Nova Scotia ExaminationsMathematics and Advanced Mathematics 122011–2012Information Guide

Table of contentsIntroduction.1Overview of NSE Mathematics 12/Advanced Mathematics 12.1Curriculum Links and Rationale.2Tables of Specification.3Exam ConstructionSpecification TablesExplanation of Cognitive LevelsItem Bank SubmissionsSecurity.12Administration of Examination.13Pre-administrationDuring tions.15Adaptations.16Scoring and Reporting.17Procedures for Requestion a Re-Score of a Nova Scotia ExaminationExam Scoring Norms for Constructed ResponseAppendices.20A: Outcomes as outlined in the Atlantic Canada MathematicsCurriculum – Mathematics 12/Advanced Mathematics 12.21B: Outcomes assessed on NSE – Mathematics 12.24C: Outcomes assessed on NSE – Advanced Mathematics 12.26D: Formula Sheet – Mathematics 12.29E: Formula Sheet – Advanced Mathematics 12.30F: List of Mathematical Terms and Concepts.31G: Graphing Calculator Use.33H: How to clear the memory on graphing calculators.38

IntroductionThe purpose of this Information Guide is to provide teachers with information about the NovaScotia Examinations (NSE) in Mathematics 12 and Advanced Mathematics 12. There is currentlyno NSE in Mathematics Foundations 12. Teachers are encouraged to share the contents of the guide,particularly the sample questions and answers, with their students.Overview of NSE Mathematics 12/Advanced Mathematics 12The NSE in Mathematics 12/ Advanced Mathematics 12 is designed to evaluate student achievementin relation to the outcomes for Mathematics 12/ Advanced Mathematics 12 prescribed by the AtlanticCanada Mathematics Curriculum. The examination results contribute 30% to students’ final coursemark.All students registered in Mathematics 12/ Advanced Mathematics 12 are eligible to write thecorresponding NSE. Students who have an Individual Program Plan (IPP) in Mathematics, andtherefore engage with a different set of mathematics outcomes, do not participate in the examination.The NSE Mathematics Advisory Group, comprised of senior high school mathematics teachersrepresenting all school boards in Nova Scotia, assists in the development of examinations. Theadvisory group, under the guidance of department staff, follows the examination developmentprocedures outlined in the Nova Scotia Examination Development Model. As well, an examinationreview team, comprising experienced senior high mathematics teachers, reviews and approvesthe final examination forms for each administration. All processes, examination development,administration, marking, and reporting are facilitated by the Evaluation Services Division of theDepartment of Education.The mathematics examination is constructed according to precise specifications. Questions arewritten to match outcomes in the curriculum and these are field-tested with students in Nova Scotiaschools. Field-test results are then analyzed.Examination questions, either in selected response or constructed response format, are used toevaluate students’ mathematics thinking and understanding in relation to quadratics, exponentials,geometry, and probability.The examination is scored by Mathematics 12/ Advanced Mathematics 12 teachers at regionalmarking sessions led by Evaluation Services Division staff. Standards for marking are set by theAdvisory Group. Results are published in the Minister’s Report to Parents.NSE – Mathematics 12/Advanced Mathematics 12Information Guide1

Curriculum Links and RationaleThe document for the Atlantic Canada Mathematics Curriculum,Mathematics 12 and Advanced Mathematics 12, articulates thecurriculum for both of these courses. The document provides theteachers of each mathematics course with information to planfor instruction. Teachers must carefully follow the curriculum ascontained therein to design learning experiences for their students.The NSE in Mathematics 12 and Advanced Mathematics 12 are designed to reflect the tablesof specifications in this guide (see page 3). The tables are aligned to reflect the Atlantic CanadaMathematics Curriculum. The outcomes listed in the appendices of this guide have been used toconstruct both examinations. Appendix B outlines those outcomes that may be addressed on theMathematics 12 examination and Appendix C outlines those that may be addressed on the AdvancedMathematics 12 examination. Note that the examinations may not assess all outcomes listed in theappendices.Some examination questions will assess students’ understanding of an individual outcome, whileother questions will assess a number of outcomes. The examinations are comprised of a varietyof question types including selected response and constructed response, requiring both short andextended answers. Questions are developed to assess students’ performance at different cognitivelevels (low, moderate, and high complexity).The Information Guide will be revised as needed to reflect changes in the examination process.Teachers will be notified as soon as possible when any changes occur. A copy of this guide is postedon the Evaluation Services Division website .2NSE – Mathematics 12/Advanced Mathematics 12Information Guide

