Mathematics - Newfoundland And Labrador
MathematicsApplied Mathematics 2202Interim EditionCurriculum Guide2012
CONTENTSContentsAcknowledgements. iiiIntroduction.1Background .1Beliefs About Students and Mathematics .1Affective Domain .2Goals For Students .2Conceptual Framework for 10-12 Mathematics.3Mathematical Processes .3Nature of Mathematics .7Essential Graduation Learnings .10Outcomes and Achievement Indicators . 11Program Organization .12Summary .12Assessment and Evaluation . .13Assessment Strategies .15Instructional FocusPlanning for Instruction .17Teaching Sequence .17Instruction Time Per Unit .17Resources .17General and Specific Outcomes.18Outcomes with Achievement IndicatorsUnit 1: Surface Area .19Unit 2: Drawing and Design .41Unit 3: Volume and Capacity .65Unit 4: Interpreting Graphs .83Unit 5: Banking and Budgeting .97Unit 6: Slope .133Unit 7: Trigonometry .147AppendixOutcomes with Achievement Indicators Organized by Topic .159References .171APPLIED MATHEMATICS 2202 CURRICULUM GUIDE - INTERIM 2012i
CONTENTSiiAPPLIED MATHEMATICS 2202 CURRICULUM GUIDE - INTERIM 2012
ACKNOWLEDGEMENTSAcknowledgementsThe Department of Education would like to thank the Western and Northern Canadian Protocol (WNCP)for Collaboration in Education, The Common Curriculum Framework for K-9 Mathematics - May 2006and The Common Curriculum Framework for Grades 10-12 - January 2008, reproduced and/or adapted bypermission. All rights reserved.We would also like to thank the provincial Grade 11 Mathematics curriculum committee and the followingpeople for their contribution:Joanne Hogan, Program Development Specialist – Mathematics, Division of ProgramDevelopment, Department of EducationDeanne Lynch, Program Development Specialist – Mathematics, Division of ProgramDevelopment, Department of EducationSharon Whalen, Program Development Specialist – Division of Student Support Services,Department of EducationJohn-Kevin Flynn, Test Development Specialist – Mathematics/Science, Division ofEvaluation and Research, Department of EducationNicole Bishop, Mathematics Teacher – Prince of Wales Collegiate, St. John’sJohn French, Mathematics Teacher – Corner Brook Regional High, Corner BrookDean Holloway, Mathematics Teacher – Heritage Collegiate, LethbridgeJane Kelly, Instructional Resource Teacher – Booth Memorial, St. John’sJoanne McNeil, Mathematics Teacher – Ascension Collegiate, Bay RobertsSandra Rennie, Instructional Resource Teacher – St. Lawrence Academy, St. LawrenceJoanne Riggs Moulton, Instructional Resource Teacher – Marystown Central High,MarystownJane Sears, Instructional Resource Teacher – Mount Pearl Central High, Mount PearlTanya Warford, Mathematics Teacher – Valmont Academy, King’s PointEvery effort has been made to acknowledge all sources that contributed to the development of this document.Any omissions or errors will be amended in future printings.APPLIED MATHEMATICS 2202 CURRICULUM GUIDE - INTERIM 2012iii
ACKNOWLEDGEMENTSivAPPLIED MATHEMATICS 2202 CURRICULUM GUIDE - INTERIM 2012
INTRODUCTIONINTRODUCTIONBackgroundThe curriculum guidecommunicates highexpectations for students.The Mathematics curriculum guides for Newfoundland and Labradorhave been derived from The Common Curriculum Framework for 1012 Mathematics: Western and Northern Canadian Protocol, January2008. These guides incorporate the conceptual framework for Grades10 to 12 Mathematics and the general outcomes, specific outcomesand achievement indicators established in the common curriculumframework. They also include suggestions for teaching and learning,suggested assessment strategies, and an identification of the associatedresource match between the curriculum and authorized, as well asrecommended, resource materials.Beliefs AboutStudents andMathematicsStudents are curious, active learners with individual interests, abilitiesand needs. They come to classrooms with varying knowledge, lifeexperiences and backgrounds. A key component in developingmathematical literacy is making connections to these backgrounds andexperiences.Mathematicalunderstanding is fosteredwhen students build ontheir own experiences andprior knowledge.Students learn by attaching meaning to what they do, and they needto construct their own meaning of mathematics. This meaning is bestdeveloped when learners encounter mathematical experiences thatproceed from the simple to the complex and from the concrete to theabstract. Through the use of manipulatives and a variety of pedagogicalapproaches, teachers can address the diverse learning styles, culturalbackgrounds and developmental stages of students, and enhancewithin them the formation of sound, transferable mathematicalunderstandings. Students at all levels benefit from working with avariety of materials, tools and contexts when constructing meaningabout new mathematical ideas. Meaningful student discussions provideessential links among concrete, pictorial and symbolic representationsof mathematical concepts.The learning environment should value and respect the diversityof students’ experiences and ways of thinking, so that students feelcomfortable taking intellectual risks, asking questions and posingconjectures. Students need to explore problem-solving situations inorder to develop personal strategies and become mathematically literate.They must come to understand that it is acceptable to solve problemsin a variety of ways and that a variety of solutions may be acceptable.APPLIED MATHEMATICS 2202 CURRICULUM GUIDE - INTERIM 20121
