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TESTINFORMATION

BOOKLET

Massachusetts Department of Elementary and Secondary Education

Massachusetts

Tests for

Educator

Licensure®

47 Middle School Mathematics

MA-SG-FLD047-08

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Int roduct ion

Purpose of the Test Information Booklets

The Test Information Booklets are designed to help familiarize candidates with the format and contentof the tests of the Massachusetts Tests for Educator Licensure® (MTEL®) program. Familiarity with

test format and content is one component of test preparation. The primary means of preparing forthe tests for educators seeking PreKindergarten to grade 12 licenses is coursework in the field of studyat the candidate's higher education institution. For candidates seeking vocational technical licenses,the primary means of preparation is the candidate's educational and work experience background, andfor adult basic education licenses, the primary approach is adequate study in the content defined by theLicensure of Adult Basic Education Teachers and Preparation Program Approval and other statepolicies.

Education and arts and sciences faculty at Massachusetts higher education institutions with approvededucator preparation programs may also find the information in these booklets useful as they helpcandidates prepare for the MTEL.

Key Features of the Test Information Booklets

• Suggestions for preparing to take the test

• Examples of MTEL test item formats and ways to approach the test items

• Sample test items and responses

• What to expect on the day of the test

• Information on score reporting and interpreting a score report

Further Information

Information about MTEL registration and test administration procedures is available in the currentversion of the MTEL registration bulletin. The registration bulletin may be viewed or downloaded on the

Internet at www.mtel.nesinc.com or www.doe.mass.edu/mtel.

If after reviewing the registration bulletin you have further questions, please contact the offices listed onthe next page.

Copyright © 2013 Pearson Education, Inc. or its affiliate(s). All rights reserved.Evaluation Systems, Pearson, P.O. Box 226, Amherst, MA 01004

Massachusetts Tests for Educator Licensure and MTEL are trademarks of the Massachusetts Department of Elementary andSecondary Education and Pearson Education, Inc. or its affiliate(s).

Pearson and its logo are trademarks, in the U.S. and/or other countries, of Pearson Education, Inc. or its affiliate(s).

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LICENSURE

Educators enrolled in an Academic (PreK–12) Massachusetts higher education institution educator preparationprogram or educators seeking a Vocational Technical or Adult Basic Education license should refer to the relevantprogram section of the current registration bulletin for further information.

You may apply for an educator license at any time, before or after taking the MTEL. However, candidatesseeking their first Massachusetts license must achieve a passing score on the Communication and Literacy

Skills test before their license applications can be reviewed.

CONTACT: MASSACHUSETTS DEPARTMENT OF ELEMENTARY AND SECONDARY EDUCATIONOFFICE OF EDUCATOR LICENSURE75 PLEASANT STREETMALDEN, MA 02148-4906

TELEPHONE: (781) 338-6600 TTY-NET RELAY: (800) 439-2370

Attendants are available 2:00 p.m.–5:00 p.m. eastern time, Monday through Friday, excluding holidays. Pleasecheck the web site (www.doe.mass.edu/licensure) for up-to-date information on hours of operation.

24-HOUR AUTOMATED LINE: Call (781) 338-3000 and then press "2" to obtain the status of an existinglicense application.

WALK-IN SERVICE: 8:45 a.m.–4:45 p.m., Monday through Friday, excluding holidays

WEB SITE: LICENSING INFORMATION: www.doe.mass.edu/licensure and then select theappropriate license category. At this site you will find regulations, guidelines, and applications for

Academic (PreK–12), Vocational Technical Education, and Adult Basic Education licenses. Access "Licensure Help" and apply for a license online through the ELAR system for Academic(PreK–12) and Vocational Technical Education licenses. First-time applicants must create anELAR profile to apply online.

WEB SITE: MTEL PROGRAM INFORMATION: www.doe.mass.edu/mtel At this site you will find MTEL notices, a list of test preparation resources available to the public, andfrequently asked questions.

TESTING

If you have questions about:

Test registration Test session procedures Changes to test center or date Score reports

CONTACT: MASSACHUSETTS TESTS FOR EDUCATOR LICENSUREEVALUATION SYSTEMSPEARSON

P.O. BOX 660 WEB SITE: www.mtel.nesinc.com AMHERST, MA 01004-9013

TELEPHONE: (413) 256-2892 FAX: (413) 256-7077

(866) 565-4894 (toll free)

Operators are available 9:00 a.m. –5:00 p.m. eastern time, Monday through Friday, excluding holidays. Automated Information System is available 24 hours daily.

If you have questions about your computer-based test appointment:

CONTACT: PEARSON VUE

WEB SITE: www.pearsonvue.com/mtel

TELEPHONE: (800) 989-8532

Operators are available 8:00 a.m.–8:00 p.m. eastern time, Monday through Friday, excluding holidays.

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Table of Con tents

Overview of the Tests............................................................................................................................ 1

Description of the Tests ....................................................................................................................... 1 Development of the Tests.................................................................................................................... 4

Structure of the Tests........................................................................................................................... 4

How to Prepare for the Tests................................................................................................................ 6

Using the Test Objectives .................................................................................................................... 6Developing a Study Outline ................................................................................................................. 7 Identifying Resources .......................................................................................................................... 7

Approaching the Test Items................................................................................................................ 12

Multiple-Choice Item Formats............................................................................................................ 12 Open-Response Item Formats........................................................................................................... 15 Scoring of Open-Response Items...................................................................................................... 16

The Day of the Test.............................................................................................................................. 18

Preparing for the Test ........................................................................................................................ 18 Test-Taking Tips ................................................................................................................................ 18

After the Test........................................................................................................................................ 20

Score Reporting ................................................................................................................................. 20 Interpreting Your Score Report.......................................................................................................... 20

Field-Specific Information....................................................................................................................21

Test Overview Chart ...........................................................................................................................23

Sample Test Items..............................................................................................................................26 Answer Key and Sample Response ................................................................................................... 36Test Objectives ...................................................................................................................................39

Readers should be advised that this booklet, including many of the excerpts used herein,is protected by federal copyright law.

Test policies and materials, including but not limited to tests, item types, and item formats, are subject

to change at the discretion of the Massachusetts Department of Elementary and Secondary Education.

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Overview of th e Tests

The Massachusetts Tests for Educator Licensure (MTEL) program includes a Communication andLiteracy Skills test, PreKindergarten to grade 12 subject matter tests, a Vocational Technical Literacy

Skills Test, and an Adult Basic Education test. See www.mtel.nesinc.com for a complete list of MTELtests currently offered. Information on what tests must be taken may be obtained from the

Massachusetts Department of Elementary and Secondary Education's web site, www.doe.mass.edu.