Tables of SpecificationsExamination ConstructionNova Scotia Examinations in Mathematics 12 and Advanced Mathematics 12 are constructed inaccordance with tables of specifications and the Nova Scotia Assessment Development Model. Theyinclude questions (items) that have met the following criteria: rigorous content review by the provincial mathematics examination advisorygroup for alignment with outcomes as listed in the appendices and forpossible bias and construction flaws; field-testing under monitored conditions in Mathematics 12 and AdvancedMathematics 12 classrooms; statistical analysis of the students’ responses following the field-testing todetermine levels of difficulty, validity, and reliability of each question.Specification TablesThe following table provides the approximate weightings of each unit on the examinations andis based on the recommendations for time allotment found in the Atlantic Canada MathematicsCurriculum for Mathematics 12 and Advanced Mathematics 12.Table 1UNITQuadraticsExponential GrowthCircle GeometryProbabilityPercentage of the examination Percentage of the examinationMath 12Advanced Math 1230% – 40%30% – 35%30% – 40%30% – 35%10% – 15%20% – 25%10% – 15%10% – 15%NSE – Mathematics 12/Advanced Mathematics 12Information Guide3

Table 2 outlines the construction of each examination according to question format, includingselected-response and constructed-response questions. The selected-response questions offer thestudent four choices, three of which are plausible distractors, and one that is the correct response.Constructed-response questions may require the solution of a problem or a written response at any ofthe three cognitive levels.Table 2Question Format# ofPercentage of Cognitive LevelsQuestions ExaminationSelected response (multiple choice)35 35%1 and 2Constructed response (short answerand extended response)14 – 18 65% *1, 2, and 3*The exam does not necessarily add to 100 points.Table 3 outlines the construction of each examination according to three levels of questioncomplexity: low complexity (level 1), moderate complexity (level 2), and high complexity (level 3).Table 3Cognitive LevelLow complexity(level 1)Moderate complexity(level 2)High complexity(level 3)4ApproximateWeighting30%50%20%NSE – Mathematics 12/Advanced Mathematics 12Information Guide

Explanation of Cognitive LevelsQuestions on the NSE are developed to assess students’ performance at three cognitive levels.Cognitive levels indicate the type of intellectual process required to respond to each question. Thisguide includes the marking scheme for constructed-response questions so that you may familiarizeyourself with scoring as it is done at regional sessions.Low Complexity Questions (Level 1)Low complexity questions will require students to recall and recognize previously learned conceptsand principles. Students may demonstrate the use of routine procedures to solve a problem. Studentsare not be required to develop an original method for solving a problem. This level may includerecognition or recall of terminology, formulae, algorithms, graphs, geometric figures, properties, andtheorems. Questions at this level include key words such as: identify, compute, recall, recognize,find, use, what, list, define, and name.The following are some examples of what a low-complexity question might require a student to do: recall or recognize a fact, term, or propertyrecognize an example of a conceptcompute a sum, difference, product, or quotientperform a specific proceduresolve a one-step word problemretrieve information from a graph, table, figure or functionExamples:Selected-response questionGiven y ax2 bx c. The value of brepresents2aa) the minimum or maximum value{outcome C31}Pb) the ‘x’ value of the vertexc) the coordinates of the vertex d) the y-interceptIn this example the student needs to recall or recognize a fact.NSE – Mathematics 12/Advanced Mathematics 12Information Guide5