INTRODUCTIONAffective DomainTo experience success,students must learn to setachievable goals and assessthemselves as they worktoward these goals.A positive attitude is an important aspect of the affective domain andhas a profound impact on learning. Environments that create a sense ofbelonging, encourage risk taking and provide opportunities for successhelp develop and maintain positive attitudes and self-confidence withinstudents. Students with positive attitudes toward learning mathematicsare likely to be motivated and prepared to learn, participate willinglyin classroom activities, persist in challenging situations and engage inreflective practices.Teachers, students and parents need to recognize the relationshipbetween the affective and cognitive domains, and attempt to nurturethose aspects of the affective domain that contribute to positiveattitudes. To experience success, students must learn to set achievablegoals and assess themselves as they work toward these goals.Striving toward success and becoming autonomous and responsiblelearners are ongoing, reflective processes that involve revisiting,asssessing and revising personal goals.Goals ForStudentsMathematics educationmust prepare studentsto use mathematicsconfidently to solveproblems.The main goals of mathematics education are to prepare students to: use mathematics confidently to solve problems communicate and reason mathematically appreciate and value mathematics make connections between mathematics and its applications commit themselves to lifelong learning become mathematically literate adults, using mathematics tocontribute to society.Students who have met these goals will:2 gain understanding and appreciation of the contributions ofmathematics as a science, philosophy and art exhibit a positive attitude toward mathematics engage and persevere in mathematical tasks and projects contribute to mathematical discussions take risks in performing mathematical tasks exhibit curiosity.APPLIED MATHEMATICS 2202 CURRICULUM GUIDE - INTERIM 2012
MATHEMATICAL PROCESSESCONCEPTUALFRAMEWORKFOR 10-12MATHEMATICSMathematicalProcesses Communication [C] Connections [CN] Mental Mathematicsand Estimation [ME] Problem Solving [PS] The chart below provides an overview of how mathematical processesand the nature of mathematics influence learning outcomes.There are critical components that students must encounter in amathematics program in order to achieve the goals of mathematicseducation and embrace lifelong learning in mathematics.Students are expected to: communicate in order to learn and express their understanding connect mathematical ideas to other concepts in mathematics, toeveryday experiences and to other disciplines demonstrate fluency with mental mathematics and estimation develop and apply new mathematical knowledge through problemsolvingReasoning [R] develop mathematical reasoning Technology [T] Visualization [V]select and use technologies as tools for learning and for solvingproblems develop visualization skills to assist in processing information,making connections and solving problems.This curriculum guide incorporates these seven interrelatedmathematical processes that are intended to permeate teaching andlearning.APPLIED MATHEMATICS 2202 CURRICULUM GUIDE - INTERIM 20123
MATHEMATICAL PROCESSESCommunication [C]Students need opportunities to read about, represent, view, write about,listen to and discuss mathematical ideas. These opportunities allowstudents to create links between their own language and ideas, and theformal language and symbols of mathematics.Students must be able tocommunicate mathematicalideas in a variety of waysand contexts.Communication is important in clarifying, reinforcing and modifyingideas, attitudes and beliefs about mathematics. Students should beencouraged to use a variety of forms of communication while learningmathematics. Students also need to communicate their learning usingmathematical terminology.Communication helps students make connections among concrete,pictorial, symbolic, oral, written and mental representations ofmathematical ideas.Connections [CN]Through connections,students begin to viewmathematics as useful andrelevant.Contextualization and making connections to the experiencesof learners are powerful processes in developing mathematicalunderstanding. When mathematical ideas are connected to each otheror to real-world phenomena, students begin to view mathematics asuseful, relevant and integrated.Learning mathematics within contexts and making connections relevantto learners can validate past experiences and increase student willingnessto participate and be actively engaged.The brain is constantly looking for and making connections. “Becausethe learner is constantly searching for connections on many levels,educators need to orchestrate the experiences from which learners extractunderstanding Brain research establishes and confirms that multiplecomplex and concrete experiences are essential for meaningful learningand teaching” (Caine and Caine, 1991, p.5).4APPLIED MATHEMATICS 2202 CURRICULUM GUIDE - INTERIM 2012
MATHEMATICAL PROCESSESMental Mathematicsand Estimation [ME]Mental mathematics andestimation are fundamentalcomponents of number sense.Mental mathematics is a combination of cognitive strategies thatenhance flexible thinking and number sense. It is calculating mentallywithout the use of external memory aids.Mental mathematics enables students to determine answers withoutpaper and pencil. It improves computational fluency by developingefficiency, accuracy and flexibility.