given, and carefully but efficiently set up the path to the correct solution. Multiple-Choice Tips Here are some additional expert tips to help you succeed on the mathematics multiple-choice questions. Look at your choices before solving. Scan the answer choices offered before you start working on a problem. This might give you an idea of the sort of answer you should be looking for and may also give you a clue about how to solve the problem. Use a process of elimination. In a multiple-choice question, you know that of the four answer choices, one is correct and three are incorrect. If you eliminate three of the choices as possible answers, then the choice that you have not eliminated must be the correct answer. 14 Praxis Study Guide for the Mathematics: Content Knowledge Test
CHAPTER 3 Use estimation. Sometimes you can save time and effort by simply estimating the correct answer. After that, you can scan the answer choices and pick the one closest to your estimate. Use good judgment when checking a calculator-derived answer. You may find that the answer you produce with the help of the calculator does not match any of the answer choices exactly. This can occur because calculator output will vary slightly depending on the method by which the output is produced. In this case, scan the choices. One of them should differ only slightly from your answer. If none of the answer choices is close to your answer, look for an error in how you solved the problem. Be Familiar with Multiple-Choice Question Types You will probably see more than one question format on a multiple-choice test. Here are examples of some of the more common question formats. 1. Complete the statement In this type of question, you are given an incomplete statement. You must select the choice that will make the completed statement correct. A quadrilateral with four right angles could be a (A) triangle (B) trapezoid (C) square (D) hexagon To check your answer, reread the question and add your answer choice at the end. Be sure that your choice best completes the sentence. The correct answer is (C). Praxis Study Guide for the Mathematics: Content Knowledge Test 15
CHAPTER 3. Which of the following This question type contains the details that must be satisfied for a correct answer, and it uses which of the following to limit the choices to the four choices shown, as this example demonstrates. When you see questions that use this phrase, there are typically several correct answers, but only one will be among the choices given. In this case, you should look at the answer choices carefully since you need only consider the choices that are given. Which of the following numbers is a multiple of 4? (A) (B) 8 (C) 14 (D) 30 The correct answer is (B). 3. Questions containing NOT, LEAST, or EXCEPT The words NOT, LEAST, and EXCEPT can make comprehension of test questions more difficult. Such questions ask you to select the choice that doesn t fit, that is different in some specified way from the other answer choices. You must be very careful with this question type because it s easy to forget that you re 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. These words are always capitalized when they appear in The Praxis Series test questions, but they are easily (and frequently) overlooked. For the following test question, determine what kind of answer you need and what the details of the question are. Which of the following is NOT a prime number? (A) 8 (B) 11 (C) 3 (D) 31 The correct answer is (A). 16 Praxis Study Guide for the Mathematics: Content Knowledge Test
CHAPTER 3 You re looking for the integer that is NOT a prime number. (A) is the correct answer all of the other choices are prime numbers. TIP It s easy to get confused while you re processing the information to answer a question with a NOT, LEAST, or EXCEPT in the question. If you treat the word NOT, LEAST, or EXCEPT as one of the details you must satisfy, you have a better chance of understanding what the question is asking. 4. Other formats New formats are developed from time to time in order to find new ways of assessing knowledge with multiple-choice questions. If you see a format you are not familiar with, read the directions carefully. Then read and approach the question the way you would any other question, asking yourself what you are supposed to be looking for and what details are given in the question that can help you find the answer. Other Useful Facts About the Test 1. You can answer the questions in any order. 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. There is no right or wrong way. Use the approach that works best for you.. There are no trick questions on the test. You don t have to find any hidden meanings or worry about trick wording. All of the questions on the test ask about subject matter knowledge in a straightforward manner. 3. Don t worry about answer patterns. There is one myth that says that answers on multiplechoice tests follow patterns. There is another myth that there will never be more than two questions with the same lettered answer following each other. There is no truth to either of these myths. Select the answer you think is correct based on your knowledge of the subject. 4. There is no penalty for guessing. Your test score for multiple-choice questions is based on the number of correct answers you have. When you don t know the answer to a question, try to eliminate any obviously wrong answers and then guess at the correct one. 5. It s OK to write in your test booklet. You can work out problems right on the pages of the booklet, make notes to yourself, mark questions you want to review later, or write anything at all. Your test booklet 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. Praxis Study Guide for the Mathematics: Content Knowledge Test 17