Constructed-response questionGiven the function y –2x2 4x – 5(a) Write the function in standard or transformational form.(2 points){outcomes C9, C31}Points awarded: 0.5 pt : algebraic manipulation 1 pt : completing the square and balancing 0.5 pt : final answer(b) What is the vertex of the parabola?(1 point)(c) What is the equation of the axis of symmetry?(1 point)The above example (a) requires a student to perform a specific procedure while parts (b) and (c)require the student to retrieve information from a given function. The scoring for (b) and (c) wouldbe all or nothing.6NSE – Mathematics 12/Advanced Mathematics 12Information Guide

Moderate Complexity Questions (Level 2)Moderate-complexity questions require students to identify and understand how the parts of aquestion are connected to the task of solving the problem or question. A question at this levelmight be a problem that is typical of, but not identical to, ones studied in class and it requires thatstudents identify and use the appropriate algorithm. At this level of complexity, students are asked totranslate, interpret, or extrapolate. Translation refers to the ability to communicate the problem andits solution. Interpretation involves making inferences, generalizations, or summaries. Extrapolationwould require the student to estimate or predict the solution from given information by identifyingtrends and tendencies. Questions at this level include key words such as: classify, organize, estimate,interpret, predict, infer, translate, generalize, summarize, problem solve, and apply.The following are some examples of what a moderate-complexity question might require a studentto do: make connections between facts, terms, properties, or operationssolve a word problem requiring multiple stepscompare figures or statementsprovide a justification for steps in a solution processinterpret a visual representationextend a patternretrieve information from a graph, table, or figure, and use it to solve a problem requiringmultiple stepsdetermine the formula for a relationship given data and conditionscompute and solve using appropriate methodsExamples:Selected-response questionTwo dice are rolled. What is the probability that both dice will land on the same number?a)136b)118Pc)16d)12In this example, the student needs to solve using appropriate methods.NSE – Mathematics 12/Advanced Mathematics 12Information Guide7

Constructed-response questionAt the Halifax airshow, a plane performs a power dive. The equation h t2 – 16t 90 expresses therelationship between height, h, in metres and time, t, in seconds.(a) What is the minimum height the plane reaches during the dive?(2 points){outcomes C1, C23, C31}Points awarded: 1.5 pt : graph with vertex clearly marked 0.5 pt : final answer(b) When will the plane be at a height of 35 metres?(2.5 points){outcomes C1, C23}Points awarded: 0.5 pt : substitution of 35 for h 1 pt : solving quadratic equation 1 pt : final answersIn this example, the student needs to make a connection between (a) the minimum height of theplane and the vertex of the given function and (b) the height of the plane and the time needed toreach this height.8NSE – Mathematics 12/Advanced Mathematics 12Information Guide

High Complexity Questions – Novel Problems (Level 3)High-complexity questions include analysis, synthesis, and evaluation. At this level of questioning,students are required to think in an abstract and sophisticated way to reason, plan, analyse, judge,and create. Questions at this level will often include key words such as: analyse, investigate,formulate, prove, derive, explain, and describe.Note: There are no level 3 questions in the selected-response portion of the examination.The following are some examples of what a high-complexity question might require a student to do: explain relations among facts, terms, properties, or operationsanalyse similarities and differences between procedures and conceptsgeneralize a patternformulate an original problem, given a situation or functionsolve a problem in more than one wayexplain and justify a solution to a problemdescribe, compare, and contrast solution methodsformulate a mathematical model for a complex situationanalyse the assumptions made in a mathematical modelanalyse or produce a deductive argumentprovide a mathematical justificationuse concepts taught at prior levels to solve a novel problemExamples:Constructed-response questionIs 2x, 2x 2, 2x 4 a geometric sequence? Explain your reasoning.(3 points){outcomes C4, C29}Points awarded: 2 pt : calculatingratios 1 pt : conclusionThis example requires students to use prior knowledge to solve a novel problem.NSE – Mathematics 12/Advanced Mathematics 12Information Guide9

tCreate a problem that could be modelled by the equation P 5 ( 2 )10 .(2 points){outcomes A5, C2}Points awarded: 0.5 pt : initial amount0.5 pt : doubling0.5 pt : 10 units of time0.5 pt : reasonable contextThis example requires students to formulate an original problem, given a function.10NSE – Mathematics 12/Advanced Mathematics 12Information Guide