All tests included in the program are criterion referenced; that is, they are designed to measure acandidate's knowledge in relation to an established standard of competence (criterion) rather than inrelation to the performance of other candidates. Each MTEL test is also objective-based; that is, itis based on content stated in a set of test objectives for the test field. The purpose of the tests is tohelp identify those candidates who have the appropriate level of knowledge and skills judged to beimportant for educators in Massachusetts public schools.

The MTEL tests are explicitly aligned with the relevant Massachusetts licensure regulations and thestate's standards in the Massachusetts curriculum frameworks. These documents are available at

www.doe.mass.edu; the MTEL test objectives are available at www.mtel.nesinc.com .

Description of the Tests

All tests in the MTEL program include both multiple-choice and open-response test items. The numberof multiple-choice and open-response test items varies across the tests. Refer to the table below forinformation about types and numbers of test items on each type of test.

Test Type Test Description

Communication and LiteracySkills (01)

Reading subtest This subtest includes 42 multiple-choice items. The multiple-choicesection counts for 100 percent of the candidate's total subtest score.

Writing subtest This subtest includes 35 multiple-choice items, seven (7) short-answersentence correction items, and two open-response items—a summaryexercise and a composition exercise. The summary exercise requirescandidates to spend about 20–30 minutes to prepare an adequateresponse; the composition exercise requires candidates to spend about45–60 minutes to prepare an adequate response. The multiple-choiceand short-answer item section counts for 50 percent of the candidate'sscore, the summary exercise counts for 15 percent of the candidate'stotal test score, and the composition exercise counts for 35 percent of thecandidate's total subtest score.

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Subject Matter Tests

Academically Advanced (52)Biology (13)Business (19)Chemistry (12)

Early Childhood (02)Earth Science (14)English (07)Foundations of Reading (90)General Science (10)Health/Family and Consumer

Sciences (21)History (06)Latin and Classical Humanities (15)Mathematics (09)Mathematics (Elementary) (53)Mathematics (Middle School) (47)Mathematics/Science (Middle

School) (51)

Middle School Humanities (50)Music (16)Physical Education (22)Physics (11)Political Science/Political

Philosophy (48)Reading Specialist (08)Speech (44)Technology/Engineering (33)Theater (45)Visual Art (17)

These subject matter tests include 100 multiple-choice items and twoopen-response items requiring candidates to spend about 20–30 minuteseach to prepare an adequate response. The multiple-choice sectionaccounts for 80 percent and the open-response section counts for20 percent of the candidate's total test score.

Subject Matter Test

General Curriculum (03)

Multi-subject subtest This subtest includes 55 multiple-choice items and one open-responseitem requiring candidates to spend about 20–30 minutes to prepare anadequate response. The multiple-choice section counts for 90 percentand the open-response section counts for 10 percent of the candidate'ssubtest score.

Mathematics subtest This subtest includes 45 multiple-choice items and one open-responseitem requiring candidates to spend about 20–30 minutes to prepare anadequate response. The multiple-choice section counts for 90 percentand the open-response section counts for 10 percent of the candidate'ssubtest score.

Subject Matter Test

Dance (46)

The Dance test includes 55 multiple-choice items and two open-responseitems requiring candidates to spend about 60 minutes each to prepare an

adequate response. The multiple-choice items count for 75 percent andthe open-response items count for 25 percent of the candidate's total testscore.

Subject Matter Test

English as a Second Language (54)

The English as a Second Language test includes 100 multiple-choiceitems and four open-response items, all of which require oral responses.The multiple-choice items count for 80 percent of the total test score andthe open-response items count for 20 percent of the total test score.

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Language Subject Matter Tests

Chinese (Mandarin) (29)French (26)German (27)Italian (30)

Portuguese (32)Russian (31)Spanish (28)

French and SpanishThe language tests in French and Spanish include approximately100 multiple-choice items and two open-response items, one of whichrequires a written response and the other an oral response. The multiple-choice items count for approximately 70 percent and the open-responseitems count for approximately 30 percent of the candidate's total testscore.

German and Italian The language tests in German and Italian include approximately55 multiple-choice items and two open-response items, one of whichrequires a written response and the other an oral response. The multiple-choice items count for approximately 67 percent and the open-responseitems count for approximately 33 percent of the candidate's total testscore.

Chinese (Mandarin), Portuguese, and Russian The language tests in Chinese (Mandarin), Portuguese, and Russian

include approximately 55 multiple-choice items and short-answer itemsand four open-response items, one of which requires an oral response.The multiple-choice items and short-answer items count forapproximately 33 percent and the open-response items count forapproximately 67 percent of the candidate's total test score.

Adult Basic Education (55) This subject matter test includes 100 multiple-choice items and two open-response items requiring candidates to spend about 20–30 minutes eachto prepare an adequate response. The multiple-choice section accountsfor 85 percent and the open-response section counts for 15 percent ofthe candidate's total test score.

Vocational Technical Literacy

Skills Test (91)

Reading subtest

This subtest includes 60 multiple-choice items. The multiple-choicesection counts for 100 percent of the candidate's total subtest score.

Writing subtest This subtest includes 40 multiple-choice items and two open-responseitems—a summary exercise and a composition exercise. The summaryexercise requires candidates to spend about 20–30 minutes to preparean adequate response; the composition exercise requires candidates tospend about 45–60 minutes to prepare an adequate response. Themultiple-choice item section counts for 55 percent of the candidate'sscore; the summary and composition exercises count for 45 percent ofthe candidate's total subtest score.

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Development of the Tests

The development process for the Massachusetts Tests for Educator Licensure involves numeroussteps. During the development of the test objectives, the Massachusetts Regulations for EducatorLicensure and Preparation Program Approval , the Massachusetts curriculum frameworks, and otherkey state and national documents were consulted. Massachusetts educators participated throughoutthe development process. The objectives for each test field were reviewed and validated by anadvisory panel of licensed and practicing Massachusetts public school educators and higher education

institution faculty. A content validation survey was conducted using randomly sampled, practicingMassachusetts public school educators and higher education institution faculty members in educatorpreparation programs and in the arts and sciences. Each survey participant reviewed the testobjectives in his or her field to ensure that the objectives are important to the job of a Massachusettseducator and reflect the subject matter knowledge necessary for entry-level teaching in Massachusettspublic schools. The test items were aligned with both the test objectives and the educational materialsrelevant to Massachusetts, including the state's licensure regulations and the curriculum frameworks.The test items were reviewed and verified by Massachusetts educators for inclusion on the test.

An individual's performance on a test is evaluated against an established standard or qualifying score.The qualifying score for each test is established by the Massachusetts Commissioner of Elementaryand Secondary Education and is based in part on the professional judgments and recommendations of

Massachusetts educators. Candidates who do not achieve the qualifying score on a test may retake it.Candidates who have met the qualifying score on a test are not required to retake it.

Structure of the Tests

While the content covered by each test in the MTEL program is different, the structure that underliesthe content of the tests is similar. In general, the structure is as follows:

• The content of each subject matter test field is organized into three to nine subareas. Thesesubareas define the major content domains of the test.

