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Nova Scotia ExaminationsMathematics and Advanced Mathematics 122011–2012Information 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