Item Bank SubmissionsTeachers are encouraged to submit test items of all types for consideration by the Nova ScotiaExamination Advisory Group for Mathematics 12 and Advanced Mathematics 12.Send materials to:Lennie Comeau, Mathematics Evaluation CoordinatorEvaluation Services DivisionNova Scotia Department of EducationPO Box 578Halifax, NSB3J 2S9or e-mail to [email protected] – Mathematics 12/Advanced Mathematics 12Information Guide11

SecurityNova Scotia Examinations are secure. This means that all examination materials must be sent toyour regional marking site as soon as possible as directed by your Board Assessment Coordinator.The materials include all Student Booklets, both used and unused. All examination materials arenumbered and personalized, and each booklet sent to a school is tracked. No part of the examination,including student work, is to be reproduced in any form or by any means, electronic or mechanical,including photocopying, recording or by any other information storage or retrieval system. Inaddition, teachers must not make use of the exam questions in their teaching.Securing the NSE is critical to ensuring that the evaluation of student achievement is valid and fair.Users of the examination results draw conclusions about the ability of students based on the scoresthe students achieve.The evaluation of student achievement in relation to the selected learning outcomes on theseexaminations is premised on the students’ encountering the tasks for the first time. Any priorexposure compromises the validity of the conclusions drawn about student ability. Because theDepartment of Education will use assessment items from past (secured) examinations in futureexaminations, all involved must do their part to secure these examinations.The use of particular examination questions on a subsequent examination is an important part ofensuring that different examinations render reliable and valid information about student achievementover time. Through the use of one or more anchor questions, two different Mathematics examinationscan be equated, meaning that we can calculate the degree to which one examination is easier orharder than another, and then make appropriate adjustments to equate the two administrations. In thisway, we can assert with greater confidence that changes in results over a period of time represent realchanges in the standard of student performance and not variation in the examinations themselves.12NSE – Mathematics 12/Advanced Mathematics 12Information Guide

Administration of ExaminationThe 2011-2012 administration dates for NSE Mathematics 12 and Advanced Mathematics 12 areJanuary 26 and June 15, 2012. In addition to this Information Guide, the following materials relatingto the administration of the examinations are distributed to schools along with examinations in theweek prior to the date of writing: Nova Scotia Examination (NSE) Packing Slip Mathematics 12/Advanced Mathematics 12(used to verify the materials sent to the school and to account for materials returned to theregional marking site)Student lists with corresponding booklet numbersQuality Control Declaration (to be completed by the School Assessment Coordinatordeclaring that the examinations have been secured prior to the examination)Instructions to Teachers (invigilating directions)Instructions on clearing graphing calculator memoryNote: The School Assessment Coordinator should open the box(es) of examination materials assoon as possible after receipt and check that booklets match the school list.Pre-Administration Two months prior to the date of the examination, teachers and the School AssessmentCoordinator must consult regarding the number of examinations required for each course aswell as the types and numbers of examinations required in alternate formats (Braille,large print, audio CD).Teachers ensure that students have been informed of what they will need in advance of theexamination: an HB pencil, a graphing calculator, and a ruler.NOTE: Schools should be able to provide graphing calculators to students who do nothave their own.The School Assessment Coordinator ensures that exams are scheduled according to the datesin the provincial assessment schedule.The School Assessment Coordinator ensures that students with special needs will beaccommodated.The School Assessment Coordinator verifies the correctness and number of materials sent bythe department.The School Assessment Coordinator discusses exam protocol and specific instructions withinvigilators, and distributes “Instructions to Teachers” sheet (see above).The School Assessment Coordinator that the examination venue does not display materialthat might advantage students in writing the examination.The School Assessment Coordinator maintains security of the examinations and ensuresthat neither students nor teachers have access to the examinations until the morning of theadministration date.NSE – Mathematics 12/Advanced Mathematics 12Information Guide13