• Each subarea includes one or more test objectives. The test objectives are broad, conceptualstatements, written in language that reflects some of the skills, knowledge, and understandingthat an educator needs in order to teach the test field in Massachusetts public schools.

• Each test objective is clarified and further described by a descriptive statement, providing

examples of the types of content covered by the test objective.

• Each multiple-choice item is written to assess understanding of a test objective.

• Each open-response item is written to assess understanding of one or more test objectives.

• The number of test objectives within a given subarea generally determines the proportion ofmultiple-choice items that will address that subarea on a test. In other words, subareas witha greater number of test objectives will generally receive more emphasis (i.e., have a greaternumber of corresponding multiple-choice items) on the test than those with fewer testobjectives. The number of test items and subarea proportions for each MTEL test are indicated

in the Test Overview Chart found in the field-specific section of the Test Information Booklet forthe field.

The chart on the following page illustrates the relationship among subareas, test objectives, descriptivestatements, and test items in a typical test field.

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Organization of Content Within a Test Field

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How to Prepare for the Tests

This section of the Test Information Booklet offers suggestions on how to prepare for a test in theMTEL program. Candidates planning to take a test may find the suggestions helpful. However, thisbooklet is not intended to address all possible strategies for preparation.

Using the Test Objectives

The test objectives may be useful to candidates preparing to take one of the Massachusetts Testsfor Educator Licensure. The test objectives are based on Massachusetts licensure regulations andother state policies. They reflect subject matter knowledge that the Massachusetts Department ofElementary and Secondary Education has determined to be important for educators entering teachingin Massachusetts public schools. The content described in the test objectives is typically learnedthrough college-level coursework and other educational experiences at Massachusetts educatorpreparation institutions.

The test objectives are organized in such a way that each element of the test objectives can be helpfulin one or more aspects of test preparation. Test objectives for each of the tests can be obtained from

the MTEL web site at www.mtel.nesinc.com .

Review the test objectives. The first step in preparing for one of the Massachusetts Tests forEducator Licensure is to identify the information the test will cover. The test objectives can be a helpfulstudy tool because they define the content that will be assessed by the test you are preparing to take.

Read the entire set of test objectives for your test field to get a general picture of the material the testcovers. Remember that the number of test objectives within a subarea broadly reflects the emphasisgiven to that subarea on the test; the proportion of the test allocated to the subarea is listed in the TestOverview Chart in the field-specific section of this booklet. In general, subareas with greaterproportions will include a greater number of test objectives and will be covered by more multiple-choiceitems on the test.

Focus your studies. When you have become familiar with the test objectives, make a list of those testobjectives about which you feel you know the least. A suggested approach for outlining the content ofthe test objectives and identifying those on which you need to concentrate is presented below, underthe heading "Developing a Study Outline." This approach can help you set priorities for your studytime. You should leave time to review the content of all test objectives, both the familiar and the lessfamiliar ones. Keep in mind that you may improve your total test score by improving your performancein any subarea of the test. So, any additional preparation that you undertake may contribute toimproving your total test score.

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Review the sample test items. This guide contains examples of the formats of multiple-choice itemsand open-response items that you may encounter on the test. It also presents sample test items andcorrect responses. The sample items are designed to illustrate the nature of the test items; they shouldnot be used as a diagnostic tool to determine your individual strengths and weaknesses.

Develop study techniques. It is usually best to approach preparing for a test with a study plan inmind, together with a schedule for accomplishing what you need to do to feel prepared. You may also

wish to obtain from either your school's library or a bookstore one of the many books available on studyskills.

For some people, study groups are particularly helpful; you may want to form or join a study group withothers who will be taking the test at the same time. Find out if your school sponsors study groups.Study groups are more effective once you have identified the test objectives with which you are lessfamiliar.

A gradual study pattern over an appropriate period of time can be effective for learning unfamiliar ordifficult content. In the last few days before the test, take time to review those topics with which youfeel most comfortable and avoid "cramming"—trying to learn too much new material quickly.

Use subarea information. If you are retaking a test, the information from your score report thatindicates test subarea-level strengths and weaknesses may further help you determine areas foradditional preparation. You may be able to improve your performance on the test by studying in thoseareas in which you did well in addition to studying those areas for which your performance was weak.

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App roaching the Test I tems

Multiple-Choice Item Formats

The purpose of this section is to describe and give examples of multiple-choice item formats that youmay see on the Massachusetts Tests for Educator Licensure. This section also suggests possible

ways to approach thinking about and answering the multiple-choice items. The approaches describedin this section are intended as suggestions; you may want to try using them on the sample itemsincluded in this Test Information Booklet. However, these approaches are not intended to replace test-taking strategies with which you are already comfortable and that work for you.

The multiple-choice items in the MTEL program are designed to assess subject matter knowledgedescribed in the set of test objectives for each test field. You will be expected to demonstrate morethan your ability to recall factual information; you may be asked to think critically about information,analyze it, apply it, consider it carefully, compare it with other knowledge you have, or make a judgmentabout it.

You may see several multiple-choice item formats in the MTEL. Not all types of test items that may

appear on a test are illustrated. Actual test items are not necessarily identical in wording or format tothe sample items in this booklet. Not every test will include all the item formats discussed in thissection, but it is in your best interest to be prepared to answer all types of items in the event that theyare included on the test(s) you elect to take. On the following pages are examples of the most commonmultiple-choice item formats and descriptions of approaches you might use to answer the items.

For each multiple-choice item there are four response options, one of which is the best response of thechoices given. Each multiple-choice item counts equally toward a candidate's total multiple-choicesection score. There is no penalty for guessing.

The tests may include a number of test items that do not count toward a candidate's score becausethey are being tried out for future administrations of the MTEL. Approximately 15 percent of the

multiple-choice item section of each test form includes items that are nonscorable. Nonscorable itemsare included so that data can be collected on their psychometric characteristics for possible use asscorable items on future test forms.

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Mult iple-Choice Item Format One: The Single Test Item

In the single test-item format, a problem is presented as a direct question or an incomplete statement,and four response options (A, B, C, and D) appear below the question. The following sample test itemfrom a General Curriculum test is an example of this type. (The * indicates the best response of thechoices given.)

Example of multiple-choice item format one: a single test item

A town planning committee must decide

how to use a 110-acre piece of land.

The committee sets aside 20 acres of the

land for watershed protection and an

additional 46 acres for recreation. What

percentage of the land is set aside for

either watershed protection or recreation?

A. 19%

B. 44%

*C. 60%

D. 66%

For test items in this format, a suggested approach is to read the entire item carefully and critically.Think about what it is asking and the situation it is describing. It is best to read and evaluate all

four response options to find the best answer; you should not stop at the first answer that seemsreasonable. After reading all four response options, eliminate any obviously wrong answers, and selectthe best choice from the remaining answers. There is no penalty for guessing.