CHAPTER 3 Smart Tips for Taking the Test 1. Put your answers in the right bubbles. It seems obvious, but be sure that you are filling 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.. Skip the questions you find extremely difficult. There are sure to be some questions that you think are hard. Rather than trying to answer these on your first pass through the test, leave them blank and mark them in your test booklet so that you can come back to them later. 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 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. 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 and then reread the question to be sure the answer you have selected really answers the question being asked. Remember that 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. Check 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 filled in the bubble on the answer sheet 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 SAT, the GRE, or other similar-looking (but in fact very different!) tests. It doesn t matter on this test 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, your actual score doesn t matter, as long as it is above the minimum required score. With your score report you will receive a booklet entitled Understanding Your Praxis Scores, which lists the passing scores for your state. 18 Praxis Study Guide for the Mathematics: Content Knowledge Test
Chapter 4 Study Topics
CHAPTER 4 This chapter is intended to help you organize your preparation for the Praxis Mathematics: Content Knowledge test and to give you a clear indication about the depth and breadth of the knowledge required for success on the test. You are not expected to be an expert on all aspects of the knowledge and skills statements that follow. You should understand the major concepts and procedures associated with each statement. Virtually all accredited undergraduate mathematics programs address the majority of these topics. When you find a skill or topic that is unfamiliar or fuzzy to you, you ll need to find out more. Consult materials and resources, including lecture and seminar notes, from all your mathematics course work. You should be able to match up specific topics with what you have covered in your courses. You may also, at times, want to refer to supplementary books or Web sites that cover the material. In addition, you should seek assistance from a professor or mentor teacher if you are stuck. Try not to be overwhelmed by the volume and scope of knowledge and skills in this guide. An overview such as this does not offer you a great deal of context. Although a specific term may not seem familiar as you see it here, you might find you can understand it when applied to a real-life situation. Many of the items on the actual Praxis test will provide you with a context in which to apply to these topics or terms, as you will see when you look at the practice questions. The Content Knowledge Test The questions on the Mathematics: Content Knowledge test can all be solved by using skills and abilities from ten different areas. A summary of the skills and abilities required in each of these areas is listed below. A. Algebra and Number Theory Demonstrate an understanding of the structure of the natural, integer, rational, real, and complex number systems and the ability to perform the basic operations (+,,, and ) on numbers in these systems Compare and contrast properties (e.g., closure, commutativity, associativity, distributivity) of number systems under various operations Demonstrate an understanding of the properties of counting numbers (e.g., prime, composite, prime factorization, even, odd, factors, multiples) Solve ratio, proportion, percent, and average (including arithmetic mean and weighted average) problems 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 0 Praxis Study Guide for the Mathematics: Content Knowledge Test
CHAPTER 4 Solve and graph systems of equations and inequalities, including those involving absolute value Interpret algebraic principles geometrically Recognize and use algebraic representations of lines, planes, conic sections, and spheres Solve problems in two and three dimensions (e.g., distance between two points, the coordinates of the midpoint of a line segment) B. Measurement Make decisions about units and scales that are appropriate for problem situations involving measurement; use unit analysis Analyze precision, accuracy, and approximate error in measurement situations Apply informal concepts of successive approximation, upper and lower bounds, and limit in measurement situations C. Geometry 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 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, and volume of two- and threedimensional figures, and of regions and solids that are combinations of these figures Solve problems involving reflections, rotations, and translations of geometric figures in the plane D. Trigonometry Define and use the six basic trigonometric relations using degree or radian measure of angles; know their graphs and be able to identify their periods, amplitudes, phase displacements or shifts, and asymptotes. Apply the law of sines and the law of cosines. Apply the formulas for the trigonometric functions of x, x, x, x y, and x y; prove trigonometric identities Solve trigonometric equations and inequalities Convert between rectangular and polar coordinate systems Praxis Study Guide for the Mathematics: Content Knowledge Test 1