During Administration Teachers/invigilators ensure students are under supervision at all times.Teachers/invigilators ensure students work independently at all times.Teachers/invigilators allow up to three hours to write the examinations.Students retain their examination and stay in the examination room for at least one hour afterthe administration has begun (or longer if so required by school examination procedures).Each student receives a personalized Student Booklet, a personalized Student Response Formfor recording responses to selected response questions, and a personalized Formula Booklet.Teachers/invigilators ensure that students use a pencil to complete the selected responsesection of the Student Response Form.Students are to do all their work in the Student Booklet, work not done in the Student Bookletwill NOT be evaluated.Teachers/invigilators collect all examination materials, including scrap paper and FormulaBooklets from students before the students leave the examination room, all materials must beaccounted for.Teachers/invigilators do not read to students or discuss examination questions with students.Students work at their own pace; however, they should take note of the suggested times givenfor each portion of the examination, one hour for the selected response section and two hoursfor the constructed response section.Post-Administration 14In the case of a student for whom adaptations were made in his or her writing of theexamination, the teacher must indicate so on the front cover and insert a copy of theadaptations checklist pertaining to assessment from the student’s cumulative record file insidethe front cover or the examination.Teachers write in and bubble each student’s term mark on their corresponding StudentResponse Form. Teachers do this for all students currently registered in Mathematics 12 orAdvanced Mathematics 12 even if the student did not write the examination.As soon as possible following the completion of the examination, teachers must return tothe School Assessment Coordinator all student booklets (used and unused) and the SchoolAssessment Coordinator all student booklets (used and unused). The School AssessmentCoordinator accounts for (and if necessary follows up on) all materials sent to the school,sign the Quality Control Declaration, and packages the required materials. Instructions forposting or pick-up will be sent to schools by the Board Assessment Coordinator.The Nova Scotia Examinations are secure. Therefore all booklets received by the school,including the student booklets (used and unused) must be accounted for and returned to thedepartment. Under no circumstances is reproduction of any part of the examination, includingstudent work, permitted.NSE – Mathematics 12/Advanced Mathematics 12Information Guide

Eligibility/ExemptionsEligibility – Mathematics 12All students registered in Mathematics 12 will write the NSE Mathematics 12 on the dates specifiedin the provincial assessment schedule. Students studying Mathematics 12 by correspondence willalso write on the specified dates.Students who are on Individual Program Plans relating to Mathematics and students enrolled in theInternational Baccalaureate Mathematics course(s) are working within outcome frameworks thatmay differ from those of the Atlantic Canada Curriculum. These students will be evaluated usingother approved forms of assessment and will not write NSE Mathematics 12.Eligibility – Advanced Mathematics 12All students registered in Advanced Mathematics 12 will write the NSE Advanced Mathematics12 on the dates specified in the provincial assessment schedule. Students studying AdvancedMathematics 12 by correspondence will also write on the specified dates.Students who are on Individual Program Plans relating to Mathematics and students enrolled in theInternational Baccalaureate Mathematics course(s) are working within outcome frameworks thatmay differ from those of the Atlantic Canada Curriculum. These students will be evaluated usingother approved forms of assessment and will not write NSE Advanced Mathematics 12.ExemptionsThe principal, in consultation with the student and/or parent/guardian, may grant an exemption toan individual student in the case of illness, bereavement, or other exceptional circumstances, insuch cases the student’s mark will be determined by the Mathematics or Advanced Mathematics 12teacher in consultation with the principal. Exceptional circumstances are determined on a case-bycase basis as professional judgement and consultation are required.Exemptions are not granted on the basis of how challenging the examination might be for aparticular student. For example, an international student who is enrolled in Mathematics 12 andseeks a course credit in Mathematics 12 must write the examination even if the teacher believes thelanguage competence of that student might not be sufficient to allow success on the examination.The examination assesses the learning outcomes of the course, and it is a requirement for coursecompletion.NSE – Mathematics 12/Advanced Mathematics 12Information Guide15