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Mult iple-Choice Item Format Two: Test Items w ith Stimulu s Material

Some test items are preceded by stimulus material to which the test items relate. Some examples ofstimulus material that may be included on a test are maps, charts, tables, graphs, reading passages,and descriptions of classroom situations. In some cases, there is only one test item related to thestimulus provided. In other cases, two or more test items are related to a single stimulus. Eachstimulus is preceded by a direction line indicating how many items are related to that stimulus. Thefollowing sample from an English test includes an excerpt from a poem as the stimulus for one sample

test item. (The * indicates the best response of the choices given.)

Example of multiple-choice item format two: a test item with stimulus material

Use the excerpt below from a poem by

Queen Elizabeth I of England to answer

the question that follows.

I grieve and dare not show my discontent,

I love and yet am forced to seem to hate,

I do, yet dare not say I ever meant,

I seem stark mute but inwardly do prate.

I am and not, I freeze and yet am burned,

Since from myself another self I turned.

In this work, the poet constructs meaning

by using:

A. understatement to emphasize the

speaker's sense of incompleteness.

B. allusion to elevate the speaker's

situation.

C. onomatopoeia to heighten the dramatic

effect.

* D. antithesis to dramatize the speaker's

inner conflict.

The following approach is suggested when responding to test items with stimulus material. First,examine the stimulus. Note how the information is presented. In this test item, the stimulus is a poeticexcerpt and must be read carefully and slowly to gain a sense of the meaning of the words and their

artistic use in the passage.

Now you are prepared to address the sample test item associated with this stimulus. As you considereach response option, look back at the stimulus to determine which literary device is chiefly used in theexcerpt. The device that is most clearly used to construct meaning in the excerpt is antithesis, the juxtaposition of contrasting ideas. Therefore, D is the correct response.

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Open-Response Item Formats

Typically, the Massachusetts Tests for Educator Licensure include one or more open-response itemsfor which you are asked to prepare a written response. These items are designed to measure thedepth and breadth of the candidate's knowledge in the subject area, the candidate's understanding offundamental concepts of the discipline, and the candidate's familiarity with field-specific methodologies.In language tests, the open-response items generally assess speaking, writing, listening, reading,and/or cultural understanding.

The open-response items may be designed to require the integration of knowledge from one or severalsubareas and to be comparable to each other in terms of information presented, the type of responserequired, and the difficulty of the items from one test form to the next. Responses to each open-response item typically require about 20 to 30 minutes to complete. For the open-response items thatrequire written responses, each item will generally include:

1. contextual or background information that presents the topic of the open-response item; and

2. one or more specific directions or assignments that advise you of the elements that you areexpected to provide in your response.

The following open-response item from a Political Science/Political Philosophy (48) test illustratesthe components of a typical open-response item. The first paragraph introduces the topic of theassignment—the checks and balances that regulate the exercise of power among the three branchesof the U.S. government. The second paragraph, and accompanying bulleted instructions, describes thewriting task candidates must undertake and the elements they must include in their response.

Sample open-response item

Read the information below; then complete the exercise that follows.

The U.S. Constitution creates a government of checks and balances in which legislative,executive, and judicial authority reside in separate branches of the government.

Using your knowledge of the U.S. government, write a response in which you:

• describe the powers that are given to each of the three branches of the government underthe Constitution; and

• discuss how these powers enable each branch of the government to check the activitiesof each of the other two branches.

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Scoring of Open-Response Items

Open-response items are scored holistically according to standardized procedures, using scoringscales that have been approved by the Massachusetts Department of Elementary and SecondaryEducation. In judging the overall effectiveness of each response, scorers use scoring scales thatdescribe varying levels of performance.

Open-response items are scored holistically by two or more qualified educators during scoring sessions

held after each test administration. Scorers are oriented before the scoring session to ensure that allresponses are scored according to standardized procedures.

Performance characteristics. A set of performance characteristics, similar to the following, guidescorers in considering responses to the open-response items. The performance characteristics arecontained in the test directions for the open-response items. It may be helpful for you to be aware ofthese performance characteristics as you prepare for and take the MTEL. The actual performancecharacteristics used for your field may be found in the practice tests (available for some fields) and

MTEL Faculty Guide available at www.mtel.nesinc.com .

Performance

Characteristics Definitions

Purpose The extent to which the response achieves the purpose of theassignment.

Subject MatterKnowledge

Accuracy and appropriateness in the application of subjectmatter knowledge.

Support Quality and relevance of supporting details.

Rationale Soundness of argument and degree of understanding of thesubject matter.

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Scoring scale. The scoring scale below, which is related to the performance characteristics for thetests, is used by scorers in assigning scores to responses to the open-response items. Note that thisscoring scale is typical of the scoring scale used for many tests of the MTEL. The actual score scaleused for your field may be found in the practice tests (available for some fields) and MTEL Faculty

Guide available at www.mtel.nesinc.com .

Score

Point Score Point Description

4

The "4" response reflects a thorough knowledge and understanding of the

subject matter.• The purpose of the assignment is fully achieved.• There is substantial, accurate, and appropriate application of subject matter

knowledge.• The supporting evidence is sound; there are high-quality, relevant examples.• The response reflects an ably reasoned, comprehensive understanding of the topic.

3

The "3" response reflects an adequate knowledge and understanding of the

subject matter.• The purpose of the assignment is largely achieved.

• There is a generally accurate and appropriate application of subject matterknowledge.

• The supporting evidence is adequate; there are some acceptable, relevantexamples.

• The response reflects an adequately reasoned understanding of the topic.

2

The "2" response reflects a limited knowledge and understanding of the subject

matter.• The purpose of the assignment is partially achieved.• There is a limited, possibly inaccurate or inappropriate application of subject matter

knowledge.• The supporting evidence is limited; there are few relevant examples.

• The response reflects a limited, poorly reasoned understanding of the topic.

1

The "1" response reflects a weak knowledge and understanding of the subject

matter.• The purpose of the assignment is not achieved.• There is little or no appropriate or accurate application of subject matter knowledge.• The supporting evidence, if present, is weak; there are few or no relevant examples.• The response reflects little or no reasoning about or understanding of the topic.

A response to an open-response item is designated "unscorable" if it is unrelated to the assigned topic,illegible (for hand-written responses), not in the appropriate language, of insufficient length to score, ormerely a repetition of the assignment. If there is no response to an open-response item, it isdesignated "blank."

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The Day of the Test

This section provides suggestions for preparing for the experience of the test as well as strategies you maywish to employ while taking the test.

Preparing for the TestThis section offers tips that you can use to prepare for the test day.

Start early. • Make sure you leave plenty of time to get to the test center.

• Report to the test center 30 minutes before your scheduled testing time.