CHAPTER 4 E. Functions 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. Find an appropriate family of functions to model particular phenomena (e.g., population growth, cooling, simple harmonic motion) Determine properties of a function such as domain, range, intercepts, symmetries, intervals of increase or decrease, discontinuities, and asymptotes Use the properties of trigonometric, exponential, logarithmic, polynomial, and rational functions to solve problems 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 Interpret representations of functions of two variables, such as three-dimensional graphs, level curves, and tables F. Calculus 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 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) Show that a particular function is continuous; understand the relationship between continuity and differentiability Numerically approximate derivatives and integrals Use standard differentiation and integration techniques Analyze the behavior of a function (e.g., find relative maxima and minima, concavity); solve problems involving related rates; solve applied minima-maxima problems Demonstrate understanding of and ability to use the Mean Value Theorem and the Fundamental Theorem of Calculus Demonstrate an intuitive understanding of integration as a limiting sum that can be used to compute area, volume, distance, or other accumulation processes Determine the limits of sequences and simple infinite series G. Data Analysis and Statistics Organize data into a suitable form (e.g., construct a histogram and use it in the calculation of probabilities) Know and find the appropriate uses of common 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) Analyze data from specific situations to determine what type of function (e.g., linear, quadratic, Praxis Study Guide for the Mathematics: Content Knowledge Test
CHAPTER 4 exponential) would most likely model that particular phenomenon; use the regression feature of the calculator to determine the curve of best fit; interpret the regression coefficients, correlation, and residuals in context Understand and apply normal distributions and their characteristics (e.g., mean, standard deviation) Understand how sample statistics reflect the values of population parameters and use sampling distributions as the basis for informal inference Understand the differences among various kinds of studies and which types of inferences can legitimately be drawn from each Know the characteristics of well-designed studies, including the role of randomization in surveys and experiments H. Probability Understand the concepts of sample space and probability distribution and construct sample spaces and distributions in simple cases Understand the concepts of conditional probability and independent events; understand how to compute the probability of a compound event Compute and interpret the expected value of random variables in simple cases (e.g., fair coins, expected winnings, expected profit) Use simulations to construct empirical probability distributions and to make informal inferences about the theoretical probability distribution I. Matrix Algebra 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) Scalar multiply, add, subtract, and multiply vectors and matrices; find inverses of matrices Use matrix techniques to solve systems of linear equations Use determinants to reason about inverses of matrices and solutions to systems of equations Understand and represent translations, reflections, rotations, and dilations of objects in the plane by using sketches, coordinates, vectors, and matrices J. Discrete Mathematics 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) 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 Determine whether a binary relation on a set is reflexive, symmetric, or transitive; determine whether a relation is an equivalence relation Praxis Study Guide for the Mathematics: Content Knowledge Test 3