“Even more important than performing computational procedures orusing calculators is the greater facility that students need—more thanever before—with estimation and mental math” (National Council ofTeachers of Mathematics, May 2005).Students proficient with mental mathematics “. become liberated fromcalculator dependence, build confidence in doing mathematics, becomemore flexible thinkers and are more able to use multiple approaches toproblem solving” (Rubenstein, 2001, p. 442).Mental mathematics “. provides the cornerstone for all estimationprocesses, offering a variety of alternative algorithms and nonstandardtechniques for finding answers” (Hope, 1988, p. v).Estimation is used for determining approximate values or quantities orfor determining the reasonableness of calculated values. It often usesbenchmarks or referents. Students need to know when to estimate, howto estimate and what strategy to use.Estimation assists individuals in making mathematical judgements andin developing useful, efficient strategies for dealing with situations indaily life.Problem Solving [PS]Learning through problemsolving should be the focusof mathematics at all gradelevels.Learning through problem solving should be the focus of mathematicsat all grade levels. When students encounter new situations and respondto questions of the type, “How would you know?” or “How couldyou .?”, the problem-solving approach is being modelled. Studentsdevelop their own problem-solving strategies by listening to, discussingand trying different strategies.A problem-solving activity requires students to determine a way to getfrom what is known to what is unknown. If students have already beengiven steps to solve the problem, it is not a problem, but practice. Atrue problem requires students to use prior learning in new ways andcontexts. Problem solving requires and builds depth of conceptualunderstanding and student engagement.Problem solving is a powerful teaching tool that fosters multiple,creative and innovative solutions. Creating an environment wherestudents openly seek and engage in a variety of strategies for solvingproblems empowers students to explore alternatives and developsconfident, cognitive mathematical risk takers.APPLIED MATHEMATICS 2202 CURRICULUM GUIDE - INTERIM 20125
MATHEMATICAL PROCESSESReasoning [R]Mathematical reasoninghelps students thinklogically and make sense ofmathematics.Mathematical reasoning helps students think logically and make senseof mathematics. Students need to develop confidence in their abilitiesto reason and justify their mathematical thinking. High-order questionschallenge students to think and develop a sense of wonder aboutmathematics.Mathematical experiences in and out of the classroom provideopportunities for students to develop their ability to reason. Studentscan explore and record results, analyze observations, make and testgeneralizations from patterns, and reach new conclusions by buildingupon what is already known or assumed to be true.Reasoning skills allow students to use a logical process to analyze aproblem, reach a conclusion and justify or defend that conclusion.Technology [T]Technology contributesto the learning of a widerange of mathematicaloutcomes and enablesstudents to exploreand create patterns,examine relationships,test conjectures and solveproblems.Technology contributes to the learning of a wide range of mathematicaloutcomes and enables students to explore and create patterns, examinerelationships, test conjectures and solve problems.Technology can be used to: explore and demonstrate mathematical relationships and patterns organize and display data extrapolate and interpolate assist with calculation procedures as part of solving problems decrease the time spent on computations when other mathematicallearning is the focus reinforce the learning of basic facts develop personal procedures for mathematical operations create geometric patterns simulate situations develop number sense.Technology contributes to a learning environment in which the growingcuriosity of students can lead to rich mathematical discoveries at allgrade levels.6APPLIED MATHEMATICS 2202 CURRICULUM GUIDE - INTERIM 2012
NATURE OF MATHEMATICSVisualization [V]Visualization “involves thinking in pictures and images, and the abilityto perceive, transform and recreate different aspects of the visual-spatialworld” (Armstrong, 1993, p. 10). The use of visualization in the studyof mathematics provides students with opportunities to understandmathematical concepts and make connections among them.Visualization is fosteredthrough the use of concretematerials, technologyand a variety of visualrepresentations.Visual images and visual reasoning are important components ofnumber, spatial and measurement sense. Number visualization occurswhen students create mental representations of numbers.Being able to create, interpret and describe a visual representation ispart of spatial sense and spatial reasoning. Spatial visualization andreasoning enable students to describe the relationships among andbetween 3-D objects and 2-D shapes.Measurement visualization goes beyond the acquisition of specificmeasurement skills. Measurement sense includes the ability todetermine when to measure, when to estimate and which estimationstrategies to use (Shaw and Cliatt, 1989).Nature ofMathematics Change Constancy Number Sense Patterns Relationships Spatial Sense UncertaintyMathematics is one way of trying to understand, interpret and describeour world. There are a number of components that define the natureof mathematics and these are woven throughout this curiculum guide.The components are change, constancy, number sense, patterns,relationships, spatial sense and uncertainty.ChangeIt is important for students to understand that mathematics is dynamicand not static. As a result, recognizing change is a key component inunderstanding and developing mathematics.Change is an integral partof mathematics and thelearning of mathematics.Within mathematics, students encounter conditions o
of mathematics at all grade levels. Mental mathematics is a combination of cognitive strategies that enhance fl exible thinking and number sense. It is calculating mentally without the use of external memory aids. Mental mathematics enables students to determine answers without paper and pencil. It improves computational fl uency by developing
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