Dress comfortably. • Wear layers of clothing that you can easily remove or add on (e.g., sweater, sweatshirt). This

will allow you to make personal adjustments for fluctuations in room or body temperature.

• Wear soft-soled shoes so that you will not disturb other candidates when you leave your seat.

Arriving at the test center. • Any personal items that you bring to the test center must be stored in a locker outside of the

testing room during the test. This includes watches, wallets, and purses in addition to prohibitedmaterials described on the MTEL web site.

• Do not bring any unauthorized aids or prohibited materials, as listed under "Test Center Rules"on the MTEL web site, with you to the test center. If you do bring these materials, you will berequired to leave them outside the testing area, as designated.

Test-Taking Tips

The following tips for taking standardized tests are offered as suggestions that may contribute to yoursuccess and confidence during the test session.

Watch the tutorials.• Take the Computer-Based Testing Tutorial on the MTEL web site before the day of your test to

learn how to navigate through a computer-based test (CBT), including how to indicate answersand how to use CBT features.

• Take the Pearson Professional Centers Tour on the MTEL web site to preview a visit to aPearson Professional Center, including a view of the typical test room and other test centerareas, to gain advance knowledge of the test center experience.

Follow directions. • At the beginning of the session and throughout the test, follow all directions carefully. This

includes instructions and written directions given to you by the test center administrators andwritten directions in the test itself.

• The test will contain general directions for the test as a whole and specific directions forindividual test items or groups of test items.

• If you do not understand something about the directions, raise your hand and ask a test centeradministrator.

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Pace your work. • Each test session is four hours long. The test session is designed to allow sufficient time for

sign-in and communication of directions before testing begins and for completion of the test bycandidates. Some candidates may finish testing well before the scheduled ending time of thetest session.

• You may find that you need less time than the four hours in a test session, but be prepared tostay for the entire time.

• Do not make any other commitments for this time period that may cause you to work morequickly than you should.

• Do not spend a lot of time on a test item that you cannot answer promptly; skip that item andmove on.

Read carefully. • Read the directions and the test items carefully.

• Read all response options.

• Remember that the test items call for the "best answer." Do not choose the first responseoption that seems reasonable; read and evaluate all choices to find the best answer.

• Read the test items closely so that you understand what they ask.

• Do not skim the test items in an effort to save time; you may misread key words and select thewrong answer or spend more time than needed on a test item. For example, if a test item callsfor an approximate answer and you skip over that detail, you could waste time performing a longcomputation.

• Read the test items, but don't read into them. The test items are designed to be straightforward,not tricky.

Determine the "best answer" and guess wisely. • As you read through the multiple-choice item response options, try to find the best answer. If

you cannot quickly determine the best answer, try to eliminate as many of the options as

possible. Then, guess among the remaining answer choices.

• Your score on each test will be based on the number of multiple-choice items you haveanswered correctly, in combination with your score(s) on the open-response items.

• There is no penalty for incorrect multiple-choice item answers; therefore, it is better to guessthan not to respond at all.

Check your accuracy. • Use any remaining time at the end of the test session to check your work.

• Go back to the test items that gave you difficulty and verify your work on them.

Review carefully the open-response items.• Be sure to read and respond to each part of the items.

• Write legibly (if answers are being written by hand). Your responses to the open-responseitems will be evaluated by at least two qualified scorers. It is important for the scorers to be ableto read your response.

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After the Test

Score Reporting

After you have taken a test in the MTEL program, you will receive a score report indicating:

• whether or not you met the qualifying score on the test;• your total test score, if you did not meet the qualifying score; and

• information about your performance on the subareas/skills of the test.

Your score report is available to you on the Internet for 45 days, beginning at 10:00 p.m. eastern timeaccording to the schedule on the MTEL web site, as a PDF document which you may view, print, andsave for your records.

If you register online, you may request during the registration process that your score report be sentto you by e-mail. Score report e-mails are sent beginning at 5:00 p.m. eastern time on the score reportdate.

If you register by mail or telephone, your score report will also be sent to you by U.S. mail on thescore report date.

Your score report will be sent to the Department of Elementary and Secondary Education andautomatically added to your licensure file once you apply for a license, based on your social securitynumber or Department-issued identification number. Your score report will be sent automatically tothe Massachusetts institution(s) you indicated during registration. In addition, if you give explicitpermission, your educator preparation institution will also receive the last five digits of your socialsecurity number.

Interpreting Your Score Report

Each MTEL test has its own qualifying score (i.e., passing score) set by the MassachusettsCommissioner of Elementary and Secondary Education. In order to report test scores in a consistentway, the scores for all the tests are reported on the same scale. Test results are reported as scores ina range of 100 to 300, with a score of 240 representing the qualifying, or passing, score. Therefore, atotal test scaled score of 240 or higher is required to meet the qualifying score for each test.

Your total test score is based on your performance on all sections of the test. Your multiple-choicescore and scores on open-response items are combined to obtain your total score, reported on the100 to 300 scale. The scaled score for the multiple-choice items is obtained from the number of testitems answered correctly. The scaled score for the open-response items is obtained from the scoresassigned to the candidate's written or oral responses. If a candidate does not attempt an open-response item on the test, no points are contributed to the candidate's score for that test item.

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M iddle School M athematics (F ield 47)

Test Overview Chart

Formulas

Sample Test Items

Answer Key and Sample Response

Test Objectives

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Test Overview Char t:

M iddle School Mathematics (47)

SubareasApproximateNumber ofMultiple-

Choice Items

Number ofOpen-

Response

Items

I. Number Sense and Operations 18–20

II. Patterns, Relations, and Algebra 30–32

III. Geometry and Measurement 21–23

IV. Data Analysis, Statistics, and Probability 14–16

V. Trigonometry, Calculus, and Discrete Mathematics 12–14

VI. Integration of Knowledge and Understanding2

The Middle School Mathematics test is designed to assess the candidate's knowledge of the subject matter

required for the Massachusetts Mathematics (Level: 5–8) license. This subject matter knowledge is delineated in

the Massachusetts Department of Elementary and Secondary Education's Regulations for Educator Licensure and

Preparation Program Approval (7/2001), 603 CMR 7.06, "Subject Matter Knowledge Requirements for

Teachers."

The Middle School Mathematics test assesses the candidate's proficiency and depth of understanding of the

subject at the level required for a baccalaureate minor (minimum of 24 semester hours). Candidates are typically

nearing completion of or have completed their undergraduate work when they take the test.

The multiple-choice items on the test cover the subareas as indicated in the chart above. The open-response itemsmay relate to topics covered in any of the subareas and will typically require breadth of understanding of the

mathematics field and the ability to relate concepts from different aspects of the field. Responses to the open-

response items are expected to be appropriate and accurate in the application of subject matter knowledge, to

provide high-quality and relevant supporting evidence, and to demonstrate a soundness of argument and

understanding of the mathematics field.