CHAPTER 4 Use finite and infinite arithmetic and geometric sequences and series to model simple phenomena (e.g., compound interest, annuity, growth, decay) Understand the relationship between discrete and continuous representations and how they can be used to model various phenomena Use difference equations, vertex-edge graphs, trees, and networks to model and solve problems Mathematical Process Categories Mathematical Problem Solving Solve problems that arise in mathematics and those involving mathematics in other contexts Build new mathematical knowledge through problem solving Apply and adapt a variety of appropriate strategies Mathematical Reasoning and Proof Select and use various types of reasoning and methods of proof Make and investigate mathematical conjectures Develop and evaluate mathematical arguments and proofs Mathematical Connections Recognize and use connections among mathematical ideas Apply mathematics in context outside of mathematics Demonstrate an understanding of how mathematical ideas interconnect and build on one another Mathematical Representation Select, apply, and translate among mathematical representations to solve problems Use representations to model and interpret physical, social, and mathematical phenomena Create and use representations to organize, record, and communicate mathematical ideas Use of Technology Use technology as an aid to understanding mathematical ideas Use technology appropriately as a tool for problem solving 4 Praxis Study Guide for the Mathematics: Content Knowledge Test
Chapter 5 Mathematics: Content Knowledge Test Practice Questions
CHAPTER 5 Now that you have studied the content topics and have worked through strategies relating to the Content Knowledge test, you should answer the following practice questions. This set of 40 practice questions is not a full-length test, but you will probably still find it helpful to simulate actual testing conditions: have a pencil and graphing calculator available, and give yourself about 95 minutes to work on the questions. You can use the answer sheet provided if you wish. Keep in mind that the test you take at an actual administration will have different questions and you will have 10 minutes to complete 50 questions. You should not expect your level of performance to be exactly the same as when you take the test at an actual administration, since numerous factors affect a person s performance in any given testing situation. When you have finished the practice questions, you can score them and read the explanations of correct answers in chapter 6. 6 Praxis Study Guide for the Mathematics: Content Knowledge Test
TEST NAME Mathematics: Content Knowledge (0061) Time 10 Minutes 50 Questions
78853 0890 TF79E00 Printed in U.S.A. Q573-06 1. NAME Enter your last name and first initial. Omit spaces, hyphens, apostrophes, etc. Last Name (first 6 letters) F I A A A A A A A B B B B B B B C C C C C C C D D D D D D D E E E E E E E F F F F F F F G G G G G G G H H H H H H H I I I I I I I J J J J J J J K K K K K K K L L L L L L L M M M M M M M N N N N N N N O O O O O O O P P P P P P P Q Q Q Q Q Q Q R R R R R R R S S S S S S S T T T T T T T U U U U U U U V V V V V V V W W W W W W W X X X X X X X Y Y Y Y Y Y Y Z Z Z Z Z Z Z DO NOT USE INK Use only a pencil with soft black lead (No. or HB) to complete this answer sheet. Be sure to fill in completely the oval that corresponds to the proper letter or number. Completely erase any errors or stray marks. Answer Sheet C. YOUR NAME: (Print) Last Name (Family or Surname) First Name (Given) M. I. 3. DATE OF BIRTH Month Day Jan. 4. SOCIAL SECURITY NUMBER Feb. MAILING ADDRESS: Mar. 0 0 0 (Print) P.O. Box or Street Address Apt. # (If any) 0 0 0 0 0 0 0 0 April 1 1 1 1 1 1 1 1 1 1 1 May City State or Province June 3 3 3 3 3 3 3 3 3 3 3 July 4 4 Country Zip or Postal Code 4 4 4 4 4 4 4 4 Aug. 5 5 5 5 5 5 5 5 5 5 TELEPHONE NUMBER: ( ) ( ) Home Sept. Business Oct. 6 7 6 6 6 6 6 6 6 6 6 7 7 7 7 7 7 7 7 7 Nov. 8 8 SIGNATURE: TEST DATE: 8 8 8 8 8 8 8 8 Dec. O Q S13456789T 9 9 9 9 9 9 9 9 9 9 O 5. CANDIDATE ID NUMBER 6. TEST CENTER / REPORTING LOCATION 7. TEST CODE / FORM CODE 8. TEST BOOK SERIAL NUMBER S Center Number Room Number Q 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 9. TEST FORM 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 Center Name 10. TEST NAME 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 5 5 5 5 5 5 5 5 City State or Province 5 5 5 5 5 5 5 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 7 7 7 7 7 7 7 7 Country 7 7 7 7 7 7 7 8 8 8 8 8 8 8 8 9 9 9 9 9 9 9 9 Copyright 009 Educational Testing Service. All rights reserved. Educational Testing Service, ETS, and the ETS logo are registered trademarks of Educational Testing Service (ETS) in the United States and other countries. The Praxis Series is a trademark of ETS. 8 8 8 8 8 8 8 9 9 9 9 9 9 9 751670 1 3 4 PAGE 1
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CHAPTER 5 NOTATION ( ab, ) x : a x b [ ab, ) x : a x b ( ab, ] x : a x b [ ab, ] x : a x b gcd m, n lcm m, n greatest common divisor of two integers m and n least common multiple of two integers m and n x greatest integer m such that m x m k modn 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) 1 f inverse of an invertible function f (not the same as 1 f ) lim f( x ) x a right-hand limit of f x ; limit of f x as x approaches a from the right lim f( x ) x a left-hand limit of f x ; limit of f x as x approaches a from the left 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 S T union of sets S and T S T intersection of sets S and T DEFINITIONS A relation on a set S is reflexive if x x for all x S symmetric if x y y x for all x, y S transitive if x y and y z x z for all x, y, z S antisymmetric if x y and y x x y for all x, y S An equivalence relation is a reflexive, symmetric, and transitive relation. 30 Praxis Study Guide for the Mathematics: Content Knowledge Test