AdaptationsCertain students will require adaptations in order to allow them to demonstrate their abilities inrelation to learning outcomes. These adaptations should in no way change or modify the learningoutcomes of the course, but rather provide for the long–term and short–term needs of the students byfurnishing them with alternate ways to show that they have met the outcomes. Decisions regardingadaptations will be made at the school level and will reflect those adaptations documented inthe student’s cumulative record file as being needed during assessment periods, so long as theseadaptations do not compromise or alter the validity of the examination.For those students where adaptations were made for the writing of the NSE Mathematics 12 or NSEAdvanced Mathematics 12, a copy of a list of relevant adaptations from the student’s cumulativerecord file must be inserted in the inside front cover of the student’s examination booklet.Additional information on adaptations and the NSEs is available online at http://plans.ednet.ns.ca16NSE – Mathematics 12/Advanced Mathematics 12Information Guide

Scoring and ReportingRegional LevelAs of January 2008, all NSE Mathematics 12 and Advanced Mathematics 12 examinations arescored at regional marking sessions under the guidance of the professional staff of EvaluationServices Division. Results are returned to Board Assessment Coordinators for distribution to schoolswithin two to three days of the completion of all scoring.Regional sessions are led by experienced mathematics teachers that attend a three day trainingsession at the Department of Education. All regional sites are connected by Internet using videoconferecing software. Should any revisions be made to the marking guide, all sites are consulted.Professional staff from the Evaluation Services Division are also on site in each region to assistsession leaders.Provincial LevelGiven that all examinations are marked regionally under the guidance of trained teachers andEvaluation Services Division staff, there is no sampling for central marking. Results are reportedbased on the census scoring that occurs regionally.NSE – Mathematics 12/Advanced Mathematics 12Information Guide17

Procedures for requesting a re-score of a Nova Scotia ExaminationA request for the re-scoring of a provincially-scored examination is to be made to the schoolprincipal by a parent/guardian of the student or by the student him/herself. Deadlines for makingsuch a request for the 2010-2011 NSE Mathematics 12 and Advanced Mathematics 12 examinationsare February 10, 2012 for January examinations and July 3, 2012 for June examinations.The principal will explain to the person making the request, in person or by telephone conversation,that the student examination will not be made available for viewing. The principal will explain thereasoning for this: that the NSE is a secured assessment instrument, and that there is no evidence onthe student paper itself of how the examination was scored (so that all scoring is blind). The principalwill further convey to the person(s) requesting the re-scoring information pertaining to: the tracking of the examination from the point of student writing to the receipt of theexamination mark by the school; the procedures used in the scoring of the NSE to ensure reliable, accurate and fair scoring ofstudent work; the outcome of the re-scoring request, that being the determination of the re-scoring will befinal and the results will replace those of the original scoring of the examination.The principal will then have the person requesting the re-scoring confirm their request for and theacceptance of the outcomes of the re-scoring by their signing in the required space the Request forRe-scoring form. The principal will complete the Request for Re-scoring form with all the requisiteinformation.The principal will forward the completed Request for Re-scoring forms to the Board AssessmentCoordinator by February 10, 2012 or July 3, 2012.The Board Assessment Coordinator will forward all requests from his/her board for any NSE to theappropriate coordinator in Evaluation Services by February 14, 2012 or July 4, 2012.An expert panel of three teachers will convene on February 16, 2012 and July 5, 2012 to re-score theexaminations. All

NSE - Mathematics 12/Advanced Mathematics 12 Information Guide 2 Curriculum Links and Rationale The document for the Atlantic Canada Mathematics Curriculum, Mathematics 12 and Advanced Mathematics 12, articulates the curriculum for both of these courses. The document provides the teachers of each mathematics course with information to plan