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Candidates taking the Middle School Mathematics test (field 47) will be provided with the formulasshown below at the test administration.

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Sample Test I tems:


Calculators will NOT be provided for the Middle School Mathematics test (field 47).

1. Use the sequence of steps below to answer the questionthat follows.

4.26 × 2.2 =

426 ×

1100 ×

22 ×

110

= 426 ×

1

100 × 22 ×110

= 426 ×

22 ×

1100 ×

110

= (426 × 22) ×

1

100 ×110

= (426 × 22) ×

1

1000

The sequence of steps above could be used to answer whichof the following questions?

A. How can a decimal be converted to a fraction reduced tolowest terms?

B. Where should the decimal point be placed in the productof two decimals?

C. Why is it necessary to invert and multiply when solvingproblems involving fractions?

D. How is scientific notation used to multiply decimals?

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Sample Test Items: M iddle School Mathematics

27

2. Which of the following situations represents 225 ÷

14?

A. A 225-acre lot needs to be plowed. If four workers split

the plowing evenly, how many acres will each person

plow?

B. One side of a one-fourth-square-foot rectangle is

225 feet. How long is the other side?

C. Alix sawed off one fourth of a 225-yard log. How many

yards did Alix saw off?

D. Terry picked 225 pounds of berries and put them into one-

quarter-pound containers. How many containers of

berries did Terry fill?

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Sample Test I tems: M iddle School Mathematics

28

3. Use the diagram below to answer the question that follows.

Square value Subtract from two

Input ( ) x

Output ( )y

Which of the following graphs represents the function described in the above diagram?

A. B.

x

y

x

y

C. D.

x

y

x

y

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4. Use the graph below to answer the question that follows.

100

90

80

70

60

50

40

30

20

10

0

Length of Eruption, (min)

T i m e U n t i l N e x t E r u p t i o n ,

( m i n

)

t

0.5 1 1.5 2 2.5 3 3.5 4 4.5 5

The graph shown above represents the relationship betweenthe length of a geyser's eruption, B, and the time until the nexteruption, t . Which of the following linear equations bestmodels the data?

A. t + 12B = 33

B. 2t – 3B = –66

C. 12B – t = –33

D. 3B + 2t = 66

5. In solving the quadratic equation x 2 + 14 x – 4 = –30 bycompleting the square, the first step is to add 4 to bothsides of the equation. The second step is to:

A. add 49 to both sides of the equation.

B. factor x from the binomial x 2 + 14 x .

C. factor the number 2 from 14, –4, and 30.

D. take the square root of both sides of the equation.

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6. Use the geometric proof below to answer the question that follows.

Given: Circle P ;

mPQ is a median of PRS

Prove: RPQ ≅ SPQ

R

P

S

Q

Statements Reasons

1. Circle P ; mPQ is a median of PRS 1. Given

2. mRQ ≅ mQS 2. Definition of median

3. mPR and mPS are radii of P . 3. Definition of radius

4. mPR ≅ mPS 4. All the radii of a circle are congruent.

5. ___ ? 5. ___ ?

6. RPQ ≅ SPQ 6. SSS postulate

Which of the following statements and reasons would be mostappropriate in step 5 of this proof?

A. Statement Reason

∠QPR ≅ ∠QPS mPQ ⊥ mRS

B. Statement Reason P is the center of the Definition of a circlecircle.

C. Statement Reason ∠PSQ ≅ ∠PRQ Properties of isosceles

triangle

D. Statement Reason

mPQ ≅ mPQ Reflexive property ofcongruence

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7. Use the diagram below to answer the question thatfollows.

Square corners, 5 units on a side, are removed from a 15-unit-by-25-unit rectangular piece of cardboard. The sides of thecardboard are then folded to form an open box. Which of thefollowing is the volume, in cubic units, of the box?

A. 375

B. 625

C. 1000

D. 1125

8. Which of the following situations best represents a randomsampling?

A. Ask every tenth person coming out of a health spa how

many times a week they exercise to determine how oftenpeople in the town exercise.

B. Survey students in advanced biology classes todetermine the average amount of time students ina certain school study each week.

C. Find the heights of all boys in a senior gym class todetermine the average height of all boys in the school.

D. Count the number of chocolate chips in every fifth cookieto determine the average number of chocolate chips per

cookie in a bag of cookies.

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9. Use the diagram below to answer the question thatfollows.

x

y

(0, 8) (4, 8)

(0, 0) (4, 0)

A point is randomly selected within the rectangle shown in thediagram above. What is the probability that the y -coordinate ofthe point is less than or equal to 2?

A.18

B.29

C.14

D.12

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10. The graph of the function f( x ) = 2 x 2 – 6 x + 4 has its minimum

value at the point

3

2 , –

12 . Which of the following statements

must be true?

A. The derivative of f( x ) is zero when x is equal to32 .

B. The value of f( x ) approaches a limit of zero as x

approaches32 .

C. The graph of f( x ) shifts from a concave downward shapeto a concave upward shape.

D. The slope of the line tangent to f( x ) at x = 32 is equal

to –12.

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DIRECTIONS FOR THE OPEN-RESPONSE ITEM ASSIGNMENTS

This section of the test consists of two open-response item assignments. You will be asked toprepare a written response of approximately 150–300 words, or 1–2 pages, for each assignment.

Read the assignments carefully before you begin your responses. Think about how you willorganize your responses. You may use the erasable sheet(s) to make notes, write an outline, orotherwise prepare your responses. However , your f inal response to each assignm ent mus t beeither:

1) typed into the on-screen response box,

2) written on a response sheet and scanned using the scanner provided at your workstation, or

3) provided using both the on-screen response box (for typed text) and a response sheet (forcalculations or drawings) that you will scan using the scanner provided at your workstation.

Instructions for scanning your response sheet(s) are available by clicking the “Scanning

Help” button at the top of the screen.

As a whole, your response to each assignment must demonstrate an understanding of theknowledge of the field. In your response to each assignment, you are expected to demonstrate thedepth of your understanding of the subject area by applying your knowledge rather than by merelyreciting factual information.

Your responses to the assignments will be evaluated based on the following criteria.

PURPOSE: the extent to which the response achieves the purpose of the assignment

SUBJECT KNOWLEDGE: appropriateness and accuracy in the application of subject knowledge

SUPPORT: quality and relevance of supporting evidence

RATIONALE: soundness of argument and degree of understanding of the subject area

The open-response item assignments are intended to assess subject knowledge. Your responsesmust be communicated clearly enough to permit valid judgment of the evaluation criteria byscorers. Your responses should be written for an audience of educators in this field. The finalversion of each response should conform to the conventions of edited American English. Yourresponses should be your original work, written in your own words, and not copied or paraphrasedfrom some other work.

Be sure to write about the assigned topics. Remember to review your work and make any changesyou think will improve your responses.

Any time spent responding to an assignment, including scanning the response sheet(s), is part ofyour testing time. Monitor your time carefully. When your testing time expires, a pop-up messagewill appear on-screen indicating the conclusion of your test session. Only response sheets that arescanned before you end your test or before time has expired will be scored. Any response sheetthat is not scanned before testing ends will NOT be scored.

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11. Use the information below to complete the exercise that follows.

A company is considering two bonus-plan options for its employees for the next 20 years. Thetwo options are explained in the following chart.

Option 1: Receive $2 the first year. Every year thereafter receive twice the bonus amountof the previous year.

Option 2: Receive $200 the first year. Every year thereafter receive $200 more than thebonus amount of the previous year.

Use your knowledge of exponential and linear functions to develop a response in which youanalyze the bonus received each year during a 20-year period under each option. In yourresponse:

• create a data table representing the bonus received each year over a 12-year period foreach option;

• graph the data from both tables on the same coordinate grid and connect the data with theline or curve that best fits the data;

• compare the bonus plans over the 12-year period, including a discussion of the significanceof the point of intersection of the two graphs;

• explain what type of function, exponential or linear, models each option;

• find equations that describe each option; and

• identify an expression that represents the difference between the bonuses received underthe two options in the twentieth year.

Be sure to show your work and explain the reasoning you use in analyzing and solving thisproblem.

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Answer Key and Sample Response:


QuestionNumber

CorrectResponse

Test Objective*

1. B Understand principles and operations related to integers, fractions, decimals, percents,ratios, and proportions.

2. D Understand and solve problems involving integers, fractions, decimals, percents, ratios,and proportions.

3. B Understand properties of functions and relations.

4. C Understand properties and applications of linear relations and functions.

5. A Understand properties and applications of quadratic relations and functions.

6. D Understand the principles of Euclidean geometry and use them to prove theorems.

7. A Solve problems involving three-dimensional shapes.

8. D Understand descriptive statistics and the methods used in collecting, organizing,reporting, and analyzing data.

9. C Understand the fundamental principles of probability.

10. A Understand the conceptual basis of calculus.

*Each test objective is clarified and further described by a descriptive statement, which provides examples of the

types of knowledge and skills covered by the test objective. The test objectives for the Middle School

Mathematics test begin on page 39.

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Answer Key and Sample Response: M iddle School M athematics

37

The sample response below reflects a strong knowledge and understanding of the subject matter.

The data tables representing the bonuses received each year over a 12-year period for thetwo options are shown below.

Option 1

Year 1 2 3 4 5 6 7 8 9 10 11 12

Bonus (dollars)

2 4 8 16 32 64 128 256 512 1024 2048 4096

Option 2

Year 1 2 3 4 5 6 7 8 9 10 11 12

Bonus (dollars)

200 400 600 800 1000 1200 1400 1600 1800 2000 2200 2400

The data from the tables can be graphed on a coordinate grid as shown below.

P

x 4 8 12

1000

2000

3000

4000

Number of Years

B o n u s ( D o l l a r s )

y

option 1

option 2

(continued)

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Answer Key and Sample Response: M iddle School M athematics

38

(continued from previous page)

The two graphs intersect at point P, where they both pay the same bonus. For all years beforethis point (approximately years 1 through 11), option 2 yields a higher bonus. For all years afterthis point (years 12 and beyond), option 1 yields a higher bonus.

The bonus plan offered in option 1 increases by a constant factor of 2 and therefore ismodeled by an exponential equation. The exponential equation that models this bonus plan is y = 2x . The bonus plan offered in option 2 increases by a constant rate of $200 each year andtherefore is modeled by a linear equation. The linear equation that models this bonus plan is y = 200x .

In the 20th year, the bonus offered by option 1 exceeds that offered by option 2 by220 – 4000 dollars.

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Test Objectives:


Massachusetts Tests for Educator Licensure

FIELD 47: MIDDLE SCHOOL MATHEMATICS

TEST OBJECTIVES

Subarea

Multiple-Choice Range of

Objectives

ApproximateTest

Weighting

I. Number Sense and Operations 01–04 15%

II. Patterns, Relations, and Algebra 05–10 25%

III. Geometry and Measurement 11–15 18%

IV. Data Analysis, Statistics, and Probability 16–17 12%

V. Trigonometry, Calculus, and DiscreteMathematics

18–20 10%

80%

Open-Response

VI. Integration of Knowledge and Understanding 21 20%

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Test Objectives: M iddle School Mathematics

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SUBAREAS:

NUMBER SENSE AND OPERATIONSPATTERNS, RELATIONS, AND ALGEBRA

GEOMETRY AND MEASUREMENTDATA ANALYSIS, STATISTICS, AND PROBABILITY

TRIGONOMETRY, CALCULUS, AND DISCRETE MATHEMATICSINTEGRATION OF KNOWLEDGE AND UNDERSTANDING

NUMBER SENSE AND OPERATIONS [15%]

0001 Understand the structure of numeration systems and multiple representations of numbers.

For example: place value; number bases (e.g., base 2, base 10); order relations;relationships between operations (e.g., multiplication as repeated additions); number factorsand divisibility; prime and composite numbers; prime factorization; multiple representationsof numbers (e.g., physical models, diagrams, numerals); and properties of early numerationsystems (e.g., Mayan, Mesopotamian, Egyptian).

0002 Understand principles and operations related to integers, fractions, decimals, percents,ratios, and proportions.

For example: order of operations; identity and inverse elements; associative, commutative,and distributive properties; absolute value; operations with signed numbers; multiplerepresentations (e.g., area models for multiplication) of number operations; analyzingstandard algorithms for addition, subtraction, multiplication, and division of integers andrational numbers; number operations and their inverses; and the origins and development ofstandard computational algorithms.

0003 Understand and solve problems involving integers, fractions, decimals, percents, ratios, andproportions.

For example: solving a variety of problems involving integers, fractions, decimals, percents(including percent increase and decrease), ratios, proportions, and average rate of change;and using estimation to judge the reasonableness of solutions to problems.

0004 Understand the properties of real numbers and the real number system.

For example: rational and irrational numbers; properties (e.g., closure, distributive,associative) of the real number system and its subsets; operations and their inverses; thereal number line; roots and powers; the laws of exponents; scientific notation; using numberproperties to prove theorems (e.g., the product of two even numbers is even); and problemsinvolving real numbers and their operations.

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PATTERNS, RELATIONS, AND ALGEBRA [25%]

0005 Understand and use patterns to model and solve problems.

For example: making conjectures about patterns presented in numeric, geometric, or tabularform; representing patterns and relations using symbolic notation; identifying patterns of

change created by functions (e.g., linear, quadratic, exponential); and usingfinite and infinite series and sequences (e.g., Fibonacci, arithmetic, geometric) to modeland solve problems.

0006 Understand how to manipulate and simplify algebraic expressions and translate problemsinto algebraic notation.

For example: the nature of a variable; evaluating algebraic expressions for a given value of avariable; the relationship between standard computational algorithms and algebraic processesexpressing direct and inverse relationships algebraically; expressing one variable in terms ofanother; manipulating and simplifying algebraic expressions; solving equations; and usingalgebraic expressions to model situations.

0007 Understand properties of functions and relations.For example: the difference between functions and relations; the generation andinterpretation of graphs that model real-world situations; multiple ways of representingfunctions (e.g., tabular, graphic, verbal, symbolic); properties of functions and relations(e.g., domain, range, continuity); piecewise-defined functions; addition, subtraction, andcomposition of functions; and graphs of functions and their transformations [e.g., therelationships among f( x ), f( x + k ), and f( x ) + k ].

0008 Understand properties and applications of linear relations and functions.

For example: the relationship between linear models and rate of change; direct variation;graphs of linear equations; slope and intercepts of lines; finding an equation for a line;methods of solving systems of linear equations and inequalities (e.g., graphing, substitution);

and modeling and solving problems using linear functions and systems.

0009 Understand properties and applications of quadratic relations and functions.

For example: methods of solving quadratic equations and inequalities (e.g., factoring,completing the square, quadratic formula, graphing); real and complex roots of quadraticequations; graphs of quadratic functions; quadratic maximum and minimum problems; andmodeling and solving problems using quadratic relations, functions, and systems.

0010 Understand properties and applications of exponential, polynomial, rational, and absolutevalue functions and relations.

For example: problems involving exponential growth (e.g., population growth, compoundinterest) and decay (e.g., half-life); inverse variation; modeling problems using rationalfunctions; properties and graphs of polynomial, rational, and absolute value functions; andthe use of graphing calculators and computers to find numerical solutions to problemsinvolving exponential, polynomial, rational, and absolute value functions.

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GEOMETRY AND MEASUREMENT [18%]

0011 Understand principles, concepts, and procedures related to measurement.

For example: using appropriate units of measurement; unit conversions within and amongmeasurement systems; problems involving length, area, volume, mass, capacity, density,

time, temperature, angles, and rates of change; problems involving similar plane figuresand indirect measurement; the effect of changing linear dimensions on measures of length,area, or volume; and the effects of measurement error and rounding on computed quantities(e.g., area, density, speed).

0012 Understand the principles of Euclidean geometry and use them to prove theorems.

For example: the nature of axiomatic systems; undefined terms and postulates of Euclideangeometry; relationships among points, lines, angles, and planes; methods for provingtriangles congruent; properties of similar triangles; justifying geometric constructions; provingtheorems within the axiomatic structure of Euclidean geometry; and the origins anddevelopment of geometry in different cultures (e.g., Greek, Hindu, Chinese).

0013 Apply Euclidean geometry to analyze the properties of two-dimensional figures and to solveproblems.

For example: using deduction to justify properties of and relationships among triangles,quadrilaterals, and other polygons (e.g., length of sides, angle measures); identifying planefigures given characteristics of sides, angles, and diagonals; the Pythagorean theorem;special right triangle relationships; arcs, angles, and segments associated with circles;deriving and applying formulas for the area of composite shapes; and modeling and solvingproblems involving two-dimensional figures.

0014 Solve problems involving three-dimensional shapes.

For example: area and volume of and relationships among three-dimensional figures(e.g., prisms, pyramids, cylinders, cones); perspective drawings; cross sections (including

conic sections) and nets; deriving properties of three-dimensional figures from two-dimensional shapes; and modeling and solving problems involving three-dimensionalgeometry.

0015 Understand the principles and properties of coordinate and transformational geometry.

For example: representing geometric figures (e.g., triangles, circles) in the coordinateplane; using concepts of distance, midpoint, slope, and parallel and perpendicular lines toclassify and analyze figures (e.g., parallelograms); characteristics of dilations, translations,rotations, reflections, and glide-reflections; types of symmetry; properties of tessellations;transformations in the coordinate plane; and using coordinate and transformational geometryto prove theorems and solve problems.

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DATA ANALYSIS, STATISTICS, AND PROBABILITY [12%]

0016 Understand descriptive statistics and the methods used in collecting, organizing, reporting,and analyzing data.

For example: constructing and interpreting tables, charts, and graphs (e.g., line plots, stem-

and-leaf plots, box plots, scatter plots); measures of central tendency (e.g., mean, median,mode) and dispersion (e.g., range, standard deviation); frequency distributions; percentilescores; the effects of data transformations on measures of central tendency and variability;evaluating real-world situations to determine appropriate sampling techniques and methodsfor gathering and organizing data; making appropriate inferences, interpolations, andextrapolations from a set of data; interpreting correlation; and problems involving linearregression models.

0017 Understand the fundamental principles of probability.

For example: representing possible outcomes for a probabilistic situation; countingstrategies (e.g., permutations and combinations); computing theoretical probabilitiesfor simple and compound events; using simulations to explore real-world situations;

connections between geometry and probability (e.g., probability as a ratio of two areas);and using probability models to understand real-world phenomena.

TRIGONOMETRY, CALCULUS, AND DISCRETE MATHEMATICS [10%]

0018 Understand the properties of trigonometric functions and identities.

For example: degree and radian measure; right triangle trigonometry; the law of sines andthe law of cosines; graphs and properties of trigonometric functions and their inverses;amplitude, period, and phase shift; trigonometric identities; and using trigonometric functionsto model real-world periodic phenomena.

0019 Understand the conceptual basis of calculus.For example: the concept of limit; the relationship between slope and rates of change; howthe derivative relates to maxima, minima, points of inflection, and concavity of curves; therelationship between integration and the area under a curve; modeling and solving basicproblems using differentiation and integration; and the development of calculus.

0020 Understand the principles of discrete/finite mathematics.

For example: properties of sets; recursive patterns and relations; problems involvingiteration; properties of algorithms; finite differences; linear programming; properties ofmatrices; and characteristics and applications of graphs and trees.

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INTEGRATION OF KNOWLEDGE AND UNDERSTANDING [20%]

In addition to answering multiple-choice items, candidates will prepare written responses to questionsaddressing content from the preceding objectives, which are summarized in the objective anddescriptive statement below.

0021 Prepare an organized, developed analysis on a topic related to one or more of the following:number sense and operations; patterns, relations, and algebra; geometry and measurement;data analysis, statistics, and probability; and trigonometry, calculus, and discrete

mathematics.

For example: presenting a detailed solution to a problem involving one or more of thefollowing: place value, number base, and the structure and operations of number systems;application of ratios and proportions in a variety of situations; properties, attributes, andrepresentations of linear functions; modeling problems using exponential functions; thederivative as a rate of change and the integral as area under the curve; applications of planeand three-dimensional geometry; and design, analysis, presentation, and interpretation of astatistical survey.

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47_MS MATH